AN INVOLUTION SATISFYING PARTICLE-IN-CELL METHOD

By

Stephen Robert White

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Computational Mathematics, Science, and Engineering—Doctor of Philosophy

2025

ABSTRACT

This thesis presents a new Particle-in-Cell method for numerically simulating plasmas under the

Vlasov-Maxwell system. Maxwell’s equations and the Newton-Lorentz force are both recast in

terms of vector and scalar potentials under the Lorenz gauge. This results in a set of decoupled

wave equations governing the potentials and a Hamiltonian system with a generalized momentum

formulation governing the particles. The Particle-in-Cell framework for solving a plasma system

requires two main components, a method for updating the fields and a method for updating the

particles of the system.

The first part of this thesis introduces the Method of Lines Transpose, or MOLT, as a way of

solving partial differential equations in general and the wave equation in particular. Additionally it

introduces a new particle pusher, the Improved Asymmetrical Euler Method, that is a modification

of a previously existing method. We deploy these two techniques in the Particle-in-Cell framework.

In this section in particular MOLT employs a dimensional splitting algorithm, solving a set of one

dimensional boundary value problems using a Green’s function. This will all be done using one

particular temporal discretization scheme, the first order Backward Difference Formula. Numerical

results are shown to give evidence for the quality of these techniques, though it is noteworthy that

the combination of this wave solver and particle pusher does not satisfy the Lorenz gauge condition,

nor does it satisfy the involutions of Maxwell’s equations, otherwise known as Gauss’s laws.

The second part of this thesis fills this lacuna, suggesting two ways for doing so. First it will

consider theory to connect satisfaction of the continuity equation with satisfaction of the Lorenz

gauge. It will consider in particular a way of satisfying this theory with multi-dimensional Green’s

functions, eschewing the dimension splitting of the first part. It will additionally consider the solution

of the boundary value problems via other numerical techniques such as the Fast Fourier Transform

or Finite Difference approach, ultimately choosing these for simplicity. The second approach will

consider a gauge correction technique. It will be shown that both of these preserve the gauge, but

the first method will additionally satisfy the involutions of Maxwell’s equations. In a similar manner

to the first part, it will do so using the first order Backward Difference Formula as the temporal

discretization scheme. Numerical evidence will be given to support the theory developed.

The third part of this thesis will generalize the theory connecting the satisfaction of the continuity

equation with satisfaction of the Lorenz gauge and, in most cases, with Gauss’s Laws. It will extend

this theory to not only all orders of the Backward Difference Formulation, but to a family of second

order time centered methods, arbitrary stage diagonally implicit Runge-Kutta methods, and all orders

of Adams-Bashforth methods. In all but the diagonally implicit Runge-Kutta methods, Gauss’s laws

will be shown to be satisfied if the Lorenz gauge is. Once again numerical evidence will be given to

support this.

Finally some future projects will be suggested to capitalize on this work.

Copyright by
STEPHEN ROBERT WHITE
2025

To those who wouldn’t let me quit.

v

ACKNOWLEDGMENTS

It has been quite the journey. Five years ago in the height of a pandemic I began the long process

that has resulted in what you are holding in your hands. It is a very safe thing to say that I would

not be here without a number of people. First and foremost I would like to thank my advisor, Dr.

Andrew Christlieb, whose mentorship, teaching, patience, and kindness have made my experience

here at the CMSE better than I could have hoped.

I am additionally indebted to my committee members. Dr. Zhang taught my first plasma

physics class under the extraordinary circumstances of remote education with poise and clarity. His

encouragement that indeed I could understand this material when I was fairly certain I could not was

not unappreciated. The techniques I picked up in Dr. O’Shea’s computational astrophysics class

have born fruit, but I would be remiss if I did not also express gratitude for his advice on finishing my

PhD strong and without losing my sanity. Lastly, Dr. Verbonceour has provided numerous insights

that have worked their way into this thesis, both through in-person conversations as well as his 2005

review paper, which probably reigns supreme as the paper to which I have returned the most.

Graduate school is hardly composed only of one’s committee. I also am very grateful to my

labmates and fellow CMSE graduate students, Nic, Cole, Zenas, Gina, Arash, Mandela, Carolyn,

Alejandro, Bowen, as well as Jamal from over at ME, all of whom all have my gratitude for their

company along this journey. Dr. Nathan Haut especially has my thanks for all the shared midday

lunches our second and third year while students were still trickling back into the office. In particular

I would like to thank Dr. William Sands, an alumni of the Christlieb group, who guided me through

my first two years of graduate school and continued his support up to now. The work composing this

thesis was done directly in collaboration with him and Dr. Christlieb, and it is not an exaggeration

to say that his friendship and mentorship made this thesis possible. I am additionally grateful to

Drs. Sining Gong and Shiping Zhou for their help in understanding the collision operators and finite

element formulation of MOLT. Sungmin, whom I have met nearly every week since my third year

to talk about everything from international politics to baseball, has my thanks for the many great

lunches and conversations. My girlfriend, Cassie, whom I met here at the CMSE, has shown me

vi

more patience and kindness throughout our time together than I deserve.

Outside the CMSE I also find myself surrounded by good people. My professors at Spring Arbor

were not only my teachers, but role models and mentors. Dr. Wu, with whom I have fond memories

of not only classes but also road trips to programming competitions; Dr. Wegner, who showed

me just how beautiful and elegant math could be; Dr. Hill, whose friendship and mentorship have

persisted even after college; and Dr. Brewer, whose memory I carry with me; thank you all.

My friends, fellow philosophy enthusiasts, and podcasters at The Problem With Reading (available

on all your favorite podcast platforms), Brevin and Sam, whose friendship formed over a fellow love

of ideas has been a constant source of joy. The weekly “philosophy nights,” hosted by Livingston

Garland, have resulted in yet another group of friends I have come to cherish. These, too, have my

thanks for their constant encouragement along these last five years.

I am deeply thankful to my parents, Jerry and Polly White, who instilled in me both the love of

learning and the willingness to face life’s challenges that are required to tackle something like graduate

school; to my brothers, Aaron and Justin, who never hesitate to offer a word of encouragement or

ego-deflation, whichever the need may be; and to my grandparents, who have been a constant source

of joy and warmth that I have had the rare gift of enjoying up to my early 30s.

I wanted to quit my first year; four people stopped me. The first was my then-roommate Livingston

Garland, who, as he was finishing his own graduate school race, took the time to encourage me

to keep going at the start of mine. The second was Brevin Anderson, would conduct daily video

chats for lunch in which we would, for a half hour, attempt to forget our respective graduate school

struggles and enjoy a bit of food and company. The third was Curtis Underwood, also a daily visitor

for morning coffee, who when told I wanted to quit responded that there was no shame in being

defeated by such a challenge, but there was shame in quitting. The best friends are the ones who

tell you what you need to hear rather than what you wish to hear. Finally, there was “Ryan” Kyle

Mero, who when told the same quoted Thucydides, “Surely the bravest among us is the one with the

clearest vision of what lies ahead, danger and glory alike, yet notwithstanding goes forth.”

It is a blessed man to have one friend such as the these. I have many.

vii

Sic transit gloria mundi

viii

PREFACE

A Canticle For Leibowitz [1] tells the haunting story of a world that has deliberately forgotten its

scientific heritage in the aftermath of nuclear armageddon. In a world torn apart by the “flame

deluge,” the survivors turned on the scientists with the accusation that it was their hands that created

these instruments of war, that they were the ones who scorched the world, and as such the world is

better off without them. In the midst of the mindless carnage and destruction, the titular Leibowitz,

himself a scientist at an unnamed lab in the American Southwest, founds an order of monks who are

dedicated to preserving the scientific heritage of a world that no longer wants it. As time passes, the

monastic order continues to faithfully transcribe the strange symbols and concepts of a knowledge

they no longer understand, believing, with good reason, that these texts are important and even

sacred. As such, they shelter them and await the arrival of someone who will understand their worth

and treat them with the respect and care they deserve.

Thankfully, no such armageddon has taken place, but nonetheless this story serves as a meditation

on the nature of science (a word that really just means knowledge, stemming from the Latin scientia)

and the scientific enterprise, and how without some sort of reverence it is easy to forget its worth.

As we begin this exploration into one (extremely) small component of the scientific venture, it is

worth remembering just how long it has taken to get here. Millennia of philosophers and scientists

have passed their flame on to us. Now it is our turn.

ix

TABLE OF CONTENTS

CHAPTER 1

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ENFORCING THE LORENZ GAUGE .
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GENERALIZATION .
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CHAPTER 6

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NONDIMENSIONALIZATION OF EQUATIONS .

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APPENDIX B

DERIVING THE PARTICLE EQUATIONS OF MOTION FROM
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DERIVING THE BOLTZMANN EQUATION FROM FIRST PRIN-
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APPENDIX D

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APPENDIX E

PROVING VILLASENOR AND BUNEMAN’S SCHEME CON-
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xi

CHAPTER 1

INTRODUCTION

1.1

Introduction

Charged particles in motion constitute a current. A current induces an electromagnetic field.

An electromagnetic field accelerates and rotates charged particles. As one can imagine, this system

becomes quite complicated with only a handful of particles, and the typical plasma system is

composed of trillions of such particles. Modern study of this phenomenon is predicated upon being

able to accurately and efficiently simulate these particles, no small challenge.

The purpose of this thesis is to consider a new method of simulating plasmas using a kinetic

method known as Particle-in-Cell (PIC). In this chapter we will first briefly consider what a plasma

is and why we are so interested in studying this phenomenon. We will then explore some of the

physical laws governing it in greater detail. Once we have a better understanding of the problem

space we are navigating, we will then go over a history of plasma science in general and PIC in

particular, examining how these studies have evolved over time. The remainder of the thesis will be

dedicated to the work with which I (alongside others) have been engaged.

In Chapter 2 we will develop a novel PIC method. This method considers Maxwell’s equations

under the Lorenz gauge, which considers the electric and magnetic fields in terms of a vector and

scalar potential, and these decoupled into a set of heterogeneous wave equations. We approach these

wave equations using the Method of Lines Transpose under the first order Backward Difference

Formula. Having discretized in time, we then dimensionally split the spatial operator and solve

the resulting one dimensional boundary value problem with a Green’s function. The particles are

updated using a modification of the Asymmetrical Euler Method. We will compare the numerical

results of this method with the flagship PIC scheme that employs the Yee grid together with the

Boris push, showcasing a number of improvements. It will be noted that, despite a number of

advantages, this method does not preserve the Lorenz gauge condition, nor does it satisfy Gauss’s

laws for electricity and magnetism, unlike the Yee-Boris method.

In Chapter 3, we will take a look at how this method may be modified to fill this lacuna. Theory

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will be developed to show that the charge conservation is intimately connected with satisfaction of

the Lorenz gauge. Moreover, this theory will prove that satisfaction of the Lorenz gauge will in turn

enforce satisfaction of Gauss’s laws. Eschewing the one dimensional Green’s function for reasons

that we will discuss, we consider a multidimensional Green’s function as well as other numerical

techniques such as the Fast Fourier Transform or Finite Difference method. We will also consider a

gauge correction technique that will manually keep the Lorenz gauge to machine precision without

concern for the continuity equation. Numerical evidence will be given to support this theory.

In Chapter 4, we will generalize the properties established in Chapter 3 to a wider family of time

discretizations. Not only will we generalize to all orders of the Backward Difference Formulation,

but also to a family of second order time centered methods, arbitrary stage diagonally implicit

Runge-Kutta methods, and all orders of Adams-Bashforth methods. All of these methods will be

shown to satisfy the Lorenz gauge if the continuity equation is satisfied. All but the diagonally

implicit Runge Kutta methods will be shown to satisfy Gauss’s laws if the Lorenz gauge is satisfied.

Numerical evidence will showcase this theory.

Finally, in Chapter 5 we will sketch out some future lines of inquiry this work unlocks. Chapter

6 will bring us home with some concluding remarks.

The three main chapters, 2, 3, and 4, are based on the works [2, 3, 4]. respectively, done by

myself in collaboration with Dr. Andrew Christlieb and Dr. William Sands.

I have included a number of appendices that are dedicated to explaining in greater detail concepts

that are already very well established. All too often when something is well established the details

are brushed aside, putting the novice looking to understand it in a difficult position, so it is my hope

that anyone looking to learn about this method will find these helpful. I have no wish to pass off this

work as my own, but rather to provide an easily accessible source for those interested, as well as

elaborate on some steps I personally found difficult to follow. I hope it helps the reader.

And now to the matter at hand.

2

1.2 A Very Brief History of Plasmas

1.2.1 Etymology

The word “plasma” will be used to designate that portion of an arc-type discharge in

which the densities of ions and electrons are high but substantially equal.

With these words, in a footnote no less [5], Irving Langmuir gave a label to the fourth state

of matter and the vast majority of matter in the observable universe. His collaborator in this now

famous paper, Lewi Tonks, would later go on to describe the interaction the two had that led to this

term [6]:

Langmuir came into my room in the General Electric Research Laboratory one day and

said, “Say, Tonks, I’m looking for a word. In these gas discharges we call the region in

the immediate neighborhood of the wall or an electrode a ‘sheath,’ and that seems to be

quite appropriate; but what should we call the main part of the discharge?...”

My reply was classic: “I’ll think about it, Dr. Langmuir.”

The next day Langmuir breezed in and announced, “I know what we’ll call it! We’ll call

it the ‘plasma’” The image of blood plasma immediately came to mind; I think Langmuir

even mentioned blood. In the light of the contemporary state of our knowledge, the

choice seemed very apt.

The word itself, “plasma,” is a Greek term meaning “something molded or created,” itself coming

from an older Greek word, “plassein,” meaning “to mold” or “to spread thin.” It then migrated to

Latin and took on the meaning “mold or matrix in which anything is cast or formed to a particular

shape,” and then eventually came to be associated with blood, and finally with the phenomenon

Langmuir and Tonks studied. But what is this state of matter?

1.2.2 State of Matter

There are four fundamental states of matter in the universe, these being solid, liquid, gas, and

plasma. An intuitive way of understanding plasmas, as well as states of matter in general, is in

understanding the strength of the bonds of their constituent particles [7]. A solid is a material in

which the constituent molecules are bonded together, resulting in a rigid form. When energy is

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applied to a solid, eventually it reaches its melting point, the bonds are broken, but attraction from

intermolecular forces remains, and the material becomes a liquid. More energy applied results in its

reaching its boiling point, in which the attraction between the neutral particles is insufficient to keep

them in place, and the material becomes a gas. When further energy is injected into the system, the

neutral particles themselves begin to come apart, the electrons leave their atoms, and the material

becomes ionized, resulting, finally, in a plasma.

Solid

Liquid

Gas

Plasma

Figure 1.1 Changes in states of matter as energy is added.

It is in this last state of matter that we are most interested.

As Langmuir said, the ions and electrons are many, but they are for the most part equal, which

results in macroscopic neutrality.1 The distance at which this macroscopic neutrality breaks down is

called the Debye length, and is denoted 𝜆𝐷. A perturbation disturbing this macroscopic neutrality

will cause electrons (the lighter of the two species) to rush to fill the void. Of course, in so doing

they encounter each other, and the neutrality is broken yet again, causing them to rush away. This

oscillation is called the plasma frequency, and is denoted 𝜔 𝑝. Collisions between electrons and

neutral particles will reduce this oscillation amplitude over time. A much more thorough discussion

of these parameters may be found in [7]. We will go over more details on those who derived these

physical parameters in Section 1.4.2.

1.2.3 Motivation

Though perhaps not encountered by us humans in our day-to-day lives quite as much as the

other three (after all, we walk upon solid earth, drink liquid water, and inhale gaseous air, but we

do not encounter plasmas in an analogous manner), this phase of matter composes over 99% of

the observable universe. One example we see every day, and depend upon for our very existence,

the sun, is a plasma. The solar wind emitted therefrom is a plasma, as are the particles the earth’s

1This is perhaps an oversimplification. Plenty of plasmas are not macroscopically neutral, in fact one of our examples
in Section 2.4 will be a beam of particles of a single species. It is under the assumption of macroscopic neutrality that
we get the following parameters.

4

magnetosphere catches from this wind. We observe the Aurora Borealis and Australis2 because

of the plasma that forms in the ionosphere when solar ultraviolet and x-ray radiation ionize the

atmospheric particles. Beyond our solar system there are countless examples, and ones of a far

more exotic nature, such as nebula, supernovae, pulsars, accretion disks about black holes contain

plasmas, the list goes on [7].

This is all fascinating, and of course learning more about the universe is its own reward. However,

what of the more practical applications of this material? One of the more urgent examples would be

the Carrington event, in which a massive Coronal Mass Ejection (CME) struck the earth in 1859.

The Aurora Borealis reached all the way down to the Caribbean, and multiple telegraph machines

exploded. This is a relatively benign event in the mid 1800s (aside from some terrified telegraph

operators), but it does not take a very active imagination to imagine the devastation a similar event

would wreak upon the globe in the 2000s. It is of vital importance to have the capabilities to monitor

and predict this sort of plasma phenomenon. And while PIC is a kinetic model more suited to

smaller, bounded domains, there are a number of works that combine PIC with other methods such

as magnetohydrodynamics (MHD) for modelling solar phenomena [8, 9].

One perhaps less urgent, though certainly no less exciting, use case would be plasma fusion

energy. In December 2022 Lawrence Livermore successfully achieved the first igniting fusion

plasma in a laboratory and produced 1.37 MJ of fusion energy [10, 11]. While there is still a long

way to go to perfect the process and make it competitive with fossil fuel et al, this is a very significant

step to more abundant, and more clean, energy.

Of more common, though admittedly less bombastic, examples there is a plethora. The manufac-

turing industry has come to rely heavily on plasmas, as they have the ability to modify materials and

surfaces in very unique ways, such as being able to etch a .2-𝜇m wide, 4-𝜇m deep trench into a piece

of silicon, which is an indispensable requirement for integrated circuits. Plasmas are also used to

harden metals, which has increased the durability of the machine tools of the manufacting industry,

also of which the medical industry has found useful for the creation of artificial joints [7, 12]. Hall

2Named for the Roman goddess of the dawn, Aurora, and the Greek gods of the North and South winds, Boreas and

Auster, respectively.

5

thrusters [13], A6 magnetrons [14], next generation high power microwave sources [15, 16], and

cross field amplifiers [17], all use plasmas, and all have nontrivial geometries that need resolving.

1.3 Problem Formulation

1.3.1 Introduction

We now have some understanding of what plasmas are and are motivated to consider more

carefully how they behave. Thankfully this phenomena has been well studied for some time (to be

explored further in Sections 1.4.1 and 1.4.2). First we will consider the laws governing a plasma,

how electromagnetic waves propagate and how a charged particle behaves in these waves. We will

then take a closer look at Particle-in-Cell (PIC) and consider how we may simulate plasmas using

this method.

1.3.2 Problem Formulation

The laws governing a collection of particles of species 𝛼 may be described as the evolution of

the particle probability density function, 𝑓𝛼, in phase space, where phase space is both location in

physical space and velocity. 𝑓𝛼 is a seven dimensional function, consisting of three spatial, three

velocity, and one temporal dimension. Of course, x and v are themselves functions of 𝑡. In the

absence of collisions, the distribution does not change over time with regards to the total time

derivative, D

D𝑡 . We apply this derivative, use the chain rule, and see

D 𝑓𝛼
D𝑡

=

≡

𝜕 𝑓𝛼
𝜕𝑡
𝜕 𝑓𝛼
𝜕𝑡

·

+

+

𝜕v
𝜕𝑡

𝜕x
𝜕𝑡

𝜕 𝑓𝛼
𝜕x
+ v · ∇ 𝑓𝛼 + a · ∇𝑣 𝑓𝛼.

·

𝜕 𝑓𝛼
𝜕v

Setting this equal to zero gives us the Vlasov equation,

𝜕 𝑓𝛼
𝜕𝑡

+ v · ∇ 𝑓𝛼 + a · ∇𝑣 𝑓𝛼 = 0.

This is the Boltzmann equation in the absence of collisions:

𝜕 𝑓𝛼
𝜕𝑡

+ v · ∇ 𝑓𝛼 + a · ∇𝑣 𝑓𝛼 =

(cid:19)

(cid:18) 𝛿 𝑓𝛼
𝛿𝑡

.

𝑐𝑜𝑙𝑙

See Appendix C for details on how to derive this from first principles.

6

(1.1)

(1.2)

(1.3)

Here v is velocity and a is acceleration of a particle species 𝛼. For our purposes we will be

examining collisionless plasmas, ie (1.2).

Now, for the governing force we have the Lorentz equation of motion governing the motion of a

particle in electric (E) and magnetic (B) fields:

𝑑p
𝑑𝑡

= F = 𝑞 (E + v × B)

(1.4)

This is relativistically correct if p = 𝛾𝑚v. We have the relativistic form of Newton’s second law:3
𝑑p
𝑑𝑡

= 𝛾𝑚a.

F =

(1.5)

With the Lorentz force we can rewrite the Vlasov equation as the Vlasov-Maxwell equation

𝜕 𝑓𝛼
𝜕𝑡

+ v · ∇ 𝑓𝛼 +

𝑞
𝛾𝑚

(E + v × B) · ∇𝑣 𝑓𝛼 = 0.

The electric and magnetic fields must satisfy Maxwell’s equations:

,

∇ · E =

𝜌
𝜖0
∇ · B = 0,

∇ × E = −

,

𝜕B
𝜕𝑡
𝜕E
𝜕𝑡

1
𝑐2

∇ × B =

+ 𝜇0J.

(1.6)

(1.7a)

(1.7b)

(1.7c)

(1.7d)

Here 𝜌 is the total charge density, J is the total current density, 𝜖0 is the free space permittivity, 𝜇0
is the free space permeability, and 𝑐 = 1√𝜖0 𝜇0
[32], and Chapter 1 of [7]. For a more thorough history, see also Sections 1.4.1 and 1.4.2. Finally,

is the speed of light.4 For more details see Chapter 6 of

see also Appendix A for nondimensionalizing both the field equations and the particle equations

of motion. It is simulating the systems governed by the five above equations with which we are

most concerned. The standard Yee-Boris scheme is well equipped to handle these equations, but the

method we introduce in this thesis will require us to consider what they look like in terms of vector

and scalar potentials.

3If 𝛾 is a function of 𝑡, which in most interesting cases it is, we need to apply the chain rule, seeing
(cid:16) 𝑑𝛾
𝑑𝑡 v + 𝛾 𝑑v

𝑑𝑡 [𝛾v] =
. Conveniently, as 𝑣 → 𝑐, the first term dominates (eg a quick calculation shows that 𝑣 = .99995𝑐 yields

(cid:17)

𝑑p
𝑑𝑡 = 𝑚 𝑑

𝑚
𝛾 ≈ 100, many orders of magnitude lesser).

𝑑𝑡

4SI units.

7

1.3.3 Maxwell’s Equations Under the Lorenz Gauge

We will have need to have Maxwell’s equations written in terms of a vector and scalar potential.

As such, we follow the work of Jackson Chapter 6 [32] to convert them into this form.

We have Maxwell’s equations, (1.7a) - (1.7d). From Gauss’ Law we know ∇ · B = 0, and as

such we also know B = ∇ × A, given the vector identity ∇ · (∇ × A) = 0. We plug this into Faraday’s

Law (1.7c)

∇ ×

(cid:18)

E +

(cid:19)

𝜕A
𝜕𝑡

= 0

Another vector identity, ∇ × (∇𝜙) = 0 implies

E +

𝜕A
𝜕𝑡

= −∇𝜙

(1.8)

(1.9)

This gives us alternative definitions of the magnetic and electric fields in terms of vector and

scalar potentials:5

B = ∇ × A,

E = −

𝜕A
𝜕𝑡

− ∇𝜙.

Using these definitions, Gauss’s Law for Electricity becomes

∇ · E =

(cid:18)

=⇒ ∇ ·

−

(cid:19)

=

− ∇𝜙

=⇒ Δ𝜙 +

(∇ · A) = −

𝜌
𝜖0
𝜕A
𝜕𝑡
𝜕
𝜕𝑡

𝜌
𝜖0
𝜌
𝜖0

(1.10a)

(1.10b)

(1.11)

5These are in fact implicit in Maxwell’s equations, and were derived by Maxwell in [23, 26, 28]. To be clear,
B = ∇ × A is not original to him, it was already explored by Thomson, Neumann, Weber, and Kirchhoff in the early to
mid 1800s, and he explicitly borrows this from Thomson [33]. It is unclear if E = −∇𝜙 − 𝜕A
𝜕𝑡 is a Maxwell original. In
any case, they may be derived from physical principles or from Maxwell’s equations.

8

Ampere’s Law becomes

∇ × B =

1
𝑐2

𝜕E
𝜕𝑡

+ 𝜇0J
𝜕
𝜕𝑡

1
𝑐2

(cid:18) 𝜕A
𝜕𝑡
𝜕2A
𝜕𝑡2

−

=⇒ ∇ × (∇ × A) = −

(cid:19)

+ ∇𝜙

+ 𝜇0J

=⇒ ∇(∇ · A) − ΔA = −

1
𝑐2

=⇒ − ΔA +

1
𝑐2

𝜕2A
𝜕𝑡2

+ ∇(∇ · A) +

=⇒

1
𝑐2

𝜕2A
𝜕𝑡2

− ΔA + ∇

(cid:18)

∇ · A +

1
𝑐2

𝜕
𝜕𝑡
𝜕
𝜕𝑡
(cid:19)

1
𝑐2
1
𝑐2
𝜕𝜙
𝜕𝑡

= 𝜇0J

∇𝜙 + 𝜇0J

∇𝜙 = 𝜇0J

(1.12)

These are coupled, and as such quite gnarly to solve. However, there is an exploitable property in
𝜕𝜓
𝜕𝑡 ,

the definition of the potentials. Note if we define some other potentials A(cid:48) := A+∇𝜓 and 𝜙(cid:48) := 𝜙−
and consider B(cid:48) := ∇ × A(cid:48) and E(cid:48) := −∇𝜙(cid:48) − 𝜕A(cid:48)

𝜕𝑡 , we see

E(cid:48) = −∇

(cid:18)

𝜙 −

B(cid:48) = ∇ × (A + ∇𝜓) = ∇ × A + ∇ × ∇𝜓 = ∇ × A = B.
𝜕𝜓
𝜕𝑡

[A + ∇𝜓] = −∇𝜙 + ∇

𝜕A
𝜕𝑡

𝜕𝜓
𝜕𝑡

𝜕𝜓
𝜕𝑡

𝜕
𝜕𝑡

− ∇

−

−

(cid:19)

= −∇𝜙 −

(1.13)

𝜕A
𝜕𝑡

= E.

(1.14)

This means we have one remaining degree of freedom, which we pin down via the Lorenz6 gauge

∇ · A +

1
𝑐2

𝜕𝜙
𝜕𝑡

= 0.

This leaves us with a set of decoupled heterogeneous wave equations:

1
𝑐2

1
𝑐2

𝜕2𝜙
𝜕𝑡2
𝜕2A
𝜕𝑡2

− Δ𝜙 =

𝜌
𝜖0

− ΔA = 𝜇0J.

(1.15)

(1.16)

(1.17)

It is important to remember that we have introduced the requirement of enforcing the Lorenz

gauge condition, this acting as our connection back to Maxwell’s equations. Now, of course the

choice of this particular gauge condition is not necessary. We could have chosen the Coulomb

gauge, ∇ · A = 0, the Weyl gauge, 𝜙 = 0, or any other gauge, but we are interested in working with

the Lorenz gauge and exploiting some of the properties that come with solving the above wave

equations.

6Named after Danish physicist Ludvig Lorenz, not to be confused with the Dutch physicist, Hendrik Lorentz, after

whom the Lorentz force, F = 𝑞 (E + v × B) is named. (A common mistake, even [32] makes it.)

9

1.3.4 The Particle Equations of Motion Under the Lorenz Gauge

We wish to derive the equations of motion in terms of the vector and scalar potentials. To do so,

we have two options. We could first begin with the relativistic Lagrangians for a free particle and a

particle interacting in a field and derive from there, obtained from Jackson [32]. Alternatively, we

could simply take the Lorentz force and plug in the E and B fields and see what comes out, as does

Sands [34]. The latter, though more tedious, follows the same logic as deriving the field equations

from Maxwell’s equations, so it makes sense for us to follow in its steps. See Appendix B for the

derivation of the equations of motion from the relativistic Lagrangian.

Consider the Lorentz force, (1.4). We plug in the definitions and observe:

F = 𝑞 (E + v × B)
𝜕A
𝜕𝑡

−∇𝜙 −

= 𝑞

(cid:18)

(cid:19)

+ v × (∇ × A)

We use a vector identity to simplify things:

v × (∇ × A) = ∇ (A · v) − (v · ∇) A.

Plugging this in yields

F = 𝑞

(cid:18)

−∇𝜙 −

𝜕A
𝜕𝑡

+ (∇ (A · v) − (v · ∇) A)

(cid:19)

.

We denote the total convective derivative

So we have

𝑑A
𝑑𝑡

:=

𝜕 𝐴
𝜕𝑡

+ (v · ∇) A.

F = 𝑞

(cid:18)

−∇𝜙 −

𝑑A
𝑑𝑡

+ ∇ (A · v)

(cid:19)

.

Using Newton’s second law (1.5), we shuffle some terms around and see

𝑑
𝑑𝑡

(p + 𝑞A) = 𝑞 (−∇𝜙 + ∇ (A · v))

We use another vector identity to see

∇ (A · v) = (∇v) · A + (∇A) · v.

10

(1.18)

(1.19)

(1.20)

(1.21)

(1.22)

(1.23)

(1.24)

We know v is a function of time, so ∇v = 0. Denote the canonical momentum

P := p + 𝑞A.

We also know from Newton’s second law and this definition

v =

1
𝛾𝑚

(P − 𝑞A)

Combining all of this yields

(cid:18)

= 𝑞

𝑑P
𝑑𝑡

−∇𝜙 +

1
𝛾𝑚

(∇A) · (P − 𝑞A)

(cid:19)

.

Lastly, we know from

𝑑x
𝑑𝑡 = v and (1.26)

𝑑x
𝑑𝑡

=

1
𝛾𝑚

(P − 𝑞A)

Thus, for every particle 𝑖, evaluating 𝜙 and A at x𝑖, we have arrived at our update equations:

(P𝑖 − 𝑞A) ,

=

1
𝛾𝑖𝑚𝑖
(cid:18)

𝑑x𝑖
𝑑𝑡
𝑑P𝑖
𝑑𝑡

= 𝑞𝑖

−∇𝜙 +

(∇A) · (P𝑖 − 𝑞𝑖A)

(cid:19)

.

1
𝛾𝑖𝑚𝑖

We note

(cid:115)

𝛾 =

1 +

1
(𝑚𝑐)2

(P − 𝑞A)2

(Appendix B justifies this identity.) This yields our final set of update equations:

𝑑x𝑖
𝑑𝑡

=

(cid:113)

𝑑P𝑖
𝑑𝑡

=

(cid:113)

𝑐2 (P𝑖 − 𝑞A)
𝑐2 (P𝑖 − 𝑞𝑖A)2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2
𝑐2 (∇A) · (P𝑖 − 𝑞𝑖A)
𝑐2 (P𝑖 − 𝑞𝑖A)2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2

− 𝑞𝑖∇𝜙

(1.25)

(1.26)

(1.27)

(1.28)

(1.29)

(1.30)

(1.31)

(1.32)

(1.33)

1.3.4.1 Hamiltonian Systems

The above is a Hamiltonian system in disguise. Appendix B shows this explicitly by deriving the

equations of motion first from a Lagrangian and then converting from a Lagrangian to a Hamiltonian,

which turns out to be

(cid:113)

H (x, P) =

𝑐2 (P − 𝑞A)2 + (cid:0)𝑚𝑐2(cid:1) 2 + 𝑞𝜙.

(1.34)

11

Here A and 𝜙 are functions of x. This Hamiltonian represents the total energy in the system, and the

updates to the x and P variables are computed as follows:

𝑑𝑥𝑖
𝑑𝑡
𝑑𝑃𝑖
𝑑𝑡

=

𝜕H
𝜕𝑃𝑖

,

= −

𝜕H
𝜕𝑥𝑖

.

(1.35)

(1.36)

If we plug our Hamiltonian in we find this produces the update equations (1.32)-(1.33). This

Hamiltonian is noteworthy in particular due to the nature of the variables, in that it is a non-separable

Hamiltonian. A Hamiltonian H is said to be separable if it can be written in the form

H (x, P) = K (P) + U (x) ,

Where K and U denote the kinetic and potential energy of the system, respectively. In contrast,

the Hamiltonian for the relativistic VM system considered in this work is non-separable because it

contains a momentum-dependent potential of the form

H (x, P) = K (P) + U (x, P) .

This causes challenges in preserving the symplectic nature of the Hamiltonian as it evolves over

time. More of this will be discussed in Section 1.4.4.2.

1.3.5 The Particle-in-Cell Algorithm

Plasmas consist of many charged particles, all of which contributed to the fields which, in turn,

contribute to the movement of the particles. This presents two problems. The first: the number of

physical particles in any system of interest is far more than any computer could hope to simulate

in a reasonable amount of time. The second: to model the movement of one particle amidst 𝑛,
one must make 𝑛 operations. Doing this 𝑛 times yields an O (cid:0)𝑛2(cid:1) operation for each timestep of

the simulation, which is prohibitively expensive for large 𝑁. Particle-in-Cell (PIC), and its closely

related cousin, Cloud-in-Cell (CIC), was developed in response to these challenges. In response to

the first challenge, PIC eschews simulating each physical particle, instead grouping a large number

12

of particles into a single so-called macroparticle with a weight corresponding to the number of

physical particles it represents.7

This answers the first challenge, but not the second. Even drastically reducing 𝑛 physical particles

to 𝑁 macroparticles, where 𝑁 (cid:28) 𝑛, computing the interactions between each macroparticle would
be computationally intractable. PIC answers the second challenge by mapping each particle to a 𝑘 𝑑

mesh, where 𝑘 (cid:28) 𝑁 and 𝑑 is the number of spatial dimensions we are considering, and use this

mesh to dictate the movements of the particles.8 The interaction between the Eulerian mesh and

the Lagrangian particles depends on the shape function 𝑆, which acts as a map between the two.

The resulting scheme gives us a O (𝑘 2) or even O (𝑘 log(𝑘)) update scheme, depending on our wave

solver.

There are some schemes in which we will place the the particle locations and velocities at
2 , where Δ𝑡 is our timestep (see Figure Figure 1.3). This allows us to leapfrog

staggered timesteps Δ𝑡

the velocity and position updates, using the fields to update velocities, velocities to update positions

and fields, and so on (see Figure 1.2). Given an initial condition for both the fields and the particles,

we interpolate the fields to the particles and use our particle update scheme to push the particles.

This motion generates a current J, and the new location provides us a new charge density 𝜌, which

we interpolate back to the mesh, which in turn informs our wave solvers and we update accordingly.

We represent our particle distribution function as a summation of shape functions multiplied

7These two techniques are practically very similar, with their main difference residing in how they consider the
macroparticle. PIC considers it as a discrete particle it interpolates to a mesh, while Cloud-in-Cell considers the particle
as the center of a cloud of particles whose shape dictates the value given to the mesh. This, practically, results in a
very similar algorithm, though it does result in some key differences, especially when considering nonuniform meshes
and the weights of the particles near the boundary. If 𝑓 is represented by a summation of 𝛿 functions over phase
space, ie (cid:205)𝑛
𝛿 (x − x𝑖) 𝛿 (v − v𝑖), PIC interpolates the 𝛿 function in space using a shape function 𝑆, while CIC relaxes
𝑖=1
the 𝛿 function in space to a shape function 𝑆 [35]. Both frameworks result a new distribution function with 𝑁 (cid:28) 𝑛
macroparticles and the substitution of a shape function for the spatial component of the distribution function.

8The reader may justifiably wonder why a 𝑘 2 operation is fine when an 𝑁 2 or 𝑛2 is not. Each approximation
represents an order of magnitude. Obviously this will vary with each simulation, this thesis is concerned with method
development rather than application, and so 𝑁 is often on the order of ten thousand while 𝑘 is at most in the low hundreds.
A code like EMPIRE [36] may have a mesh with thousands or millions of nodes, millions or billions of macroparticles,
simulating trillions of physical particles. In short, if 𝑁 particles give a computationally vexing amount of particles
interactions to compute, 𝑛 is practically impossible.

13

Integration of
Equations of
Motion
F𝑖 → v𝑖 → x𝑖

Interpolation of
Fields to Particles
(cid:0)E 𝑗 , B 𝑗 (cid:1) → F𝑖

Interpolation of
Macroparticles to
Mesh
(x𝑖, v𝑖) → (cid:0)𝜌 𝑗 , J 𝑗 (cid:1)

Integration of Field
Equations on Mesh
(cid:0)𝜌 𝑗 , J 𝑗 (cid:1) → (cid:0)E 𝑗 , B 𝑗 (cid:1)

Figure 1.2 The basic Particle-in-Cell engine.

v𝑛− 3

2

x𝑛−1

v𝑛− 1

2

x𝑛

v𝑛+ 1

2

x𝑛+1

Figure 1.3 The basic Leapfrog scheme. v𝑛+ 1
v𝑛− 1

2 , and so on.

2 updates from v𝑛− 1

2 and x𝑛

, x𝑛+1 updates from x𝑛

and

against a delta function (thus we are actually tracking macroparticles, not individual particles),

𝑓𝛼 (x, v, 𝑡) =

𝑁 𝛼(cid:213)

𝑝

𝑆(x − x𝑝)𝛿(v − v𝑝).

(1.37)

We wish to acquire 𝜌 and/or J to inform our wave solvers. Integrating over all of these gives us

the number of particles:

∫

∫

𝑁 =

Ω𝑥

Ω𝑣

𝑓𝛼 (x, v, 𝑡)𝑑v𝑑x.

We can thus approximate 𝜌 and J:

𝜌(x) =

J(x) =

𝑁
(cid:213)

𝑝=1
𝑁
(cid:213)

𝑝=1

𝑞 𝑝𝑆(x − x𝑝)

𝑞 𝑝v𝑝𝑆(x − x𝑝)

14

(1.38)

(1.39)

(1.40)

The shape function can be arbitrary, the most common being the linear spline function for a

particle 𝑝 at 𝑥 𝑝:

𝑆(𝑥 − 𝑥 𝑝) =

|𝑥 − 𝑥 𝑝 |
Δ𝑥

,

1 −

0,

0 ≤ |𝑥 − 𝑥 𝑝 | ≤ Δ𝑥,

|𝑥 − 𝑥 𝑝 | > Δ𝑥.

(1.41)





We must be careful here, if computing the current, a naive multiplication of charge by velocity

at a particular timestep may lead to a violation of the continuity equation:

𝜕 𝜌
𝜕𝑡

+ ∇ · J = 0.

(1.42)

Linear maps are not charge conserving for Yee, thus the shape functions of Villasenor and

Buneman [37] are needed to preserve the continuity equation. See Appendix E for more details

including the proof of charge conservation. For MOLT, no such modification is necessary, and

linear weights are computed (see Figure 1.4 for a simplified 2D map). If we consider a particle at

location x, with the bottom lefthand node being at location X, and the cell having dimensions d, we
let 𝑤𝑖 := x𝑖−X𝑖
𝑑𝑖

and compute the weights as follows:

𝑄1 = 𝑞𝑖 (1 − 𝑤1)(1 − 𝑤2)(1 − 𝑤3)

𝑄2 = 𝑞𝑖 (𝑤1)(1 − 𝑤2)(1 − 𝑤3)

𝑄3 = 𝑞𝑖 (1 − 𝑤1)(𝑤2)(1 − 𝑤3)

𝑄4 = 𝑞𝑖 (𝑤1)(𝑤2)(1 − 𝑤3)

𝑄5 = 𝑞𝑖 (1 − 𝑤1)(1 − 𝑤2)(𝑤3)

𝑄6 = 𝑞𝑖 (𝑤1)(1 − 𝑤2)(𝑤3)

𝑄7 = 𝑞𝑖 (1 − 𝑤1)(𝑤2)(𝑤3)

𝑄8 = 𝑞𝑖 (𝑤1)(𝑤2)(𝑤3).

We then compute 𝜌 =

𝑄
𝑉 , where 𝑉 is the volume of the cell.9 See Figure 1.4 for a 2D version of

this. See also Appendix F for a more in depth exploration into high order interpolation schemes and

9This becomes a more complicated with a nonuniform mesh, which [38] derives. In Section 5.2 we briefly consider
nonuniform meshes as the subject of future inquiry, but other than this the only mesh of interest is uniform, so the
uniform mapping described above and in Appendix F will suffice.

15

B-splines.

𝑤1

𝜌 𝑗,𝑘+1

𝜌 𝑗+1,𝑘+1

(1 − 𝑤1 ) 𝑤2

𝑤1𝑤2

Δ𝑦

𝑝

(1 − 𝑤1 ) (1 − 𝑤2 )

𝑤1 (1 − 𝑤2 )

𝜌 𝑗,𝑘

Δ𝑥

𝑤2

𝜌 𝑗+1,𝑘

Figure 1.4 A linear interpolation of particle 𝑝.

Using the above method allows for an computationally efficient method for the modeling of

plasmas. We have the framework, now we must start putting the pieces together.

1.3.6 Conclusion

In this section we considered first the basic equations governing a plasma. The Vlasov equa-

tion (1.6) governs the particle distribution function over phase space over time, evolving over the
𝑞
𝛾𝑚 (E + v × B). This in turn indicates that we are

characteristics described by v and a = 1

𝛾𝑚 F =

interested in considering how the fields and particles themselves are evolved. The fields are governed

by Maxwell’s equations (1.7) and the particles are governed by the equations of motion described

by Newton (1.5) and Lorentz (1.4). These were then cast in terms of vector and scalar potentials

under the Lorenz gauge. Finally, we began considering the technique with which this dissertation is

concerned, that of Particle-in-cell, or PIC, and how it takes both of these components, fields and

particles, into consideration as it updates the plasma in question. Now that we have an idea of the

16

task at hand, we can begin a brief historical inquiry into the progress made thus far.

1.4 Literature Review

A classical problem confronting any historical inquiry is the question, “How far back do you

wish to go?” Should we only consider PIC methods developed in the last ten years? This hardly

seems just. Should we go back as far as Euclid developing his geometric theories? This hardly

seems practical.10 As such, we are in the unhappy position of having to choose where to begin and

simply take what came before as a given. This hardly seems fair to the giants whose shoulders we

are now taking for granted,11 but time and paper command these constraints. As such, we will begin

with a brief exploration into the work done on electromagnetics in the 1800s, and then move quickly

into the early explorations into plasma science that occurred in the early 1900s. Without these two

remarkable scientific achievements, we would having nothing to simulate. After this, we will begin

a much more thorough inspection into the history of plasma simulations using PIC beginning in the

1940s and continuing to the present time.

1.4.1 Early Electromagnetics

It is impossible to study plasmas without at least briefly considering electromagnetics, the

study of the underlying fields governing the particle equations of motion. Faraday’s concept of

expressing magnetism as lines of force, particularly in Part III of Experimental Researches in

Electricity [20],12,13 was one of the main inspirations for James Clerk Maxwell to begin considering

10Though David Foster Wallace somehow manages to accomplish this in his absolute tour de force historical inquiry

into Georg Cantor in [39]. But he also had 300+ pages to do so and we do not.

11We will be effectively ignoring Newton, Coulomb, Ampère, Ohm, and reducing Faraday to a series of footnotes.
12Maxwell cites Series XXXVIII of the Experimental Researches as well as Phil. Mag. 1852 in his On Faraday’s
Lines of Force [23]. There are two journals, Philosophical Magazine and Journal of Science and Philosophical
Transactions of the Royal Society of London, both of which have articles by Faraday published in 1852 [21, 22]. The
Philosophical Magazine article is a continuation of the Transactions article. The Transactions article itself is the 28th
(XXVIII) of 29 articles compiled in the three volumes of Experimental Researches in Electricity [18, 19, 20]. Volume
III is the last in the set and it only goes to series XXIX. Now, there are chapters in each series that were continuous
throughout (eg series XXVI had chapters 32 and 33, series XXVII continued to chapter 33), but these leave off at 37,
and besides, are labeled using Arabic numerals rather than Roman. There are some articles at the end of Volume III that
are not a part of a series, the first three of which have to do with lines of magnetic force (the second of which is the
Philosophical Magazine [21] that is the continuation of the Transactions article [22], ie Series XXVIII. I am somewhat
hesitant to accuse Maxwell of even this small a typo, however, a number of typos have been cataloged, for example
Boltzmann [40] did find a number of other errors in this work, so it is not beyond the pale that this may be an instance of
one.

13Part III may get special attention, but he references throughout Faraday’s corpus, and does so almost every other
page. It is also worth noting he repeatedly cites Professor William Thomson (better known as Lord Kelvin) almost as

17

how magnetism, electricity, and charge and current density were intimately connected. Faraday

had already developed the concept of magnetism as “lines of magnetic force as representations of

magnetic power, not merely in the points of quality and direction, but also in quantity.” This can be

easily visualized by spilling some iron filings in the presence of a magnet, and, interestingly enough,

an electric current. He takes pains to point out that this line of force means no more and no less than

just that, giving the strength and direction of the magnetic force, as he is reluctant to tie this to a

specific physics (§3175), though he does give a nod to the possibility of modeling this field as either

a fluid or a magnetic center of action. In any case, Maxwell takes his lines of force with the analogy

of an incompressible fluid14 in On Faraday’s Lines of Force [23], in which he considered magnetism

as an array of infinitely tiny tubes through which a fluid flowed exerting force. The purpose of this

paper was mostly to give a mathematical framework for what Faraday had discovered. In the section

entitled “Electro-magnetism” of this paper he notes an interesting observation, that the law observed

by Ampère, ∇ × H = J, results in ∇ · J = 0, the continuity equation for closed circuits. Given the

assumption of incompressibility, this limits the study of magnetism to steady currents, breaking

down on the introduction of any variation over time.

It is in a series of papers in the 1860s that he fully develops a unified theory of light and

electromagnetism [24, 25, 26, 27],15 and it is in these papers that he solved this issue. Proposition

XIV in [26] is where the correction to Ampère’s law was made, he adds “a variation of displacement,”

much as Faraday. Ironically, though the two had sincere admiration for the other, Thomson was skeptical of Maxwell’s
conclusions to his death [33].

14Anticipated by Faraday as well in §3073.
15It is worth noting that Parts I and II both appear in the same volume of Philosophical Magazine, but Part II is itself

split into two separate sections. The ending of the first part fails to mention this.

18

noting that this is the exact same as a current.16 If

then the identity ∇ · ∇ × v = 0 ∀ v ∈ R𝑛

is exploited to show

∇ × B = 𝜇0J,

∇ · J = 0.

This only satisfies the continuity equation if the current is constant, in other words

(1.43)

(1.44)

𝜕 𝜌
𝜕𝑡 = 0.

Maxwell in this proposition adds 1
𝑐2

𝜕E
𝜕𝑡 to the righthand side to get the electrodynamic version of

Ampère’s law:17

∇ × B = 𝜇0J +

1
𝑐2

𝜕E
𝜕𝑡

,

(1.45)

From Gauss’s Law for Electricity, ∇ · E =

𝜌
𝜖0

(also known at the time), when we apply the divergence

to (1.45) we get

∇ · J +

𝜕 𝜌
𝜕𝑡

= 0.

(1.46)

This was all developed before the standardization of variables; there was no real use of vectors E or

B to represent the electric and magnetic fields, nor was there any convenient notation such as ∇· or

∇× to represent the divergence or curl, and the units of Maxwell are somewhat difficult to track to

the point where John Arthur advises, wisely, to not worry too much about it [43]. All work was done

component-wise and with different variables even for components of the same fields. Thankfully

Feynman makes very short and accessible work of this in Chapter 18 of [44], as does Jackson in

Chapter 6 of [32], and Griffiths in Chapter 7 of [42].

16Despite an emphasis placed by many on the aesthetic nature of a more symmetric set of equations this change
induced, Bork [41] finds no compelling evidence to suggest this was what motivated Maxwell, who was seemingly more
interested in the satisfaction of the continuity equation and the nature of the displacement current as being as physical a
current as the conduction current. Interestingly, Griffiths [42] claims that the satisfaction of the continuity equation was
more accidental, a “happy dividend” rather than the actual intent. True, Maxwell does explicitly says in Prop. XIV, “To
correct the equations (9) of electric currents for the effect due to the elasticity of the medium,” with (9) being ∇ × H = J,
but then immediately shows that this satisfies the continuity equation, so it seems somewhat up for grabs if he was only
𝜕D
motivated by the interpretation of
𝜕𝑡 as another current or by satisfying the continuity equation. Griffiths cites Bork,
but Bork himself lists the two reasons Maxwell had as satisfying the continuity equation and treating the variation of the
displacement as a physical current, in that order, so it is unclear if Maxwell was more concerned in considering the
displacement current as a physical current, correcting the continuity equation, or both equally.

17This is a somewhat simplified version of the events. For brevity we are smuggling in the assumption that the
permittivity and permeability are respectively 𝜖0 and 𝜇0, which is the only scenario in which this current work is
interested, though by no means the only interesting one.

19

It was in Part III of this work [26] as well that Maxwell derived the speed of propagation of

electromagnetic waves through the aether (Proposition XVI), which he immediately noted was less

than 1.5% off from the speed of light found by Fizeau. He was quick to identify this similarity and

conclude, “we can scarcely avoid the inference that light consists in the transverse undulations of

the same medium which is the cause of electric and magnetic phenomena” (emphasis in original).

Maxwell would go on to refine and compile this work into A Dynamical Theory of the Electro-

magnetic Field [28]. He finally expanded this into the two volume work A Treatise on Electricity

and Magnetism [29]. A more thorough discussion on how this developed can be found in [41, 43].

Far from his only work, he published prolifically on a variety of topics, including gas dynamics,

fluid mechanics, the nature of color and colorblindness, thermodynamics, and topics in philosophy

including the nature of free will and paradox [30, 31]. He died at age 49 in 1879, and left the world

with a treasure trove of scientific discovery.

The work of Maxwell sent waves throughout the scientific community, who wasted no time

in dissecting this work and examining the implications it held. [43] does a good job discussing

the history of this. Oliver Heaviside is credited with taking Maxwell’s equations, written in [29]

in quaternion form, and rewriting them in the vector notation with which we are familiar today,

doing so in Article 30 of [45]. Willard Gibbs introduced the ∇ operator to vector calculus in [46].18

Though he was mostly interested in discussing math rather than electromagnetics, it worth noting he

briefly mentions Maxwell’s curl equations on pages 195 and 196 in [47]. To emphasize the novelty

of this, the first of the equations he emphasizes the equivalence by the following line:

∇ × E = − (cid:164)B,

curl E = − (cid:164)B.

Amusingly enough, he is more brief with the other equation,

∇ × H = 4𝜋C.

(1.47)

(1.48)

We see here the standardization of the electric and magnetic fields taking place, but the total

current is still described, understandably enough, as C.

18Maxwell did use the nabla in A Treatise on Electricity and Magnetism, but that was in the context of quaternions.

He and Gibbs both define the nabla in the same exact way though, ∇ = i 𝜕

𝜕𝑥 + j 𝜕

𝜕𝑦 + k 𝜕
𝜕𝑧 .

20

Heinrich Hertz, a student of Helmholtz, is the next major player to consider Maxwell’s theory. He

got his start though by attempting to derive Maxwell’s theory from a different foundation, attempting

to avoid what was seen as a rickety argument justifying the addition of the displacement current

𝜕D
𝜕𝑡 to

Ampère’s law. Taking a page from Ampère, who stated that the magnetic field induced by a current

is the same nature as that generated by an “ordinary” magnet, otherwise known as the principle

of the “unity of magnetic force,” Hertz thought this wise and applied this principle to the electric

field generated by a changing magnetic field, claiming it is of the same nature as that generated

by electrostatic charges. This was the principle of the “unity of electric force.” While brilliant, it

was dependent on some assumptions on how current is induced that were far from secure. He then

turned to what would inadvertently make him the legend in not only the scientific community, but

would possibly have the greatest impact on communication technology to his time. He was the one

who gave experimental verification of Maxwell’s theory, detailed in his book [48].19 It was in this

work that he managed to not only prove that Maxwell’s equations correct, as his identification with

the speed of electromagnetic waves and the speed of light,20 but he also effectively invented the

radio [33].

A few years after Hertz gave physical proof for Maxwell’s theory came Hendrik Lorentz, after

whom the equation of motion of a charged particle in an electric and magnetic field is named.

Though the now famous equation of motion is derivable directly from Maxwell’s work, it would

take some time for this force to be codified. Heaviside for example was able to identify the correct

equation of motion for a magnetic field [50],21 but it was not until Hendrik Lorentz in [52]22 that

19Dedicated to his advisor, Helmholtz, and with a forward by Lord Kelvin. It is quite astonishing seeing the

relationships between these legends.

20Larmor said of this, “the discoveries of Hertz left no further room for doubt that the physical scheme of Maxwell...

constituted a real formulation of the underlying unity in physical dynamics.” [49], quoted in [33].

21This may be found in Chapter 50 of [51]. Interestingly, equations that looks awfully similar to the Lorentz factor
begin appearing in this paper. Given a particle with charge 𝑞, velocity 𝑢 along the 𝑧 axis, then the tensor representing
𝑞
𝑢 sin(𝜃) = 𝑐𝐸𝑢𝜈, with 𝜃 being the angle r makes with the
the magnetic force at distance 𝑟 from this particle is 𝐻 =
𝑟 2
axis, and 𝜈 = sin(𝜃). We are given 𝜇0𝑐𝑉 2 = 1, where 𝑐 is the permittivity that enforces this in a medium such that 𝑉 is
the speed of propagation (𝑉 is 𝑣 in the original text, here switched to the capital letter to reduce confusion between 𝑣
, where 𝜇 = cos 𝜃, which has a term that looks
(vee) and 𝜈 (nu)). We begin seeing a term 𝐻 =
(cid:16)

. Even more interesting, he acknowledges that this only works for 𝑢2/𝑉 2 (cid:28) 1, and

1 − 𝜈 𝑢2
𝑉 2

1 + 𝜇 𝑑
𝑑 𝜇

𝑞𝑢𝜈
𝑟 2

(cid:17) −1/2

(cid:17) (cid:16)

(cid:16)

(cid:17)

eerily reminisce of 𝛾 =
that it all breaks down when 𝑢2/𝑉 2 → 1.

1 − 𝑢2
𝑐2

22The translation of the title is Attempt of a Theory of Electrical and Optical Phenomena in Moving Bodies,

21

the full Lorentz force would be established:

F = 𝑞 (E + v × B) .

(1.49)

It is also worth noting the Lorentz had been doing serious work with relativity, considering

how a body moving relative to the stationary aether’s frame of reference would change, developing

what is now known as the Lorentz transform. He still considered the aether as the “true” frame of

reference, with everything moving relative to it.

It was slightly over a decade until Einstein would take the work of Maxwell, including the work

he had done in deriving the speed of light with regards to vacuum permittivity and permeability

in (136) of [26], and revolutionize the world of physics with his theory of special relativity [53].

That connection between electromagnetism and light turns out to be more profound than anyone had

anticipated. Maxwell’s equations, like the speed of light, are true for any reference frame. There is

something about the very nature of spacetime to which Maxwell’s equations speak.

1.4.2 Early Plasmas

The above is concerned with abstract electromagnetic fields. But what happens when charged

particles are placed in these fields? What about the fields generated by these particles, how do they

impact the other particles within the system? Scientists were eager to explore this burgeoning field.

We first go back slightly in time, Brown [54] identifies Crookes in the late 1800s as the prophet

of plasmas. In an address delivered to the British Association for the Advancement of Science in

1879 [55], Crookes announced the onset of what he viewed as a fourth state of matter in a rather

captivating way, “[w]e have actually touched the border land [sic] where Matter and Force seem

to merge into one another.” It is worth noting that he himself cites Faraday as the true forerunner

of this fourth state of matter, quoting him,23 “If we conceive a change as far beyond vaporisation

[sic] as that is above fluidity, and then take into account also the proportional increased extent of

alteration as the changes rise, we shall perhaps, if we can form any conception at all, not fall far short

and the full translation to English may be found online at https://en.wikisource.org/wiki/Translation:
Attempt_of_a_Theory_of_Electrical_and_Optical_Phenomena_in_Moving_Bodies

23I am unable to find this quote directly in Faraday’s corpus, Crookes cites Dr. Bence Jones’s Life and Letters of

Faraday.

22

of Radiant Matter; and as in the last conversion many qualities were lost, so here also many more

will disappear.” This assertion stemmed from his work with glow discharges that, when described

in [56], sound quite similar to that which was described by Langmuir in [57].

Several scientists would continue this work. Lodge was the scientist who identified the cyclotron

radius and frequency [58].24 Lord Rayleigh began the conception of the plasma frequency in [59],

that of a stationary set of ions among which an equal number of electrons oscillate, and this was

codified in its modern form in [5]. Debye and Hückel in [60] identified the Debye length, interestingly

enough referring to it by the inverse square: 𝜒2. Langmuir and Tonks would use both of these

constants in their seminal work Oscillations in Ionized Gases [5], which coined the term, “plasma.”

While Boltzmann was the one who codified gaseous dynamics including collisions [61, 62] (English

translation [63]), Landau developed what would become the standard operator for plasma collisions

in the 1930s [64],25 which is a simplified form of the Boltzmann collision operator, considering

only grazing collisions with small angle deflections. Vlasov would take the works of Boltzmann,

Langmuir, Tonks, Rayleigh, and Landau and, in 1938, publish what would become known as the

Vlasov equation [69], translated to English later in 1967 [70].26

Presently we have a system of charged particles that, according the the laws of Maxwell (though

24In his A Short History of Gaseous Electronics, Brown [54] attributes Lodge as the one who identified the cyclotron
frequency. In his bibliography, he specifically cites page 36 of [58]. It is unclear if he is attributing this page to
where Lodge identifies the frequency, given page 36 contains a rather lovely description of the current conception of
electrons, or “electrions” or “electrical ghosts”, but nothing about a frequency or even a magnetic field. The next chapter,
specifically page 42, is where he goes into a discussion of a charged particle in a static magnetic field and identifies the
radius of this particle. Of course, contained in the radius is the frequency (see Chapter 2 of [7] for a clearer explanation).
It is entirely possible Brown simply had a different edition of Electrons than the one I found, no edition is given in his
bibliography.

25An English translation may be found in chapter 24 of [65] It’s worth noting there is a 1937 article [66] published in
a different journal, but the book simply cites both as the original work it is translating, and the titles are both extremely
similar, one is in Russian and one is in German. Both (Google) translate to The Kinetic Equation in the Case of Coulomb
interaction, which is close to the title of the chapter, the only difference being the “transport” is used in place of “kinetic.”
Needless to say I am no Russian or German speaker, but given the plethora of references to both “transport” and “kinetic”
English translations of the term in the literature (eg Kunz uses “transport” [67] while Bobilev, Potapenko, and Chuyanov
(themselves writing from Moscow) [68] use “kinetic” when citing Landau) and the similarity of the terms themselves in
physics contexts, it seems safe to conclude these are referencing the same concept. Dr. Matthew Pauly at MSU has
suggested that “kinetic” is indeed the better translation.

26Frustratingly, while there are many references to his 1945 monograph Теория колебательных свойств
электронного газа и её приложения, or Theory of Vibrational Properties of an Electron Gas and Its Applications, as
his allegedly more popular and thorough explanation of the subject. I am unable to locate this monograph and, despite
several references to the title itself, unable to even find out what journal or publisher it was.

23

we now see that the discovery of these laws was a group effort), generate electromagnetic fields

when they move, forming a current. By Lorentz we also know that a charged particle is accelerated

and twisted by these fields. Truly a complicated system, and one that will require a considerable

amount of effort to model. But, it was at this time that the first concepts of modern computing were

beginning to be introduced to the scientific community. In 1936 Turing would introduce a formal

language for discussing computing [71]27 and the problems that may be solved by this process.28

As it turns out, the simulation of plasmas by a technique known as Particle-in-cell (PIC) is one such

problem that may be computed. It is this topic, the main focus of the thesis, to which we now turn.

1.4.3 Early PIC

Birdsall [73] traces the first numerical simulations of plasmas as far back as the 1940s, citing

Professor Hartree and Phyllis Nicolson who, using Coulomb’s Law, computed the orbits of around

30 interacting electrons in a magnetron with a simplified 1D projection of the electric field in 1943

[74].29 This was accomplished using only a hand calculator! This model included the electric and

magnetic fields, as well as the effect of space charge. Hartree would later go on to study electron

flow in a one-dimensional diode in a like manner in 1950 [75]. As such, the first physics computers

were Hartree and Nicolson.

The first great breakthrough for what we now think of as computers occurred in 1956 by Tien

and Moshman [76], simulating noise in a high-frequency diode. In the classic textbook by Hockney

and Eastwood [77], they pay tribute to this paper in a manner worth quoting in full:

At about the time the first digital computers became available, and perhaps the earliest

simulation on such computers, was the study of noise in a high-frequency diode by Tien

and Moshman of Bell Laboratories. They used a Univac I to simulate a one-dimensional

diode with about 360 electrons sheets and a physical time step of 2 ps. About 3000

27Of course, at the time “computing” was only the act of calculation, the idea of our modern computers was

completely foreign.

28He published roughly at the same time as Church [72], tackling the same problem from very different angles. He
modified his paper to acknowledge this, proving the equivalence of their approaches (“computability” from Turing and
“effective calculability” from Church).

29Birdsall cites Buneman and Dunn’s 1966 work, specifically page 36. I have copied their citation, Science J. vol. 2,
pp 34-43, but it is worth noting that I was only able to locate a NASA archived document of the same name. This work
does discuss this remarkable accomplishment, but on page 21, not 36.

24

time steps were taken, each of which required between 25 and 40 s of computer time.

The total computing task must rank as one of the most impressive achievements of

early device modeling, requiring as it did some 25 h of reliable computing on a first

generation vacuum-tube computer with a mercury delay-line store of only about 1000

numbers.

These 1000 numbers (or words to use the language of the original paper) were only capable of

storing 11 decimal digits plus the sign. 500 of these words were used to store the actual program,

the rest of them were dedicated to storing position and velocity, with each word being split into

two pieces, the first six digits containing the spatial information and the last containing the sign

and magnitude of the velocity (recall this is 1D). If you squint at this, it is the first warm plasma

simulation. Buneman, the student of Hartree, would write the first paper intentionally modeling a

plasma in 1959 [78], which studied the cold two stream instability problem, the problem in which

electrons are moved amongst immobile ions that are representing as a neutralizing background.

Buneman found that instabilities would emerge if electron drift velocity rises above .9𝑘 𝐵𝑇, where

𝑘 𝐵 is the Boltzmann constant and 𝑇 is the absolute temperature. Shortly following this was Dawson

[79], who set up a similar system in which electrons were injected into a neutralizing background

of ions. He considered series of “sheets” of electrons and how they would act when perturbed,

finding they behaved as harmonic oscillators if they didn’t cross each other. In representing these

sheets as perfectly elastic, upon colliding he would simply have the particles exchange velocities,

and in so doing he could represent them as a sequence of pendulums. In his 2005 review paper,

Verbonceour [80] also identifies these as being the root to what is now known as Particle-in-Cell

(PIC). Dawson and Smith in 1963, in modeling two warm species obtained results confirming theory

five years before a laboratory was able to do so [73, 81]. It is now that we begin to see the trend of

computation emerging. Until now, labwork would be the manner in which theory was confirmed,

now computation would begin both confirming and refining theory.

Until this point, Birdsall [73] identifies all the models as being one-dimensional, electrostatic,

collisionless, mesh-free, and simulating “disk” particles, or particles with zero thickness. Now we

25

reach the part of the story in which these limitations begin to be shrugged.

Throughout the 1960s, we see the 1D model get pushed to its absolute limits, with 1 spatial

dimension being complemented with 2 and 3 velocity dimensions, and this being done to great effect.

However, eventually the need came to extend to multiple spatial dimensions. With this need came

the grid based approach, as solving Coulomb’s law became prohibitive with the increased domain

and thus the increased count of particles (and even in 1D this can become prohibitively expensive).

The domain is discretized into a finite set of grid nodes, and various techniques are used to compute

the fields only at these locations (more on this later). It was the work of Hockney [82], his student

Burger [83], as well as Dunn and Ho [84], that pushed this grid based method. In computing charge

or current from particles, a nearest-grid-point method would be used, simply totaling up the number

of particles that are in the vicinity of a node and counting that as the charge in the area.

It is important to note that these are all viewing the electric and magnetic fields as static, with the

ions in the background creating enough of a neutralizing presence such that we can simply ignore

the derivatives of the electric and magnetic fields. But this is obviously an approximation, and

eventually the need to have electromagnetic codes became apparent. One of the first attempts at

such was a 1D-3V code by Langdon and Dawson [85] in 1967, which computes the longitudinal

field 𝐸𝑥 by solving Poisson’s equation, and finds the transverse fields through Maxwell’s equations

(see [35], chapter 6 for more details). This is a somewhat involved process, and still leaves the user

in one spatial dimension.

The need to consider electromagnetic conditions in multiple dimensions gave rise to one of the

heroes of our story, applied mathematician Kane Yee, who developed one of the leading methods

of solving Maxwell’s Equations over time in 1965 [86], which soon came to be known as the

Yee scheme (see Appendix D.1 for more details). While Yee was only concerned with simulating

Maxwell’s equations without particles, soon the plasma community would put this scheme to good

work, with Buneman [87] in 1968 and Morse and Nielson [88] in 1971. Later on Langdon would go

on to develop an implicit field solver [89] in 1985.

PIC would become fully codified in the 1970s [80]. This decade began with another large step in

26

the right direction. It was Langdon and Birdsall who identified the idea of a “finite-particle shape”

in which a particle was no longer understood as a point particle represented by charge multiplied

by a Dirac delta function, 𝑞𝛿(x − x(cid:48)), but rather as a collection of particles, or cloud, and as such a

shape function multiplied by a charge, so 𝑞𝑆(x − x(cid:48)) [90]. We are no longer representing individual

particles, but rather clouds of particles bundled together in a single macroparticle that we evolve

over time. This increases our capacity to simulate many particles, resulting in a flexible method.

Additionally, being able to modify shape functions in different dimensions will be used to preserve

the continuity equation, which will be of vital importance. While this is a CIC concept, as mentioned

previously this is easily mappable to a PIC framework, where 𝑆 is viewed as the interpolant of a

discrete point charge rather than the physical nature of what the macroparticle represents.

Also during this time, a number of accurate 3d particle pushers were developed, the most famous

of which was developed by Boris in 1970 [91], but another, less well known, was also developed

by Buneman [92]. The Boris push is further explored in Appendix D.2. These methods, though

accurate and reliable, are not implicit and thus have the setback of requiring a timestep of 𝜔 𝑝Δ𝑡 < 2.

We must wait until the 80s for implicit particle integrators to come onto the scene and dodge this

issue (though not without their own drawbacks, they required timesteps in proportion with their

maximum wavelength, 𝑘𝑣Δ𝑡 < 1), when in 1981 several papers came out on this topic, Mason [93],

Friedman, Landon, and Cohen [94], and Denavit [95]. Mason and Denavit both approached this

problem by introducing fluid moment equations for mass and momentum, where Friedman, Langdon,

and Cohen elected to explicitly approximate the future density, linearize it, and compute the its

correction in the advanced field (described in more detail in [96]). See [97] for a good overview of

the methods in 1985, [35], p 205 for a brief summary and [98], chapter 8 for a more exhaustive one.

More will be said about implicit PIC in Section 1.4.4.1.

The 1970s saw the PIC field finally come together, the results of which produced the seminal

texts of Hockney and Eastwood [77] and Birdsall and Langdon [35] being written in 1981 and 1985,

respectively.30 There was a rather large problem confronting the community, however, in that most

30Dr. John Verboncoeur at MSU noted that the work undergirding these texts was concurrent, with [35] being

published later due to Langdon’s efforts to complete the analysis of the implicit model.

27

of the literature was concerned with periodic boundary problems, which does not capture device

simulation. This was the problem Lawson [99] sought to solve, providing the first paper discussing

non-periodic boundary conditions, though he is quick to point out that many had been working on

non-periodic boundaries and as such he is simply giving a voice to this work ([80] also points out

that this paper was the result of The Plasma Device Workshop at Berkeley and featured Birdsall,

Crystal, Kuhn, and Lawson).

We now see how most of the limitations Birdsall elucidated have been eliminated. All but one,

that is, we still are relying on collisionless plasmas. Collisions remained beyond reach for these

techniques, a fact which [35] has a rather frank take on:

It has often been stated that to obtain “collisionless” behavior, the collision times must

be longer than the length of the run. Fortunately this is overly pessimistic. What

appears to be closer to the truth is that collision times should exceed, e.g., instability

exponentiation times and trapping times.

Now, at first glance this indicates that we have no way of taking collisions into account. While

strictly speaking this is true, as we are not including any sort of overt collision operator, at long

range (ie > Δ𝑥) the fields induced by the particles will influence their trajectories, thus a sort of

Coulomb “collision” will take place.

Finally, collisions were introduced using Monte Carlo methods (MC) by individuals such as

Burger, Takizuka, Hirotada, Vahedi, Surendra, Morey, Birdsall, et al [100, 101, 102, 103, 104, 105].

For our purposes we are dealing with collisionless plasmas, so we will not be going into details, but

as this represented the last of the hurdles Birdsall identified, it is worth at least noting the individuals

who first leaped it. We will give collisions a slightly closer look as a future line of inquiry in Section

5.5, some more will be said then.

1.4.4 Later PIC

We have now come to the end of the histories provided by Birdsdall [73] in 1991 and Verboncoeur

[80] in 2005. What does the field look like now? There have been a few recent developments that

are worth looking over.

28

Any PIC solver has two challenges confronting it, solving the fields (represented by Maxwell’s

equations), and solving the particles (represented by the Newton-Lorentz system). As such we will

first examine what has been done to push the particle solvers forward, and then turn to the field

solvers.

1.4.4.1 Implicit PIC

An explicit method carries with it the requirement of keeping its timestep below whatever the

Courant–Friedrichs–Lewy (CFL) limit of the method is. It is typically much more intuitive to

implement and much less costly (per step at least), and therefore it is often the default approach,

most textbooks will present the explicit approach to PIC as the PIC approach. Ripperda et al [106]

do an excellent job comparing five classic explicit, semi-implicit, and implicit particle methods, one

of which, the Boris method, we explore in some detail in Appendix D.2. In the historical discussion,

the implicit field and particle solvers that began development in the 1980s were mentioned [93, 94],

but these have serious drawbacks, as both technological and scholarly development was at a place

such that approximations such as linearizing and lagging were required, which resulted in energy

conservation errors, which in turn leads to numerical heating and cooling [107].

Lapenta and Markidis [108, 109] in 2011 developed a method, which is implicit, can have

electromagnetic phenomena, and is energy conserving. While [108] is not charge conserving, [109]

adds a correction step that regulates charge conservation, reducing error in Gauss’s law. The error

that comes with the lack of charge conservation can be reduced with a technique called divergence

cleaning [110, 111] (see also [112] for a slightly more niche cleaning technique), however, there is

still the issue of light wave dispersion errors [113]. Due to its semi-implicit nature, however, it still

has a CFL to satisfy, and as such is struggles with speed, requiring careful optimization. Lapenta

would later go on in 2017 [114] to combine the approaches of [96] and [115] which resulted in

an energy conserving and unconditionally stable scheme that only increases computation time in

the field solver, the particle solver remains the same as that of an explicit pusher. However, in so

doing charge and momentum conservation were again sacrificed. Again, this can be mitigated by

divergence cleaning, though of course this increases computation time, and, despite the increased

29

adherence to these particular physical laws, reduced accuracy on the whole. Also in 2017 Siddi,

Lapenta, and Gibbon developed a mesh free method for a plasma that recovered the Darwin limit

of Maxwell’s Equations [116]. Out of this work came the asymmetrical Euler method (AEM) of

pushing particles. This is a semi-implicit method which is of particular interest for our purposes, as

it is a particle pusher in terms of the scalar and vector potential rather than the electric and magnetic

fields, which will be of great use when we discuss solutions to Maxwell’s equations in terms of

these potentials (see Section 2.3.1).

Another line of inquiry began with Chacón, Chen, and Barnes in 2011 [107], in which they

developed a fully implicit, energy and charge conserving, electrostatic PIC code using a Jacobian-free

Newton-Krylov (JFNK) [117] solver and employing Cohen et al’s substepping algorithm [118],

which uses a small timestep to move particles and a larger timestep to solve the fields. They were not

exactly momentum conserving; however, their adaptive particle substepping combined with spatial

smoothing keep the errors small. They followed this up in 2013 [119], in which they extend this to

general mapped meshes, still keeping electrostatic and 1D, but outlining how they may extend to

multiple dimensions. In 2014, Chen and Chacón extend this to the Darwin model [120], bringing

this to 2D in 2015 [121] and to curvilinear coordinates in 2016 [122]. Chen, Chacón et al [113]

have more recently (2020) combined the leap-frog method for the fields and the Crank-Nicolson

method for the particle equations, resulting in an exactly energy and charge conserving relativistic

electromagnetic PIC scheme which also overcomes the light wave dispersion errors from which

Lapenta’s method suffers, bringing them back down to the levels that come with explicit PIC. In

the same year, Ricketson and Chacón [123] developed an asymptotically preserving implicit Crank-

Nicolson based method which not only tracks full-orbit motion on small timescales, but also recovers

all first order guiding center drifts and the correct gyroradius on large timescales. Koshkarove et

al [124] built upon this in 2022, noting that, though remarkable, it was done with a brute force

generalized minimal residual method (GMRES) JFNK solver, which is rather expensive. They

managed to remove the JFNK component and replace it with Picard iteration for a 3-4 times speedup.

Chen and Chacón in 2023 developed an implicit, charge and energy conserving PIC method that

30

preserves all drifts, though at the sacrifice that it is back to electrostatic [125]. Very recently, these

authors regained a significant amount of ground, developing a fully implicit local energy and charge

conserving scheme for both the electrostatic and Darwin electromagnetic limit [126].31

1.4.4.2 Structure Preserving Particle Methods

Any Hamiltonian system, such as a charged particle in a field, acts on a symplectic manifold,

so it is desireable for any numerical update scheme to preserve this symplectic structure [131,
132, 133, 134]. What is a symplectic update? Given a vector in phase space z ∈ R2𝑛
matrix 𝐴 ∈ R2𝑛×2𝑛

is called volume preserving if its determinant is unity, a necessary but not

, an update

sufficient condition for it being symplectic, which is when 𝐴𝑇 𝐽 𝐴 = 𝐽,32 where 𝐽 = (cid:169)
(cid:173)
(cid:173)
(cid:171)

This translates over to a Hamiltonian system when we combine the x and P (typically denoted q and

(cid:170)
(cid:174)
(cid:174)
(cid:172)

.33

0

−𝐼𝑛

𝐼𝑛

0

p, respectively) into a single z := (p, q) vector and writing

𝑑z
𝑑𝑡

= 𝐽−1∇H (z).

We see this perfectly captures the Hamiltonian behavior of (1.35)-(1.36). While symplecticity is not

strictly speaking necessary for accurate behavior, it is certainly desirable, as it ensures one aspect of

the physical nature of the plasma.

Several lines of inquiry have followed this direction of research. Squire et al [136] developed a

single spacetime discrete Lagrangian using discrete exterior calculus. It is worth noting they also

preserved Gauss’s law through this, specifically by equating gauge invariance of the action with

31The Darwin model, developed by Charles Galton Darwin, (the grandson of the grandfather of modern evolutionary
biology, Charles Robert Darwin) [127], first considered in a coding perspective by Nielson and Lewis in 1976 [128],
effectively reduces Maxwell’s equations to a set of elliptic equations. This is justified for quasistatic systems, where the
characteristic length of the domain is far greater than the characteristic time, ie ¯𝑣/𝑐 ≡ 𝜖 (cid:28) 1 such that 𝜖 is significant,
but 𝜖 2 is not. Effectively this results in the reduction of Maxwell’s equations to a set of elliptic equations. [129] gives a
good overview of how to derive this system, building on the work of [130].

32This is the symplectic equivalent to 𝐴 being orthogonal, 𝐴𝑇 𝐴 = 𝐴𝑇 𝐼 𝐴 = 𝐼. Orthogonality implies the identity 𝐼 is

preserved, symplecticity that 𝐽 is preserved.

33Note that 𝐽2 = −𝐼, which should bring to mind the imaginary unit. In fact, “symplectic” is simply “complex” with
the Greek root “syn-” substituted for the Latin root “con-”, both meaning “with,” while the proto-European root “-plek”
is where we get “to plait.” Not to be confused with “simplex” with the Proto-Indo-European root “sem-” meaning
“one” or “together with” (this all may be found in the Online Etymology Dictionary). Coined by Weyl in [135], “The
name ‘complex group’ formerly advocated by me in allusion to line complexes, as these are defined by the vanishing of
antisymmetric bilinear forms, has become more and more embarrassing through collision with the word ‘complex’ in the
connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective ‘symplectic.’”

31

conserving Gauss’s law and current conservation. Xiao et al [137] would also consider a Lagrangian

that preserved symplectic structure while allowing for a variety of smoothing functions to minimize

numerical noise. He et al [138, 139]34 would develop a Hamiltonian splitting method, decomposing

the Hamiltonian functional into separate components (first five and then reducing it to four, though at

the cost of preserving 𝐾-symplecticity rather than general symplecticity.), solving them in a manner

preserving symplectic structure, and then recombining the results. Having laid this theoretical

groundwork, they would move on to apply this to PIC in a finite element formulation in [142]. This

was an explicit update done in non-canonical form, which Xiao et al then extended to a higher order

in [143] and generalized it to relativistic settings in [144]. Qin et al [145] were able to obtain a

discrete Hamiltonian formulation of the Vlasov-Maxwell system by discretizing a canonical Poisson

bracket. Their approach used the symplectic Euler, which resulted in a semi-implicit method that

required 3 × 3 matrix inversion for each particle, bordering on an explicit update time. Xiao and

Qin have extended the non-canonical geometric PIC scheme to curvilinear coordinates in [146].

Wang et al [147] introduced a structure-preserving relativistic PIC code in canonical coordinates,

generalizing the work of [145].35

Shadwick, Evstatiev, and Stamm have a sequence of papers in which they explore a variational

approach of Low’s Lagrangian [148] to develop a structure-preserving PIC code. They do so in a

variety of ways deriving a non-canonical [149] and canonical [150] Hamiltonian, integrating the

original Lagrangian in Cartesian [151] and cylindrical coordinates [152], and finally developing a

symplectic integrator for a Fourier transformed canonical Hamiltonian [153]. In all cases, energy is

conserved.

A research group out of Max Planck developed a finite element PIC code relying on the semi-

discrete Poisson bracket they derived [154]. This is the generalization of [142]. This geometric solver

was proven to conserve charge and conserve total energy just as well or better than the FDTD-Boris

scheme. This line of inquiry was continued in [155] in which they developed a semi-implicit method

34These are both commonly cited as their arxiv preprints [140, 141] in the early literature, hence the anachronisms.
35It is worth noting throughout all this, within the “et al”, with the exception of [147], Hong Qin is a constant (and

even then, [147] explicitly says Qin’s work is the basis).

32

that conserves energy but at the cost of not satisfying Gauss’s law. They additionally developed as

a second method that conserves both energy while satisfying Gauss’s law, but requires nonlinear

iteration to solve the coupled particle and field equations. This nonlinear iteration is, of course,

more computationally expensive, but conserves charge and satisfies Gauss’s laws for magnetism and

electricity. This is further extended to allow for a wider variety of structure-preserving discretizations

and particle shapes in [156]. Lastly, a geometric PIC method has been developed by Pinto and

Pagés [157] that is known to conserve charge and energy, doing so by taking Lapenta’s semi-implicit

energy-conserving particle pusher [114] and adding a predictor/corrector step. This avoids the

nonlinear solve required in [155].

An interesting capstone to this history is the work of Glasser and Qin in [158]. In this piece they

discuss the relationship between gauge invariance and charge conservation in both the Lagrangian

and Hamiltonian formulation of the Vlasov-Maxwell system, noting that much of the above research

has noticed this phenomenon, but never proven it.

Symplectic integration methods for this non-separable Hamiltonians such as ours are generally

limited to fully-implicit Runge-Kutta type methods [131, 159, 160], which can become prohibitively

expensive for systems with many simulation particles. As an example, the simplest method among

this class of algorithms is the second-order implicit midpoint rule. Recently, an explicit, symplectic

approach with fractional time steps was presented by Tao in [161] that extends phase space by

duplicating variables and prescribes a certain mixing operator to keep these copies “close” together;

however, the numerical experiments they presented did not consider problems with self-fields, so

over time, these copies can drift apart and can lead to certain non-physical behavior (see, e.g., Figures

4.4 and 4.5 in [34]). Additionally, the duplication of phase space variables also applies to the fields

associated with each set of particle data. This makes the approach computationally demanding in

terms of memory usage [162]. Qin et al have shown the possibility of explicit symplectic methods for

product separable (as opposed to sum-separable) Hamiltonians [163, 164], with product separability

defined as either H (q, p) = 𝑞𝑖K (p) or 𝑝𝑖U (q).36 This works fine for a non-relativistic Hamiltonian,

36It is important to note that a Hamiltonian can be composed of multiple sub-Hamiltonians. So if H (q, p) =

(cid:205)𝑖 𝑞𝑖K (p) + (cid:205)𝑖 𝑝𝑖U (q) this would qualify as product-separable.

33

but our relativistic Hamiltonian does not qualify. We will therefore attempt to find a balance between

computational cost, numerical feasibility, and accuracy.

1.4.4.3 Finite Difference Time Domain

In order to represent the fields in this, or in any, method, we must use the curl equations to update

the fields while satisfying the divergence equations. The former is a fairly straightforward task, the

latter is the rub. A common technique for solving partial differential equations (PDEs) is the Finite

Difference (FD) method, which employs Taylor expansions to discretize a domain over space and/or

time, and solves this discretized problem using a variety of numerical techniques (see [165] for a

more thorough examination).37 We must take care, however, if we were to simply Taylor expand the

curl equations in space and time we would find ourselves quickly violating the divergence equations.

This is where the brilliance of the Yee scheme, mentioned above in Section 1.4.3 and discussed

in more detail in Appendix D, comes to play, by staggering the fields and their components (see

Figure D.1 in Appendix D) the divergence equations are automatically satisfied. Initially this was

just used to model electromagnetic waves, but the applications to plasmas quickly became apparent,

and the marriage of the Finite Difference Time Domain (FDTD) to PIC came swiftly. This was not

without some struggle, however. While Yee preserves the divergence equations for electromagnetic

waves, the standard shape functions for most PIC, when combined with Yee, do not satisfy the

continuity equation, which leads to a violation of Gauss’s Law. Now, this need not necessarily

cause too much alarm. As will be shown in Chapter 2, errors in Gauss’s Law do not necessarily

result in a simulation’s particles behaving nonphysically, especially if a bound remains on the error.

Nonetheless, this is something we must be wary of, and in some cases must address. Villasenor and

Buneman [37] developed a charge preserving map that reduces the order of the shape functions but

in so doing preserves continuity (see Appendix E for proof of this feature). Divergence cleaning has

already been mentioned [110, 111, 112] and is a good, albeit somewhat costly, manner of handling

this.

37I am also indebted to Dr. Ameeya Kumar Nayak of the Indian Institute of Technology in Roorkee for his excellent
Youtube series on the Finite Difference Method. This series was my first taste of numerical analysis of PDEs, and it is
one I recommend all first year students review before their Numerical PDEs course.

34

Typical FDTD methods (see [166] for a thorough discussion) are second order in time and space,

though Greenwood et al [167] put forward a fourth order extension to remove Cerenkov radiation, a

numerical dispersion error resulting from high energy particles moving faster than the numerical

waves can propagate. To be clear, they also stress that this is computationally more vexed, and

as such an easier alternative is to just filter the erroneous waves out as shown in Godfrey [168],

Friedman et al [169], and Rambo et al [170]. Hesthaven also has a good review of higher order

methods [171].

One problem that arises with FDTD, and Yee in particular, is the grid’s requirement of uniformity.

While Δ𝑥 can be different than Δ𝑦, changing the two values over space naïvely results in the staggering

of the mesh being thrown off, which in turn results in the violation of the divergence equations. This

means something as simple as rotating a 2d grid, keeping the x and y lines as they are, becomes a

very thorny problem, as the question of what to do with the boundaries arises immediately. The more

straightforward solution is to simply staircase the boundaries, which preserves the structure of the

grid, but does introduce errors (see Section 5.2). The dispersion errors were studied by Cangellaris

and Wright [172], while Verboncoeur [173] looked at the modal content. It is shown that this

staircasing (or sugarcubing in 3d) introduces continuum waves that cannot be resolved on the mesh.

The shorter frequency waves can be dispensed with, but the longer frequency waves are nontrivial

to remove without knowledge of the physical modes. Even disregarding this phenomenon, with this

representation you have an inaccurate grid for the domain if it is supposed to be a perfect rectangle.

This inaccuracy, as well as the aliasing, can be mitigated by increasing the mesh refinement (though

even so, problems will persist), but the CFL for the Yee scheme is such that doing so will make the

simulation run for much longer.

There have been some attempts at remedying this, such as the Contour Path (CP-)FDTD, which

recasts the FDTD algorithm in terms of surface and contour integrals, but this is plagued by late

time instabilities. Railton and Schneider [174] examined a few different approaches have been made

to fix this. Railton and Craddock [175] developed a method that removed the instability by adding a

term to the electric field updates. Anderson et al [176] in 1996 developed a method in which cut

35

cells’ values were modified in proportion to the fraction of their area that was part of the boundary.

Lastly comes Dey, Mittra, and Chebolu in 1997. First in 2d [177] and subsequently in 3d [178],

they found that stability was achieved by distorting cells to bring them within the domain, but only

changing the magnetic update equations, leaving the electric alone. This does come at the price of

reducing the timestep by roughly 70%. [174] found this to be the best alternative as of 1999. See

[179] for an full discussion on CP-FDTD.

More recently Enquist et al [180] found that, by modifying the coefficients of the update equations,

they could have both embedded boundaries and keep energy conservation, the structure of the Yee

grid, and the normal CFL. This is only possible for angles of the nature arctan (1/𝑛) or arctan (𝑛),

where 𝑛 ∈ Z. They do rush to point out that one can extend this to arbitrary angles by approximating

the boundary by piecewise sections keeping this angle requirement.

Within the last few years Law et al [181] developed a FDTD grid with embedded boundaries

by implementing a fourth order Yee grid and a Correction Function Method (CFM), originally

developed to solve Poisson problems with interface jumps [182]. This carries with it a serious

computation cost, as the CFM is a minimization problem that must be solved. Additionally, the

scheme, when directly applied for Perfectly Electrically Conducting boundary conditions would

lead to an ill-posed minimization problem for the CFM, something Law and Nave would go on to

address in [183].

1.4.4.4 Finite Element and PIC

FDTD is intuitive and fairly easy to code, but has several setbacks, among which reside geometry

(discussed above, and note this is just for a rotated rectangular grid), the related difficulty with

refining patches of your domain, the difficulty finite difference has in dealing with discontinuities

(see [165], p 231 for a brief discussion of this), and a restrictive CFL that makes overall refinement

difficult. The Finite Element Method (FEM) is an alternative to the more common FD method to

solving PDEs. FEM, instead of discretizing by Taylor expanding and solving on a finite set of nodes,

forms a variational problem of which it seeks to minimize the energy. From this setup it breaks

the domain into an arbitrary set of subdomains called “elements,” which themselves are typically

36

low-order piecewise polynomial, and approximates the solution on these elements by solving the

resulting system of equations. The advantage here is that the domain may be subdivided as much as

desired, moreover, the refinement need not be uniform. The geometric advantage lies not only in the

nonuniform refinement capabilities, but also by the nature of the element shape. Finite difference

requires its nodes to be organized in uniform lines along the Cartesian axes, whereas the elements in

FEM may take any orientation or shape (triangles being the most common, but far from the only

shape). Additionally, FEM gives a much clearer account of its error bounds. [184] gives a good

examination of the method overall.

Much of the origins of FEM and PIC is wrapped up in Richard True’s research into the so-

called “electron gun” device [185, 186, 187, 188, 189, 190], in which FEM was employed to solve

Poisson’s equation in an electrostatic PIC scenario.38 Additional work was done with Zaki et al

[193, 194] for general Vlasov-Poisson problems in the 1D setting. This FEM work was extended

to electromagnetics first for unstructured meshes around the boundaries (ie a uniform grid39 other

than the boundary elements themselves) [195, 196] and then extended to a general FEM formulation

[197]. Sonnendrücker et al in 1995 [198] introduced a FEM approach to the Darwin limit. Gauss’s

Law for electricity is, however, not satisfied by this, and as such they employ both two divergence

cleaning methods, one a projection in the spirit of the two that we have seen before [110, 111],

and the other a hyperbolic divergence clean found in [199]. Under this scheme we are now solving

elliptic equations, which have no CFL, and so the bound for solving the field equations is gone. They

then combine this field solve with a Boris push. Eastwood et al [200] at the same time developed a

more efficient FEM method that was easily parallelizable.

FEM would soon become a standard approach to PIC, with Discontinuous Galerkin (DG)

becoming the standard bearer of this method (see [201] for a more thorough discussion of DG). In

May 1999, the first international conference on DG was held in Rhode Island, during which the

first paper on approximating Maxwell’s equations was written by Kopriva, Woodruff, and Hussaini

38True’s work has carried on in the MICHELLE codebase [191, 192].
39The motivation here is that a full unstructured requires more bookkeeping, which is memory intensive. The authors
acknowledged in these works that the advent of high performance computing would eventually undo this requirement,
and indeed it did.

37

[202]. After them, Hesthaven and Warburton in 2002 [203] continued to pioneer this particular

technique. Hesthaven [171] gives a good summary of the state of FEM-PIC up to this point in 2003,

including DG, though he is quick to lament the lack of progress made, attributing it to a combination

of difficult technical questions (eg what elements to use, variation form) and the failure of the more

simple FEM formulations. Since then, much work has been done with FEM in general and DG

in particular. Jacobs and Hesthaven in 2006 [204] developed the first FEM-PIC combination not

limited by the Darwin model (see [205] for their benchmarks). Movahhedi et al developed the

first Finite Element Time Domain (FETD) Maxwell solver that implements both the alternating

direction implicit (ADI) and Crank-Nicolson techniques in 2007, both of which are implicit in

time and therefore unconditionally stable [206]. Cheng et al would developed a DG solver for the

Vlasov-Maxwell system [207] which conserved mass and energy (though conservation of energy

requires a careful choice of the numerical flux). Their setup was for semi-discrete total energy

conservation, Cheng, Christlieb, and Zhong [208] developed solvers that in a fully discrete setting

preserve mass and energy for arbitrary time and spatial DG discretizations for the Vlasov-Ampere

system in 2014 and then generalize it back up to the Vlasov-Maxwell system [209]. Very recently

Pinto et al [210] developed a generalized FEM-PIC method which is charge preserving on arbitrary

grids. See Ramachandran et al [211] for a full review. [179] additionally gives a good discussion on

FEM-PIC.

O’Connor, Crawford, et al [212, 213] developed a set of benchmark tests specifically for FEM-

PIC (though it should be noted that these test cases may be abstracted for FDTD-PIC, and in fact we

will see a few of them shortly).

As with all algorithms, there is no free cake, and while FEM does an excellent job handling

geometry and establishing rigorous error bounds, it is noteworthy that it is computationally expensive,

most FEM methods requiring the inversion of a mass matrix at each step. This can be mitigated by

techniques such as preconditioning, but only so much can be done. DG does dodge this by allowing

the elements to be discontinuous at the interfaces, instead taking a page from the Finite Volume

Method (FVMs) and using a numerical flux function to transmit information between the surfaces.

38

It is also worth mentioning that it is difficult to implement, in the words of Hesthaven, “The success

of the finite difference methods for many problems combined with its simplicity also made the finite

element formulation less attractive” [171].

1.4.4.5 Miscellaneous Approaches to the Yee Grid

Zhen, Chen, and Zhang [214] introduced the first unconditionally stable numerical solver of

Maxwell’s equations using the alternating direction implicit (ADI) technique in 1999, stabilizing a

traditional Yee grid. ADI was pioneered by Douglas, Gunn, Peaceman, Pearcy, and Rachford [215,

216, 217, 218, 219, 220] in the 1950s and 60s. Zhen, Chen, and Zhang swiftly followed this success

up with a full 3d code in 2000 [221]. In their words, “the time step used in the simulation is no

longer restricted by stability but by accuracy of the algorithm.” Inspired by this success, Lee and

Fornberg improved upon it by adding a Crank-Nicolson split-step (CNS) method in 2003 [222],

following it up with another improvement in 2004 [223] by way of special time-step sequences [224],

Richardson extrapolation [225], and deferred correction [226].

A different take on FDTD is the finite difference frequency domain (FDFD) is a Fourier trans-

formed Yee grid. It is a flexible method, fairly intuitive, great for resolving fields with sharp

resonances, accurate, and stable [227]. This technique has much of the strengths of the Yee grid, but

instead of stepping forward in time, it solves a linear system in frequency space, Fourier transforming

back to time after. For a full discussion, see [228]. While this completely avoids the issue of

timesteps entirely, it is important to note that this does not necessarily equate to a speedup. The

sparse linear system that results is massive. Additionally, while it incorporates all the strengths of

Yee, it also carries with it its geometrical weaknesses.

1.4.4.6 The Method of Lines (Transpose)

𝑇
We now come to another one of the heroes of our story, the Method of Lines Transpose (MOL

),40

a technique developed by Rothe in 1930 [229]. Of historical interest is the fact that Rothe referred

to this as the Method of Lines (MOL), however, this is not the current situation. What is now

referred to as MOL is a related technique which which dates back to at least the early 1960s in a

𝑇
40As appropriate as MOL

may be notationally speaking, we will be using MOLT from here on out.

39

paper published in the USSR by Sarmin and Chudov in 1963 [230]. The basic idea is, given a PDE

that evolves over time, semidiscretize it in space and then solve each timeline as an initial value

problem (IVP) using whatever method you wish, eg Runge-Kutta (as Sarmin and Chudov did with

the wave equation). Cho [231] begins her history of MOL in 1998, in which Eyre used second-order

centered FD in space to discretize the spatial domain of the Cahn Hilliard (CH) equation [232].

Liu and Shen [233] approach the CH and related Allen-Cahn (AC) equation, but instead opt to

use a spectral deferred correction approach instead, resulting in unconditional stability. Christlieb

et al [234] propose implicitly stepping in time, generating the solution with a conjugate gradient

method. Jones [235] would go on to examine the functionalized CH equation with, among other

things, MOL, applying exponential time differencing [236], which involves discretizing via Fourier

spectral method (or, in the case of non-periodic boundaries, transforming to some other basis such

as Chebyshev polynomials), decomposing the linear and nonlinear parts, transforming the nonlinear

parts back to space, evaluating at the grid points, and then transforming back to spectral space. The

resulting system of stiff ODEs may then be solved in a variety of methods. Further searching reveals

a plethora of different techniques to solve whatever system of ODEs result from the spatial/spectral

discretization, for example Yuan [237] in 1999 describes the parametric finite difference method,

the Karantovitch method (the semi-discrete counterpart of the Ritz method), and the Finite Element

method, developed in 1990 [238]41, which itself is derived from the Karantovitch method. See Yuan

[237] for more details up to 1999, Cho [231] for up to 2016.

MOLT is, unsurprisingly, simply MOL except transposed, discretizing in time rather than space,

solving each discrete timestep as a system of boundary value problems (BVP). The solution at this

timestep will then act as a source term for the subsequent time step. In other words, our update

equation becomes something akin to

L (cid:2)𝑢𝑛+1(cid:3) (𝑥) = ¯𝑆(𝑥, 𝑡),

(1.50)

where L is a linear differential operator and ¯𝑆 is a linear combination of the source term 𝑆(𝑥, 𝑡) of

the original PDE and the previous timesteps of 𝑢. This BVP may be solved in any manner provided

41Frustratingly, this paper is cited by Yuan as the progenitor of the FE approach, but is unavailable online

40

it leaves the solution stable and within desired accuracy.

There is a gap between Rothe’s paper in 1930 and the 1970s when interest in this technique

increased. Kačur published several papers on this in the 1970s [239, 240, 241], where his primary

interest is in nonlinear parabolic problems. Martensen at the same time applied Rothe to a number

of problems which are of interest to us, solving Maxwell’s equations [242] and the wave equation in

several dimensions [243]. Finally, both individuals presented on Rothe’s technique at a conference

in Berlin in 1986, Martensen using it to solve the Cauchy problem for Burger’s equation [244] and

Kačur staying true to form and primarily addressing nonlinear parabolic problems [245]. Amusingly,

neither men appear in the other’s bibliographies.

There is one technique worth discussing before we consider a Green’s function approach to

solving these BVPs, that of Bruno and Lyon’s Fourier Continuation - Alternating Direction (FC-AD).

Bruno and Lyon developed the FC-AD method in 2010 [246, 247]. This technique is based on the

ADI method of approaching MOLT, combined with Fourier continuation, which was developed

by Bruno, Boyd, Han, and Pohlman in the 2000s [248, 249]. ADI is used to semi-discretize in

time, while the spatial domain is solved spectrally. Now, if the domain is nonuniform, this would

result in Gibbs phenomenon. Adding Fourier continuation avoids this obstacle. This results in a

least squares problem that [249] used a singular value decomposition to solve. However, noting the

computational complexity of this, and that they are in a 1D context, they use a modification involving

Gram polynomials that results in a faster algorithm. As such, FC-AD is able to handle complicated

geometry while maintaining its unconditional stability. In addition, it has O (𝑁 log (𝑁)) runtime and

brings with it a very impressive order convergence, eg solving the wave equation with fourth-order

in time, sixth-order in space accuracy and completely eliminating dispersion error. It is worth noting

that ADI typically struggles with hyperbolic problems, but this FC(Gram) method enables them to

do so with remarkable accuracy and efficiency. They extended this to handle nonuniform coefficients

[250], and to compressible Navier-Stokes and non-linear acoustics [251]. [252] focused on waves,

particularly electromagnetic phenomena, tweaking their method to consider a frequency and time

hybrid integral equation method, and again finding they could eliminate numerical dispersion and

41

handle nontrivial geometries.42 Additional emphasis was placed on parallelization possibilities.

Recent work has been done to replace the Gram polynomials with Hermite polynomials, resulting

in elimination of expensive precompute of the extension data and gives provable error bounds

[253].43 So the ADI method is employed to semi-discretize in time, and the Fourier Continuation

by means of Gram polynomials is used to solve the BVPs. The main takeaway here though is that

the dimensions are split in the ADI methodology, which brings with it its own errors, but greatly

reduces the complexity of the problem. This key insight will yield dividends shortly.

Back to solving BVPs. For our purposes we are interested in solving this sequence of BVPs that

come with MOLT via an analytic inversion of L with a Green’s function 𝐺:

𝑢𝑛+1(𝑥) =

∫

Ω

𝐺 (𝑥; 𝑥(cid:48)) ¯𝑆(𝑥(cid:48))𝑑𝑥(cid:48).

(1.51)

This technique has been used across a number of fields, including acoustics [254], fluid dynamics

[255, 256, 257], diffusion, Allen-Cahn, and Cahn-Hilliard [258, 259, 231], and, of particular interest

to us, electromagnetics [260]. In particular, we are interested in using this method to solve a set of

wave equations described in Section 1.3.3.

Now, Cho [231] points out that, if we wished, we could simply form a linear system out of this

using quadrature

𝑢𝑛+1(𝑥 𝑗 ) =

∫

Ω

𝐺 (𝑥 𝑗 ; 𝑥(cid:48)) ¯𝑆(𝑥(cid:48))𝑑𝑥(cid:48) ≈

𝑁
(cid:213)

𝑖=1

𝐺 (cid:0)𝑥 𝑗 ; 𝑥𝑖(cid:1) 𝑓𝑖Δ𝑥𝑖, 𝑗 = 1, ..., 𝑁.

(1.52)

However, though straightforward enough, this is computationally infeasible for even moderately
sized grids, being O (cid:0)𝑁 2(cid:1). There are two ways of approaching this challenge we will go over. The

Fast Multipole Method (FMM) and the dimensional splitting approach.

The FMM was developed first by Greengard and Rokhlin [261] to evaluate the Coulomb potential

and force fields in a system of particles. Greengard would later go on to extend this to other potential

functions with Huang [262]. This method was used by Kropinksi and Quaife to evaluate the modified

Helmholtz equation using a Green’s function [263], which they would then go on to use in solving

42Frustratingly, the promised Part II has not been published as of April 2025.
43Under review as of April 2025.

42

diffusion problems [258] using the MOLT framework. Kropinski would go on to extend this to a

Navier-Stokes solver in [257]. One issue with this method, however, is how it fits in with modern

computing. FMM is predicated on receiving information from the entire domain, meaning parallel

processing is a vexed challenge at best, as communication between processes will be intensive.

FC-AD introduced the idea of dimension splitting, however. The combination of operator

splitting with an analytic inversion a la Green’s function has been used extensively by the Christlieb

group. Causley et al [264] and the thesis of Cho [231], both in the 2010s, explored its applications

to parabolic problems. Hyperbolic problems were more thoroughly explored by Causley, Christlieb,

Groningen, Ong, and Yaman. [265] introduced the method as a solution to the wave equation using

ADI. [266] followed improved upon this by replacing the Lax correction it used by more accurate

quadrature. [267] extended MOLT to a family of wave solvers of 2𝑃 order accuracy that run in
O (cid:0)𝑃𝑑 𝑁 (cid:1) steps per timestep, where 𝑁 is the number of grid points in one direction and 𝑑 is the

number of dimensions, and explored its performance with complex geometries. Finally, [268]

expanded the boundary conditions in 1 and 2D, including outflow. It also explored embedded

boundaries for Neumann and Dirichlet boundary conditions. It is worth noting that this itself

helps with computational efficiency, computing a sequence of one dimensional integrals is faster

than computing a multi-dimensional system, however, even here improvements have been made.

[269] developed a fast convolution algorithm that was not only computationally efficient, but also

easily parallelizeable, requiring only the nearest neighbor (or “halo”) nodes to pass information,

unlike that of FMM. We will discuss this method more thoroughly in Section 2.2. We then see it

extended to PIC methods in Wolf, Causley, Christlieb, and Bettencourt [270], going on to form the

backbones of the dissertations of Wolf [271] and Sands [34]. One may also see a prototype of a

MOLT-PIC code in [272, 273], both of which used a Green’s function to solve Poisson’s equation

to acquire the potential and implemented this using a treecode. Very recently there has been work

in comparing the simulations of the plasma simulations done by the FDTD-PIC and MOLT with

Improved Asymmetrical Euler Method [2, 3, 4], using, among others, the toy problems developed in

[212], and it is to this we will devote the bulk of the study.

43

1.5 Conclusion

In this chapter we considered first what a plasma is and why it is we are interested in this form

of matter. We then turned to consider the basic equations governing a plasma. The Vlasov equation

(1.6) governs the particle distribution function over phase space over time, and relies on particle

location in phase space as well as field information to update itself. The fields are governed by

Maxwell’s equations, (1.7), and the particles are governed by the equations of motion described by

Newton (1.5) and Lorentz (1.4). We discussed how all of these may be cast in terms of vector and

scalar potentials under the Lorenz gauge. Finally, we began examining the technique with which this

dissertation is concerned, that of Particle-in-Cell, or PIC, and how it takes both of these components,

fields and particles, into consideration as it updates the plasma in question.

After this discussion, we turned to review what those who came before developed, as it is their

shoulders upon which this thesis stands. It is clear that the study of plasmas and the computational

simulation of them, despite only having existed for slightly over a century (less for the simulation

science), has burgeoned into a rich field with an extensive history and literature.

We are now motivated by some practical examples of plasmas in nature, informed of the laws

governing these plasmas we wish to accurately simulate, and inspired by the scholars that came

before us. We now turn to consider a new PIC method.

44

CHAPTER 2

A NEW PARTICLE-IN-CELL METHOD

The above formulated PIC algorithm has two primary requirements: a wave solver and a particle

pusher. In this chapter, we first give a very brief review of the equations we are interested in solving

in Section 2.1. We then describe the algorithm used for wave propagation in Section 2.2 as well as

the particle pusher in Section 2.3. After establishing these we will examine the numerical results in

Section 2.4, comparing them to a standard PIC method which uses the Yee grid as a wave solver and

the Boris push as the particle solver. Details on the Yee wave solver and Boris push may be found in

Appendix D. We conclude with a brief summary in Section 2.5. This chapter is based on the work

published by myself in collaboration with Dr. Andrew Christlieb and Dr. William Sands [2]. My

contributions in this work was the development of the Yee-Boris code against which we compared

our novel method, as well as the implementation of the sheath problem using our method.

2.1 Problem Review

The vector and scalar potential formulation of Maxwell’s equations and the Lorentz force has

already been discussed thoroughly in Section 1.3. Here we simply repeat the equations to save the

reader from the hassle of flipping back and forth between chapters. Firstly, we have the Lorentz

Force which governs the motion of a charged particle in an electric (E) and magnetic (B) field:

𝑑p
𝑑𝑡

= F = 𝑞 (E + v × B)

(2.1)

This is relativistically correct if p = 𝛾𝑚v. We have the relativistic form of Newton’s second law:
𝑑p
𝑑𝑡

= 𝛾𝑚a.

F =

(2.2)

The electric and magnetic fields must satisfy Maxwell’s equations:

,

∇ · E =

𝜌
𝜖0
∇ · B = 0,

∇ × E = −

∇ × B =

+ 𝜇0J.

,

𝜕B
𝜕𝑡
𝜕E
𝜕𝑡

1
𝑐2

45

(2.3a)

(2.3b)

(2.3c)

(2.3d)

Under the Lorenz gauge, the above may be rewritten in terms of a scalar (𝜙) and vector (A)

potential:

𝑑x𝑖
𝑑𝑡

=

(cid:113)

𝑑P𝑖
𝑑𝑡

=

(cid:113)

𝑐2 (P𝑖 − 𝑞𝑖A)
𝑐2 (P𝑖 − 𝑞𝑖A)2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2
𝑞𝑖𝑐2 (∇A) · (P𝑖 − 𝑞𝑖A)
𝑐2 (P𝑖 − 𝑞𝑖A)2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2

− Δ𝜙 =

,

𝜌
𝜖0

− ΔA = 𝜇0J,

+ ∇ · A = 0,

𝜕2𝜙
𝜕𝑡2
𝜕2A
𝜕𝑡2
𝜕𝜙
𝜕𝑡

1
𝑐2

1
𝑐2
1
𝑐2
𝜕 𝜌
𝜕𝑡

+ ∇ · J = 0.

,

− 𝑞𝑖∇𝜙,

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)






Equations (2.4) and (2.5) are the particle equations of motion, (2.6) and (2.7) are the scalar and

vector potential wave equations, (2.8) is the Lorenz gauge condition we must satisfy, and (2.9) is the

continuity equation we will ideally satisfy as well. The nondimensionalization of these equations

may be found in Appendix A.

Now that we are properly refreshed on the equations governing the system we wish to simulate,

we now need two main components to do so, a wave solver and a particle pusher. We now turn to

consider the wave solver.

2.2 Numerically Solving the Wave Equation

We begin with a brief discussion of integral equations in Section 2.2.1, which is helpful for

introducing the dimensionally-split methods considered in this paper. The semi-discrete form of the

solver considered in this work is presented in Section 2.2.2.1, and its stability is discussed in Section

2.2.2.2. A solution is formulated in terms of one-dimensional operators that can be inverted using

the methods discussed in Section 2.2.3. We then discuss the methods used to obtain derivatives in

2.2.4 and demonstrate the application of boundary conditions in Section 2.2.5. These derivatives

will be shown in Section 2.4.1 to have the same temporal and spatial convergence rates as the fields.

46

2.2.1 Integral Equation Methods and Green’s Functions

Integral equation methods are a powerful class of techniques used to solve boundary value

problems (BVPs) in a range of applications, and has been discussed in Section 1.4.4.6. Such

methods allow one to write an explicit solution of a PDE in terms of a Green’s function (see

Appendix G for more details). While explicit, this solution can be difficult or impossible to evaluate,

so numerical quadrature is used to evaluate these terms. Layer potentials can then be introduced in

the form of surface integrals to adjust the solution to satisfy the prescribed boundary data [274]. We

illustrate these features with an example that is the basis for the method presented in this work.

Suppose that we are solving the following modified Helmholtz equation

(cid:18)

I −

(cid:19)

Δ

1
𝛼2

𝑢(x) = 𝑆(x),

x ∈ Ω,

(2.10)

where Ω ⊂ R𝑛

and I is the identity, Δ is the Laplacian in R𝑛

, 𝑆 is a source, and 𝛼 ∈ R is a parameter.

While this method can be broadly applied to other elliptic PDEs, equation (2.10) is of interest to us

because it can be obtained from the time discretization of a parabolic or hyperbolic PDE. In this

case, the source function can include additional time levels of 𝑢 and the parameter 𝛼 = 𝛼(Δ𝑡) is

connected to the time discretization of this problem.

To apply a Green’s function method to equation (2.10), one first identifies the function 𝐺 (x, y)

that solves

(cid:18)

I −

(cid:19)

Δ

1
𝛼2

𝐺 (x, y) = 𝛿 (x − y) ,

x, y ∈ R𝑛,

(2.11)

over free-space, with 𝛿 (x − y) being the Dirac delta distribution. The construction of fundamental

solutions is extensively tabulated for many different operators, including the modified Helmholz

operator [275].1 For details on how this is derived for the modified Helmholtz equation, see Appendix

G. The fundamental solution 𝐺 (x, y), which solves (2.11) can be used to build a solution to the

original problem (2.10). First, let 𝑢 be a solution of the problem (2.10). Multiplying the equation

(2.11) by 𝑢, integrating over Ω, and applying the divergence theorem (or integration by parts in the

1[275] has the Helmholtz operator, not the modified one, but the logic is the same.

47

one-dimensional case) leads to the integral identity

𝑢(x) =

∫

Ω

𝐺 (x, y)𝑆(y) 𝑑𝑉y +

1
𝛼2

∫

(cid:18)

𝜕Ω

𝐺 (x, y)

𝜕𝑢
𝜕n

− 𝑢(y)

(cid:19)

𝜕𝐺
𝜕n

𝑑𝑆y,

(2.12)

where we have used the assumption that the function 𝑢 solves the PDE (2.10). Since the volume

integral term does not enforce boundary conditions, the surface integral contributions involving 𝑢

are replaced with an ansatz of the form

𝑢(x) =

∫

Ω

𝐺 (x, y)𝑆(y) 𝑑𝑉y +

∫

(cid:18)

𝜕Ω

𝜎(y)𝐺 (x, y) + 𝛾(y)

(cid:19)

𝜕𝐺
𝜕n

𝑑𝑆y,

(2.13)

where 𝜎(y) and 𝛾(y) are the single- and double-layer potentials, which are used to enforce the

boundary conditions. The choice of names is reflected by the behavior of the Green’s function

associated with each of the terms. The Green’s function itself is continuous, but its derivative will

have a “jump.” Based on the boundary conditions, one selects either a single or double layer form as

the ansatz for the solution. The single-layer is used in the Neumann problem, while the double-layer

is chosen for the Dirichlet problem.

The field solver considered in this work factors the multi-dimensional Laplacian into a product

of one-dimensional operators, which are inverted in a dimension-by-dimension manner using the

one-dimensional form of (2.13) [265]. The resulting methods solve for something that looks like a

layer potential, with the key difference being that the linear system is now only a small, 2 × 2 matrix,

which can be inverted by hand, rather than with an iterative method. Each integral over the one-

dimensional line segment can be computed using a lightweight, recursive, fast summation method.

This approach allows us to accommodate geometry with ease, as the domain can be represented

using one-dimensional line segments whose termination points coincide with the geometry [276].

2.2.2 Description of the Wave Solver

Unlike the high-order time accurate methods from Causley et al [267], which are based on succes-

sive convolution, we discretize the time derivatives of the wave equation using a first-order accurate

backward difference formula (BDF). Again, we wish to emphasize that the spatial discretization in the

fully-discrete method is either fourth or fifth-order accurate, so we retain high-order spatial accuracy.

Brief sections concerning the stability and phase error analyses of the semi-discrete method are

48

presented. We briefly discuss the splitting technique that reduces multi-dimensional problems into a

sequence of one-dimensional updates and show how boundary conditions are applied.

2.2.2.1 The Semi-discrete BDF Scheme

To derive the BDF wave solver, we start with the equation

1
𝑐2

𝜕𝑡𝑡𝑢 − Δ𝑢 = 𝑆(x, 𝑡),

(2.14)

where 𝑐 is the wave speed and 𝑆 is a source function. Then, using the notation 𝑢(x, 𝑡𝑛) = 𝑢𝑛
, we can
apply a three-point backwards finite-difference stencil for the second derivative about time level 𝑡𝑛+1

to obtain

𝑢𝑛+1 − 2𝑢𝑛 + 𝑢𝑛−1
Δ𝑡2
where Δ𝑡 = 𝑡 𝑘 − 𝑡 𝑘−1, for any 𝑘, is the grid spacing in time. Evaluating the remaining terms in
equation (2.14) at time level 𝑡𝑛+1, and inserting the above difference approximation, we obtain

(cid:12)
(cid:12)
(cid:12)𝑡=𝑡𝑛+1

+ O (Δ𝑡),

𝜕𝑡𝑡𝑢

=

1
𝑐2Δ𝑡2

(cid:16)

𝑢𝑛+1 − 2𝑢𝑛 + 𝑢𝑛−1(cid:17)

− Δ𝑢𝑛+1 = 𝑆𝑛+1(x) + O (Δ𝑡) ,

which can be rearranged to obtain the semi-discrete equation

(cid:18)

I −

(cid:19)

Δ

1
𝛼2

𝑢𝑛+1 =

(cid:16)

2𝑢𝑛 − 𝑢𝑛−1(cid:17)

+

1
𝛼2

𝑆𝑛+1(x) + O

(cid:19)

(cid:18) 1
𝛼3

, 𝛼 :=

1
𝑐Δ𝑡

.

(2.15)

We note that the source term is treated implicitly in this method, which creates additional com-

plications if the source function 𝑆 depends on 𝑢. This necessitates some form of iteration, which

increases the cost of the method.

2.2.2.2 Stability and Dispersion Analysis of the Semi-discrete BDF Scheme

We now analyze the stability of the first-order semi-discrete BDF scheme given by equation

(2.15). Suppose that the solution 𝑢(x) takes the form of the plane wave given by

𝑢𝑛 (x) = 𝑒𝑖(k·x−𝜔𝑡𝑛) ≡ 𝜆𝑛𝑒𝑖k·x.

Substituting this ansatz into equation (2.15) and ignoring contributions due to sources, we obtain

the polynomial equation

(cid:16)

1 + 𝑧2(cid:17)

𝜆2 − 2𝜆 + 1 = 0.

49

In the above equation, we have defined the real number 𝑧2 = |k|2/𝛼2 for simplicity. The roots of this

polynomial are a pair of complex conjugates that can be written as

𝜆± =

1
1 + 𝑧2

(cid:16)

1 ± 𝑖(cid:112)𝑧2(cid:17)

,

which satisfy the condition |𝜆±| ≤ 1 for any Δ𝑡. This shows that the amplitude of the plane wave

does not grow in time, so the scheme is unconditionally stable.

The phase error introduced by the semi-discrete scheme can also be determined by first noting

that 𝑡𝑛 = 𝑛Δ𝑡, so that

𝜆 = 𝑒−𝑖𝜔Δ𝑡 ⇐⇒ 𝜔 = −

1
𝑖Δ𝑡 log (𝜆) .

(2.16)

Then, we insert the factor 𝜆+ into equation (2.16) and expand the resulting expression into a series

about 𝑧 = 0, which gives

𝜔+ = −

= −

1
𝑖Δ𝑡 log (𝜆+) ,
(cid:18)
𝑧2
1
−𝑖𝑧 −
𝑖Δ𝑡

2

+

𝑖𝑧3

3

+

𝑧4

4

+ O (𝑧5)

(cid:19)

,

𝑧 (cid:28) 1.

Since 𝑧 = |k|/𝛼 = 𝑐|k|Δ𝑡, the last expression can be further simplified to

𝜔+ = 𝑐|k| −

𝑖𝑐2|k|2Δ𝑡
2

−

𝑐3|k|3Δ𝑡2
3

+

𝑖𝑐4|k|4Δ𝑡3
2

+ O (𝑐5|k|5Δ𝑡4).

Since the analytical dispersion relation for the plane wave solution is 𝜔 = 𝑐|k|, the phase error is

𝜔 − 𝜔+ =

𝑖𝑐2|k|2Δ𝑡
2

+ O

(cid:16)

𝑐3|k|3Δ𝑡2(cid:17)

.

Moreover, since the leading order term in the error is imaginary, this mode decays with time, which

introduces dissipation into the scheme. Similar behavior is observed with the factor 𝜆−, so we

exclude it from the discussion.

2.2.2.3 Splitting Method Used for Multi-dimensional Problems

The semi-discrete equation (2.15) is a modified Helmholtz equation of the form (2.10). Instead

of appealing to (2.13), which formally inverts the multi-dimensional modified Helmholtz operator,

50

we apply a factorization into a product of one-dimensional operators. For example, the factorization

used in two-spatial dimensions is given by

𝛼2I − Δ = 𝛼2

(cid:18)

I −

1
𝛼2

𝜕𝑥𝑥

(cid:19) (cid:18)

I −

(cid:19)

𝜕𝑦𝑦

1
𝛼2

+

1
𝛼2

𝜕𝑥𝑥𝜕𝑦𝑦,

≡ 𝛼2L𝑥L𝑦 +

1
𝛼2

𝜕𝑥𝑥𝜕𝑦𝑦,

where L𝑥 and L𝑦 are one-dimensional operators and the last term represents the splitting error

associated with the factorization step. Note that the coefficient of the splitting error is 1/𝛼2 = O (Δ𝑡2),

which will not affect the global truncation error for the time discretization considered in this paper.

However, for time discretizations higher than second-order accuracy, one must address the splitting

error. Dropping the error terms, the semi-discrete equation (2.15) can be written more compactly as

L𝑥L𝑦 (cid:2)𝑢𝑛+1(cid:3) (x) = 2𝑢𝑛 (x) − 𝑢𝑛−1(x) +

1
𝛼2

𝑆𝑛+1(x).

(2.17)

Next, we discuss the procedure used to invert the one-dimensional operators used in the above

factorization.

2.2.3 Inverting One-dimensional Operators

The choice of factoring the multi-dimensional modified Helmholtz operator means we now have

to solve a sequence of one-dimensional BVPs of the form

(cid:18)

I −

(cid:19)

𝜕𝑥𝑥

1
𝛼2

𝑤(𝑥) = 𝑓 (𝑥),

𝑥 ∈ [𝑎, 𝑏],

(2.18)

where [𝑎, 𝑏] is a one-dimensional line and 𝑓 is a new source term that can be used to represent a

time history or an intermediate variable constructed from the inversion of an operator along another

direction. We will show the process by which one obtains the general solution to the problem (2.18),

deferring the application of boundary conditions to Section 2.2.4. Further, this section also discusses

the construction of spatial derivatives.

51

2.2.3.1 Integral Solution

Since the BVP (2.18) is linear, its general solution can be expressed using the one-dimensional

analogue of equation (2.12):

𝑤(𝑥) =

∫ 𝑏

𝑎

𝐺 (𝑥, 𝑦) 𝑓 (𝑦) 𝑑𝑦 +

1
𝛼2

(cid:2)𝐺 (𝑥, 𝑦)𝜕𝑦𝑢(𝑦) − 𝑢(𝑦)𝜕𝑦𝐺 (𝑥, 𝑦)(cid:3)

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑦=𝑏

,

𝑦=𝑎

where the free-space Green’s function in one-dimension (derived in Appendix G) is

𝐺 (𝑥, 𝑦) =

𝑒−𝛼|𝑥−𝑦|.

𝛼

2

(2.19)

(2.20)

The solution (2.19) requires the derivatives of the Green’s function near the boundary. Noting that

𝜕𝑦𝐺 (𝑥, 𝑦) =

𝛼

2

−

𝛼

2






𝑒−𝛼(𝑥−𝑦),

𝑥 ≥ 𝑦,

𝑒−𝛼(𝑦−𝑥),

𝑥 < 𝑦,

after taking limits, we find that

𝜕𝑦𝐺 (𝑥, 𝑦) =

𝑒−𝛼(𝑥−𝑎),

𝛼

2

𝜕𝑦𝐺 (𝑥, 𝑦) = −

𝑒−𝛼(𝑏−𝑥).

𝛼

2

lim
𝑦→𝑎

lim
𝑦→𝑏

Combining these limits with (2.19), we obtain the general solution

𝑤(𝑥) =

𝛼

2

∫ 𝑏

𝑎

𝑒−𝛼|𝑥−𝑦| 𝑓 (𝑦) 𝑑𝑦 + 𝐴𝑒−𝛼(𝑥−𝑎) + 𝐵𝑒−𝛼(𝑏−𝑥).

(2.21)

The constants 𝐴 and 𝐵 are determined by boundary conditions and serve the same purpose as the

layer potentials in (2.13). Further, we identify the general solution (2.21) as the inverse of the

one-dimensional modified Helmholtz operator. In other words, we define L−1
𝑥

so that

𝑤(𝑥) = L−1
𝑥
∫ 𝑏

𝛼

[ 𝑓 ] (𝑥),

≡

2

𝑎

𝑒−𝛼|𝑥−𝑦| 𝑓 (𝑦) 𝑑𝑦 + 𝐴𝑒−𝛼(𝑥−𝑎) + 𝐵𝑒−𝛼(𝑏−𝑥),

≡ I𝑥 [ 𝑓 ] (𝑥) + 𝐴𝑒−𝛼(𝑥−𝑎) + 𝐵𝑒−𝛼(𝑏−𝑥).

52

(2.22)

(2.23)

(2.24)

Section 2.2.4 will make repeated use of definitions (2.22)-(2.24) in the construction of spatial

derivatives and the application of boundary conditions. The integral operator I𝑥 [ 𝑓 ] (𝑥) is evaluated

as

where the integrals

I𝑥 [ 𝑓 ] (𝑥) =

𝑥 [ 𝑓 ] (𝑥) + I 𝐿
I 𝑅

𝑥 [ 𝑓 ] (𝑥)

(cid:17)

.

(cid:16)

1
2

I 𝑅
𝑥 [ 𝑓 ] (𝑥) ≡ 𝛼

I 𝐿
𝑥 [ 𝑓 ] (𝑥) ≡ 𝛼

∫ 𝑥

𝑎
∫ 𝑏

𝑥

𝑒−𝛼(𝑥−𝑦) 𝑓 (𝑦) 𝑑𝑦,

𝑒−𝛼(𝑦−𝑥) 𝑓 (𝑦) 𝑑𝑦,

(2.25)

(2.26)

(2.27)

are computed with a recursive fast summation method, to which we now turn.

2.2.3.2 Fast Convolution Algorithm

Computing the above integral may be accomplished in a very efficient manner, and has been

well explored as discussed in Section 1.4.4.6. We exploit a useful property of both left and right

moving convolutions:2

𝑥 [𝑢] (𝑥) = I 𝐿
I 𝐿

𝑥 [𝑢] (𝑥 − 𝛿𝐿) 𝑒−𝛼𝛿𝐿 + J 𝐿
𝑥

[𝑢] (𝑥), J 𝐿

𝑥 [𝑢] (𝑥) := 𝛼

𝑥 [𝑢] (𝑥) = I 𝑅
I 𝑅

𝑥 [𝑢] (𝑥 + 𝛿𝑅) 𝑒−𝛼𝛿𝑅 + J 𝑅
𝑥

[𝑢] (𝑥), J 𝑅

𝑥 [𝑢] (𝑥) := 𝛼

∫ 𝑥

𝑥−𝛿𝐿

𝑢 (𝑦) 𝑒−𝛼(𝑥−𝑦) 𝑑𝑦,

(2.28)

∫ 𝑥+𝛿𝑅

𝑥

𝑢 (𝑦) 𝑒−𝛼(𝑦−𝑥) 𝑑𝑦.

(2.29)

This is easily proven:

I 𝐿
𝑥 [𝑢] (𝑥) = 𝛼

= 𝛼

= 𝛼

∫ 𝑥

𝑒−𝛼(𝑥−𝑦)𝑢(𝑦)𝑑𝑦

𝑎

∫ 𝑥−𝛿𝐿

𝑎

∫ 𝑥−𝛿𝐿

𝑎

𝑒−𝛼(𝑥−𝑦)𝑢(𝑦)𝑑𝑦 + 𝛼

∫ 𝑥

𝑥−𝛿𝐿

𝑒−𝛼(𝑥−𝑦)𝑢(𝑦)𝑑𝑦

𝑒−𝛼(𝑥−𝛿𝐿−𝑦)𝑢(𝑦)𝑒−𝛼𝛿𝐿 𝑑𝑦 + 𝛼

∫ 𝑥

𝑥−𝛿𝐿

𝑒−𝛼(𝑥−𝑦)𝑢(𝑦)𝑑𝑦

(2.30)

= I 𝐿

𝑥 [𝑢] (𝑥 − 𝛿𝐿) 𝑒−𝛼𝛿𝐿 + J 𝐿

𝑥 [𝑢] (𝑥).

2It is worth mentioning different papers will put this in a slightly different way. For example [268] uses a change of
variables to reorient the integrals to range from 0 to 𝛿{ 𝐿,𝑅}. This takes the 𝛿{ 𝐿,𝑅} notation from [265], but leaves the
integral oriented around 𝑥 as in [270, 269]. Its coefficients are 𝛼 rather than 𝛼/2 due to how the original I operator is
defined.

53

The right moving operator follows likewise. Typically 𝛿𝐿 and 𝛿𝑅 are quite small, in our case they

are the Δ𝑥 grid spacing. They need not be uniform, so long as all the vertices properly match up (ie

the mesh is a tensor product of a line of non-necessarily uniform nodes). We discretize according to

the grid nodes:

J 𝐿
𝑗

:= 𝛼

J 𝑅
𝑗

:= 𝛼

∫ 𝑥 𝑗

𝑒−𝛼 (cid:0)𝑥 𝑗 −𝑦(cid:1)𝑢(𝑦)𝑑𝑦,

𝑥 𝑗 −1
∫ 𝑥 𝑗+1

𝑥 𝑗

𝑒−𝛼 (cid:0)𝑦−𝑥 𝑗 (cid:1)𝑢(𝑦)𝑑𝑦.

We compute J {𝐿,𝑅}

𝑗

using quadrature as follows:

𝑗 ≈ 𝑃𝑢 (cid:0)𝑥 𝑗 (cid:1)) + 𝑄𝑢 (cid:0)𝑥 𝑗−1
J 𝐿

(cid:1) − 𝑅 (cid:0)𝑢 (cid:0)𝑥 𝑗+1

(cid:1) − 2𝑢 (cid:0)𝑥 𝑗 (cid:1) + 𝑢 (cid:0)𝑥 𝑗−1

(cid:1)(cid:1) ,

𝑗 ≈ 𝑃𝑢 (cid:0)𝑥 𝑗 (cid:1)) + 𝑄𝑢 (cid:0)𝑥 𝑗+1
J 𝑅

(cid:1) − 𝑅 (cid:0)𝑢 (cid:0)𝑥 𝑗+1

(cid:1) − 2𝑢 (cid:0)𝑥 𝑗 (cid:1) + 𝑢 (cid:0)𝑥 𝑗−1

(cid:1)(cid:1) .

The coefficients are defined as follows, with 𝜈 := 𝛼Δ𝑥 𝑗 and 𝑑 = 𝑒−𝜈

:

𝑃 = 1 −

𝑄 = −𝑑 +
1 − 𝑑
𝜈2

𝑅 =

,

1 − 𝑑
𝜈
1 − 𝑑
𝜈
1 + 𝑑
2𝜈

−

,

.

(2.31)

(2.32)

(2.33)

(2.34)

(2.35)

(2.36)

(2.37)

For deriving these weights, see Appendix H.

Importantly, the only information needed is nearest-neighbor, or a “halo” of nodes. If we have an
error tolerance 𝜖 (cid:28) 1, we need only constrain 𝛼 such that 𝑒−𝛼𝐿𝑚 ≤ 𝜖, where 𝐿𝑚 = min𝑖 (Δ𝜂𝑖) , 𝜂 ∈
{𝑥, 𝑦, 𝑧} is the smallest length between nodes. ie −𝛼 ≤ log(𝜖)

𝐿𝑚 . So long as this is preserved, we need

only consider the halo of neighbors, making this easy to parallelize and even more efficient [269].

2.2.4 Methods for the Construction of Spatial Derivatives

To set the stage for the ensuing discussion, note that the semi-discrete update for the first-order

BDF method, in one-spatial dimension, can be obtained by combining (2.21) with the semi-discrete

equation (2.15). Defining the operand

𝑅(𝑥) = 2𝑢𝑛 (𝑥) − 𝑢𝑛−1(𝑥) +

1
𝛼2

𝑆𝑛+1(𝑥),

54

we obtain the update

𝑢𝑛+1(𝑥) =

𝛼

2

∫ 𝑏

𝑎

𝑒−𝛼|𝑥−𝑦| 𝑅(𝑦) 𝑑𝑦 + 𝐴𝑒−𝛼(𝑥−𝑎) + 𝐵𝑒−𝛼(𝑏−𝑥),

≡ I𝑥 [𝑅] (𝑥) + 𝐴𝑒−𝛼(𝑥−𝑎) + 𝐵𝑒−𝛼(𝑏−𝑥),

(2.38)

(2.39)

where we have used I𝑥 [·] to denote the convolution integral which is not to be confused with the

identity operator.

In order to enforce conditions on the derivatives of the solution, we will also need to compute

a derivative of the update (2.38) (equivalently (2.39)). For this, we observe that the dependency

for 𝑥 appears only on analytical functions, i.e., the Green’s function (kernel) and the exponential

functions in the boundary terms. To differentiate (2.38) we start with the definition (2.25), which

splits the integral at the point 𝑦 = 𝑥 and makes the kernel easier to manipulate. Then, using the

fundamental theorem of calculus, we can calculate derivatives of (2.26) and (2.27) to find that

(cid:16)

𝑑
𝑑𝑥
𝑑
𝑑𝑥

I 𝑅
𝑥 [ 𝑓 ] (𝑥)

(cid:17)

=

(cid:18)

∫ 𝑥

𝛼

𝑒−𝛼(𝑥−𝑦) 𝑓 (𝑦) 𝑑𝑦

(cid:19)

= −𝛼I 𝑅

𝑥 [ 𝑓 ] (𝑥) + 𝛼 𝑓 (𝑥),

(cid:16)

I 𝐿
𝑥 [ 𝑓 ] (𝑥)

(cid:17)

=

𝑒−𝛼(𝑦−𝑥) 𝑓 (𝑦) 𝑑𝑦

(cid:19)

= 𝛼I 𝐿

𝑥 [ 𝑓 ] (𝑥) − 𝛼 𝑓 (𝑥).

𝑑
𝑑𝑥
𝑑
𝑑𝑥

(cid:18)

𝛼

𝑎
∫ 𝑏

𝑥

(2.40)

(2.41)

These results can be combined according to (2.25), which provides an expression for the derivative

of the convolution term:

(cid:16)

𝑑
𝑑𝑥

I𝑥 [ 𝑓 ] (𝑥)

(cid:17)

=

𝛼

(cid:16)

2

− I 𝑅

𝑥 [ 𝑓 ] (𝑥) + I 𝐿

𝑥 [ 𝑓 ] (𝑥)

(cid:17)

.

Additionally, by evaluating this equation at the ends of the interval, we obtain the identities

(cid:16)

(cid:16)

𝑑
𝑑𝑥
𝑑
𝑑𝑥

I𝑥 [ 𝑓 ] (𝑎)

I𝑥 [ 𝑓 ] (𝑏)

(cid:17)

(cid:17)

= 𝛼I𝑥 [ 𝑓 ] (𝑎),

= −𝛼I𝑥 [ 𝑓 ] (𝑏),

(2.42)

(2.43)

(2.44)

which are helpful in enforcing the boundary conditions. The relation (2.42) can be used to obtain

a derivative for the solution at the new time level. From the update (2.39), a direct computation

reveals that

𝑑𝑢𝑛+1
𝑑𝑥

=

(cid:16)

𝛼

2

−I 𝑅

𝑥 [𝑅] (𝑥) + I 𝐿

𝑥 [𝑅] (𝑥)

(cid:17)

− 𝛼𝐴𝑒−𝛼(𝑥−𝑎) + 𝛼𝐵𝑒−𝛼(𝑏−𝑥).

(2.45)

55

Notice that no additional approximations have been made beyond what is needed to compute I 𝑅
𝑥
and I 𝐿

𝑥 . These terms are already evaluated as part of the base method. For this reason, we think

of equation (2.45) as an analytical derivative. The boundary coefficients 𝐴 and 𝐵 appearing in

(2.45) will be calculated in the same way as the update (2.39), and are discussed in the remaining

subsections. This treatment ensures that the discrete derivative will be consistent with the conditions

imposed on the solution variable.

2.2.5 Applying Boundary Conditions

The boundary coefficients 𝐴 and 𝐵 appearing in (2.45) will be calculated in the same way as

the update (2.39), and are discussed in the remaining subsections utilizing the work of [266]. Our

numerical experiments are only concerned with Dirichlet and Periodic boundary conditions, and so

in the following sections we will derive the constants for these conditions, for Neumann and Outflow

see [267, 268, 34].

2.2.5.1 Dirichlet Boundary Conditions

Suppose we are given the function values along the boundary, which is represented by the data

𝑢𝑛+1(𝑎) = 𝑔𝑎

(cid:16)

𝑡𝑛+1(cid:17)

,

𝑢𝑛+1(𝑏) = 𝑔𝑏

(cid:16)

𝑡𝑛+1(cid:17)

.

If we evaluate the BDF-1 update (2.39) at the ends of the interval, we obtain the conditions

(cid:16)

𝑡𝑛+1(cid:17)

(cid:16)

𝑡𝑛+1(cid:17)

𝑔𝑎

𝑔𝑏

= I𝑥 [𝑅] (𝑎) + 𝐴 + 𝜇𝐵,

= I𝑥 [𝑅] (𝑏) + 𝜇𝐴 + 𝐵,

where we have defined 𝜇 = 𝑒−𝛼(𝑏−𝑎). This is a simple linear system for the boundary coefficients 𝐴

and 𝐵, which can be inverted by hand. Proceeding, we find that

𝐴 =

𝐵 =

𝑔𝑎 (cid:0)𝑡𝑛+1(cid:1) − I𝑥 [𝑅] (𝑎) − 𝜇 (cid:0)𝑔𝑏 (cid:0)𝑡𝑛+1(cid:1) − I𝑥 [𝑅] (𝑏)(cid:1)
1 − 𝜇2
𝑔𝑏 (cid:0)𝑡𝑛+1(cid:1) − I𝑥 [𝑅] (𝑏) − 𝜇 (cid:0)𝑔𝑎 (cid:0)𝑡𝑛+1(cid:1) − I𝑥 [𝑅] (𝑎)(cid:1)
1 − 𝜇2

,

.

56

2.2.5.2 Periodic Boundary Conditions

Periodic boundary conditions are enforced by taking

𝑢𝑛+1(𝑎) = 𝑢𝑛+1(𝑏),

𝜕𝑥𝑢𝑛+1(𝑎) = 𝜕𝑥𝑢𝑛+1(𝑏).

Enforcing these conditions through the update (2.39) and its derivative (2.45), using the identities

(2.43)-(2.44), leads to the system of equations

(1 − 𝜇) 𝐴 + (𝜇 − 1)𝐵 = I𝑥 [𝑅] (𝑏) − I𝑥 [𝑅] (𝑎),

(𝜇 − 1) 𝐴 + (𝜇 − 1)𝐵 = −I𝑥 [𝑅] (𝑏) − I𝑥 [𝑅] (𝑎),

with 𝜇 = 𝑒−𝛼(𝑏−𝑎). The solution of this system, after some simplifications is given by

𝐴 =

𝐵 =

I𝑥 [𝑅] (𝑏)
1 − 𝜇
I𝑥 [𝑅] (𝑎)
1 − 𝜇

,

.

2.3 Numerically Solving the Particle Equations of Motion

With the field solver in place, we now must consider the particle pusher component. We

couple this field solver to a standard particle description of the particles using linear spline function

representation, briefly discussed in Section 1.3.5 and more thoroughly explored in Appendix F.

We then discuss a method developed by Gibbon et al [116] and an improvement to this method

in Section 2.3.1. An algorithm which couples the time integration method for particles with the

proposed field solvers is also presented. We conclude with a brief summary in Section 2.3.2.

2.3.1 The Base Method and an Improvement

2.3.1.1 The Asymmetrical Euler Method

A time integration method suitable for non-separable Hamiltonian systems was mentioned in

Section 1.4.4, [116], which developed mesh-free methods for solving the Vlaxov-Maxwell system in

the Darwin limit. Their adoption of a generalized Hamiltonian model for particles was motivated by

the numerical instabilities associated with time derivatives of the vector potential in this particular

57

limit, which effectively sends the speed of light 𝑐 → ∞. The resulting approach for the particles,

which is essentially identical to the system (2.4)-(2.5), trades additional coupling of phase space for

numerical stability through the elimination of this time derivative. They proposed a semi-implicit

method, dubbed the asymmetrical Euler method (AEM), which has the form

x𝑛+1
𝑖

P𝑛+1
𝑖

= x𝑛

𝑖 + v𝑛

𝑖 Δ𝑡,
(cid:32)

= P𝑛

𝑖 + 𝑞𝑖

− ∇𝜙𝑛+1 + ∇A𝑛+1 · v𝑛
𝑖

(cid:33)

Δ𝑡,

v𝑛+1
𝑖

≡

(cid:113)

𝑐2 (cid:0)P𝑛+1

𝑖 − 𝑞𝑖A𝑛+1(cid:1)
𝑖 − 𝑞𝑖A𝑛+1(cid:1) 2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2

𝑐2 (cid:0)P𝑛+1

.

(2.46)

(2.47)

(2.48)






This method, which is globally first-order accurate in time, proceeds by, first, performing an explicit
update of the particle positions using (2.46). Next, with the new positions x𝑛+1 and the old velocity
, we obtain the charge density 𝜌𝑛+1 and an approximate current density ˜J𝑛+1 which are used to
v𝑛

evolve the fields under the BDF-1 discretization. We note that the use of v𝑛
in the construction of
˜J𝑛+1 is consistent with a first-order approximation of the true current density J𝑛+1. Finally, once the
fields are updated, the generalized momentum P𝑛+1 and its corresponding velocity v𝑛+1 are updated

according to equations (2.47) and (2.48), respectively.

2.3.1.2 An Improved Asymmetrical Euler Method

One of the issues with the AEM, which was discussed in the previous section, concerns the

explicit treatment of velocity in the generalized momentum equation for problems with magnetic

fields. In such cases, this update resembles the explicit Euler method, which is known to generate

artificial energy when applied to Hamiltonian systems. We offer a simple modification for such

problems in an effort to increase the accuracy and reduce such energy violations. If the update for the

generalized momentum equation (2.47) were treated implicitly with a backward Euler discretization,

then we would instead compute

(cid:32)

P𝑛+1
𝑖

= P𝑛

𝑖 + 𝑞𝑖

− ∇𝜙𝑛+1 + ∇A𝑛+1 · v𝑛+1

𝑖

(cid:33)

Δ𝑡.

Unfortunately, this approach necessitates iteration on P𝑛+1

𝑖

(through v𝑛+1

𝑖

), which we are trying to

avoid, given that the lion’s share of the compute cost in typical PIC codes centers around particle

58

updates. Instead, with the aid of a Taylor expansion, we linearize the velocity about time level 𝑡𝑛

so

that

v𝑛+1
𝑖

Δ𝑡 + O (Δ𝑡2),

𝑑v𝑛
𝑖
𝑑𝑡
𝑖 − v𝑛−1
𝑖

,

= v𝑛

𝑖 +

≈ 2v𝑛

≡ v∗
𝑖 .

While this treatment is not symplectic, the numerical results presented in section 2.4.2 for a particle

in a magnetic field indicate that the improved accuracy from the linear correction manages to tame

the otherwise significant energy increase introduced by the AEM. Therefore, in problems with

magnetic fields, we shall, instead, use the modified update

(cid:32)

P𝑛+1
𝑖

= P𝑛

𝑖 + 𝑞𝑖

− ∇𝜙𝑛+1 + ∇A𝑛+1 · v∗
𝑖

(cid:33)

Δ𝑡,

𝑖 = 2v𝑛
v∗

𝑖 − v𝑛−1
𝑖

,

as an improvement to the generalized momentum update (2.47). Since this approach is used to

evolve particles in the electromagnetic examples considered in this work, its integration with the

PIC lifecycle is presented in Algorithm 2.1. Henceforth, we shall call refer to this as the improved

asymmetrical Euler method (IAEM).

2.3.2 Conclusion

In this section we proposed new PIC methods for the numerical simulation of plasmas. To this

end, we combined methods for fields and their derivatives, which were introduced in section 2.2,

with time integration methods for non-separable Hamiltonian systems. A high level description

of the particle method was presented. In the next section, we present results from the numerical

experiments conducted with Algorithm 2.1 in comparison with the Yee and Boris method. First,

we establish the refinement properties of the field solver and methods for derivatives. Then, we

demonstrate the performance of the proposed PIC methods in several key test problems involving

plasmas with varying complexity.

59

Algorithm 2.1 Outline of the PIC algorithm with the improved asymmetric Euler method (IAEM)

Perform one time step of the PIC cycle using the improved asymmetric Euler method.
𝑖 ), as well as the fields (cid:0)𝜙0, ∇𝜙0(cid:1) and A0, ∇A0
1: Given: (x0
2: Initialize v−1
3: while stepping do

𝑖 , P0
𝑖 = v𝑖 (−Δ𝑡) using a Taylor approximation.

𝑖 , v0

4:

Update the particle positions with

x𝑛+1
𝑖

= x𝑛

𝑖 + v𝑛

𝑖 Δ𝑡.

5:

6:
7:
8:

Using the position data x𝑛+1

𝑖

and velocity data v𝑛

𝑖 , compute the current density

˜J𝑛+1 = J𝑛+1 + O (Δ𝑡).

, compute the charge density 𝜌𝑛+1.
Using the position data x𝑛+1
Compute the potentials and their derivatives at time level 𝑡𝑛+1 using the BDF-1 field solver.
Evaluate the Taylor corrected particle velocities

𝑖

𝑖 = 2v𝑛
v∗

𝑖 − v𝑛−1
𝑖

.

9:

Calculate the new generalized momentum according to

P𝑛+1
𝑖

= P𝑛

𝑖 + 𝑞𝑖

(cid:16)

− ∇𝜙𝑛+1 + ∇A𝑛+1 · v∗
𝑖

(cid:17)

Δ𝑡.

10:

Convert the new generalized momenta into new particle velocities with

v𝑛+1
𝑖

=

(cid:113)

𝑐2 (cid:0)P𝑛+1

𝑖 − 𝑞𝑖A𝑛+1(cid:1)
𝑖 − 𝑞𝑖A𝑛+1(cid:1) 2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2

𝑐2 (cid:0)P𝑛+1

.

11:

Shift the time history data and return to step 4 to begin the next time step.

2.4 Numerical Results

This section presents numerical results that demonstrate the proposed methods for fields and

particles that comprise the formulation adopted in this work. First, we establish the convergence

properties of the BDF field solver and methods for evaluating spatial derivatives. The proposed

methods are demonstrated using boundary conditions that will be considered in the applications

involving plasmas. Once the refinement properties of the field solver are established, we focus on

applications to plasmas. We begin with a single particle example involving cyclotron motion before

moving to more complex problems involving self-fields. After benchmarking the time integration

methods used for the generalized momentum formulation, we apply the proposed PIC methods to a

60

suite of electrostatic and electromagnetic test problems.

2.4.1 Numerical Experiments for Field Solvers

In this section we show results from the refinement studies involving the BDF-1 field solver

and the proposed methods for computing spatial derivatives. Results for space and time refinement

experiments are presented in the setting of two spatial dimensions. We only consider periodic and

homogeneous Dirichlet boundary conditions, as these are relevant to the plasma examples considered

in this work. In both examples, we consider the linear inhomogeneous scalar wave equation

1
𝑐2

𝜕2𝑢
𝜕𝑡2

− Δ𝑢 = 𝑆(𝑥, 𝑦).

(2.49)

To study the convergence of the methods, we use the method of manufactured solutions.

2.4.1.1 Periodic Boundary Conditions

We first consider the case of the problem (2.49) subject to two-way periodic boundary conditions

on the domain [0, 2𝜋]2. The manufactured solution for this problem is taken to be

𝑢(𝑥, 𝑦, 𝑡) = 𝑒−𝑡 sin(𝑥) cos(𝑦),

(2.50)

which determines the initial condition, as well as the source function 𝑆(𝑥, 𝑦, 𝑡). From this solution,

one can also obtain the corresponding partial derivatives in space. To initialize the multi-step method,

we use the exact solution, since it is available.

For the space refinement experiment, we varied the spatial mesh in each direction from 16 points

to 512 points. To keep the temporal error in the methods small during the refinement, we applied the

methods for 10 time steps using a step size of Δ𝑡 = 10−4. The refinement plot presented in Figure

2.1a indicates fifth-order accuracy in space for the solution as well as the derivatives, which is the

expected order of accuracy for the quadrature rule used by the methods. We note that the derivatives

in the methods begin to level-off as the error approaches 10−11 due to a larger error constant arising

from the differentiation process.

In the temporal refinement study, the solution is computed until a final time of 𝑇 𝑓 = 1 using a

fixed 512 × 512 spatial mesh. We refine by successively doubling the number of time steps from

𝑁𝑡 = 8 until 𝑁𝑡 = 2, 048. The results of the temporal refinement study are presented in Figure 2.1b,

61

in which all methods, including those for the derivatives, display the expected first-order convergence

rate in time.

(a) Space refinement

(b) Time refinement

Figure 2.1 Refinement in space (left) and time (right) of the solution and its spatial derivatives for the
two-dimensional periodic example 2.4.1.1 obtained with the BDF-1 method. Errors are measured
using the ℓ∞-norm. We observe fifth-order convergence in space and first-order accuracy in time for
the solution and its derivatives. In each plot, we provide a reference corresponding to the expected
rate of convergence.

2.4.1.2 Dirichlet Boundary Conditions

For the next test, we, again, consider the scalar wave equation (2.49) with homogeneous Dirichlet

boundary conditions on the domain [0, 2𝜋]2. Here we use the manufactured solution

𝑢(𝑥, 𝑦, 𝑡) = 𝑒−𝑡 sin(𝑥) sin(𝑦),

(2.51)

which is used to provide the initial data for the multi-step method.

We performed the spatial refinement study by varying the number of mesh points in each direction

from 16 points to 512 points. Again, to keep the temporal error in the methods small while space

is refined, we applied the methods for only 10 time steps with a step size of Δ𝑡 = 10−4. The same

remark about small time step sizes mentioned in the space refinement experiment for the periodic

case applies here, as well (see section 2.4.1.1). The space refinement plots in Figure 2.2a indicate

that the methods refine to fourth-order in space rather than fifth-order in space. This is due to the

62

way in which we construct the stencil points in the ghost regions near the boundary of the domain. In

our implementation, we use extrapolation with fourth-order accuracy to provide these stencil points.

In the temporal refinement study, the solution is computed until a final time of 𝑇 𝑓 = 1. We use a

fixed 512 × 512 spatial mesh and the number of time steps in each case is successively doubled from

𝑁𝑡 = 8 until 𝑁𝑡 = 2, 048. The results of the temporal refinement study are presented in Figure 2.2b,

in which all methods, including those for the derivatives, display the expected first-order convergence

rate. The behavior is essentially identical to the results obtained for the periodic problem presented

in Figure 2.1b.

(a) Space refinement

(b) Time refinement

Figure 2.2 Refinement in space (left) and time (right) of the solution and its spatial derivatives for the
two-dimensional Dirichlet problem 2.4.1.2 obtained with the BDF-1 method. Errors are measured
using the ℓ∞-norm. The methods refine to fourth-order accuracy in space, since we use fourth-order
extrapolation to provide the ghost points. All methods refine to the expected first-order accuracy in
time.

2.4.2 Plasma Test Problems

This section presents numerical results showcasing the proposed methods for fields in PIC

applications. The benchmark PIC methods used in the comparisons implement conservative charge

weighting for electrostatic problems and conservative current weighting [37] for electromagnetic

problems. The electrostatic problems use the FFT to solve Poisson’s equation, while the electromag-

netic problems use the staggered FDTD grid introduced by Yee [86]. First, we consider a single

particle moving through known fields. We then focus on applying the methods to problems involving

63

fields that respond to the motion of the particles, such as the two-stream instability as well as more

challenging simulations of plasma sheaths and particle beams. In particular, the last problem we

consider is the Mardahl beam problem [112], which is a popular benchmark problem for relativistic

beams. The implementation used to obtain our results is based on the non-dimensionalization in

Appendix A. All physical constants used in this non-dimensionalization are based on SI units, and,

in all examples (with the exception of the heating tests), we show the results in normalized units.

The relevant parameters used to set up each of the test problems are provided in dimensional form,

so that the results can be more easily reproduced and compared with other methods.

2.4.2.1 Motion of a Charged Particle

We first compare the time integration methods for non-separable Hamiltonians with the well-

known Boris method [91]. This is a natural first step before applying the method to problems with

dynamic “self-fields” that respond to particle motion. Here, we consider a simple model for the

motion of a single charged particle that is given by

𝑑x
𝑑𝑡

= v,

𝑑v
𝑑𝑡

=

𝑞
𝑚

(E + v × B) .

We use electro- and magneto-static fields here and suppose that the magnetic field lies along the

ˆz unit vector

B = 𝐵0ˆz, E = 𝐸 (1) ˆx + 𝐸 (𝑦) ˆy + 𝐸 (𝑧) ˆz,

where 𝐵0 is a constant. Again, component-based definitions have been used for the fields E =
𝐸 (1), 𝐸 (2), 𝐸 (3)(cid:17)
(cid:16)

𝐵(1), 𝐵(2), 𝐵(3)(cid:17)

. Consequently, we have that

and B =

(cid:16)

v × B = 𝑣 (2) 𝐵0 ˆx − 𝑣 (1) 𝐵0 ˆy,

so the full equations of motion are

𝑑𝑥 (1)
𝑑𝑡
𝑑𝑥 (2)
𝑑𝑡
𝑑𝑥 (3)
𝑑𝑡






= 𝑣 (1),

= 𝑣 (2),

= 𝑣 (3),

𝑑𝑣 (1)
𝑑𝑡
𝑑𝑣 (2)
𝑑𝑡
𝑑𝑣 (3)
𝑑𝑡

𝑞
𝑚
𝑞
𝑚
𝑞
𝑚

=

=

=

64

(cid:16)

(cid:16)

𝐸 (1) + 𝑣 (2) 𝐵0

(cid:17)

𝐸 (2) − 𝑣 (1) 𝐵0

(cid:17)

,

,

𝐸 (3).

We can then use the linear momentum p = 𝑚v to obtain

𝑑𝑥 (1)
𝑑𝑡
𝑑𝑥 (2)
𝑑𝑡
𝑑𝑥 (3)
𝑑𝑡






=

=

=

1
𝑚

1
𝑚

1
𝑚

𝑝 (1),

𝑝 (2),

𝑝 (3),

𝑑𝑝 (1)
𝑑𝑡
𝑑𝑝 (2)
𝑑𝑡
𝑑𝑝 (3)
𝑑𝑡

(cid:18)

(cid:18)

= 𝑞

= 𝑞

𝐸 (1) +

𝐸 (2) −

(cid:19)

𝑝 (2) 𝐵0

(cid:19)

𝑝 (1) 𝐵0

,

,

1
𝑚

1
𝑚

= 𝑞𝐸 (3).

Using the potentials 𝜙 and A ≡

(cid:16)

𝐴(1), 𝐴(2), 𝐴(3)(cid:17)

, one can compute the electric and magnetic

fields via (1.10). The time-independence of the magnetic field for this problem implies that 𝜕𝑡A = 0,

so that E = −∇𝜙. For this problem, we use

𝜙 = −𝐸 (1)𝑥 − 𝐸 (2) 𝑦 − 𝐸 (3) 𝑧.

Moreover, the magnetic field contains only a z-component, which implies that it can be written as

B = (0, 0, 𝐵0) = (0, 0, 𝜕𝑥 𝐴(2) − 𝜕𝑦 𝐴(1)).

As the choice of functions for gauges is not unique, it suffices to pick

𝐴(1) ≡ 0,

𝐴(2) = 𝐵0𝑥,

𝐴(3) ≡ 0.

In summary, the non-zero values and required derivatives for the potentials are given by

−𝜕𝑥 𝜙 = 𝐸 (1), −𝜕𝑦𝜙 = 𝐸 (2), −𝜕𝑧𝜙 = 𝐸 (3),

𝐴(2) = 𝐵0𝑥,

𝜕𝑥 𝐴(2) = 𝐵0,

which results in the simplified equations of motion for the Hamiltonian system

𝑑𝑥 (1)
𝑑𝑡

𝑑𝑥 (2)
𝑑𝑡
𝑑𝑥 (3)
𝑑𝑡






=

=

=

1
𝑚

1
𝑚
1
𝑚

𝑃(1),

𝑑𝑃(1)
𝑑𝑡

= 𝑞𝐸 (1) +

(cid:34)

𝑞
𝑚

(cid:16)

𝑃(2) − 𝑞𝐵0𝑥 (1)(cid:17)

𝐵0

(cid:35)

,

,

𝑑𝑃(2)
𝑑𝑡

= 𝑞𝐸 (2),

(cid:16)

𝑃(2) − 𝑞𝐵0𝑥 (1)(cid:17)
𝑑𝑃(3)
𝑑𝑡

𝑃(3),

= 𝑞𝐸 (3).

The setup for the test consists of a single particle with mass 𝑚 = 1 and charge 𝑞 = −1 whose

initial position is at the origin of the domain x(0) = (0, 0, 0). Initially, the particle is given non-zero

65

Figure 2.3 Trajectories for the single particle test obtained using the Boris method [91], the AEM
[116], and the IAEM. The particle rotates about a static magnetic field which points in the 𝑧-
direction. Also shown is the time history of the Hamiltonian generated by each of the methods
which is measured relative to the initial data. In particular, the AEM shows a growth in the overall
energy causing the gyroradius to increase. This behavior is not observed in the improved method.

momenta in the 𝑥 and 𝑧 directions so as to generate so called “cyclotron” motion. We choose the

initial momenta to be p(0) = P(0) = (0.01, 0, 0.01). The strength of the magnetic field in the

𝑧 direction is selected to be 𝐵0 = 1, and we ignore the contributions from the electric field, so

that E = (0, 0, 0). Each method is run to a final time of 𝑇 𝑓 = 300, using a total of 104 time steps,

so that Δ𝑡 = 0.03. The position of the particle is tracked through time and plotted as a curve in

three-dimensions. In Figure 2.3, we compare the particle trajectories and the relative error in the

Hamiltonian obtained with each of the methods. We note that the gyroradius for the AEM increases

66

Figure 2.4 Refinement study for the trajectory of a single particle obtained using the Boris method
[91], AEM [116], and the IAEM that uses a Taylor correction. Errors are measured in the ℓ∞-norm
against a reference solution obtained using the Boris method. Even though the AEM with the Taylor
correction remains globally first-order accurate in time, its improvement over the AEM is quite
apparent (roughly an order of magnitude).

over time because the method is not volume-preserving. Over time, this causes the total energy

to increase, as substantiated by the growth of the Hamiltonian. In contrast, we see that the simple

correction used in the IAEM reduces this behavior; however, the correction does not completely

eliminate this growth in the case of longer simulations, as the truncation errors accumulate over

time.

Next, we perform a refinement study of the methods to examine their error properties using the

same experimental parameters from the cyclotron test. We reduce the final time to 𝑇 𝑓 = 30 and

measure the errors with the ℓ∞ norm using a reference solution computed with 106 time steps, so that

Δ𝑡 = 3.0 × 10−5. The test successively doubles the number of steps, starting with 100 steps and uses,

at most, 1.28 × 104 steps. The results of the refinement study are shown in Figure 2.4. Despite the

fact that both the base and improved versions of the AEM refine to first-order accuracy, we see that

the Taylor correction decreases the error in the base method by roughly an order of magnitude. For

coarser time step sizes, the improved method has errors that are (in some sense) comparable to the

Boris method, which is second-order accurate. Of course, the second-order method will outperform

both versions of the AEM as the time step decreases.

67

2.4.2.2 The Cold Two-stream Instability

We consider the motion of “cold” streams of electrons restricted to a one-dimensional periodic

domain by means of a sufficiently strong (uniform) magnetic field in the two remaining directions.

Ions are taken to be uniformly distributed in space and sufficiently heavy compared to the electrons

so that their motion can be neglected in the simulation. The ions, which remain stationary, act as a

neutralizing background against the dynamic electrons. The electron velocities are represented as a

sum of two Dirac delta distributions that are symmetric in velocity space:

𝑓 (𝑣) =

1
2

𝛿(𝑣 − 𝑣𝑏) +

𝛿(𝑣 + 𝑣𝑏).

1
2

(2.52)

The stream velocity 𝑣𝑏 > 0 is set according to a drift velocity whose value ultimately controls the

interaction of the streams. A slight perturbation in the electron velocities is then introduced to force

a charge imbalance, which generates an electric field that attempts to restore the neutrality of the

system. This causes the streams to interact or “roll-up,” which corresponds to regions of trapped

particles.

In order to describe the models used in the simulation, let us denote the components of the
(cid:17)

(cid:16)

(cid:16)

(cid:17)

position and momentum vectors for particle 𝑖 as x𝑖 ≡

𝑥 (1)
𝑖

, 𝑥 (2)
𝑖

, 𝑥 (3)
𝑖

and P𝑖 ≡

𝑃(1)
𝑖

, 𝑃(2)
𝑖

, 𝑃(3)
𝑖

,

respectively. Then, the equations for the motion of particle 𝑖 assume the form

𝑑𝑥 (1)
𝑖
𝑑𝑡

=

1
𝑚𝑖

𝑃(1)
𝑖

,

𝑑𝑃(1)
𝑖
𝑑𝑡

= −𝑞𝑖𝜕𝑥 𝜙.

The potential 𝜙 and its derivative 𝜕𝑥 𝜙 are obtained by solving a two-way wave equation for the scalar

potential:

1
𝑐2

𝜕𝑡𝑡 𝜙 − 𝜕𝑥𝑥 𝜙 =

𝜌
𝜖0

.

(2.53)

As this is an electrostatic problem, the gauge condition can be safely ignored. In the limit where the

thermal velocity 𝑉 of the particles become well-separated from the nondimensionalized speed of

light (𝜅 = 𝑐/𝑉 (cid:29) 1), one instead solves the Poisson equation

−𝜕𝑥𝑥 𝜙 =

𝜌
𝜖0

.

68

(2.54)

Using asymptotic analysis, it can be shown that the error incurred by adopting the Poisson model

for 𝜙 is O (1/𝜅) [271].

We establish the efficacy of the proposed algorithms for time stepping particles and evolving

fields by comparing with well-known methods. The setup for this test problem consists of a spatial

mesh defined on the interval [−10𝜋/3, 10𝜋/3], in units of the Debye length 𝜆𝐷, which is discretized

using 128 total grid points with periodic boundary conditions. The final time for the simulation is

taken to be 𝑇 𝑓 = 100 in units of 𝜔−1

𝑝𝑒 with 4,000 time steps being used to evolve the system. The

plasma is represented with a total of 30,000 macroparticles, consisting of 10,000 ions and 20,000

electrons. As mentioned earlier, the positions of the ions and electrons are taken to be uniformly

spaced along the grid. Ions remain stationary in the problem, so we set their velocity to zero. The

construction of the streams begins by first splitting the electrons into two equally sized groups whose

respective (non-dimensional) drift velocities are set to be ±1 in units of the thermal velocity 𝑉. To

generate an instability we add a perturbation to the electron velocities of the form

𝛿𝑣(𝑥) = 𝜖 sin

(cid:18) 2𝜋𝑘 (𝑥 − 𝑎)
𝐿𝑥

(cid:19)

.

Here, 𝜖 = 5.0 × 10−4 controls the strength of the perturbation, 𝑘 = 1 is the wave number for the

perturbation, 𝑥 is the position of the particle (electron), 𝑎 is the left-most grid point, and 𝐿𝑥 is the

length of the domain. In a more physically realistic simulation, the perturbation would be induced by

some external force, which would also result in a perturbation of the position data for the particles.

Such a perturbation of the position data requires a self-consistent field solve to properly initialize the

potentials. In our simulation, we assume that no spatial perturbation is present, so that the fields are

identically zero at the initial time step. The plasma parameters used in the non-dimensionalization

for this test problem are displayed in Table 2.1. Note that in this configuration, the normalized speed

of light is 𝜅 = 50, and the corresponding normalized permittivity is 𝜎1 = 1. This configuration

adequately resolves the plasma Debye length (≈ 6 cells/𝜆𝐷), the angular plasma period (≈ 40

steps/𝜔−1

𝑝𝑒), and the particle CFL < 1, which are typically used to ensure stability in explicit PIC

methods.

69

Parameter
Average number density ( ¯𝑛) [m−3]
Average temperature ( ¯𝑇) [K]
Debye length (𝜆𝐷) [m]

Inverse angular plasma frequency (𝜔−1

𝑝𝑒) [s/rad]

Thermal velocity (𝑣𝑡ℎ = 𝜆𝐷𝜔 𝑝𝑒) [m/s]

Value
7.856060 × 101
2.371698 × 106
1.199170 × 104
2.000000 × 10−3
5.995849 × 106

Table 2.1 Table of the plasma parameters used in the two-stream instability example.

We study the structure of the instability using the Poisson model (2.54) and the wave model

(2.53) for the scalar potential using the AEM. We remark that the Taylor corrected version of the

AEM was not considered in this problem because the contributions from the magnetic field are

ignored. The particular initial condition for this problem leads to a special case in which the AEM is

equivalent to leapfrog integration. Since the problem is initially charge neutral, there is no electric

field at time 𝑡 = 0. This further implies that there is no modification to the particle velocities in

the step that normally generates the initial offset required by the leapfrog method. Plots comparing

the evolution of the electron streams, obtained with both models, are presented in Figure 2.5. Each

figure contains two colors which designate the groups of electrons in the initial condition (2.52).

Here, the blue electrons move to the right, while those in red move to the left. We see that the

behavior at early times is nearly identical, but the trapping regions at later times are quite different.

In particular, the structures produced by the wave model tend to be more elongated than those of the

Poisson model. This is a likely consequence of the finite speed of propagation in the wave model,

where the potential responds more slowly to an imbalance in charge.

Next, we compare the growth rates in the electric fields produced by the Poisson and wave

models against theoretically predicted growth rates. Applying basic linear response theory (see,

e.g., Chapter 18 of [7]) with the distribution (2.52) one obtains the cold dispersion relation for the

Vlasov-Poisson system

𝜔4 − 2𝜔2 (cid:16)

𝜔2

𝑝𝑒 + 𝑘 2𝑣2
𝑏

(cid:17)

+ 𝑘 2𝑣2
𝑏

(cid:16)

𝑘 2𝑣2

𝑏 − 2𝜔2
𝑝𝑒

(cid:17)

= 0.

(2.55)

While the dispersion relation for a warm problem could also be considered, its evaluation is slightly

more complicated. We remark that cold problems cannot be not be adequately represented in

70

(a) AEM (Poisson)

(b) AEM (Wave)

(c) AEM (Poisson)

(d) AEM (Wave)

(e) AEM (Poisson)

(f) AEM (Wave)

(g) AEM (Poisson)

(h) AEM (Wave)

Figure 2.5 We plot the electrons in phase space obtained with the Poisson (left column) and wave
(right column) model for the two-stream instability problem, at different times, in units of 𝜔−1
𝑝𝑒. The
two colors represent the groups of electrons in the initial condition (2.52). The blue electrons move
to the right, while those in red move to the left. Results were obtained using the AEM for time
integration. The IAEM, which applies the Taylor correction, is not considered here because the
contributions from the magnetic fields are ignored. The FFT is used to compute the scalar potential
(as well as its derivative) in the Poisson model, while the BDF-1 field solver is used to compute the
scalar potential (as well as its derivative) in the wave model. We see that there are clear differences in
the structure of the trapping regimes at later times, with the wave model producing a more elongated
trapping region due to the finite speed of propagation.

71

mesh-based discretizations, which is a key advantage offered by a particle method. Additionally,

cold problems eliminate artifacts introduced by sampling methods during the initialization phase.

Substituting the normalized quantities 𝑣𝑏 = 1, 𝜔 𝑝𝑒 = 1, and 𝑘 = 2𝜋/𝐿𝑥 in the dispersion relation

yields the growth rate Im(𝜔) ≈ 0.2760. In Figure 2.6, we compare the growth rate of the electric

fields produced by both methods using a variety of CFL numbers against the analytical growth rate.

Again, we see identical results among both methods in the linear regime, but subtle differences at

the later times when the assumptions of the linear response theory become invalid. This is true even

when the CFL > 1 is used in the proposed BDF-1 method.

(a) AEM (Poisson) with variable CFL

(b) AEM (Wave) with variable CFL

Figure 2.6 We compare the growth rate in the ℓ2-norm of the electric field for various methods
against an analytical growth rate for the two-stream example obtained from linear response theory.
Plot (a) combines the AEM with an FFT-based Poisson solver, while plots (b) and (c), instead, use
a wave model for the scalar potential that is solved using the proposed BDF-1 approach. Plot (c)
demonstrates the impact of the CFL number on the growth rate in the electric field. Using this
particular experimental configuration, the analytical growth rate is given by Im(𝜔) ≈ 0.2760. The
AEM, when combined with each of the models, correctly follows the predicted growth rate. We
also observe excellent agreement with the analytical growth rate even when a CFL > 1 is used for
the proposed BDF-1 solver. At 𝑡 ≈ 40, the assumptions of the linearization become invalid, so the
prediction becomes inaccurate.

2.4.2.3 Numerical Heating

We perform a numerical heating experiment to study the relationship between the Debye length

𝜆𝐷 and the mesh resolution Δ𝑥. A general rule of thumb for standard explicit PIC simulations is that

the mesh spacing should satisfy Δ𝑥 < 𝜆𝐷/4 to slow the growth of the instability. Such artifacts can

72

be interpreted as an aliasing error, as the mesh cannot resolve the charge separation in the plasma.

This can be problematic for purely dispersive field solvers, e.g., the FDTD method. In contrast,

the BDF-1 field solver proposed in this work is more diffusive, which can be more effective at

reducing the growth of these aliasing errors, especially over long time intervals. We remark that

numerical heating effects can be suppressed by the use of high-order particle shape functions or even

eliminated by an energy-conserving scheme [108, 109, 107, 277]. Since this problem is typically

studied using periodic domains, the former approach creates no complication. However, we only

consider low-order maps, such as the charge conserving map [37] and the linear shape function

(1.41), as we are interested in the performance of the methods on bounded domains.

The setup for this problem is slightly different from the two-stream example discussed earlier.

Here, we provide, as input, a Debye length 𝜆𝐷 and a thermal velocity 𝑣𝑡ℎ, which can be used

to calculate the average number density ¯𝑛 and macroscopic temperature ¯𝑇 for the plasma. The

remaining parameters can be derived from these values and are shown in Table 2.2. The normalized

speed of light for both the electrostatic and electromagnetic problems is 𝜅 = 50, and the normalized

permittivity is 𝜎1 = 1. For the electromagnetic problem, the normalized permeability obtained with

these experimental parameters is 𝜎2 = 4.0×10−4. Here we consider both electrostatic (1D-1V/1D-1P)

and electromagnetic (2D-2V/2D-2P) configurations that consist of ions and electrons in a periodic

domain. The spatial domain for the electrostatic case is [−25𝜆𝐷, 25𝜆𝐷], while the electromagnetic

case uses [−25𝜆𝐷, 25𝜆𝐷]2. In both cases, the spatial domain is refined by successively doubling the

number of mesh points from 16 to 256 in each dimension. The simulations use 106 time steps with

a final time of 𝑇 𝑓 = 1, 000 angular plasma periods. Although the CFL number is quite small in the

electrostatic case, where larger time steps can be used, we are primarily interested in a consistent

comparison with the electromagnetic case. In this limit, the CFL must be ≤ 1 for the FDTD for

stability. In the electrostatic simulation, we use 5, 000 macroparticles for each species, and increase

this to 250, 632 for the electromagnetic simulation. As before, we assume that the ions remain

stationary, since they are heavier than the electrons. Electrons are given uniform positions in space

and their velocities are obtained by sampling from a Maxwellian distribution using the parameters

73

in Table 2.2. We make the problem current neutral by splitting the electrons into two equally sized

groups whose velocities differ only in sign. A drift velocity is not used in these tests. To ensure

consistency across the runs, we also seed the random number generator prior to sampling.

Parameter
Average number density ( ¯𝑛) [m−3]
Average temperature ( ¯𝑇) [K]
Debye length (𝜆𝐷) [m]

Inverse angular plasma frequency (𝜔−1

𝑝𝑒) [s/rad]

Thermal velocity (𝑣𝑡ℎ = 𝜆𝐷𝜔 𝑝𝑒) [m/s]

Value
1.129708 × 1014
2.371698 × 106
1.0 × 10−2
1.667820 × 10−9
5.995849 × 106

Table 2.2 Table of the plasma parameters used in the numerical heating examples.

We monitor heating during the simulations by computing the variance in the components of the

electron velocities, since this is connected to the temperature of a Maxwellian distribution. In the

one-dimensional case, the variance data at a given time step is converted to a temperature (in units

of Kelvin) using the relation

¯𝑇 =

𝑚𝑒𝑉 2
𝑘 𝐵

Var(𝑣 (1)),

where we have used “Var” to denote variance and 𝑉 is the normalization used for velocity. Similarly,

for the two-dimensional case, we compute the average of the variance for each component of the

velocity, which is converted to a temperature (in units of Kelvin) using

𝑚𝑒𝑉 2 (cid:16)

Var(𝑣 (1)) + Var(𝑣 (2))

2𝑘 𝐵

(cid:17)

.

¯𝑇 =

We use the superscripts in the above metrics to refer to the individual velocity components across all

of the particles. The factor of two is used to average the variance among these components. When

assessing the temperatures produced by different methods, we rescale the temperatures so they have

the proper units of Kelvin. This allows us compare the different methods in a more realistic setting

in which we might be interested in comparing the raw temperatures predicted by different methods.

The models used in the electrostatic tests are identical to the ones presented for the two-stream

instability example, so we shall skip these details for brevity. In the case of the electromagnetic

74

experiment, the particle equations in the non-relativistic Hamiltonian formulation are

𝑑𝑥 (1)
𝑖
𝑑𝑡
𝑑𝑥 (2)
𝑖
𝑑𝑡
𝑑𝑃(1)
𝑖
𝑑𝑡
𝑑𝑃(2)
𝑖
𝑑𝑡






=

=

1
𝑚𝑖

1
𝑚𝑖

,

,

(cid:16)

𝑖 − 𝑞𝑖 𝐴(1)(cid:17)
𝑃(1)

(cid:16)

𝑖 − 𝑞𝑖 𝐴(2)(cid:17)
𝑃(2)
(cid:20) (cid:16)

𝑞𝑖
𝑚𝑖

= −𝑞𝑖𝜕𝑥 𝜙 +

𝜕𝑥 𝐴(1)(cid:17) (cid:16)

𝑖 − 𝑞𝑖 𝐴(1)(cid:17)
𝑃(1)

(cid:16)

𝜕𝑥 𝐴(2)(cid:17) (cid:16)

+

= −𝑞𝑖𝜕𝑦𝜙 +

(cid:20) (cid:16)

𝑞𝑖
𝑚𝑖

𝜕𝑦 𝐴(1)(cid:17) (cid:16)

𝑖 − 𝑞𝑖 𝐴(1)(cid:17)
𝑃(1)

+

(cid:16)

𝜕𝑦 𝐴(2)(cid:17) (cid:16)

𝑖 − 𝑞𝑖 𝐴(2)(cid:17) (cid:21)
𝑃(2)
𝑖 − 𝑞𝑖 𝐴(2)(cid:17) (cid:21)
𝑃(2)

,

.

The contributions from the fields are obtained by solving a system of wave equations for the potentials,

which take the form

𝜕𝑡𝑡 𝜙 − 𝜕𝑥𝑥 𝜙 − 𝜕𝑦𝑦𝜙 =

𝜌,

1
𝜖0

𝜕𝑡𝑡 𝐴(1) − 𝜕𝑥𝑥 𝐴(1) − 𝜕𝑦𝑦 𝐴(1) = 𝜇0𝐽 (1),

𝜕𝑡𝑡 𝐴(2) − 𝜕𝑥𝑥 𝐴(2) − 𝜕𝑦𝑦 𝐴(2) = 𝜇0𝐽 (2).

1
𝑐2
1
𝑐2
1
𝑐2






To establish the heating properties of the proposed methods in an electromagnetic setting, an

identical experiment is performed using a standard FDTD-PIC approach in which the equations of

motion for the particles are expressed in terms of E and B. For this example, these equations take

the form

𝑑𝑥 (1)
𝑖
𝑑𝑡

𝑑𝑥 (2)
𝑖
𝑑𝑡

= 𝑣 (1)
𝑖

,

= 𝑣 (2)
𝑖

,

𝑑𝑣 (1)
𝑖
𝑑𝑡

𝑑𝑣 (2)
𝑖
𝑑𝑡

(cid:32)

(cid:32)

𝑞𝑖
𝑚𝑖

𝑞𝑖
𝑚𝑖

=

=

𝐸 (1) + 𝑣 (2)

𝑖 𝐵(3)

𝐸 (2) − 𝑣 (1)

𝑖 𝐵(3)

(cid:33)

,

(cid:33)

,






and are evolved in a leapfrog format through the Boris method [91]. Since we have restricted the

system to two spatial dimensions, we retain the curl equations for the transverse electric (TE) mode,

namely

𝜕𝑥 𝐸 (2) − 𝜕𝑦𝐸 (1) = −𝜕𝑡 𝐵(3),

− 𝜕𝑧 𝐵(2) = 𝜇0𝐽 (1) +

− 𝜕𝑥 𝐵(3) = 𝜇0𝐽 (2) +

1
𝑐2
1
𝑐2

𝜕𝑡 𝐸 (1),

𝜕𝑡 𝐸 (2),






which are discretized using the staggered FDTD mesh [86].

75

The results of the numerical heating experiments can be found in Figures 2.7 - 2.9. Figure

2.7 presents the result for the one-dimensional case, in which, we consider both Poisson and wave

equation models for the scalar potential. Figure 2.8 shows the results for the electromagnetic heating

experiment in which we compare the FDTD-PIC method and the IAEM. For the one-dimensional

electrostatic experiments, we observe significant differences in the heating behavior due to the

choice of models used for the scalar potential. We can clearly see the rate of heating is far more

significant when the Poisson model is used instead of the wave equation. This is due to the finite

speed of propagation offered by the wave model, which causes the potential to respond more slowly

to variations in the charge density 𝜌. We find this to be true even in cases where the Debye length

would normally be considered underresolved by practitioners, resulting in far less severe fluctuations

in temperature. The use of the leapfrog time integration scheme produces nearly identical behavior,

so we do not include these results. In the electromagnetic case, we see that the proposed method

demonstrates mesh-independent heating properties, with notably smaller temperature fluctuations

across the grid configurations, over many plasma periods. For example, over 1,000 plasma periods,

the relative increase in temperature across all meshes is < 0.1%. In contrast, the benchmark FDTD-

PIC approach displays significant fluctuations in the temperature over a smaller time interval (100

versus 1, 000 angular plasma periods), even in cases where the grid spacing resolves the Debye

length. These results indicate that the new method permits the use of a much coarser grid than current

explicit particle methods for bounded domains. We also include some plots of conserved quantities

for the electromagnetic case, namely, the total mass and the residual in the Lorenz gauge condition

in Figure 2.9 obtained with the proposed method. We observe reasonable mass conservation and

control of the gauge error over many plasma periods despite the fact that we are not enforcing the

gauge condition. Figure 2.10 considers the same heating experiment for a variety of CFLs. We

observe that larger CFL numbers slightly increase the sensitivity of heating with the mesh refinement.

However, of the cases considered, we find that the coarsest mesh changes the electron temperature

by < 1% after 1, 000 angular plasma periods when the CFL = 8.

76

(a) AEM (Poisson)

(b) AEM (Wave)

Figure 2.7 Average electron temperature as a function of the number of angular plasma periods for
the electrostatic heating experiments with Poisson (left) and wave (right) models for the potential
using the AEM [116]. Since the magnetic field is ignored in the model, the AEM and its improved
version are identical. Fields and their derivatives in the Poisson model are constructed using the
FFT, while the wave model uses the BDF-1 solver for the fields and their derivatives. The results
for the Poisson model suggest that heating can be reduced if we use ≈ 2.54 grid cells per plasma
Debye length. This is quite close to the usual rule of thumb which recommends ≈ 4 grid cells per
Debye length. In contrast, the wave model shows less severe fluctuations in temperature, even in
cases where the grid does not adequately resolve the Debye length.

(a) FDTD-PIC

(b) IAEM with BDF-1

Figure 2.8 Average electron temperature as a function of the number of angular plasma periods for
the electromagnetic heating experiments. The plot on the left was obtained with the FDTD-PIC
method, while the plot on the right uses the IAEM and the proposed BDF-1 field solver. Since the
IAEM does not conserve energy, it will, over time, generate additional sources of energy causing
the simulation to heat even if the plasma Debye length is sufficiently resolved; however, the results
indicate that fluctuations in the temperature are not substantial, even in cases where the Debye length
would normally be considered underresolved by the mesh. In contrast, the FDTD-PIC method
displays more significant fluctuations over a smaller time window, even when the Debye length is
resolved by the mesh. Note the differences in the magnitude of the electron temperature between the
plots as well as the number of angular plasma periods in each case.

77

(a) Mass conservation

(b) Lorenz gauge error (ℓ2)

Figure 2.9 Change in the total mass (left) and the error in the Lorenz gauge (right) for the BDF-1 field
solver with the IAEM in the 2D-2V heating experiment. The change in the total mass is measured
relative to its value at the initial time. Since the gauge condition is satisfied by the initial data for
the problem, its value is initially zero, so we, instead, measure errors in the absolute sense using
the ℓ2-norm. The relative error in the Hamiltonian H (total energy) is not presented for brevity,
as this information can be inferred from the temperature plot in Figure 2.8. The proposed method
demonstrates reasonable mass conservation, and the gauge condition appears to be controlled over
many plasma periods despite the absence of a cleaning method.

2.4.2.4 Plasma Sheath

Plasma sheaths are a fundamental concept in the physics of plasmas and can be simulated using

PIC methods. The study of sheaths was pioneered by Langmuir [57], who described an ionized gas

contained within a glass apparatus in a rather captivating manner:

[W]hen a current of a few milliamperes from a hot cathode is flowing in a glass tube

containing mercury vapor saturated at room temperature, the voltage being above about

20 volts, the tube is largely filled with the characteristic green-blue glow of the mercury

discharge, but the glow does not quite reach the walls. A dark space separates the glow

from the walls, as if the glow were being repelled by the glass.

The formation of sheaths is not an uncommon event, with two examples being the insertion of a

conducting probe [7] and a basic matrix sheath with uniform ion charge density, which occurs in

a DC discharge. Such discharges can be created using a pulsed negative electrode voltage during

plasma immersion ion implantation [12].

In our computational model of a sheath, a macroscopically neutral plasma is deposited in a

two-dimensional box with perfectly electrically conducting (PEC) walls, which have zero tangential

78

(a) CFL = 2

(b) CFL = 4

(c) CFL = 8

Figure 2.10 We note that increasing the CFL increases the sensitivity of heating to mesh refinement.
A CFL of 2 demonstrates reasonable resilience to this sensitivity, close to the phenomenon to that
of the refinement study with CFL 1 (Figure 2.8), though not quite as detatched. When a CFL of 4
or 8 is introduced, we see that the heating becomes sensitive to the mesh refinement, though even
under worst case scenario of 𝜆𝐷/Δ𝑥 = 0.3 (16 × 16) with a CFL of 8 we observe approximately
0.6% increase in heat over the course of 1000 angular plasma periods.

components in their electric fields. As the problem is charge neutral, the electron drift velocity

causes some of the electrons to move towards the wall. When an electron comes into contact

with a PEC wall, it is effectively neutralized by its associated image charge (of opposite sign),

which results in a cancellation of the electric field on the surface. Hence, it is removed from the

simulation. Consequently, as the lighter and hotter electrons are eliminated from the domain, the

charge imbalance forms a potential well that draws the electrons back in towards the heavier and

cooler (stationary) ions. When the electrons rush back to the center of the box, they repel each

79

other, and the process begins anew, forming a “breathing” pattern over time. The loss of the hotter

electrons to the wall results in the formation of a potential well, which, in turn, forms a sheath

near the domain boundaries. In the domain of the sheath, quasi-neutrality no longer holds on the

scale of the initial Debye length. In other words, the Debye length varies substantially between the

quasi-neutral interior and the sheath region [278]. The results of the numerical heating experiment

have clear computational implications to the study of sheaths, as one needs to ensure that the mesh

appropriately resolves the smallest Debye length set by the high density regions. Therefore, methods

that are less susceptible to (artificial) numerical heating would provide a clear advantage over those

that suffer from heating effects because they permit the use of a coarser mesh. Of course, one needs

to have adequate resolution of the sheath, but methods with high-order accuracy in space have the

potential to resolve the sheath using fewer mesh points.

The procedure used to setup the simulation is nearly identical to the one used for the numerical

heating experiment in section 2.4.2.3. Slight modifications to the plasma parameters are made

to emphasize sheath formation (See Table 2.3). The resulting normalized speed of light for this

problem is 𝜅 = 7.700159 × 102. Additionally, the normalized permittivity and permeability for this

problem are 𝜎1 = 1 and 𝜎2 = 1.686555 × 10−6, respectively. A neutral plasma consisting of ions

and electrons is deposited within a two-dimensional box whose dimensions are [−8𝜆𝐷, 8𝜆𝐷]2. The

equations for the fields and particles in this experiment are identical to those used in the 2D-2V

(2D-2P) electromagnetic heating experiment discussed in section 2.4.2.3. The proposed method

was compared with the popular FDTD-PIC method, which (again) adopts the TE mode convention;

however, this selection for the staggering with the FDTD mesh is motivated by the PEC boundary

conditions for the fields, which, respectively, require 𝐸 (1) to be zero along the 𝑥-axis and 𝐸 (2) to be

zero along the 𝑦-axis. Furthermore, since 𝐵(3) is cell-centered, its values along the boundary do not

need to be specified. In our implementation of the FDTD-PIC method, we extend the 𝐸 (1) field a

half-cell in the 𝑥 direction and the 𝐸 (2) field a half-cell in the 𝑦 direction. This ensures that 𝐸 (1) and

𝐸 (2) are positioned on the boundary and that 𝐵(3) remains on the interior of the domain. We note

that this is not a requirement for the numerical heating problem, which uses periodic boundaries

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and therefore has no such half-cell extensions.

Parameter
Average number density ( ¯𝑛) [m−3]
Average temperature ( ¯𝑇) [K]
Debye length (𝜆𝐷) [m]

Inverse angular plasma frequency (𝜔−1

𝑝𝑒) [s/rad]

Thermal velocity (𝑣𝑡ℎ = 𝜆𝐷𝜔 𝑝𝑒) [m/s]

Value
2.5 × 1012
1.0 × 104
4.364992 × 10−3
1.121147 × 10−8
3.893328 × 105

Table 2.3 Table of the plasma parameters used in the sheath problem.

In the first test, we perform a series of refinements to understand how the solution is impacted

as a function of spatial resolution and the number of macroparticles. More specifically, we fix the

total number of macroparticles and adjust the number of cells per dimension, taking 𝑁𝑥 = 𝑁𝑦 = 𝑁.

During the refinement, 𝑁 is successively doubled from 16 to 64. We use 250, 632 macroparticles for

each species, for all meshes, which results in slightly more than 60 macroparticles of each species per

mesh cell, in the case of a 64 × 64 mesh. We also fix Δ𝑡 = CFLΔ𝑥/𝑐, where CFL = 1/

√

2 for both

the FDTD and BDF-1 field solvers. As noted, the initial positions for ions and electrons are sampled

from a uniform distribution over the domain [−8𝜆𝐷, 8𝜆𝐷]2 with a fixed random seed across all runs.

The heavier ions are treated as stationary, so their velocities are set to zero in this test. Electrons

velocities are obtained by sampling from a Maxwellian distribution using the parameters shown

in Table 2.3. The problem is made current neutral by splitting the electrons into two equally sized

groups whose velocities differ only in sign, and the simulation is run for 60 angular plasma periods.

At each time step, we record the particle count, electron temperature, as well as the potentials and

the corresponding fields on the mesh.

Figure 2.11 is a plot of the scalar potential for both methods at a time of 50 angular plasma

periods using a 64 × 64 mesh. On the left, we plot the solution obtained with the Boris + FDTD

method, while the plot on the right was obtained using the IAEM + BDF-1 approach. At 50 angular

plasma periods, the transients arising from the sheath formation have mostly settled, and the fields

become flat in the middle of the domain. What remains is a steady breathing mode, which we

address in two ways. The first way is to choose a time snapshot were the field is flat on the interior,

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as seen in Figure 2.11. The second is to time-average the fields, which is done in Figure 2.12 that

follows. To get a sense of the sheath size, we can appeal to the well-known analytical theory of

the matrix sheath [12]. Marked in red is the analytical solution to the 1D matrix sheath calculated

from the plasma parameters from the simulation. We see that the predicted locations of the sheath

obtained with both methods are in reasonable agreement with the analytical solution.

Figure 2.11 A comparison of the scalar potential obtained with the benchmark approach (Boris +
FDTD) and the proposed scheme (IAEM + BDF-1). The formula for calculating the expected sheath
(cid:112)2𝜙0/𝑇𝑒, given in [12], is also included in the plot (marked with red lines). Here, 𝜙0
width 𝑠 = 𝜆𝐷
is the magnitude of the potential at the center of the domain, where the potential is flat. The sheaths
produced by both methods agree with the width predicted from theory.

In Figure 2.12, we plot the time-averaged potentials and fields obtained with both methods

using a 64 × 64 mesh. Solutions are time-averaged across the time interval ranging from 𝑡 = 40 to

𝑡 = 60 angular plasma periods. We observe more noise in the fields obtained with the Boris + FDTD

approach when compared to the fields computed with the IAEM + BDF-1 method. Additionally,

the Boris + FDTD method produces a clear asymmetry in the potential that is not present in the

potential obtained with the IAEM + BDF-1 method. This suggests that the breathing mode is less

adequately maintained with the Boris + FDTD method. If we look at the non-averaged solutions at

a later time, this is indeed true. For this problem, one expects a symmetric solution, which indicates

that the new method is an improvement to the benchmark approach.

In Figure 2.13, we plot the time-averaged center line potential about 𝑦 = 0. As in Figure 2.12,

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(a) IAEM + BDF-1

(b) Boris + FDTD
Figure 2.12 Time-averaged scalar potential along with the time-averaged 𝑥 and 𝑦 components of the
electric fields for the sheath problem. The breathing mode is well established by 𝑡 = 40, and the
fields are averaged over the next 20 angular plasma periods. The potential and fields produced by
the benchmark method (Boris + FDTD) are much rougher than those obtained with the proposed
method (IAEM + BDF-1).

the solution is time-averaged from 𝑡 = 40 to 𝑡 = 60 angular plasma periods. The plot on the left is

the IAEM + BDF-1 method and the one on the right is the Boris + FDTD method. The solid lines

in both plots use a total of 250, 632 macroparticles, per species, for all mesh resolutions, while the

dashed lines represent results with the number of macroparticles per cell set to 61. We see that the

solution for the IAEM + BDF-1 approach shows notably smaller fluctuations across different mesh

resolutions and is qualitatively more symmetric than the Boris + FDTD approach. These structural

features in the potential also influence the convergence properties of the method. This aspect of the

study is discussed in greater detail when we consider the time trace of the particle count and the

temperature as the sheath forms and the problem settles into a breathing mode dynamic. In Figure

2.14 we consider the same results across a variety of CFL numbers, observing that these potentials

also have superior smoothness to those from the Boris + FDTD method, but that the less refined

83

meshes do converge less quickly by CFL.

Figure 2.13 Cross-sections of the time-averaged potentials computed with both methods. Results
obtained using a fixed total number of macroparticles per species (“Count Fixed”) and a fixed number
of particles per cell (“Ratio Fixed”) are presented for comparison. Each simulation begins the time
averaging procedure after 40 angular plasma periods, at which point both methods are well into
the “breathing mode.” The cross-sections are taken about 𝑦 = 0 and are similar to those obtained
by [12]. We note that the time-averaged potentials obtained with the Boris + FDTD approach are
substantially “rougher” near the center.

(a) 16 × 16 Potential Slice

(b) 32 × 32 Potential Slice

(c) 64 × 64 Potential Slice

Figure 2.14 We display the time averaged potentials sliced across the center of the 𝑦-axis, refining
the IAEM + BDF-1 method by the CFL number. We see both ratio and count fixing have similar
behavior. Refining by CFL has little impact with the more refined 64 × 64 grid, but has some impact
on the less refined 16 × 16 and 32 × 32 grids, though it bears emphasizing that all refinements have
superior smoothness across the potential than those of the Boris + FDTD method (Figure 2.13).

Next, in Figure 2.15, we show the electron temperature (left), the macroparticle count (middle)

and electron count (right) for the range of spatial resolutions described earlier. In this study, the total

macroparticle count for each species is initially 250, 632 for all runs. We see that the temperature and

particle counts for the Boris + FDTD and IAEM + BDF-1 methods converge as the cell resolution

increases. In particular, the Boris + FDTD method converges more slowly than the IAEM+BDF-1

84

method. We note that there are two mechanisms associated with the faster convergence of the

IAEM + BDF-1 method. First, the IAEM+BDF-1 approach is high-order in space, so the sheath

will be more resolved when compared to the Boris + FDTD approach on a similar computational

mesh. Second, the Boris + FDTD approach contains a higher level of noise than the BDF-1 scheme

because the latter is dissipative, while the former is dispersive. Additional noise in the Boris + FDTD

approach is due to the current weighting scheme [37], which is used to enforce charge conservation.

We might also expect that issues such as numerical heating could be impacting the Boris + FDTD

results when the Debye length is not adequately resolved. In contrast, the IAEM + BDF-1 method

displays more robust behavior even when the Debye length is “under-resolved.” We repeat this

experiment but we fix the number of macroparticles per cell to be 61. Similar phenomena is

observed in Figure 2.16, namely, we see faster convergence in the number of electrons for the

IAEM + BDF-1 method. Further, the results for the IAEM + BDF-1 approach on the 16 × 16 mesh

and 61 macroparticles per cell are similar to those obtained with a 16 mesh and 250, 632 total

macroparticles. This is also true for the 32 × 32 mesh in the case of the IAEM + BDF-1 method.

In contrast, the results for the Boris + FDTD approach, while qualitatively similar, are not nearly

as close together. These results, together with the data presented in Figures 2.12 and 2.13, seem

to suggest that the IAEM + BDF-1 method can be used with far fewer simulation particles than

the traditional Boris + FDTD method. Figure 2.14 further indicates that the CFL number can be

increased without impacting the smoothness of this potential, though it does slightly impact the

magnitude of the potential for the less refined meshes. These features will be the study of future

work.

In Figure 2.17, we plot the electron temperatures as probability densities with a fixed total

number of macroparticles set to 250, 632. From left to right in the figure, the plots correspond to

data obtained using 16 × 16, 32 × 32, and 64 × 64 spatial meshes. The distribution function for

IAEM + BDF-1 is in red and the distribution function for Boris + FDTD is in blue. In each case,

the warmer tails of the IAEM + BDF-1 densities contain more simulation particles than the Boris +

FDTD method. This is likely due to the high-order spatial resolution of the IAEM + BDF-1, as it

85

Figure 2.15 Electron temperature and particle counts collected for the sheath experiment using
different grid resolutions and a fixed total number of macroparticles. As the mesh is refined, the
number of macroparticles per cell decreases. However, the total number of physical particles is
scaled so that the runs start in an identical manner. The data obtained with the new method (IAEM
+ BDF-1) is plotted using solid lines, while the results for the benchmark method (Boris + FDTD)
are plotted on dashed lines. In the plot of the electron counts we focus on the region in which
the potential begins to settle, highlighting the differences in the electron count. We see that the
retention of the faster electrons with the new method results in a larger electron temperature when
compared with the benchmark scheme. The observed electron temperatures in the proposed method
are consistent with the results of the electromagnetic heating experiment in Figure 2.8. We note that
the new method experiences less significant fluctuations in the temperature and physical particle
counts than the benchmark scheme, despite the variations in the number of macroparticles per cell.

Figure 2.16 Electron temperature and count data collected for the sheath experiment with different
grid resolutions. In contrast to the data presented in Figure 2.15, the number of macroparticles
is increased as the mesh is refined so that the number of macroparticles per mesh cell is identical
across the runs. The data obtained with the new method (IAEM + BDF-1) are plotted using solid
lines, while the results for the benchmark method (Boris + FDTD) are plotted on dashed lines. The
new method produces more qualitatively consistent results across the mesh resolutions than the
benchmark scheme. In particular, these results suggest that the new method self-refines at a faster
rate to the data obtained with the finest mesh than the benchmark scheme.

86

will be able to resolve the sheath with fewer points. It could also have to do with the reduced noise

and improved symmetry observed in the solution to the fields from the IAEM + BDF-1 method, as

noise or asymmetry in the breathing mode could easily push a warmer particle out of the domain.

Figure 2.17 Stacked histograms of the particle temperature distributions, at the final time, obtained
with both methods, for several different grid resolutions, with the total number of macroparticles
held fixed. The bars in red correspond to the proposed method, and blue bars correspond to the
FDTD-PIC method. While the bulk properties are similar among the approaches, the new method
retains more of the warmer electrons than the benchmark scheme, contributing to a larger overall
temperature.

Lastly, we check for any significant violations of the Lorenz gauge condition for the IAEM +

BDF-1 method. Figure 2.18 plots the ℓ2-norm of the gauge error as a function of time. We change

the spatial resolution but keep the number of macroparticles fixed at 250, 632. In these experiments,

the method maintains a bounded gauge error for all time. However, as the number of particles per

cell decreases, we observe an increase in size of the Lorenz gauge error. We note that there is a sign

to the gauge error, depending on whether it is ions or electrons. In a system that is truly equal, in the

sense of number of particles, this error cancels. We believe the increase in the error with increased

mesh resolution is simply a result of less local charge cancellation. This will also be explored as

part of our future work.

2.4.2.5 Non-relativisitic Expanding Particle Beam

We now consider an application of the proposed methods to expanding particle beams [212].

This example is well-known for its sensitivity to issues concerning charge conservation, so it is

typically considered when evaluating methods used to enforce charge conservation. While this

87

Figure 2.18 The ℓ2-norm of the residual in the Lorenz gauge for the new method using a fixed total
number of particles. Although the magnitude of the error increases as we refine the mesh, it is
reasonable given that no method is used to enforce the gauge condition.

particular example is normally solved in cylindrical coordinates, the simulations presented in this

work use a two-dimensional rectangular grid that retains the fields 𝐸 (1) and 𝐸 (2), as well as 𝐵(3). An

injection zone is placed on one of the faces of the box and injects a steady beam of particles into the

domain. The beam expands as particles move along the box due to the electric field and eventually

settles into a steady-state. Similar to the sheath problem, particles are absorbed or “collected” once

they reach the edge of the domain and are removed from the simulation. Along the boundary of

the domain, the electric and magnetic fields are prescribed PEC boundary conditions, which, in

two spatial dimensions, is equivalent to enforcing homogeneous Dirichlet boundary conditions on

the potentials 𝜙, 𝐴(1), and 𝐴(2). Since the problem is PEC, there can be no (tangential) currents or

charge on the boundary.

As discussed earlier, the FDTD method is known to preserve the involutions for Maxwell’s

equations in the absence of moving charge [166, 279]. However, this is not applicable to the examples

considered in this work. In order to update the fields in the FDTD approach, we need to map the

current density components 𝐽 (1) and 𝐽 (2) to mesh points that are collocated with 𝐸 (1) and 𝐸 (2),

respectively, according to the mesh staggering. As mentioned earlier, it is well known that the use of

bilinear maps for depositing current to the mesh results in catastrophic errors due to violations of

88

charge conservation. The resulting fields cause the charged particles to “focus” in certain regions,

leading to the appearance of striation patterns. An example of this phenomenon is presented in

Figure 2.19, which shows the formation of non-physical striations after twenty particle crossings.

However, indicators for such patterns can appear as early as two particle crossings. The FDTD

method used in the comparison implements the charge conserving map for the current outlined by

Villasenor and Buneman [37].

Figure 2.19 Striation patterns in a non-relativistic expanding beam simulation using the FDTD-PIC
method with bilinear current mappings (area weighting). Irrotational errors in the electric field
introduced by the mapping cause the particles to focus in regions of the domain.

To setup the simulation, we first create a box specified by the region [0, 1] × [−1/2, 1/2] that is

normalized by the length scale 𝐿, which corresponds to the physical distance along the 𝑥-axis of

the box. We assume that the beam consists only of electrons, which are prescribed some injection

velocity 𝑣injection through their 𝑥-components. An estimate of the crossing time for a particle can
be obtained from the injection velocity and the length of the domain, which sets the time scale 𝑇

for the simulation. The duration of the simulation is given in terms of particle crossings, which are

then used to set the time step Δ𝑡. In each time step, particles are initialized in an injection region

specified by the interval [−𝐿ghost, 0) × [−𝑅𝑏, 𝑅𝑏], where 𝑅𝑏 is the radius of the beam, and the width
of the injection zone 𝐿ghost is chosen such that

𝐿ghost = 𝑣injectionΔ𝑡.

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This ensures that all particles placed in the injection zone will be in the domain after one time step.

The positions of particles in the injection region are set according to samples taken from a uniform

distribution, and the number of particles injected for a given time step is set by the injection rate.

In each time step, the injection procedure is applied before the particle position update, so that, at

the end of the time step, the injection zone is empty. To prevent the introduction of an impulse

response in the fields due to the initial injection of particles, a linear ramp function is applied to the

macroparticle weights over one particle crossing. The methods were applied using both wide and

narrow beam configurations, whose parameters can be found in Tables 2.4 and 2.5, respectively. The

normalized speed of light in the simulations is 𝜅 = 5.995849. Using the wide beam configuration

listed in Table 2.4, we obtain 𝜎1 = 1.006865 × 10−1 and 𝜎2 = 2.762661 × 10−1 for the normalized

permittivity and permeability, respectively. For the narrow configuration provided in Table 2.5,

these parameters change to 𝜎1 = 5.061053 × 10−1 and 𝜎2 = 5.496139 × 10−2, respectively.

Parameter
Beam radius (𝑅𝑏) [m]
Average number density ( ¯𝑛) [m−3]
Physical domain length (𝐿) [m]
Injection velocity (𝑣injection) [m/s]
Injection rate (𝑟injection) [s−1]
Crossing time (𝑇) [s]

Value
8.0 × 10−3
7.8025 × 1014
1.0 × 10−1
5.0 × 107
1 × 102
2.0 × 10−9

Table 2.4 Parameters used in the setup for the non-relativistic expanding particle beam problems.
Wide beam configuration.

Parameter
Beam radius (𝑅𝑏) [m]
Average number density ( ¯𝑛) [m−3]
Physical domain length (𝐿) [m]
Injection velocity (𝑣injection) [m/s]
Injection rate (𝑟injection) [s−1]
Crossing time (𝑇) [s]

Value
8.0 × 10−3
1.5522581 × 1014
1.0 × 10−1
5.0 × 107
1 × 102
2.0 × 10−9

Table 2.5 Parameters used in the setup for the non-relativistic expanding particle beam problems.
Narrow beam configuration.

The proposed method was compared against the Boris + FDTD method using the problem config-

90

urations specified in Tables 2.4 and 2.5. Each simulation was evolved to a final time corresponding

to 3,000 particle crossings with a 128 × 128 mesh. A total of 4 × 106 time steps were used, which

gave a CFL ≈ 0.576 for the fields. In Figure 2.20, we plot the particles in the beams generated

using the IAEM + BDF-1 solver and the Boris + FDTD method. We observe excellent agreement

with the benchmark FDTD PIC method despite the first-order time accuracy of the new method.

Additionally, we find that the steady-state structure of the beam is well-preserved with the proposed

method despite the fact that charge conservation is not strictly enforced. We find that the potentials

and their spatial derivatives, which are computed using the BDF-1 wave solver are quite smooth and

do not show signs of excessive dissipation even after 3,000 particle crossings. Plots of the scalar

potential and its gradient obtained with the proposed methods are displayed in Figure 2.21. We

show this data for the wide beam configuration provided in Table 2.4, and note that the results are

quite similar for the narrow beam configuration provided in Table 2.5. While the goal of our work

is to build higher-order field solvers for plasma applications, these results are interesting from the

perspective of practicality, as they demonstrate that it is possible to obtain a solution of reasonable

quality in a fairly inexpensive manner.

2.4.2.6 The Mardahl Beam Problem

We conclude the numerical experiments with the Mardahl beam problem, which is a benchmark

relativistic beam problem proposed by Mardahl and Verboncoeur [112]. In this problem, electrons

are injected into a PEC cavity with relativistic velocities (𝑣injection = 0.967𝑐), and the number density

is relatively small, so the beam moves across the domain mostly unperturbed. Once the electrons

reach the boundary, they are removed from the simulation. A complete list of parameters for our

experimental setup, which were derived from [112], is provided in Table 2.6. The normalized

speed of light for this problem is 𝜅 = 1.034126, and the corresponding normalized permittivity

and permeabilities are 𝜎1 = 1.226639 × 103 and 𝜎2 = 7.623181 × 10−4 As with its non-relativistic

counterpart, this problem is also sensitive to violations of charge conservation [270]. It also serves

as as useful demonstration of the formulation presented in this work in the relativistic setting, which

is the state space for applications that will be considered in future work.

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(a)

(b)

Figure 2.20 We compare the proposed PIC method against the standard FDTD-PIC method for
the non-relativistic expanding beam configurations specified in Tables 2.4 (shown on the left) and
2.5 (shown on the right). In each case, the electrons positions generated by the two methods after
3,000 crossings are plotted together to track the shape of the beam. The electrons from the proposed
method are plotted in blue, while those of the FDTD-PIC method are plotted in red. We note that the
beams produced using the proposed methods remain intact after many particle crossings without the
use of a cleaning method. Moreover, the beams generated by the proposed methods show excellent
agreement with the beam profiles from the benchmark FDTD-PIC method.

Figure 2.21 The scalar potential 𝜙 and its spatial derivatives for the non-relativistic expanding beam
problem after 3,000 particle crossings. The fields shown above correspond to parameters listed in
Table 2.4. No methods are used to enforce the gauge condition in this experiment. We can see that
the proposed methods generate smooth fields for subsequent use in the particle update.

The setup for this test case is nearly identical to the non-relativistic expanding beam problems

considered in the previous section, so we shall limit the discussion here for brevity. In the original

presentation [112], the edge of the particle beam coincides with the boundary of the physical domain.

Instead, we extend the normalized domain to [0, 1] × [−1, 1], so that the edge of the beam can be

clearly seen. Furthermore, the original presentation showed simulation results up to 100 crossings

of the beam. The final time of the simulation was set to 3, 000 particle crossings and used 4 × 106

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Parameter
Beam radius (𝑅𝑏) [m]
Average number density ( ¯𝑛) [m−3]
Physical domain length (𝐿) [m]
Injection velocity (𝑣injection) [m/s]
Injection rate (𝑟injection) [s−1]
Crossing time (𝑇) [s]

Value
5.0 × 10−1
2.15299207054 × 1010
1.0 × 100
2.89899306886 × 108
1 × 102
3.44947358 × 10−9

Table 2.6 Table of the parameters used in the setup for the Mardahl beam problem.

time steps, so the fields and particles have a CFL ≈ 0.01. We remark that this number is quite small

for the BDF-1 field solver, which by the stability result shown in section 2.2.2.2, permits a much

larger time step. As discussed earlier, the principle concern of this work is the development of a

compatible formulation that can leverage the implicit wave solvers developed in previous work, e.g.,

[267, 268]. The exploration and integration of these solvers with the methods of this paper is an

open area of research. As with the non-relativistic test case, the results of the proposed method show

excellent agreement with the benchmark FDTD-PIC method, despite the fact that no method is used

to explicitly enforce the gauge condition.

Figure 2.22 We compare the proposed PIC method against the standard FDTD-PIC method for the
(relativistic) Mardahl beam problem whose configuration is specified in Table 2.6. The electron
positions generated by the two methods are plotted together after 3,000 crossings to track the shape of
the beam. The electrons from the proposed method are plotted in blue, while those of the FDTD-PIC
method are plotted in red. The beam simulated with the proposed method remains intact after
many particle crossings without the use of a cleaning method. Unlike the original paper [112], we
extended the physical domain beyond the edge of the beam to investigate its structure. Similar to the
non-relativistic problem, the beam structure obtained with the proposed method shows excellent
agreement with the profile from the benchmark FDTD-PIC method.

93

In this section, we presented a collection of numerical results for the BDF wave solver, including

applications to plasmas, where the wave solver is used to update fields in a new PIC method that is

based on a Hamiltonian formulation. First, we analyzed the methods for evolving the potentials as

well as the novel techniques for computing derivatives on the mesh. We considered several types

of boundary conditions of relevance to the plasma examples presented in this work. In each case,

the proposed methods for derivatives displayed space-time convergence rates that are identical to

those of the wave solver used for the potentials. After establishing the convergence properties of

the wave solver and the methods for derivatives, we then considered several applications involving

plasmas using the new PIC method. The accuracy of the new PIC method was confirmed through

a comparison of results obtained with standard PIC approaches. In particular, the new methods

displayed superior numerical heating properties over the benchmark Boris + FDTD method used

for electromagnetic problems. Additionally, the new method showed notable improvements in the

sheath experiment, in terms of stability and preservation of symmetries. We found this to be true

even in cases where the grid resolution is comparable to the plasma Debye length, which means

that the new method permits coarser grids to be used in simulations. Additionally, the methods

showed excellent agreement with the benchmark Boris + FDTD method for the relativistic and

non-relativistic particle beam tests without resorting to the use of a method to explicitly enforce

charge conservation. It is noteworthy, however, that the Lorenz gauge and Gauss’s Law still exhibit

nontrivial errors. These errors, though bound, are the reason why the Boris + FDTD is considered

one of the staples in the PIC community, the Yee grid satisfies Gauss’s law by very nature of its

staggering.

2.5 Conclusion

The method we have developed has a number of desirable traits. The resilience to numerical

heating, geometric flexibility, structure preserving behavior, and simple interpolation scheme all

recommend this method. However, the errors in the Lorenz gauge and Gauss’s law inherent MOTL +

IAEM method are nontrivial. Though the numerical tests presented in this chapter are not impacted

by these errors, this is far from guaranteed in other tests, not to say anything of real application

94

scenarios. The Lorenz gauge acts as our map back to Maxwell’s equations, and Gauss’s law for

electricity is a fundamental involution that must be satisfied for the system to remain physically

accurate. In the coming chapters we will explore how our method may approach this obstacle.

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CHAPTER 3

ENFORCING THE LORENZ GAUGE

The above chapter introduced a novel PIC algorithm combining the MOLT wave solver with the

Improved Asymmetrical Euler Method. This chapter extends the method by developing different

approaches to enforce the Lorenz gauge condition in the context of co-located meshes. We start

by establishing a certain time-consistency theorem, which shows, at the semi-discrete level, that

enforcing the Lorenz gauge is equivalent to satisfying the continuity equation. Using the time-

consistency result from this chapter, we first develop a map for the charge density that ensures we

exactly satisfy a semi-discrete analogue of the Lorenz gauge condition on a non-staggered grid. This

method utilizes the bilinear map of current density on the mesh, which is then coupled to a point-wise

solution to the continuity equation to compute the charge density. In this time-consistent approach,

the spatial derivatives are computed with high-order finite differences (e.g., sixth-order accuracy) or

with a spectral method. A Lagrange multiplier is introduced to enforce charge conservation globally.

When this charge density is used in the update of the scalar potential, both with and without the

Lagrange multiplier, the method effectively satisfies the Lorenz gauge condition to machine precision.

The second approach we propose is an exact map that enforces a semi-discrete continuity equation

through a source term in the integral formulation of the field solver. A notable advantage offered by

the integral solution with this particular map is that it allows for the removal of spatial derivatives

that act on the particle current density. This quantity may be “rough” if too few simulation particles

are used, as it is computed from linear combinations of low-order spline basis functions. The action

of the spatial derivatives further amplifies this. Instead, we propose a new map that eliminates the

spatial derivatives of the particle data by exploiting the structure of the integral solution for the

scalar potential. The method is ideal for integral solutions that make use of the multi-dimensional

Green’s function as in [129]. The third method we consider is a gauge-correcting method in which

bilinear mappings are used for both charge and current density on the mesh. Since the resulting

fields will not satisfy the gauge condition, we treat the vector potential solution as exact and use the

Lorenz gauge to compute a correction to the scalar potential. The vector potential and corrected

96

scalar potential are then guaranteed to satisfy the gauge condition to machine precision.

One of the major goals of this effort is to design fast algorithms that can address the challenges

posed by the geometry of experimental devices. For this reason, we focus on simple mappings that

are based on bilinear maps (or area weightings) which are more convenient in such circumstances

and offer a good balance between accuracy and efficiency. Bilinear maps produce smoother rep-

resentations of the particle data on the mesh compared to piece-wise constant maps, but still do

no require support from nodes not immediately adjacent to the particle in question. For the gauge

enforcing methods, the exact charge density mapping proposed in this work uses bilinear maps only

for the current densities, and the continuity equation is used to identify the corresponding charge

density. The gauge-correcting approach, on the other hand, uses bilinear maps for both the charge

and current densities, but the gauge error is controlled through an equation that corrects the scalar

potential.

This structure of this chapter is as follows. First, for convenience we will briefly review the

equations that we will be using in Section 3.1. In Section 3.2, we establish some important properties

of the semi-discrete formulation used for the potentials, including a time-consistency property

of the field solver, which largely motivates the methods introduced in subsequent sections. We

then introduce three techniques that enforce a semi-discrete gauge condition in Section 3.3. Some

numerical results are presented in Section 3.4 to test the methods in periodic domains. In particular,

we consider the relativistic Weibel instability and present a new test problem that is designed to

amplify the errors in the gauge condition. A brief summary of the chapter is presented in Section

3.5. This chapter is based on the work published by myself in collaboration with Dr. Andrew

Christlieb and Dr. William Sands [3]. My contribution in this work was the code development for

our test problems, developing the so-called “moving cloud” problem, as well as assisting in the

development of the theorems connecting satisfaction of the Lorenz gauge with satisfaction of the

continuity equation and Gauss’s law.

97

3.1 Problem Review

The vector and scalar potential formulation of Maxwell’s equations and the Newton-Lorentz

force have already been discussed thoroughly in Section 1.3, with the BDF-1 wave solver derived in

Section 2.2. As such, we will not belabor the point, but will simply recall that the physical system

we wish to simulate is as follows:

𝑑x𝑖
𝑑𝑡

=

(cid:113)

𝑑P𝑖
𝑑𝑡

=

(cid:113)

𝑐2 (P𝑖 − 𝑞𝑖A)
𝑐2 (P𝑖 − 𝑞𝑖A)2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2
𝑞𝑖𝑐2 (∇A) · (P𝑖 − 𝑞𝑖A)
𝑐2 (P𝑖 − 𝑞𝑖A)2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2

− Δ𝜙 =

,

𝜌
𝜖0

− ΔA = 𝜇0J,

+ ∇ · A = 0,

𝜕2𝜙
𝜕𝑡2
𝜕2A
𝜕𝑡2
𝜕𝜙
𝜕𝑡

1
𝑐2

1
𝑐2
1
𝑐2
𝜕 𝜌
𝜕𝑡

+ ∇ · J = 0.

,

− 𝑞𝑖∇𝜙,

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)






Equations (3.1) and (3.2) are the particle equations of motion, (3.3) and (3.4) are the scalar and

vector potential wave equations, (3.5) is the Lorenz gauge condition we must satisfy, and (3.6) is the

continuity equation we will ideally satisfy as well. Similar the the Lorenz gauge, we wish to satisfy

Gauss’s laws




(3.8) is satisfied by the definition of A, so it is really (3.7) that we must work to satisfy.

∇ · E =

𝜌
𝜖0
∇ · B = 0.

,

(3.7)

(3.8)

The non-dimensionalization of these equations may be found in Appendix A. The particle

equations of motion will remain constant throughout the rest of this thesis, we are only interested in

modifying how the potentials are updated. In Chapter 2 we semi-discretized the waves using the

BDF-1 method, dimensionally split the waves, and solved the resulting set of 1D boundary value

problems using a Green’s function. We will continue considering this discretization scheme, which

98

gives us the following system:

L (cid:2)𝜙𝑛+1(cid:3) = 2𝜙𝑛 − 𝜙𝑛−1 +

L (cid:2)A𝑛+1(cid:3) = 2A𝑛 − A𝑛−1 +
𝜙𝑛+1 − 𝜙𝑛
𝑐2Δ𝑡

+ ∇ · A𝑛+1 = 0,

,

𝜌𝑛+1
𝜖0
J𝑛+1,

1
𝛼2
𝜇0
𝛼2

(3.9)

(3.10)

(3.11)






The nondimensionalization of these equations may be found in Appendix A.

It is this system of two updates and an involution in which we are interested. However, we are

making one vital change to our approach. As mentioned above, in Chapter 2 we would split the

waves along dimensions and invert 1D operators. In this chapter, we will not be dimension splitting

for reasons that will become clear presently, ie L ≠ L𝑥L𝑦L𝑧. This means the integral solution will

look differently than that given in Section 2.2.3.1. Given L [𝑢] = 𝑆(x), the inverse to L will be as

follows:

∫

∫

(cid:18)

𝑢(x) =

𝐺 (x, y)𝑆(y) 𝑑𝑉y +

𝜎(y)𝐺 (x, y) + 𝛾(y)

(cid:19)

𝜕𝐺
𝜕n

𝑑𝑆y.

𝜕Ω
where 𝜎(y) is the single-layer potential and 𝛾(y) is the double-layer potential that are used to enforce

Ω

(3.12)

boundary conditions on 𝑢(x) [274]. Here, we use 𝐺 (x, y) to denote the Green’s function for the

operator L. This has several useful properties that will now be established.

The full algorithm is repeated, and slightly expanded, in Algorithm 3.1.

3.2 A Closer Look at the BDF-1 Wave Solver

3.2.1 Properties of the BDF-1 Wave Solver

In this section, we establish some structure preservation properties satisfied by the semi-discrete

potential formulation, (3.9)-(3.11). This includes a lemma and two theorems which provide a

connection relating the semi-discrete Lorenz gauge condition (3.11) to the semi-discrete continuity

equation

𝜌𝑛+1 − 𝜌𝑛
Δ𝑡

+ ∇ · J

𝑛+1 = 0.

We also discuss the connection to the satisfaction of Gauss’s Law, ∇ · E =

sense.

(3.13)

𝜌
𝜖0

, in the semi-discrete

99

Algorithm 3.1 Outline of the PIC algorithm with the improved asymmetric Euler method (IAEM)

Perform one time step of the PIC cycle using the improved asymmetric Euler method.
𝑖 ), as well as the fields (cid:0)𝜙0, ∇𝜙0(cid:1) and A0, ∇A0
1: Given: (x0
2: Initialize v−1
3: while stepping do

𝑖 , P0
𝑖 = v𝑖 (−Δ𝑡) using a Taylor approximation.

𝑖 , v0

4:

Update the particle positions with

x𝑛+1
𝑖

= x𝑛

𝑖 + v𝑛

𝑖 Δ𝑡.

5:
6:
7:
8:

9:

𝑛

, compute the current density J

𝑛+1 and v
𝑛+1, compute the charge density 𝜌𝑛+1 using a bilinear mapping.

Using x
Using x
Compute the potentials, 𝜙 and A, at time level 𝑡𝑛+1 using the semi-discrete BDF method.
Compute the spatial derivatives of the potentials at time level 𝑡𝑛+1 using the semi-discrete

𝑛+1 using a bilinear mapping.

BDF method.

Evaluate the Taylor corrected particle velocities

𝑖 = 2v𝑛
v∗

𝑖 − v𝑛−1
𝑖

.

10:

Calculate the new generalized momentum according to

P𝑛+1
𝑖

= P𝑛

𝑖 + 𝑞𝑖

(cid:16)

− ∇𝜙𝑛+1 + ∇A𝑛+1 · v∗
𝑖

(cid:17)

Δ𝑡.

11:

Convert the new generalized momenta into new particle velocities with

v𝑛+1
𝑖

=

(cid:113)

𝑐2 (cid:0)P𝑛+1

𝑖 − 𝑞𝑖A𝑛+1(cid:1)
𝑖 − 𝑞𝑖A𝑛+1(cid:1) 2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2

𝑐2 (cid:0)P𝑛+1

.

12:

Shift the time history data and return to step 4 to begin the next time step.

3.2.1.1 Time-consistency of the Semi-discrete Lorenz Gauge Formulation

We will now prove a theorem that establishes a certain time consistency property of the Lorenz

gauge formulation of Maxwell’s equations (3.3)-(3.5) under the semi-discrete BDF-1 discretization.

By time-consistent, we mean that the semi-discrete system (3.9)-(3.11) for the potentials induces

an equivalence between the gauge condition and continuity equation at the semi-discrete level. In

the treatment of the semi-discrete equations, we shall ignore effects of dimensional splittings and

instead consider the more general inverse induced by the integral solution (3.12). Note that space is

treated in a continuous manner, which means that effects due to the representation of particles on

the mesh are ignored.

100

We show that this semi-discrete system is time-consistent in the sense of the semi-discrete Lorenz

gauge. To simplify the presentation, we first prove the following lemma that connects the discrete

gauge condition (3.11) to the semi-discrete equations for the potentials given by equations (3.9) and

(3.10).

Lemma 3.2.1. The semi-discrete Lorenz gauge condition (4.48) satisfies the recurrence relation

𝜖 𝑛+1 = L−1 (cid:104)

2𝜖 𝑛 − 𝜖 𝑛−1 +

𝜇0
𝛼2

(cid:105)

,

𝜖 𝑛+1
2

where we have defined the semi-discrete residuals

𝜖 𝑛
1 =

𝜖 𝑛
2 =

𝜙𝑛 − 𝜙𝑛−1
𝑐2Δ𝑡
𝜌𝑛 − 𝜌𝑛−1
Δ𝑡

+ ∇ · A𝑛,

+ ∇ · J𝑛.

(3.14)

(3.15)

(3.16)

Proof. Again, we note that the modified Helmholtz operator L can be formally “inverted” with

the integral solution given by equation (3.12). From this, we can calculate the terms involving the

potentials in the residual (3.15). Proceeding, the equation for the scalar potential is found to be

(cid:32)

𝜙𝑛+1 = L−1

2𝜙𝑛 − 𝜙𝑛−1 +

(cid:33)

,

𝜌𝑛+1

1
𝛼2𝜖0

which can be evaluated at time level 𝑛 to yield

(cid:32)

𝜙𝑛 = L−1

2𝜙𝑛−1 − 𝜙𝑛−2 +

(cid:33)

.

𝜌𝑛

1
𝛼2𝜖0

Likewise, we take the divergence of A in equation (3.10) and invert L to see

∇ · A𝑛+1 = L−1 (cid:16)

2∇ · A𝑛 − ∇ · A𝑛−1 +

𝜇0
𝛼2

∇ · J𝑛+1(cid:17)

.

(3.17)

(3.18)

(3.19)

Substituting equations (3.17), (3.18), and (3.19) into (3.15) at time level 𝑛 + 1, we find it becomes

𝜖 𝑛+1
1 =

−

+

(cid:20)

(cid:20)

1
𝑐2

1
𝑐2

1
Δ𝑡
1
Δ𝑡

L−1

L−1

L−1 (cid:104)

1
𝛼2

2𝜙𝑛 − 𝜙𝑛−1 +

𝜌𝑛+1
𝜖0
𝜌𝑛
𝜖0
2∇ · A𝑛 − ∇ · A𝑛−1 +

2𝜙𝑛−1 − 𝜙𝑛−2 +

1
𝛼2

(cid:21)

(cid:21)

𝜇0
𝛼2

∇ · J𝑛+1(cid:105)

.

101

The linearity of the operator L may be exploited to rewrite this as

2L−1

(cid:20) 1
𝑐2

𝜙𝑛 − 𝜙𝑛−1
Δ𝑡

+ ∇ · A𝑛

(cid:21)

− L−1

(cid:20) 1
𝑐2

𝜙𝑛−1 − 𝜙𝑛−2
Δ𝑡

+ ∇ · A𝑛−1

(cid:21)

+

𝜇0
𝛼2

L−1

(cid:20) 𝜌𝑛+1 − 𝜌𝑛
Δ𝑡

(cid:21)

+ ∇ · J𝑛+1

Note that we have used the relation 𝑐2 = 1/(𝜇0𝜖0). It is clear the first two terms are 𝜖 𝑛
the second is the residual of the continuity equation at time 𝑛 + 1, 𝜖 𝑛+1

. We conclude

1

1 and 𝜖 𝑛−1

1

, and

1 = 2𝜖 𝑛
𝜖 𝑛+1

1 − 𝜖 𝑛−1

1

+

𝜇0
𝛼2

𝜖 𝑛+1
2

.

This completes the proof.

QED

With the aid of Lemma 3.2.1, we are now prepared to prove the following theorem that establishes

the time-consistency of the semi-discrete system.

Theorem 3.2.1. The semi-discrete Lorenz gauge formulation of Maxwell’s equations (3.9)-(3.11) is

time-consistent in the sense that the semi-discrete Lorenz gauge condition (3.11) is satisfied at any
discrete time 𝑡𝑛+1 if and only if the corresponding semi-discrete continuity equation (3.13) is also

satisfied.

Proof. We use a simple inductive argument to prove both directions. In the case of the forward
direction, we assume that the semi-discrete gauge condition is satisfied at any discrete time 𝑡𝑛
that the residual 𝜖 𝑛

such
1 = 0 ∀ 𝑛. Combining this with equation (3.14), established by Lemma 3.2.1, it

follows that the next time level satisfies

0 = L−1 (cid:104) 𝜇0
𝛼2

(cid:105)

.

𝜖 𝑛
2

Applying the operator L to both sides leads to 𝜖 𝑛

2 = 0, which establishes the forward direction.

A similar argument can be used for the converse. Here, we show that if the residual for the

semi-discrete continuity equation (3.13) is satisfied for any time level 𝑛, then the residual for the
discrete gauge condition also satisfies 𝜖 𝑛+1

1 = 0. First, we assume that the initial data and starting
1 = 0. Appealing to equation (3.14) with this initial data, it is clear that after a

values satisfy 𝜖 −1

1 = 𝜖 0

single time step, the residual in the gauge condition satisfies

1 = L−1 (cid:104)
𝜖 1

2𝜖 0

1 − 𝜖 −1

1 +

𝜇0
𝛼2

𝜖 0
2

(cid:105)

= L−1 (cid:104)

2(0) − (0) +

𝜇0
𝛼2

(cid:105)

(0)

= L−1 [0] = 0.

102

This argument can also be iterated 𝑛 more times to obtain the result, which finishes the proof. QED

Theorem 3.2.1 motivates, to a large extent, our choice in presenting a low-order time discretization

for the fields. By repeating the calculations above with a higher-order time discretization for the

fields, one can easily see that the residual for the gauge is non-vanishing, so additional modifications

will be needed. The extension to higher-order time accuracy is the subject of on-going work and will

be presented in a subsequent paper. While we have neglected the error introduced by the discrete

maps that transfer particle data to the mesh, the results we present for the numerical experiments in

Section 4.3 suggest that these effects are not significant. We will be using Theorem 3.2.1 to construct

maps for the charge density which satisfy the semi-discrete form of the gauge condition.

3.2.1.2 Satisfying Gauss’s Law

A benefit that comes with satisfying the semi-discrete Lorenz gauge (3.11) is the corresponding

satisfaction of Gauss’s law, ∇ · E =

𝜌
𝜖0

. We summarize this with the following theorem:

Theorem 3.2.2. If the semi-discrete Lorenz gauge (3.11) is satisfied, then Gauss’s law, ∇ · E =

𝜌
𝜖 , is

also satisfied.

Proof. Consider the backward Euler discretization of the Lorenz gauge condition in time:

1
𝑐2

𝜙𝑛+1 − 𝜙𝑛
Δ𝑡

+ ∇ · A

𝑛+1 = 0.

We know from the definition (1.10) that

𝑛+1 = −∇𝜙𝑛+1 − A

E

𝑛

𝑛+1 − A
Δ𝑡

.

Applying the divergence operator to both sides yields

∇ · E

𝑛+1 = −Δ𝜙𝑛+1 −

= −Δ𝜙𝑛+1 −

= −Δ𝜙𝑛+1 +

∇ · A

𝑛+1 − ∇ · A

𝑛

Δ𝑡

𝜙𝑛+1 − 𝜙𝑛
Δ𝑡

(cid:18)

−

1
𝑐2

(cid:19)

−

(cid:18)

−

1
𝑐2

𝜙𝑛 − 𝜙𝑛−1
Δ𝑡

(cid:19)

Δ𝑡
𝜙𝑛+1 − 2𝜙𝑛 + 𝜙𝑛−1
Δ𝑡2

.

1
𝑐2

103

From the semi-discrete update (3.9) for the scalar potential, the last line is equal to 𝜌𝑛+1/𝜖0, which

completes the proof.

QED

It is worth noting that the above argument requires that the discrete time derivative operator

applied to the Lorenz gauge be the same as the one applied to the scalar potential wave equation.

3.3 Enforcing the Lorenz Gauge

Enforcing the Lorenz gauge is an important step to keeping the system physical. As long as

errors exist in this gauge, the vector and scalar wave equations will not map back to Maxwell’s

equations, which is the original system we wish to simulate. In this section we present three methods

to enforce this condition: in section 3.3.1 a charge map exploiting the theorems proven in section 3.2,

in section 3.3.2 an extension to this charge map implementing multi-dimensional Green’s functions,

and finally in section 3.3.3 a gauge correcting method.

3.3.1 A Charge Map Utilizing Numerical Derivatives

We seek to create a map for the charge density that is consistent with a semi-discrete continuity

equation (3.13). Our goal is that this map enforce the Lorenz Gauge condition. The semi-discrete

properties established in Theorem 3.2.1 requires that the method satisfy (3.13) point-wise on the

mesh. From a practical perspective, if one goes through the derivation of Lemma 3.2.1, an important

observation is that if this Lemma is going to hold in the discrete setting, the discrete divergence

operators used to compute the solution to (3.13) must be the same discrete operator used to compute

the divergence of A. The remark in Section 3.3.1.1, which is presented later, provides further

clarification on this matter.

To create our map for the charge density, we will solve for 𝜌𝑛+1 using an update of the semi-
discrete continuity equation (3.13). The current density J𝑛+1 used in this update is obtained from the

particles using a bilinear map. To compute the discrete divergence operator, we will utilize either the

fast Fourier transform (FFT) or the sixth-order centered finite difference method (FD6). Additionally,

to make the point-wise map conservative, we introduce a Lagrange multiplier, which we now discuss.

In the construction, we make the following observation: Starting from the non-corrected point-wise

104

update

𝑖, 𝑗 = 𝜌𝑛
𝜌𝑛+1

𝑖, 𝑗 − Δ𝑡∇ · J𝑛+1
𝑖, 𝑗 ,

after summing over the periodic domain, we obtain

(cid:213)

𝜌𝑛+1
𝑖, 𝑗 =

(cid:213)

𝑖, 𝑗 − Δ𝑡 (cid:213)
𝜌𝑛

(cid:16)

∇ · J𝑛+1(cid:17)

𝑖, 𝑗

𝑖, 𝑗

𝑖, 𝑗

.

𝑖, 𝑗

(3.20)

To ensure charge conservation, we require

which further implies the condition

𝜌𝑛+1
𝑖, 𝑗 =

(cid:213)

𝑖, 𝑗

𝜌𝑛
𝑖, 𝑗 ,

(cid:213)

𝑖, 𝑗

(cid:213)

(cid:16)

∇ · J𝑛+1(cid:17)

𝑖, 𝑗

= 0.

𝑖, 𝑗

We introduce the Lagrange multiplier, 𝛾, to enforce this condition. For 2D, we define

𝛾 := −

1
𝑁𝑥 𝑁𝑦

(cid:213)

𝑖, 𝑗

(∇ · J)𝑖, 𝑗 , F :=

1
2

𝛾x,

and let J∗ := J + F be the adjusted current density. In 3D, the 1
∇ · J𝑛+1,∗(cid:17)

∇ · J𝑛+1 + ∇ · F

(cid:213)

(cid:213)

=

(cid:16)

(cid:16)

2 would become a 1
(cid:17)

𝑖, 𝑗

𝑖, 𝑗

𝑖, 𝑗

3. Correspondingly,

𝑖, 𝑗

(cid:213)

𝑖, 𝑗

(cid:213)

𝑖, 𝑗

=

=

(cid:32)

∇ · J𝑛+1 −

1
𝑁𝑥 𝑁𝑦

(cid:16)

∇ · J𝑛+1(cid:17)

−

𝑖, 𝑗

(cid:213)

𝑖, 𝑗

(cid:33)

𝑙,𝑘

(3.21)

(cid:213)

(cid:16)

∇ · J𝑛+1(cid:17)

𝑙,𝑘
(cid:16)

∇ · J𝑛+1(cid:17)

= 0.

𝑖, 𝑗

We now use 𝜌𝑛+1 and J𝑛+1,∗ in the update of 𝜙𝑛+1 and A𝑛+1.

As noted earlier, in the above expressions, the divergence ∇ · J

𝑛+1,∗
𝑖, 𝑗

is computed using either an

FFT or a sixth-order finite difference. For the FFT, we have the identity

𝜕𝑥𝑢 = F −1

𝑥

[𝑖𝑘𝑥 F [𝑢]] ,

where 𝑘𝑥 is the wavenumber. Derivatives in 𝑦 can be calculated using identical formulas. For FD6,

we use the following centered finite-difference stencil

𝜕𝑥𝑢𝑖, 𝑗 = −

1
60

𝑢𝑖−3, 𝑗 +

3
20

𝑢𝑖−2, 𝑗 −

3
4

𝑢𝑖−1, 𝑗 +

3
4

𝑢𝑖+1, 𝑗 −

3
20

𝑢𝑖+2, 𝑗 +

1
60

𝑢𝑖+3, 𝑗 ,

105

with an analogous set of coefficients for the 𝑦 derivative. It should be mentioned that in numerical

experiments, the schemes preserve the gauge condition to machine precision, with and without the
Lagrange multiplier, using either the FFT or FD6 in the update of 𝜌𝑛+1 (see Section 3.4).

In summary, Algorithm 3.1 is modified in two ways. Line 6 is changed by eliminating the bilinear
interpolation of particles to obtain 𝜌𝑛+1. Instead, 𝜌𝑛+1 is computed from the semi-discrete continuity
𝑛+1 for whatever discrete derivative operator has

equation (3.13), with line 5 supplying the value of J

been decided. Line 8 computes the derivatives acquired from line 7, which remains unchanged, and

it does so using the same discrete derivative operator, for reasons we now move on to explain.

3.3.1.1 A Remark on the Derivatives

It is worth commenting upon the nature of the way the ∇A and ∇𝜙 values are obtained in the

context of the IAEM method (Section 2.3.1.2). As stated above, the BDF method does have the
𝑛+1 and 𝜙𝑛+1 from the values of their previous

ability to analytically compute the derivatives of A

timesteps, combined with the sources J and 𝜌, respectively. There is a difficulty in this approach,

however. The proof of Theorem 3.2.1, which connects the gauge condition to the continuity equation,

is predicated upon Lemma 3.2.1, the proof of which assumes the derivative operator used on A and

J are the same. In order to assert (3.14), the following relationship must hold:

𝑛+1 = L−1 (cid:104)

A

2A

𝑛 − A

𝑛−1 +

𝑛+1(cid:105)

𝜇0
𝛼2 J

=⇒ ∇ · A

𝑛+1 = L−1 (cid:104)

2∇ · A

𝑛 − ∇ · A

𝑛−1 +

𝜇0
𝛼2

𝑛+1(cid:105)

.

∇ · J

Of course, in the continuous case, this is trivial. However, should we, for example, update 𝜌 based

on a discrete Fourier transform of ∇ · J, then this theorem does not hold unless the spatial derivative

applied to A is likewise obtained using a discrete Fourier transform. In the fully discrete setting,

Lemma 3.2.1 requires that the discrete derivative operators used to compute ∇ · A and ∇ · J have the

same form.

3.3.2 A Charge Map for the Multi-dimensional Boundary Integral Solution

The map used to construct 𝜌𝑛+1 in the previous section has a natural extension to boundary

integral formulations that evolve the potentials using multi-dimensional Green’s functions. The

solutions for the potentials in terms of the boundary integral representation (3.12) leads to the

106

equations

𝜙𝑛+1(x) =

A𝑛+1(x) =

(cid:18)

(cid:16)

𝐺 (x, y)

𝐺 (x, y)

∫

Ω

∫

Ω

2𝜙𝑛 − 𝜙𝑛−1 +

1
𝛼2

(cid:19)

𝜌𝑛+1
𝜖0

(y) 𝑑𝑉y +

∫

(cid:18)

𝜕Ω

𝜎𝜙 (y)𝐺 (x, y) + 𝛾𝜙 (y)

2A𝑛 − A𝑛−1 +

𝜇0
𝛼2

J𝑛+1(cid:17)

(y) 𝑑𝑉y +

∫

(cid:18)

𝜕Ω

𝜎A(y)𝐺 (x, y) + 𝛾A(y)

(cid:19)

𝜕𝐺
𝜕n

𝑑𝑆y,

(3.22)

(cid:19)

𝜕𝐺
𝜕n

𝑑𝑆y.

(3.23)

When particles are introduced, maps to the mesh for 𝜌𝑛+1 and J𝑛+1 generally do not satisfy the

semi-discrete continuity equation (3.13), which, by Lemma 3.2.1, also means that the semi-discrete
Lorenz gauge condition will be violated; however, if the map for 𝜌𝑛+1 can be connected to the map
for J𝑛+1, then this constraint will indeed be satisfied. Following the observation of the previous

section, we shall construct this map by solving equation (3.13) for the charge at the new time level

according to

𝜌𝑛+1 = 𝜌𝑛 − Δ𝑡 ∇ · J𝑛+1.

This discretization, which is first-order in time, is consistent with the BDF field solver considered

in this work. An obvious extension to higher-order is to include additional stencil points for the

approximation of time derivative, which will be considered at a later time. Note that at the initial

time 𝑡0 = 0, the particle data is known, and therefore, the charge and current densities 𝜌0 and J0 are

also known.

The primary objective is to define the volume integral term

∫

Ω

𝐺 (x, y) 𝜌𝑛+1(y) 𝑑𝑉y =

∫

Ω

𝐺 (x, y) 𝜌𝑛 (y) 𝑑𝑉y − Δ𝑡

∫

Ω

𝐺 (x, y)∇𝑦 · J𝑛+1(y) 𝑑𝑉y.

(3.24)

so that it is compatible with the low regularity charge and current data produced by particle simula-
tions. In PIC, we construct J𝑛+1 by scattering the current density of the particles to the mesh using

particle shape functions that typically have low regularity. In most cases, especially in bounded

domains, these maps are either piece-wise constant or linear functions, or combinations of these,

so we would like to avoid taking derivatives of this term. With the aid of the divergence theorem,

we can move the derivatives from the less regular particle data to the Green’s function 𝐺 whose

107

derivatives can be evaluated analytically. This allows us to write

∫

Ω

𝐺 (x, y)∇𝑦 · J𝑛+1(y) 𝑑𝑉y =

∫

𝜕Ω

𝐺 (x, y) J𝑛+1(y) · n (y) 𝑑𝑆y −

∫

Ω

∇𝑦𝐺 (x, y) · J𝑛+1(y) 𝑑𝑉y,

which can be combined with equation (3.24). This yields the map

∫

Ω

𝐺 (x, y) 𝜌𝑛+1(y) 𝑑𝑉y =

∫

Ω

− Δ𝑡

𝐺 (x, y) 𝜌𝑛 (y) 𝑑𝑉y
(cid:20)∫

𝐺 (x, y) J𝑛+1(y) · n (y) 𝑑𝑆y −

𝜕Ω

∇𝑦𝐺 (x, y) · J𝑛+1(y) 𝑑𝑉y

(cid:21)

,

∫

Ω

(3.25)

which, by Theorem 3.2.1, will be charge-conserving and enforce the Lorenz gauge condition in the

semi-discrete sense.

The charge map given by equation (3.25) can then be substituted into the boundary integral

solution (3.22) to give the update

𝜙𝑛+1(x) =

∫

(cid:18)

𝐺 (x, y)

2𝜙𝑛 − 𝜙𝑛−1 +

Ω
Δ𝑡
𝛼2𝜖0

+

(cid:20)∫

Ω

∇𝑦𝐺 (x, y) · J𝑛+1(y) 𝑑𝑉y −

∫

𝜕Ω

1
𝛼2

(cid:19)

𝜌𝑛
𝜖0

(y) 𝑑𝑉y +

∫

(cid:18)

𝜕Ω

𝜎𝜙 (y)𝐺 (x, y) + 𝛾𝜙 (y)

(cid:19)

𝜕𝐺
𝜕n

𝑑𝑆y

𝐺 (x, y) J𝑛+1(y) · n (y) 𝑑𝑆y

(cid:21)

.

(3.26)

Expressions for the gradients of the scalar potential ∇𝜙𝑛+1(x) can then be obtained by differentiating

equation (3.26). Since the derivatives are in x, they are transferred directly onto analytical functions.

We remark that the modifications required to achieve compatibility with dimensionally-split solvers

is non-trivial, since the divergence of the current density effectively couples the boundary conditions

between different directions. However, the map (3.26) would be well-suited for solvers that use the

full boundary integral solution [129].

Whereas in the previous approach we explicitly computed 𝜌𝑛+1 from 𝜌𝑛

and J

𝑛+1 using (3.13),

thus modifying line 6 and correspondingly 8, this approach circumnavigates this by, instead, replacing
computing 𝜌𝑛+1 entirely, cutting out line 6. We then modify lines 7 and 8 in Algorithm 3.1 both
𝑛+1 in the BDF method, and likewise
by computing 𝜙𝑛+1 using a combination of 𝜌𝑛, 𝜙𝑛, 𝜙𝑛−1, and J

with the spatial derivatives. It should be noted that the computation of A

𝑛+1 remains unchanged.

108

3.3.3 Correcting 𝜙 Using the Gauge Condition

As an alternative approach to solving the continuity equation, we assume that a bilinear map of
J𝑛+1 is exact and use the gauge condition itself to create a correction to 𝜙𝑛+1. To do this, we first use
a bilinear map to construct both the ‘‘exact’’ J𝑛+1 and an approximate 𝜌𝑛+1 called 𝜌𝑛+1
used to construct the “exact” vector potential A𝑛+1 and the approximate scalar potential 𝜙𝑛+1
correction to the scalar potential, 𝜙𝑛+1
𝐶 , that we propose makes use of the analytical derivatives as
discussed in Section 2.2.4 in the evaluation of the divergence of A𝑛+1. To start, we update the vector

𝐴 . These are

𝐴 . The

potential and scalar potential using bilinear maps for the particle data, which gives

A𝑛+1 (𝑥, 𝑦) = L−1

𝑦 L−1
𝑥

𝜙𝑛+1
𝐴

(𝑥, 𝑦) = L−1

𝑦 L−1
𝑥

(cid:20)

(cid:34)

2A𝑛 − A𝑛−1 +

2𝜙𝑛 − 𝜙𝑛−1 +

1
𝛼2

1
𝛼2

𝜇0J𝑛+1
(cid:35)

𝜌𝑛+1
𝐴
𝜖0

(cid:21)

(𝑥, 𝑦) ,

(𝑥, 𝑦) ,

where 𝜙𝑛

and 𝜙𝑛−1 are the corrected values such that A𝑛

, A𝑛−1, 𝜙𝑛

and 𝜙𝑛−1 satisfy the gauge

condition. We decompose the scalar potential as

𝜙𝑛+1 := 𝜙𝑛+1

𝐴 + 𝜙𝑛+1
𝐶 ,

(3.27)

where 𝜙𝑛+1
is the approximate potential and 𝜙𝑛+1
𝐴
𝜙𝑛+1 satisfy the gauge condition in the sense that

𝐶 is its correction. We also assume that A𝑛+1 and

1
𝑐2

𝜙𝑛+1 − 𝜙𝑛
Δ𝑡

+ ∇ · A𝑛+1 = 0.

Since this is designed to be a correction method, the discrete divergence can be evaluated using a

variety of methods, including the analytical differentiation techniques developed in Section 2.2.4.

Rearranging the expression for the gauge error, we obtain the correction

𝐶 = 𝜙𝑛 − 𝜙𝑛+1
𝜙𝑛+1

𝐴 − 𝑐2Δ𝑡

(cid:16)

𝜕𝑥 𝐴(1),𝑛+1 + 𝜕𝑦 𝐴(2),𝑛+1(cid:17)

.

Once the correction 𝜙𝑛+1
𝐶

is obtained, we then form the total potential according to the definition

(3.27) and compute its numerical derivatives for the particle advance used in the algorithm (no

significant difference was seen between FFT and FD6). The use of a numerical derivative in the
particle advance is because the terms that make up 𝜙𝑛+1

𝐶 do not sit under the operator L−1 [·].

109

This is the least intrusive of all our methods, being accomplished by simply adding a line after

8, in which we compute the correction to 𝜙 and its numerical derivatives. As will be seen in the

numerical results, by construction, this method satisfies the gauge error to machine precision. Also,

it is worth noting that if one needs the correct 𝜌, one can set up a iterative method that can compute
𝜌𝑛+1 from 𝜙𝑛+1. By Lemma 3.2.1, 𝜌𝑛+1 and J𝑛+1 will satisfy (3.13) because 𝜙𝑛+1 and A𝑛+1 satisfy
(4.48). Doing so would allow the use of analytical derivatives for 𝜙𝑛+1, as in 2.2.4, in the particle

update and would be useful in exploring problems with geometry. This will be explored in future

work.

3.3.4 Summary

In this section, we presented gauge-conserving maps that use Theorem 3.2.1 in a direct manner

to enforce the Lorenz gauge in the semi-discrete sense. The first map computes the current density
𝑛+1 using a standard bilinear map, then computes 𝜌𝑛+1 as the solution of a semi-discrete continuity

J

equation. A Lagrange multiplier was also used to make certain that the continuity equation was

enforced globally, though numerical experiments suggest this may not be necessary. The second map

we proposed is based on the multi-dimensional Green’s function and is well-suited for solvers that

leverage a boundary integral formulation, and will be investigated in future work. These maps, which

are based on semi-discrete theory, do not account for errors introduced by the spatial discretization.

Additionally, we introduced a third approach which identifies a correction to 𝜙 based on the discrete

gauge condition. The changes made to Algorithm 3.1 for all three methods are summarized in Table

3.1. In the next section, we investigate the capabilities of the first and third methods in problems

defined on periodic domains. We note that in all cases, developing appropriate boundary conditions

for bounded domains for inflow and outflow of charge that is consistent with gauge preservation is

the next important step, and this will be the subject of future work.

3.4 Numerical Results

This section contains the numerical results for the PIC method using the exact and approximate

methods to enforce the Lorenz gauge condition. We consider two different test problems with

dynamic and steady state qualities. We first consider the Weibel instability, which is a streaming

110

Method
Baseline method from Chapter 2
Numerical Derivatives FFT (or FD6)
Multi-dimensional Boundary Integral
Gauge correction for 𝜙

Summary of Changes
-
Lines 5 and 8
Lines (cid:1)6, 7, and 8
Correct 𝜙 after Line 8

Label
Bilinear Map
Charge Conserving, FFT (or FD6)
-
Gauge Correcting

Table 3.1 A summary of the methods considered in this paper. We use the method from our
previous work Chapter 2 as the baseline method in addition to three new methods. In each case,
we provide a summary of changes to Algorithm 3.1 required to implement the methods as well
as the corresponding label used to refer to each approach in the numerical results. Note that the
multi-dimensional boundary integral method is not given a label because it is not considered in the
numerical results.

instability that occurs in a periodic domain. The second example we consider is a simulation of a

non-relativistic drifting cloud of electrons against a stationary cluster of ions. In each example, we

compare the performance of the different methods by inspecting the fully discrete Lorenz gauge

condition and monitoring its behavior as a function of time. We conclude the section by summarizing

the key results of the experiments.

3.4.1 Relativistic Weibel Instability

The Weibel instability [280] is a type of streaming instability resulting from an anistropic

distribution of momenta, which is prevalent in many applications of high-energy-density physics

including astrophysical plasmas [281] and fusion applications [282]. In such circumstances, the

momenta in different directions can vary by several orders of magnitude. Initially, the strong currents

generated from the momenta create filament-like structures that eventually interact due to the growth

in the magnetic field. Over time, the magnetic field can become quite turbulent, and the currents

self-organize into larger connected networks. During this self-organization phase, there is an energy

conversion mechanism that transfers the kinetic energy from the plasma into the magnetic field,

which attempts to make the distribution of momenta more isotropic. The resulting instability creates

the formation of magnetic islands and other structures which store massive amounts of energy. In

such highly turbulent regions, this can lead to the emergence of other plasma phenomena such as

magnetic reconnection, in which energy is released from the fields back into the plasma [283].

The setup for this test problem can be described as follows. The domain for the problem is a

111

periodic box defined on [−𝐿𝑥/2, 𝐿𝑥/2] × [−𝐿 𝑦/2, 𝐿 𝑦/2] in units of the electron skin depth 𝑐/𝜔 𝑝𝑒.

The particular values of 𝐿𝑥 and 𝐿 𝑦 are carefully chosen using the dispersion relation and will be

specified later. Here, 𝑐 is the speed of light and 𝜔 𝑝𝑒 is the electron angular plasma frequency. The

system being modeled consists of two groups of infinite counter-propagating sheets of electrons and

a uniform background of stationary ions. The electrons in each group are prescribed momenta from

the ‘‘waterbag” distribution

𝑓 (P) =

1
2𝑃

(cid:16)

𝛿

𝑃(1) − 𝑃⊥

(cid:17) (cid:104)

(cid:16)

Θ

𝑃(2) − 𝑃

(cid:17)

(cid:16)

− Θ

𝑃(2) + 𝑃

(cid:17)(cid:105)

,

(3.28)

where Θ(𝑥) is the unit step function, and we choose 𝑃⊥ > 𝑃 > 0 to induce an instability. Along

the 𝑥 direction, electrons are prescribed only a drift corresponding to 𝑃⊥, and are linearly spaced

in the interval [−𝑃 , 𝑃 ) in the 𝑦 direction. We make the problem charge neutral by setting the

positions of the electrons to be equal to the ions. Electrons belonging to the first group are initialized

according to the distribution (3.28), and those belonging to the second group are defined to be

a mirror image (in momenta) of those in the first. This creates a return current that guarantees

current neutrality in the initial data. For simplicity, we shall assume that the sheets have the same

number density ¯𝑛, but this is not necessary. In fact, some earlier studies explored the structure

of the instability for interacting streams with different densities and drift velocities [284]. The

particular values of the plasma parameters used in this experiment are summarized in Table 3.2.

We remark that during the initialization phase of our implementation, we prescribe the velocities

of the particles and then convert them to conjugate momenta using P𝑖 = 𝛾𝑚𝑖v𝑖, since A = 0 at the

initial time. The electron velocities used to construct the distribution (3.28) are equivalently given

in units of 𝑐 by 𝑣⊥ = 1/2 and 𝑣 = 1/100. When these values are converted to conjugate momenta,

we obtain 𝑃⊥ ≈ 5.773888 × 10−1 and 𝑃 ≈ 1.154778 × 10−2, respectively, which are given in units

of 𝑚𝑒𝑐. Lastly, we note that the normalized permittivity and normalized permeability are given by

𝜎1 = 𝜎2 = 1, and the normalized speed of light is 𝜅 = 1.

In other papers, which numerically simulate the Weibel instability, e.g., [284, 120, 88], the

particle velocities are obtained using specialized sampling methods that mitigate the effects of noise

in the starting distribution. As pointed out in [88], this can lead to electrostatic effects, such as

112

Parameter
Average number density ( ¯𝑛) [m−3]
Average electron temperature ( ¯𝑇) [K]

Electron angular plasma period (𝜔−1

𝑝𝑒) [s/rad]

Electron skin depth (𝑐/𝜔 𝑝𝑒) [m]
Electron drift velocity in 𝑥 (𝑣⊥) [m/s]
Maximum electron velocity in 𝑦 (𝑣 ) [m/s]

Value
1.0 × 1010
1.0 × 104
1.772688 × 10−7
5.314386 × 101
𝑐/2
𝑐/100

Table 3.2 Plasma parameters used in the simulation of the Weibel instability. All simulation particles
are prescribed a drift velocity corresponding to 𝑣⊥ in the 𝑥 direction while the 𝑦 component of their
velocities are sampled from a uniform distribution scaled to the interval [−𝑣 , 𝑣 ).

Landau damping, which will alter the growth of the instability. This motivates the consideration of

the distribution (3.28), as it is easy to compare with theory and control effects from sampling noise.

For such distributions, the growth rate in the magnetic field can be calculated from the dispersion

relation [285]

1 −

𝑐2𝑘 2
𝜔2

−

𝜔2
𝑝𝑒
𝜔2𝛾

(cid:34)

𝐺 (𝛽 ) +

𝛽2
⊥
2 (cid:0)1 − 𝛽2(cid:1)

(cid:18) 𝑐2𝑘 2 − 𝜔2
𝜔2 − 𝑐2𝑘 2 𝛽2

(cid:19) (cid:35)

= 0.

Here, 𝑘 is the wavenumber in the 𝑦 direction, 𝜔 𝑝𝑒 is the electron plasma frequency and

(cid:115)

𝛾 :=

1 +

𝑃2
⊥
𝑒𝑐2
𝑚2

+

𝑃2
𝑒𝑐2

𝑚2

,

𝛽⊥ :=

𝑃⊥
𝛾𝑚𝑒𝑐

,

𝛽 :=

𝑃
𝛾𝑚𝑒𝑐

, 𝐺 (𝛽 ) :=

1
2𝛽 ln

(cid:18) 1 + 𝛽
1 − 𝛽

(cid:19)

,

with 𝑚𝑒 being the electron mass. In particular, Yoon and Davidson [285] showed that an instability

occurs if and only if the condition

𝛽2
⊥
2𝛽2

> (1 − 𝛽 ) 𝐺 (𝛽 ),

is satisfied, and that its corresponding growth rate can be directly calculated using the equation

Im(𝜔) =

1
√

2

where we have introduced the definitions

(cid:20) (cid:16)

𝑊 2

1 + 𝑊2

(cid:17) 1/2

(cid:21) 1/2

,

− 𝑊1

(3.29)

𝑊1 := 𝑐2𝑘 2 𝛽2 +

𝜔2

𝑝𝑒 𝛽2
⊥
2𝛾𝛽2

− 𝑐2 (cid:16)

0 − 𝑘 2(cid:17)
𝑘 2

, 𝑊2 := 4𝑐4𝑘 2 𝛽2 (cid:16)

0 − 𝑘 2(cid:17)
𝑘 2
(cid:33)

,

.

𝛽2
⊥

2𝛽2 (cid:0)1 − 𝛽2(cid:1) − 𝐺 (𝛽 )

𝑘0 :=

(cid:32)

𝜔2
𝑝𝑒
𝛾𝑐2

113

Figure 3.1 Growth in the magnetic field energy for the Weibel instability. We compare the growth
rate in the ℓ2-norm of the magnetic field 𝐵(3) for different methods against an analytical growth rate
predicted from linear response theory [285]. The analytical growth rate for this configuration is
determined to be Im(𝜔) ≈ 0.319734. We observe good agreement with the theoretically predicted
growth rate for each of the methods.

Figure 3.2 The gauge error for the different methods applied to the Weibel instability test problem.
Naive interpolation introduces significant gauge error, whereas the maps based on the continuity
equation or gauge correction result in substantial improvement. It should be noted that while FFT
and FD6 appear identical, they differ by O (10−14). The gauge-correcting approach produced the
smallest gauge error among the methods we considered.

Finally, we remark that the range of admissible values of 𝑘 for the instability is 0 < 𝑘 2 < 𝑘 2
0.

Using the parameters derived from Table 3.2 in an appropriately rescaled equation of the growth

rate (3.29), we find the value of 𝑘 that achieves the maximum growth is 𝑘 ≈ 5.445191. Based on

114

(a) Relative Energy Error

(b) Relative Mass Error

Figure 3.3 The relative energy error for the different methods applied to the Weibel instability
test problem. All methods show reasonably good energy conservation, though far from machine
precision, and the total mass is conserved in each approach. It is worth noting that that the start the
two charge-conserving methods do a slightly better job at preserving energy than the bilinear map
or gauge-correcting method, though all end up in the same order of accuracy.

this value, we scale the size of the (normalized) simulation domain so that 𝐿𝑥 = 𝐿 𝑦 = 1.153896

in units of 𝑐/𝜔 𝑝𝑒. The corresponding theoretical growth rate in the magnetic field associated with

this problem is Im(𝜔) ≈ 0.319734. We ran the simulation to a final time of 𝑇 = 50 in units of 𝜔−1
𝑝𝑒

with a 128 × 128 mesh. The total number of simulation particles used in the experiment was set to

1 × 106 and was split equally between ions and electrons. Each electron group, therefore, consists of

2.5 × 105 particles, and the number of particles per cell was approximately 61. In Figure 3.1, we

compare the growth rates in the ℓ2-norm of the magnetic field 𝐵(3) obtained with different methods.

We find that each of the methods, including the naive approach, show good agreement with the

theoretical growth rate predicted from equation (3.29). Additionally in Figure 3.2, we plot the error

in the gauge condition obtained with each of the methods, as well as the relative error in energy

and mass in Figure 3.3. The naive approach, which uses bilinear maps for the charge and current

densities introduces significant gauge error and increases over time. In contrast, the maps which

use either the continuity equation or the gauge correction technique show significant improvement

in reducing these errors when compared to the naive implementation. Most notably, the gauge

correction technique ensures the satisfaction of the gauge condition at the level of machine precision.

All methods conserve energy with roughly the same accuracy and mass is conserved exactly. It is

worth noting that the two charge-conserving methods conserve energy slightly better during the initial

115

period of growth, though all show similar deviations in energy at the end of the simulation. While

all methods conserve charge globally, only the hybrid FFT, FD6, and gauge-correcting methods

conserve charge in a local sense.

3.4.2 Drifting Cloud of Electrons

Given the task of minimizing the gauge error, a test problem that tends to have a high gauge

error was needed. When charged particles move through neutral space, the system is prone to gauge

errors unless properly managed. As such, we take a periodic domain and induce a potential well by

placing a grouping of ions, which are normally distributed with a standard deviation of 1/100 of the

domain length, in space and stationary, at its center. The electrons are given an identical distribution

in space, but are given a drift velocity equal to 𝑐/100. With such a drift, the electrons are able to

escape the potential well created by the ions and move throughout the domain (see Figure 3.4). The

domain, itself, is a periodic square [−1/2, 1/2]2 given in units of 𝜆𝐷, and we run the simulation to

a final time of 𝑇 = 1/2 in units of 𝜔−1

𝑝𝑒. The specific values for the simulation parameters used in

our experiment are specified in Table 3.3. The normalized permittivity and normalized permeability

are given by 𝜎1 = 1 and 𝜎2 = 1.686555 × 10−6, respectively, and the normalized speed of light is

𝜅 = 7.700159 × 102.

Parameter
Average number density ( ¯𝑛) [m−3]
Average electron temperature ( ¯𝑇) [K]
Debye length (𝜆𝐷) [m]

Electron angular plasma period (𝜔−1
𝑑 = 𝑣 (2)

Electron drift velocity (𝑣 (1)

𝑝𝑒) [s/rad]
𝑑 ) [m/s]

Electron thermal velocity (𝑣𝑡ℎ = 𝜆𝐷𝜔 𝑝𝑒) [m/s]

Value
1.0 × 1013
1.0 × 104
2.182496 × 10−3
5.605733 × 10−9
𝑐/100
3.893328 × 105

Table 3.3 Plasma parameters used in the simulation of the drifting cloud of electrons.

In Figure 3.5, we plot the relative errors for conservation of total energy and total mass for

different methods using a 64 × 64 grid. It is noteworthy that the relative energy error is superior

to that of the Weibel instability problem, though both are above machine precision. It is worth

considering that the relative energy error of the Weibel instability plateaus alongside the magnetic

116

(a) 𝑡 = 0

(b) 𝑡 = 0.05

(c) 𝑡 = 0.15

(d) 𝑡 = 0.5

Figure 3.4 A Maxwellian distribution of electrons and ions are placed on a periodic domain. The
electrons are given a thermal velocity in addition to a drift velocity v𝑑 = (𝑣 (1)
, where
2𝑐/100. We see the particles escape the well only
𝑣 (1) = 𝑣 (2) = 𝑐/100, that is the speed is (cid:107)v𝑑 (cid:107) =
to fall back in later. By 𝑡 = 0.15 the electrons, beginning their second such traversal, have become
quite dispersed throughout the domain, though the remnants of the original drift velocity can be
found in a slight beam along the diagonal (𝑡 = 0.5).

𝑑 , 𝑣 (2)

𝑑 )𝑇

√

magnitude, stabilizing at approximately 𝑡 = 0.20 plasma periods, whereas the relative energy error

of the moving cloud stays in the range of O (10−10) throughout. Again, all methods globally conserve

the total mass.

In Figure 3.6, we plot the gauge error for the drifting cloud problem and compare the proposed

methods against a naive implementation that uses bilinear maps for both the charge density and

117

current density. In each of the methods, we examined the gauge error on spatial meshes with different

resolution, with the coarsest mesh being 16 × 16 and the finest being 64 × 64. Aside from the naive

approach, we note that the gauge error for a given method increases proportionally with the resolution

of the spatial discretization. One explanation for this is that coarser spatial discretizations are less

likely to resolve the gauge error, which is generally small, compared to more refined meshes. Using

bilinear interpolation for both the charge density and current density results in a gauge error that
starts at O (cid:0)10−5(cid:1) and quickly shrinks to O (cid:0)10−7(cid:1). On the other hand, we find that using a continuity

equation to define the charge density, in combination with a bilinear mapping for the current density,
results in a much lower error which is O (cid:0)10−13(cid:1). This is true for methods which compute the

numerical derivatives using either the FFT or the FD6 method. We remark that the method used for

differentiation must be identical to the one used for the potentials, following the discussion in section

3.3.1.1. In practice, solving the system with this technique results in an accumulation of error that is

close to machine precision, yielding O (10−13) accuracy. The gauge correction technique, on the

other hand, does not accumulate this error, so we observe a much smaller error of O (10−18).

(a) Relative Energy Error

(b) Relative Mass Error

Figure 3.5 The relative energy error for the different methods applied to the Moving Cloud test
problem along with the relative mass error. Both of these plots are for a simulation with a 64 × 64
grid. The relative energy error is significantly better than that found in the Weibel instability, and
like the Weibel instability, the relative mass error is zero throughout.

3.5 Conclusion

This chapter constitutes a tactical step backwards from the remarkable method of Chapter 2. We

now, akin to the Yee grid, automatically satisfy the semi-discrete Lorenz gauge and Gauss’s law by

118

(a) Bilinear Map

(b) FFT Satisfying Continuity Equation

(c) FD6 Satisfying Continuity Equation

(d) Gauge-correcting

Figure 3.6 The gauge error of a cloud of electrons drifting into and out of a potential well. Naively
interpolated such that the charge and current densities are not consistent, we see a significant gauge
error. However, if the charge density is computed from the current density using the continuity
equation and a high order method (eg FFT) to compute the divergence of J, we see a much better
gauge error over time.

simply satisfying the semi-discrete continuity equation, but at the cost of the geometric flexibility

that comes with the Green’s function solvers. The next chapter is concerned with generalizing this

result to a variety of time discretizations. Once this is established, we will be fully regrouped, and

may consider how to retake the ground of geometry and boundaries.

119

CHAPTER 4

GENERALIZATION

The exploration in Chapter 3 was done entirely using a first-order BDF time discretization method

for the fields. In this chapter we seek to achieve higher-order temporal accuracy using a variety of

techniques, including BDF methods, central discretizations, and diagonally implicit Runge-Kutta

(DIRK) methods, and we prove that many of these methods satisfy the same properties established

in Section 3.2. For the general BDF and central difference methods, this will be fully achieved. For

the DIRK methods, we will prove that charge conservation implies satisfaction of the Lorenz gauge.

We will additionally establish that the two-stage method holds in the other direction, suggesting that

higher-order DIRK methods will also satisfy this property.

Our primary objective in this work is to prove the time-consistency property for a family of wave

solvers. As such, we restrict ourselves to problems with periodic boundary conditions, which allows

us to use a spectral discretization that can be rapidly evaluated using the FFT rather than the fast

convolution operator considered in Chapter 2. Additionally, the restriction to periodic boundary

conditions allows us to utilize high-order particle interpolation schemes, as is commonly done with

high order non-diffusive wave solvers [286, 287, 288, 289]. All wave solvers we implement will be
fully-implicit, with the source term J𝑛+1 being computed using an updated particle location x𝑛+1
and old velocity v𝑛

.

The remainder of the chapter is structured as follows. Section 4.1 gives a brief review of the

problem formulation for convenience. More details may be found in Section 1.3. More attention

will be paid to the various wave solvers we employ and proofs of the aforementioned properties in

Section 4.2, which occupies the bulk of the chapter. The particle method has already been discussed

in detail in Section 2.3.1.2, as such we will leave that out for brevity. Section 4.3 will present the

numerical experiments confirming our findings, concentrating on the same periodic problems as

in Section 3.4, and finally Section 4.4 will wrap things up with some concluding remarks. This

chapter is based on the work submitted by myself in collaboration with Dr. Andrew Christlieb

and Dr. William Sands [4]. My contribution in this work was the generalization of the theorems

120

presented in Chapter 3 to BDF-𝑘 methods, DIRK-𝑠 methods, and the Adams-Bashforth methods, as

well as the implementation of all the code.

4.1 Problem Review

To prevent the hassle of flipping back and forth between chapters to look up equations, here

we once more list out the equations with which we will be working. From Section 1.3.3 we know

Maxwell’s equations under the Lorenz gauge become

− Δ𝜙 =

,

𝜌
𝜖0

− ΔA = 𝜇0J,

+ ∇ · A = 0.

𝜕2𝜙
𝜕𝑡2
𝜕2A
𝜕𝑡2
𝜕𝜙
𝜕𝑡

1
𝑐2

1
𝑐2
1
𝑐2






Charge should be conserved, codified in the continuity equation

𝜕 𝜌
𝜕𝑡

+ ∇ · J = 0.

The involutions of Maxwell’s equations are Gauss’s laws:

,

∇ · E =

𝜌
𝜖0
∇ · B = 0.





(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

From Section 1.3.4 we know the particle equations of motion under the Lorenz gauge become

𝑑x𝑖
𝑑𝑡
𝑑P𝑖
𝑑𝑡






(P𝑖 − 𝑞A𝑖) ,

=

1
𝛾𝑖𝑚𝑖
(cid:18)

= 𝑞𝑖

−∇𝜙 +

(∇A) · (P𝑖 − 𝑞𝑖A)

(cid:19)

.

1
𝛾𝑖𝑚𝑖

(4.7)

(4.8)

The nondimensionalization of these equations may be found in Appendix A.

The particle motion will be accomplished by the Improved Asymmetrical Euler Method, de-

scribed in Section 2.3. It is in solving the wave equations while enforcing the Lorenz gauge via

conserving charge that we have interest. The precedent was set in Section 3.2, we now move to

generalize.

121

4.2 Generalizing the Properties

This section is concerned with generalizing the theorems proven in Section 3.2.1.1 to a variety

of time discretizations. First, in Section 4.2.1 we generalize from BDF-1 to BDF-𝑘. In Section 4.2.2

we extend to a family of time centered second order discretizations. In Section 4.2.3 we extend this

to 𝑠-stage Adams-Bashforth methods. Finally, in Section 4.2.4 we partially extend this to 𝑠-stage

DIRK methods.

4.2.1 The General Semi-Discrete Formulation for BDF Methods Under the Lorenz Gauge

In Section 3.2.1, we introduced a lemma and two theorems that showcased properties associated

with the first-order BDF method in a semi-discrete setting. That is, conservation of charge is satisfied

if and only if the semi-discrete Lorenz gauge condition is satisfied, and that Gauss’s law for electricity

is satisfied if said gauge condition is. In this section, we generalize these properties to any BDF

methods constructed with uniform step sizes. Vital to the original is the consistent nature of the

temporal derivatives, that is, applying the first order derivative BDF method twice results in the

second order derivative BDF method in the sense that 𝐷𝑡 [𝐷𝑡 [·]] = 𝐷2

𝑡 [·]; the generalization will

also take advantage of this. We start with a time-consistent 𝑘-step BDF formulation

𝑛
(cid:213)

𝑎𝑛−𝑖𝑢𝑖 + 𝑂 (Δ𝑡 𝑝),

𝑑𝑢
𝑑𝑡

=

1
Δ𝑡

𝑑2𝑢
𝑑𝑡2

=

1
Δ𝑡2

𝑖=𝑛−𝑘
𝑛
(cid:213)

𝑖
(cid:213)

𝑎𝑛−𝑖

𝑎𝑖− 𝑗 𝑢 𝑗 + 𝑂 (Δ𝑡 𝑝).

𝑖=𝑛−𝑘

𝑗=𝑖−𝑘

(4.9)

(4.10)

Note that BDF-1 through BDF-6 have this form.

For purposes of bookkeeping later on, we now prove a lemma regarding the structure of the

coefficients in the method, which, absorbing the Δ𝑡 terms into the coefficients, we write as

𝑛
(cid:213)

𝑎𝑛−𝑖

𝑖
(cid:213)

𝑖=𝑛−𝑘

𝑗=𝑖−𝑘

𝑎𝑖− 𝑗 𝑢 𝑗 ≡

𝑛
(cid:213)

𝑖=𝑛−2𝑘

𝐶𝑖𝑢𝑖.

(4.11)

This term arises in the right hand side of the implicit solution to semi-discrete Maxwell’s equations.

It is important that we sort out this term first, as it is embedded inside of other backwards differences

we encounter when doing the constructive proof for the theorems regarding preservation of the

gauge condition, continuity equation, and Gauss’s law.

122

Lemma 4.2.1. We consider the BDF-k method applied to 𝑢𝑛 twice:

𝑛
(cid:213)

𝑎𝑛−𝑖

𝑖
(cid:213)

𝑖=𝑛−𝑘

𝑗=𝑖−𝑘

𝑎𝑖− 𝑗 𝑢 𝑗 .

(4.12)

Considering (4.11), the coefficient 𝐶𝑖 for a term 𝑢𝑖 at the corresponding (𝑛 − 𝑖)th time level below 𝑛

takes the form




Proof. We prove by induction. The base case 𝑘 = 1 results in the sum

𝑛−𝑖
(cid:205)
𝑗=0
2𝑘−(𝑛−𝑖)
(cid:205)
𝑗=0

𝑎𝑘− 𝑗 𝑎𝑛−𝑖−𝑘+ 𝑗 ,

𝑎 𝑗 𝑎𝑛−𝑖− 𝑗 ,

𝑛 − 𝑖 ≤ 𝑘,

𝑛 − 𝑖 > 𝑘.

𝐶𝑖 :=

𝑎0𝑎0𝑢𝑛 + (𝑎0𝑎1 + 𝑎1𝑎0)𝑢𝑛−1 + 𝑎1𝑎1𝑢𝑛−2.

(4.13)

(4.14)

This clearly has the form given in equation (4.13).

For the inductive step, we assume the hypothesis is true for 𝑘 ≤ 𝑚. Next, let us consider 𝑘 = 𝑚 +1

for which we have the summation

𝑛
(cid:213)

𝑎𝑛− 𝑗

𝑗
(cid:213)

𝑎 𝑗−ℓ𝑢ℓ.

𝑗=𝑛−(𝑚+1)

ℓ= 𝑗−(𝑚+1)

We can easily peel the outer (𝑚 + 1)-th term out, which gives

𝑛
(cid:213)

𝑎𝑛− 𝑗

𝑗
(cid:213)

𝑗=𝑛−𝑚

ℓ= 𝑗−(𝑚+1)

(cid:169)
(cid:173)
(cid:171)

𝑎𝑚+1

𝑎 𝑗−ℓ𝑢ℓ(cid:170)
(cid:174)
(cid:172)

+ (cid:169)
(cid:173)
(cid:171)

𝑛−(𝑚+1)
(cid:213)

𝑗=𝑛−2(𝑚+1)

.

𝑎𝑛−(𝑚+1)− 𝑗 𝑢 𝑗 (cid:170)
(cid:174)
(cid:172)

Similarly, we then peel the inner (𝑚 + 1)-th term out to obtain

(4.15)

(4.16)

𝑛
(cid:213)

𝑎𝑛− 𝑗

𝑗
(cid:213)

𝑗=𝑛−𝑚

ℓ= 𝑗−𝑚

(cid:169)
(cid:173)
(cid:171)

𝑎𝑚+1

𝑎 𝑗−ℓ𝑢ℓ(cid:170)
(cid:174)
(cid:172)

+ (cid:169)
(cid:173)
(cid:171)

𝑛−(𝑚+1)
(cid:213)

𝑗=𝑛−2(𝑚+1)

𝑎𝑛−(𝑚+1)− 𝑗 𝑢 𝑗 (cid:170)
(cid:174)
(cid:172)

(cid:32)

+

𝑎𝑚+1

𝑛
(cid:213)

𝑗=𝑛−𝑚

𝑎𝑛− 𝑗 𝑢 𝑗−(𝑚+1)

(cid:33)

. (4.17)

To prove the lemma, we must consider three cases, namely, 𝑛 − 𝑖 < 𝑚 + 1, 𝑛 − 𝑖 = 𝑚 + 1, and

𝑛 − 𝑖 > 𝑚 + 1.

Consider 𝐶𝑖 for 𝑛 − 𝑖 < 𝑚 + 1. We know from the inductive hypothesis that the first nested sum

is equal to

𝑛−𝑖
(cid:213)

𝑗=0
𝑛−𝑖<𝑚+1

𝑎 𝑗 𝑎𝑛−𝑖− 𝑗 .

123

The second summation contains 𝑢 𝑗
summation cannot contain 𝑢𝑖
contains 𝑢𝑖

when 𝑗 − (𝑚 + 1) = 𝑖, or 𝑗 = 𝑖 + (𝑚 + 1). So

for 𝑛 − 2(𝑚 + 1) ≤ 𝑗 ≤ 𝑛 − (𝑚 + 1). Since 𝑛 − (𝑚 + 1) < 𝑖, this

. The third summation contains 𝑢 𝑗

for 𝑛 − 𝑚 ≤ 𝑗 ≤ 𝑛, and therefore

𝑛−𝑖
(cid:213)

𝐶𝑖 =

𝑎 𝑗 𝑎𝑛−𝑖− 𝑗 + 𝑎𝑛−𝑖−(𝑚+1)𝑎𝑚+1,

𝑗=0
𝑛−𝑖<𝑚+1
𝑛−𝑖
(cid:213)

𝑎 𝑗 𝑎𝑛−𝑖− 𝑗 .

=

𝑗=0

Next, consider 𝐶𝑖 for 𝑛 − 𝑖 = 𝑚 + 1. Again, from the inductive hypothesis, the first nested sum is

equal to

𝑛−𝑖
(cid:213)

𝑎 𝑗 𝑎𝑛−𝑖− 𝑗 .

𝑗=0
𝑛−𝑖<𝑚+1
In the second summation, the term containing 𝑢𝑖

the coefficients in second sum reduce to

occurs when 𝑗 = 𝑖 or 𝑗 = 𝑛 − (𝑚 + 1). Therefore,

𝑎𝑚+1

𝑛−(𝑚+1)
(cid:213)

𝑗=𝑛−2(𝑚+1)

𝑎𝑛−(𝑚+1)− 𝑗 = 𝑎𝑚+1𝑎0.

Similar, we find that the terms containing 𝑢𝑖

in third summation reduce to

𝑛
(cid:213)

𝑗=𝑛−𝑚

𝑎𝑛− 𝑗 𝑎𝑚+1𝑢 𝑗−(𝑚+1) = 𝑎0𝑎𝑚+1.

Combining these terms, we find that this is equivalent to

𝑛−𝑖
(cid:213)

𝐶𝑖 =

𝑎 𝑗 𝑎𝑛−𝑖− 𝑗 + 𝑎𝑚+1𝑎0 + 𝑎0𝑎𝑚+1,

𝑗=0
𝑛−𝑖<𝑚+1
𝑛−𝑖
(cid:213)

𝑎 𝑗 𝑎𝑛−𝑖− 𝑗 ,

=

𝑗=0

when 𝑛 − 𝑖 = 𝑚 + 1.

Consider 𝐶𝑖 for 𝑛 − 𝑖 > 𝑘 = 𝑚 + 1. We know from the inductive hypothesis that the first nested

sum is equal to (cid:205)2𝑚−(𝑛−𝑖)

𝑗=0

𝑎𝑚− 𝑗 𝑎𝑛−𝑖−𝑚+ 𝑗 for 𝑛 − 𝑖 ≥ 𝑚. We can adjust this index by one without any

change to the actual value:

124

𝑖
𝑛
𝑛 − 1
𝑛 − 2
...
𝑛 − 𝑘
...
𝑛 − 2𝑘 + 1
𝑛 − 2𝑘

𝑛 − 𝑖
0
1
2
...
𝑘
...
2𝑘 − 1
2𝑘

𝐶𝑖
𝑎0𝑎0
𝑎1𝑎0 + 𝑎0𝑎1
𝑎2𝑎0 + 𝑎1𝑎1 + 𝑎0𝑎2
...
𝑎𝑘 𝑎0 + 𝑎𝑘−1𝑎1 + · · · + 𝑎1𝑎𝑘−1 + 𝑎0𝑎𝑘
...
𝑎𝑘−1𝑎𝑘 + 𝑎𝑘 𝑎𝑘−1
𝑎𝑘 𝑎𝑘

Table 4.1 Table of coefficients for 𝑖th time level of the 𝑘th order Backward Difference method for
timestep 𝑛.

2𝑚−(𝑛−𝑖)
(cid:213)

𝑗=0

𝑎𝑚− 𝑗 𝑎𝑛−𝑖−𝑚+ 𝑗 =

2𝑚−(𝑛−𝑖)+1
(cid:213)

𝑗=1

𝑎𝑚+1− 𝑗 𝑎𝑛−𝑖−(𝑚+1)+ 𝑗

(4.18)

Given 𝑖 < 𝑛 − 𝑘 = 𝑛 − (𝑚 + 1), we know the corresponding 𝑢 𝑗

term is in both the second and

third summation term and have the coefficients 𝑎𝑚+1𝑎𝑛−(𝑚+1)−𝑖 and 𝑎𝑚+1𝑎𝑛−𝑖−(𝑚+1), respectively.

The former corresponds to 𝑎𝑚+1− 𝑗 𝑎𝑛−𝑖−(𝑚+1)+ 𝑗 for 𝑗 = 2(𝑚 + 1) − (𝑛 − 𝑖), the latter corresponds to

𝑗 = 0. We thus fill in our summation and see

2(𝑚+1)−(𝑛−𝑖)
(cid:213)

𝐶𝑖 =

𝑗=0

𝑎𝑚+1− 𝑗 𝑎𝑛−𝑖−(𝑚+1)+ 𝑗 .

(4.19)

In all cases we see the nested sum takes the following structure.

𝐶𝑖 :=

(cid:205)𝑛−𝑖
𝑗=0

𝑎 𝑗 𝑎𝑛−𝑖− 𝑗 , 𝑛 − 𝑖 ≤ 𝑘,

(cid:205)2𝑘−(𝑛−𝑖)
𝑗=0

𝑎𝑘− 𝑗 𝑎𝑛−𝑖−𝑘+ 𝑗 , 𝑛 − 𝑖 > 𝑘.





The base case and inductive step have been demonstrated, completing the proof.

An immediate consequence is the following corollary:

Corollary 4.2.1. The following identity holds:

𝑛−1
(cid:213)

𝑎𝑛−𝑖

𝑖
(cid:213)

𝑖=𝑛−𝑘

𝑗=𝑖−𝑘

𝑎𝑖− 𝑗 𝑢 𝑗 + 𝑎0

𝑛−1
(cid:213)

𝑖−𝑛−𝑘

𝑎𝑛−𝑖𝑢𝑖 =

𝑛−1
(cid:213)

𝑖=𝑛−2𝑘

𝐶𝑖𝑢𝑖.

125

(4.20)

QED

(4.21)

Proof. From Lemma 4.2.1 we know

𝑛
(cid:213)

𝑖=𝑛−2𝑘

𝐶𝑖 =

=

𝑛
(cid:213)

𝑖=𝑛−𝑘
𝑛−1
(cid:213)

𝑖=𝑛−𝑘

𝑎𝑛−𝑖

𝑎𝑛−𝑖

𝑖
(cid:213)

𝑗=𝑖−𝑘
𝑖
(cid:213)

𝑗=𝑖−𝑘

𝑎𝑖− 𝑗 𝑢 𝑗

𝑎𝑖− 𝑗 𝑢 𝑗 + 𝑎0

𝑛−1
(cid:213)

𝑗=𝑖−𝑘

𝑎𝑖− 𝑗 𝑢 𝑗 + 𝑎0𝑎0𝑢𝑛.

(4.22)

Subtracting 𝑎0𝑎0𝑢𝑛 ≡ 𝐶𝑛𝑢𝑛

from each side demonstrates the identity, concluding the proof. QED

Applying a general backward difference method equations to the vector potential, scalar potential,

gauge condition and continuity equation yields:

𝑛
(cid:213)

𝑖=𝑛−𝑘
𝑛
(cid:213)

𝑎𝑛−𝑖

𝑎𝑛−𝑖

1
Δ𝑐2

1
Δ𝑡2

1
Δ𝑐2

1
Δ𝑡2

𝑎𝑖− 𝑗 𝜙 𝑗 − Δ𝜙𝑛 =

𝜌𝑛
𝜖0

,

𝑎𝑖− 𝑗 A

𝑗 − ΔA

𝑛 = 𝜇0J

𝑛,

𝑖
(cid:213)

𝑗=𝑖−𝑘
𝑖
(cid:213)

𝑗=𝑖−𝑘

𝑖=𝑛−𝑘
𝑛
(cid:213)

1
Δ𝑡

1
𝑐2

1
Δ𝑡

𝑖=𝑛−𝑘
𝑛
(cid:213)

𝑖=𝑛−𝑘

𝑎𝑛−𝑖𝜙𝑖 + ∇ · A

𝑛 = 0,

𝑎𝑛−𝑖 𝜌𝑖 + ∇ · J

𝑛 = 0.

1
Δ𝑡

𝜖 𝑛
1 :=

𝜖 𝑛
2 :=

1
𝑐2

1
Δ𝑡

𝑖=𝑛−𝑘
𝑛
(cid:213)

𝑖=𝑛−𝑘

𝑛
(cid:213)

𝑎𝑛−𝑖𝜙𝑖 + ∇ · A

𝑛,

𝑎𝑛−𝑖 𝜌𝑖 + ∇ · J

𝑛.

(4.23)

(4.24)

(4.25)

(4.26)

(4.27)

(4.28)

where here we assume J𝑛

is given and compute 𝜌𝑛

and the fields for the particle update. Additionally,

we define the residuals

We now prove an additional lemma which will link the residuals of the semi-discrete continuity

equation and semi-discrete gauge condition.

Lemma 4.2.2. In the system (4.23) - (4.24), the residual 𝜖 𝑛
where 𝑛 − 2𝑘 ≤ 𝑖 < 𝑛.

1 is a linear combination of 𝜖 𝑛

2 and 𝜖 𝑖
2,

126

Proof. From (4.23) and (4.24), we get the update schemes

𝜙𝑛 = L−1

𝑛 = L−1

A














1
𝛼2

𝜌𝑛
𝜖0

−

𝑛−1
(cid:213)

𝑎𝑛−𝑖

𝑖
(cid:213)

𝑖=𝑛−𝑘

𝑗=𝑖−𝑘

𝑎𝑖− 𝑗 𝜙 𝑗 − 𝑎0

𝑛−1
(cid:213)

𝑖=𝑛−𝑘

𝑎𝑛−𝑖𝜙𝑖

1
𝛼2

𝜌𝑛
𝜖0

−

𝑛−1
(cid:213)

𝑎𝑛−𝑖

𝑖
(cid:213)

𝑖=𝑛−𝑘

𝑗=𝑖−𝑘

𝑎𝑖− 𝑗 A

𝑗 − 𝑎0

𝑛−1
(cid:213)

𝑖=𝑛−𝑘

𝑖
𝑎𝑛−𝑖A








,








.

(4.29)

(4.30)

We plug these into the definition of 𝜖 𝑛

1 , (4.27), and see

𝑎𝑛−𝑖𝜙𝑖 + ∇ · A

𝑛

𝜖 𝑛
1 =

1
𝑐2

1
Δ𝑡

=

1
𝑐2

1
Δ𝑡

𝑛
(cid:213)

𝑖=𝑛−𝑘
𝑛
(cid:213)

𝑖=𝑛−𝑘

𝑎𝑛−𝑖L−1

1
𝛼2

𝜌𝑖
𝜖0

−

𝑖−1
(cid:213)

𝑎𝑖− 𝑗

𝑗
(cid:213)

𝑗=𝑖−𝑘

𝑙= 𝑗−𝑘

𝑎 𝑗−𝑙 𝜙𝑙 − 𝑎0

𝑖−1
(cid:213)

𝑗=𝑖−𝑘

𝑎𝑖− 𝑗 𝜙 𝑗








𝑛−1
(cid:213)

𝑖
(cid:213)

𝑎𝑖− 𝑗 ∇ · A

𝑗 − 𝑎0

𝑎𝑛−𝑖

𝑖=𝑛−𝑘

𝑗=𝑖−𝑘

𝑛−1
(cid:213)

𝑖=𝑛−𝑘

𝑖
𝑎𝑛−𝑖∇ · A








+ L−1

𝜇0
𝛼2 J

𝑛 −








This becomes

= L−1

(cid:34) 𝜇0
𝛼2

(cid:32)

1
Δ𝑡

𝑛
(cid:213)

𝑖=𝑛−𝑘

(cid:33)(cid:35)

𝑎𝑛−𝑖 𝜌𝑖 + ∇ · J

𝑛

−

1
𝑐2

1
Δ𝑡

𝑛
(cid:213)

𝑖=𝑛−𝑘

𝑎𝑛−𝑖L−1

− L−1

𝑎𝑛−𝑖

𝑖
(cid:213)

𝑗=𝑖−𝑘

𝑛−1
(cid:213)





𝑖=𝑛−𝑘



𝑖−1
(cid:213)

𝑎𝑖− 𝑗

𝑗
(cid:213)

𝑗=𝑖−𝑘

𝑙= 𝑗−𝑘








𝑎𝑖− 𝑗 ∇ · A

𝑖 + 𝑎0








𝑎 𝑗−𝑙 𝜙𝑙 + 𝑎0

𝑖−1
(cid:213)

𝑗=𝑖−𝑘

𝑎𝑖− 𝑗 𝜙 𝑗

𝑛−1
(cid:213)

𝑖=𝑛−𝑘

𝑖
𝑎𝑛−𝑖∇ · A

.















(4.31)

(4.32a)

(4.32b)

(4.32c)

The first term is clearly L−1 (cid:2) 𝜇0

𝛼2

𝜖 𝑛
2

(cid:3). Disregarding the 1

𝑐2

Δ𝑡 coefficient and the L−1, we apply
1

127

Corollary 4.2.1 to the argument passed to L−1, and, breaking up the outermost summation, we see:

𝑛−1
(cid:213)

𝑎𝑛− 𝑗

𝑗
(cid:213)

𝑗=𝑛−𝑘

𝑙= 𝑗−𝑘

𝑎 𝑗−𝑙 𝜙𝑙 + 𝑎0

𝑛−1
(cid:213)

𝑗=𝑛−𝑘

𝑛−2
(cid:213)

𝑗=𝑛−𝑘−1

𝑛−3
(cid:213)

𝑗=𝑛−𝑘−2

𝑎𝑛−1− 𝑗

𝑎𝑛−2− 𝑗

𝑗
(cid:213)

𝑙= 𝑗−𝑘

𝑗
(cid:213)

𝑙= 𝑗−𝑘

𝑎 𝑗−𝑙 𝜙𝑙 + 𝑎0

𝑎 𝑗−𝑙 𝜙𝑙 + 𝑎0

𝑛−2
(cid:213)

𝑎𝑛− 𝑗 𝜙 𝑗 (cid:170)
(cid:174)
(cid:172)
𝑎𝑛−1− 𝑗 𝜙 𝑗 (cid:170)
(cid:174)
(cid:172)
𝑎𝑛−2− 𝑗 𝜙 𝑗 (cid:170)
(cid:174)
(cid:172)

𝑛−3
(cid:213)

𝑗=𝑛−𝑘−1

𝑗=𝑛−𝑘−2

𝑎0 (cid:169)
(cid:173)
(cid:171)

+𝑎1 (cid:169)
(cid:173)
(cid:171)

+𝑎2 (cid:169)
(cid:173)
(cid:171)
+ · · ·

𝑛−𝑘−1
(cid:213)

𝑗=𝑛−2𝑘

+𝑎𝑘 (cid:169)
(cid:173)
(cid:171)

𝑎𝑛−𝑘− 𝑗

𝑗
(cid:213)

𝑙= 𝑗−𝑘

𝑎 𝑗−𝑙 𝜙𝑙 + 𝑎0

𝑛−𝑘−1
(cid:213)

𝑗=𝑛−2𝑘

.

𝑎𝑛−𝑘− 𝑗 𝜙 𝑗 (cid:170)
(cid:174)
(cid:172)

We apply Corollary 4.2.1 to each nested summation, and we see

𝑎0

+𝑎1

(cid:16)

(cid:16)

𝐶𝑛−2𝑘 𝜙𝑛−2𝑘 + 𝐶𝑛−2𝑘+1𝜙𝑛−2𝑘+1 + · · · + 𝐶𝑛−𝑘 𝜙𝑛−𝑘 + · · · + 𝐶𝑛−1𝜙𝑛−1(cid:17)
𝐶𝑛−2𝑘 𝜙𝑛−2𝑘−1 + 𝐶𝑛−2𝑘+1𝜙𝑛−2𝑘 + · · · + 𝐶𝑛−𝑘 𝜙𝑛−𝑘−1 + · · · + 𝐶𝑛−1𝜙𝑛−2(cid:17)

+ · · ·

+𝑎𝑘

(cid:16)

𝐶𝑛−2𝑘 𝜙𝑛−3𝑘 + 𝐶𝑛−2𝑘+1𝜙𝑛−3𝑘+1 + · · · + 𝐶𝑛−𝑘 𝜙𝑛−𝑘−2 + · · · + 𝐶𝑛−1𝜙𝑛−𝑘 (cid:17)

(4.33)

(4.34)

We then group like terms with like, and find that we have a linear combination of first order BDF-k

derivatives.

128

𝐶𝑛−1

+𝐶𝑛−2

(cid:16)

(cid:16)

𝑎0𝜙𝑛−1 + 𝑎1𝜙𝑛−2 + · · · + 𝑎𝑘 𝜙𝑛−𝑘−1(cid:17)
𝑎0𝜙𝑛−2 + 𝑎1𝜙𝑛−3 + · · · + 𝑎𝑘 𝜙𝑛−𝑘−2(cid:17)

+ · · ·

+𝐶𝑛−𝑘+1
(cid:16)

+𝐶𝑛−𝑘

+𝐶𝑛−𝑘−1

(cid:16)

𝑎0𝜙𝑛−𝑘+1 + 𝑎1𝜙𝑛−𝑘 + · · · + 𝑎𝑘 𝜙𝑛−2𝑘+1(cid:17)
𝑎0𝜙𝑛−𝑘 + 𝑎1𝜙𝑛−𝑘−1 + · · · + 𝑎𝑘 𝜙𝑛−2𝑘 (cid:17)
𝑎0𝜙𝑛−𝑘−1 + 𝑎1𝜙𝑛−𝑘−2 + · · · + 𝑎𝑘 𝜙𝑛−2𝑘−1(cid:17)
(cid:16)

+ · · ·

+𝐶𝑛−2𝑘+1
(cid:16)

+𝐶𝑛−2𝑘

(cid:16)

𝑎0𝜙𝑛−2𝑘+1 + 𝑎1𝜙𝑛−2𝑘 + · · · + 𝑎𝑘 𝜙𝑛−3𝑘+1(cid:17)
𝑎0𝜙𝑛−2𝑘 + 𝑎1𝜙𝑛−2𝑘−1 + · · · + 𝑎𝑘 𝜙𝑛−3𝑘 (cid:17)

.

This may be more compactly written as

𝑛−1
(cid:213)

𝑘
(cid:213)

𝐶𝑖

𝑎 𝑗 𝜙𝑖− 𝑗

𝑖=𝑛−2𝑘

𝑗=0

(4.35)

(4.36)

A simpler process of applying Corollary 4.2.1 to the argument in (4.32c), gives similar results:

𝑛−1
(cid:213)

𝑖=𝑛−2𝑘

𝐶𝑖∇ · A𝑖

(4.37)

Multiplying the 1
𝑐2

1
Δ𝑡 coefficients we left out from (4.32b) against (4.36), we add this to (4.37) and

see (4.32b) + (4.32c) becomes





𝑖=𝑛−2𝑘


This contains the exact definitions for the gauge residuals (4.27), 𝜖 𝑖

(cid:0)𝑎 𝑗 𝜙𝑖− 𝑗 (cid:1) + ∇ · A𝑖(cid:170)
(cid:174)
(cid:172)

𝐶𝑖 (cid:169)
(cid:173)
(cid:171)

1
Δ𝑡

L−1

1
𝑐2

𝑗=0

𝑛−1
(cid:213)

𝑘
(cid:213)

.








(4.38)

1. Thus we see combining (4.32a),

(4.32b), and (4.32c) yields

𝜖 𝑛
1 = L−1

(cid:34) 𝜇0
𝛼2

𝜖 𝑛
2 +

𝑛−1
(cid:213)

𝑖=𝑛−2𝑘

(cid:35)

,

𝐶𝑖𝜖 𝑖
1

𝐶𝑖 :=

(cid:205)𝑖

𝑗=0

(cid:205)𝑟

𝑗=0





𝑎 𝑗 𝑎𝑖− 𝑗 , 𝑖 ≤ 𝑘,

𝑎𝑘− 𝑗 𝑎𝑘−𝑟+ 𝑗 , 𝑖 > 𝑘, 𝑟 := 2𝑘 − 𝑖.

129

(4.39)

We have demonstrated 𝜖 𝑛

1 is a linear combination of 𝜖 𝑛

2 and previous steps of the gauge residual.

QED

Having proven this lemma, we now can prove that any BDF method satisfies the semi-discrete

continuity equation satisfies the semi-discrete gauge condition and vice versa.

Theorem 4.2.1. Under the system (4.23)-(4.24), (4.26) is satisfied if (4.25) is, and, assuming the

initial conditions of (4.25) are satisfied, (4.25) is satisfied if (4.26) is satisfied.

Proof. Assume (4.25) is satisfied, that is, 𝜖 𝑖

1 = 0 ∀ 𝑖. From Lemma 4.2.2 we have

𝜖 𝑛
1 = L−1

(cid:34)

𝜖 𝑛
2 +

𝑛−1
(cid:213)

(cid:35)

,

𝐶𝑖𝜖 𝑖
1

𝑖=𝑛−𝑘−1
𝑛−1
(cid:213)

𝑖=𝑛−𝑘−1

(cid:35)

,

(4.40)

𝐶𝑖 (0)

=⇒ 0 = L−1

(cid:34)

𝜖 𝑛
2 +

=⇒ 0 = L−1 (cid:2)𝜖 𝑛
2

(cid:3) .

Clearly L−1 is invertible, so 𝜖 𝑛

2 = 0.

Assume both (4.26) and the initial conditions of (4.25) are satisfied. That is, 𝜖 𝑖

2 = 0∀𝑖 and 𝜖 𝑛

1 = 0

∀ 𝑛 ≤ 0. We have

(cid:34)

(cid:34)

(cid:34)

𝜖 𝑛
1 = L−1

=⇒ 𝜖 𝑛

1 = L−1

=⇒ 𝜖 𝑛

1 = L−1

𝑛−1
(cid:213)

(cid:35)

𝐶𝑖𝜖 𝑖
1

𝜖 𝑛
2 +

𝑖=𝑛−𝑘−1
𝑛−1
(cid:213)

(0) +

𝑛−1
(cid:213)

𝑖=𝑛−𝑘−1
(cid:35)

𝐶𝑖𝜖 𝑖
1

𝑖=𝑛−𝑘−1

𝐶𝑖𝜖 𝑖
1

.

,

(cid:35)

,

(4.41)

Given the initial conditions are all zero, we know 𝜖 1

1 is zero, as is 𝜖 2

1 , and so on. This logic can be

stepped to arbitrary 𝑛.

We have demonstrated (4.25) =⇒ (4.26) if (4.25) is satisfied initially, and (4.26) =⇒

(4.25).

QED

130

We have linked the semi-discrete continuity equation with the semi-discrete gauge condition

for arbitrary order BDF methods. Now we will show Gauss’s law for electricity follows from the

satisfaction of the semi-discrete gauge condition (4.25).

Theorem 4.2.2. If the semi-discrete gauge condition (4.25) is satisfied for time level 𝑛 + 1, then

∇ · E

𝑛+1 =

𝜌𝑛+1
𝜖0

.

Proof. We assume (4.25):

1
𝑐2

1
Δ𝑡

𝑛
(cid:213)

𝑖=𝑛−𝑘

𝑎𝑛−𝑖𝜙𝑖 + ∇ · A

𝑛 = 0.

By definition we have E = −∇𝜙 − 𝜕A

𝜕𝑡 , whose BDF-k time discrete form is as follows:

𝑛 = −∇𝜙𝑛 −

E

1
Δ𝑡

𝑛
(cid:213)

𝑖=𝑛−𝑘

𝑖.
𝑎𝑛−𝑖A

We take the divergence, which yields

∇ · E

𝑛 = −Δ𝜙𝑛 −

1
Δ𝑡

𝑛
(cid:213)

𝑖=𝑛−𝑘

𝑖
𝑎𝑛−𝑖∇ · A

𝑖
Using (4.25), we replace ∇ · A

with the BDF-k of 𝜙𝑖

:

∇ · E

𝑛 = −Δ𝜙𝑛 −

1
Δ𝑡

𝑛
(cid:213)

𝑖=𝑛−𝑘

𝑖
(cid:213)

1
𝑐2

1
Δ𝑡

−

𝑎𝑛−𝑖 (cid:169)
(cid:173)
(cid:171)
𝑎𝑖− 𝑗 𝜙 𝑗 − Δ𝜙𝑛

𝑖
(cid:213)

𝑗=𝑖−𝑘

𝑎𝑖− 𝑗 𝜙 𝑗 (cid:170)
(cid:174)
(cid:172)

𝑛
(cid:213)

𝑎𝑛−𝑖

𝑖=𝑛−𝑘

𝑗=𝑖−𝑘

=

=

1
𝑐2

1
Δ𝑡2

𝜌𝑛
𝜖0

.

(4.42)

(4.43)

(4.44)

(4.45)

The last step was taken by definition of the scalar potential wave equation (4.23).

We have demonstrated Gauss’s Law for electricity is satisfied, completing the proof.

QED

Remark 4.2.1. We note that all BDF methods satisfy this theorem. However, while first and second

order BDF methods are A-stable, third order BDF methods and above are A𝛼-stable. In practice,

this means that using first and second order BDF methods can be used with any CFL one wants,

while third order BDF and above need to take time steps that are greater than 𝑡𝑚𝑖𝑛 to be stable when

solving a wave equation, which has eigenvalues on the imaginary axis. This is because third order

131

BDF and above have stability diagrams that slightly move into the left half of the plane of the region

of absolute stability and then back out again. A large enough CFL will avoid this issue.

Remark 4.2.2. For our numerical experiments we use the second order BDF method, BDF-2. The

semi-discrete wave equations, Lorenz gauge condition, and continuity equation take the following

form:

𝜙𝑛+1 − 8
3

1
𝑐2

𝜙𝑛−2 + 1
9

𝜙𝑛−3

− Δ𝜙𝑛+1 =

𝜌𝑛+1
𝜖0

,

9A

𝑛−2 + 1

9A

𝑛−3

− ΔA = 𝜇0J

𝑛+1,

𝑛+1 − 8

A

1
𝑐2

𝜙𝑛 + 22
𝜙𝑛−1 − 8
9
9
((2/3)Δ𝑡)2
𝑛−1 − 8
𝑛 + 22
9 A
((2/3)Δ𝑡)2
𝜙𝑛+1 − 4
3

3A

1
𝑐2

𝜙𝑛 + 1
3
(2/3) Δ𝑡
𝜌𝑛 + 1
3
(2/3) Δ𝑡

𝜌𝑛+1 − 4
3

𝜙𝑛−1

+ ∇ · A

𝑛+1 = 0,

𝜌𝑛−1

+ ∇ · J

𝑛+1 = 0.

4.2.2 A Semi-discrete Time-centered Method for the Lorenz Gauge Formulation

To start, consider the semi-discrete treatment of the Lorenz gauge formulation, as in Chapter 3,

in which second-order approximations are used for 𝜕𝑡 and 𝜕𝑡𝑡. As we will prove in this section, the

satisfaction of the semi-discrete gauge condition as well as the involutions, (4.5) and (4.6), can be

ensured using the time-centered difference approximations
𝑢𝑛+1 − 𝑢𝑛
Δ𝑡

+ O (Δ𝑡2),

𝜕𝑡𝑡𝑢𝑛 =

𝜕𝑡𝑢𝑛+ 1

2 =

𝑢𝑛+1 − 2𝑢𝑛 + 𝑢𝑛−1
Δ𝑡2

+ O (Δ𝑡2).

A critical observation is that two applications of the first difference approximation to a discrete data

set produces the discrete second order discrete second difference in time, i.e.,

𝜕𝑡𝑡𝑢 =

𝜕𝑡𝑢𝑛+ 1

2 − 𝜕𝑡𝑢𝑛− 1
Δ𝑡

2

+ O (Δ𝑡2) =

𝑢𝑛+1 − 𝑢𝑛
Δ𝑡

𝑢𝑛 − 𝑢𝑛−1
Δ𝑡

−

Δ𝑡

+ O (Δ𝑡2).

This leads to the semi-discrete system in an implicit form

− Δ

(cid:32) 𝜙𝑛+ 3

2 + 𝜙𝑛− 1

2

2

(cid:33)

=

− Δ

(cid:18) A𝑛+1 + A𝑛−1
2

(cid:19)

= 𝜇0

2 + 𝜌𝑛− 1

2

𝜌𝑛+ 3

2

1
𝜖0
J𝑛+1 + J𝑛−1
2

,

2 + 𝜙𝑛− 1

2

𝜙𝑛+ 3

2 − 2𝜙𝑛+ 1
𝑐2Δ𝑡2
A𝑛+1 − 2A𝑛 + A𝑛−1
𝑐2Δ𝑡2
2 − 𝜙𝑛+ 1
𝑐2Δ𝑡

𝜙𝑛+ 3

2






+ ∇ · A𝑛+1 = 0,

,

(4.46)

(4.47)

(4.48)

132

a semi-implicit form






− Δ

(cid:32) 𝜙𝑛+ 3

2 + 𝜙𝑛− 1

2

2

(cid:33)

=

1
𝜖0

𝜌𝑛+ 1
2 ,

− Δ

(cid:18) A𝑛+1 + A𝑛−1
2

(cid:19)

= 𝜇0J𝑛,

2 + 𝜙𝑛− 1

2

𝜙𝑛+ 3

2 − 2𝜙𝑛+ 1
𝑐2Δ𝑡2
A𝑛+1 − 2A𝑛 + A𝑛−1
𝑐2Δ𝑡2
2 − 𝜙𝑛+ 1
𝑐2Δ𝑡

𝜙𝑛+ 3

2

+ ∇ · A𝑛+1 = 0,

and, alternatively, an explicit form

𝜌𝑛+ 1
2 ,

2 =

− Δ𝜙𝑛+ 1

1
𝜖0
− ΔA𝑛 = 𝜇0J𝑛,

2 + 𝜙𝑛− 1

2

𝜙𝑛+ 3

2 − 2𝜙𝑛+ 1
𝑐2Δ𝑡2
A𝑛+1 − 2A𝑛 + A𝑛−1
𝑐2Δ𝑡2
2 − 𝜙𝑛+ 1
𝑐2Δ𝑡

𝜙𝑛+ 3

2






+ ∇ · A𝑛+1 = 0,

(4.49)

(4.50)

(4.51)

(4.52)

(4.53)

(4.54)

where here we assume J{𝑛,𝑛+1} is given and use this to compute 𝜌𝑛+ 3
2 and the fields for the particle
update. All examples we explore in section 4.3 are implicit and compute an approximation of J𝑛+1

(see section 1.3.5 for more details).

Of particular relevance to this paper, much like Chapter 3, is a theorem that establishes an

equivalence, at the semi-discrete level, between the continuity equation

𝜌𝑛+ 3

2 − 𝜌𝑛+ 1
Δ𝑡

2

+ ∇ · J𝑛+1 = 0,

(4.55)

and the Lorenz gauge condition (4.48) (see Theorem 3.2.1). To establish similar results for these

methods, we will need to identify groupings of terms. For convenience, we shall introduce the

notation

𝜖 𝑘
1 =

𝜙𝑘+ 1

2 − 𝜙𝑘− 1
𝑐2Δ𝑡

2

+ ∇ · A𝑘 ,

𝜖 𝑘
2 =

𝜌𝑘+ 1

2 − 𝜌𝑘− 1
Δ𝑡

2

+ ∇ · J𝑘 ,

(4.56)

to simplify the presentation. Our first task is to establish the interdependence of satisfying the

gauge condition with satisfying the continuity equation. Then we can establish that if our method

is time-consistent, and the gauge condition is satisfied, then we will ensure that Gauss’s law for

electricity is satisfied. In the semi-discrete case, the derivatives are analytical, meaning Gauss’s law

133

for magnetism is satisfied by definition. In the next section, we will examine how this looks in the

fully-discrete case, and introduce the method for ensuring satisfaction of the continuity equation,

and hence we will establish that the method preserves the gauge condition and the involutions.

We now define operator notation that will be used in the proofs of the semi-discrete theorem for

the fully-implicit update. Consider now the linear wave equation

1
𝑐2

𝜕𝑡𝑡𝑢 − Δ𝑢 = 𝑆.

(4.57)

The coefficient 𝑐 > 0 and assumed to be a constant. Additionally, we denote 𝑆(x, 𝑡) as the source
function. An implicit second-order Crank-Nicolson method for 𝑢𝑛+1 centered at 𝑡 = 𝑡𝑛

is

(cid:18)

I −

(cid:19) (cid:16)

1
𝛼2

Δ

𝑢𝑛+1 + 𝑢𝑛−1(cid:17)

= 2𝑢𝑛 +

1
𝛼2

Γ𝑛,

(4.58)

where we use 𝛼 =

√

2/(𝑐Δ𝑡) and Γ𝑛 = 𝑆𝑛+1 + 𝑆𝑛−1. We define

L (·) =

(cid:18)

I −

(cid:19)

Δ

1
𝛼2

(·)

and denote the inverse as L−1(·). The inverse can be computed with any number of methods. For

our purposes, we will compute the inverse using the FFT.

Lemma 4.2.3. (Implicit method-Green’s Function) For the implicit formulation in (4.46)-(4.47), the

semi-discrete Lorenz gauge condition (4.48) satisfies

1 = −𝜖 𝑛−1
𝜖 𝑛+1

1

+ L−1 (cid:16)

2𝜖 𝑛

1 +

(cid:16)

𝜇0
𝛼2

2 + 𝜖 𝑛−1
𝜖 𝑛+1

2

(cid:17)(cid:17)

.

(4.59)

Proof. This proof is a direct construction. We start with equation (4.46). Inverting the modified

Helmholtz operator L gives the scalar potential as

𝜙𝑛+ 3

2 = −𝜙𝑛− 1

2 + L−1

which can be shifted to obtain

𝜙𝑛+ 1

2 = −𝜙𝑛− 3

2 + L−1

𝜌𝑛+ 3

2 + 𝜌𝑛− 1

2

𝜌𝑛− 1

2 + 𝜌𝑛− 3

2

(cid:17) (cid:19)

,

(cid:17)(cid:19)

.

(4.60)

(4.61)

(cid:18)

2𝜙𝑛+ 1

2 +

(cid:16)

1
𝛼2𝜖0

(cid:18)

2𝜙𝑛− 1

2 +

(cid:16)

1
𝛼2𝜖0

134

Next, we take the divergence of A in equation (4.47) and find that

(cid:16)

∇ · A𝑛+1 + ∇ · A𝑛−1(cid:17)

L

= 2∇ · A𝑛 +

𝜇0
𝛼2

(cid:16)

∇ · J𝑛+1 + ∇ · J𝑛−1(cid:17)

.

Formally inverting the operator L, we obtain the relation

∇ · A𝑛+1 = −∇ · A𝑛−1 + L−1 (cid:16)

2∇ · A𝑛 +

𝜇0
𝛼2

(cid:16)

∇ · J𝑛+1 + ∇ · J𝑛−1(cid:17)(cid:17)

.

(4.62)

With the aid of equations (4.60), (4.61), and (4.62), along with the linearity of the operator L, a

direct calculation reveals that the residual at time level is given by

𝜙𝑛+ 3

2

2 − 𝜙𝑛+ 1
𝑐2Δ𝑡
(cid:32)

(cid:32)

L−1

2𝜖 𝑛

1 +

+ ∇ · A𝑛+1 = −𝜖 𝑛−1
1 +
(cid:33)
(cid:32) 𝜌𝑛+ 3

2

2 − 𝜌𝑛+ 1
𝑐2Δ𝑡

1
𝛼2𝜖0

∇ · J𝑛+1

(cid:33)

(cid:32)

+

+

𝜇0
𝛼2

1
𝛼2𝜖0

(cid:32) 𝜌𝑛− 1

2

2 − 𝜌𝑛− 3
𝑐2Δ𝑡

(cid:33)

+

𝜇0
𝛼2

(cid:33) (cid:33)

.

∇ · J𝑛−1

From these calculations, we see that the above expression is the same (4.59), with the observation that

we have used the following relation 𝑐2 = 1/(𝜇0𝜖0). We observe that the corresponding consistent

semi-discrete continuity equation

𝜌𝑛+ 3

2 − 𝜌𝑛+ 1
Δ𝑡

2

+ ∇ · J𝑛+1 = 𝜖 𝑛+1

2

,

acts as a source for the residual, which completes the proof.

(4.63)

QED

With the aid of Lemma 4.2.3, we are now prepared to prove the following theorem that establishes

the time-consistency of the semi-discrete system.

Theorem 4.2.3. (Implicit method-Green’s Function) Assuming 𝜖 𝑛

2 = 0, ∀ 𝑛 ≤ 0, the semi-
discrete Lorenz gauge formulation of Maxwell’s equations (4.46)-(4.48) is time-consistent in the

1 = 𝜖 𝑛

sense that the semi-discrete Lorenz gauge condition (4.48) is satisfied at any discrete time if and

only if the corresponding semi-discrete continuity equation (4.63) is also satisfied.

Proof. We use a simple inductive argument to prove both directions. In the case of the forward
direction, we assume that the semi-discrete gauge condition is satisfied, so 𝜖 𝑛
1

≡ 0, ∀𝑛 ≥ 0. From

135

our assumption 𝜖 −1

2 ≡ 0. Combining this with equation (4.59) established by Lemma 4.2.3, it follows

that the next time level satisfies

0 = L−1

(cid:32) 𝜇0
𝛼2

(cid:32) 𝜌 1

2

2 − 𝜌− 1
Δ𝑡

(cid:33) (cid:33)

+ ∇ · J0

.

Applying the operator L to both sides leads to

𝜌 1

2

2 − 𝜌− 1
Δ𝑡

+ ∇ · J0 = 0.

This argument can be iterated 𝑛 times to show that

𝜌𝑛+ 1

2 − 𝜌𝑛− 1
Δ𝑡

2

+ ∇ · J𝑛 = 0,

also holds at any discrete time level 𝑛, which establishes the forward direction.

A similar argument can be used for the converse. Here, we show that if the semi-discrete

continuity equation (4.63) is satisfied for any time level 𝑛, i.e., 𝜖 𝑛
2
semi-discrete gauge condition also satisfies 𝜖 𝑛+1

1

≡ 0. First, we assume that the initial data and

≡ 0, then the residual for the

starting values satisfy 𝜖 −1

1 ≡ 𝜖 0

1

≡ 0. Appealing to equation (4.59) with this initial data, it is clear

that after a single time step, the residual in the gauge condition satisfies

(cid:32)

1 = −𝜖 −1
𝜖 1

1 + L−1

2𝜖 0

1 +

(cid:16)

𝜇0
𝛼2

2 + 𝜖 −1
𝜖 1

2

(cid:33)

(cid:17)

≡ L−1(0).

This argument can also be iterated 𝑛 more times to obtain the result, which finishes the proof. QED

Lemma 4.2.4. (Semi-Implicit method-Green’s Function) For the semi-implicit formulation in

(4.49)-(4.50), the semi-discrete Lorenz gauge condition (4.51) satisfies

1 = −𝜖 𝑛−1
𝜖 𝑛+1

1

+ L−1 (cid:16)

2𝜖 𝑛

1 +

𝜇0
𝛼2

(cid:17)

.

𝜖 𝑛
2

Proof. The proof is nearly identical to that of Lemma 4.2.3, so we exclude it.

(4.64)

QED

Theorem 4.2.4. (Semi-Implicit method-Green’s Function) Assuming 𝜖 𝑛

2 = 0, ∀𝑛 ≤ 0, the
semi-discrete Lorenz gauge formulation of Maxwell’s equations (4.49)-(4.51) is time-consistent in

1 = 𝜖 𝑛

the sense that the semi-discrete Lorenz gauge condition (4.51) is satisfied at any discrete time if and

only if the corresponding semi-discrete continuity equation (4.63) is also satisfied.

136

Proof. The proof is nearly identical to that of Theorem 4.2.3, so we exclude it.

QED

Lemma 4.2.5. (Explicit method) For the explicit formulation in (4.52)-(4.53), the semi-discrete

Lorenz gauge condition (4.54) satisfies

1 = 2𝜖 𝑛
𝜖 𝑛+1

1 − 𝜖 𝑛−1

1

(cid:16)

𝑐2Δ𝑡2(cid:17)

+

Δ𝜖 𝑛

1 +

(cid:16)

𝑐2Δ𝑡2(cid:17)

𝜇0𝜖 𝑛
2

.

(4.65)

Proof. Start with

𝜖 𝑛+1
1 =

+ ∇ · A𝑛+1,

2

𝜙𝑛+ 3

2 − 𝜙𝑛+ 1
𝑐2Δ𝑡
2 and 𝜙𝑛+ 1

and substitute equation (4.52) in for 𝜙𝑛+ 3

2 , then substitute equation (4.53) in for A𝑛+1. We

have

and

𝜙𝑛+ 3

2

2 − 𝜙𝑛+ 1
𝑐2Δ𝑡

=

−

2𝜙𝑛+ 1

2 − 𝜙𝑛− 1

2 − (𝑐2Δ𝑡2)Δ𝜙𝑛+ 1

2 + (𝑐2Δ𝑡2) 1
𝜖0

𝜌𝑛+ 1
2

(cid:16)

2𝜙𝑛− 1

2 − 𝜙𝑛− 3

𝑐2Δ𝑡
2 − (𝑐2Δ𝑡2)Δ𝜙𝑛− 1

2 + (𝑐2Δ𝑡2) 1
𝜖0

𝑐2Δ𝑡

(cid:17)

𝜌𝑛− 1
2

,

∇ · A𝑛+1 = 2∇ · A𝑛 − ∇ · A𝑛−1 − 𝑐2Δ𝑡2Δ∇ · A𝑛 + 𝑐2Δ𝑡2𝜇0∇ · J𝑛.

After grouping terms in the form of 𝜖 𝑘

1 and 𝜖 𝑘

2 as in Lemma 4.2.3, we arrive at (4.65). This completes

the proof.

QED

Theorem 4.2.5. (Explicit method) Assuming 𝜖 𝑛

2 = 0, ∀ 𝑛 ≤ 0, the explicit semi-discrete
Lorenz gauge formulation of Maxwell’s equations (4.52)-(4.54) is time-consistent in the sense that

1 = 𝜖 𝑛

the semi-discrete Lorenz gauge condition (4.54) is satisfied at any discrete time if and only if the

corresponding semi-discrete continuity equation (4.63) is also satisfied.

Proof. The proof is an identical construction to that in Theorem 4.2.3.

QED

Theorem 4.2.6. Under the assumption that 𝜖 𝑘

1 = 𝜖 𝑘

2 = 0, ∀ 𝑘 ≤ 𝑛 + 1, the implicit semi-discrete

methods (4.46)-(4.48) satisfy Gauss’s Law

∇ · E𝑛+ 1

2 =

1
𝜖0

(cid:32) 𝜌𝑛+ 3

2 + 𝜌𝑛− 1

2

2

(cid:33)

.

(4.66)

137

Proof. Starting with the identity E = −∇𝜙 − 𝜕𝑡A, we have

𝑛+ 1

2 = −∇

E

(cid:32) 𝜙𝑛+ 3

2 + 𝜙𝑛− 1

2

2

(cid:33)

− A

𝑛

𝑛+1 − A
Δ𝑡

.

Taking the divergence, we obtain

∇ · E

𝑛+ 1

2 = −Δ

(cid:32) 𝜙𝑛+ 3

2 + 𝜙𝑛− 1

2

2

(cid:33)

−

∇ · A

𝑛+1 − ∇ · A

𝑛

Δ𝑡

.

Then, from the assumption, we know that

𝜖 𝑛
1 = 0 =⇒

1
𝑐2

𝜙𝑛+ 1

2 − 𝜙𝑛− 1
Δ𝑡

2

+ ∇ · A

𝑛 = 0.

From this, we see that

∇ · E

𝑛+ 1

2 = −Δ

= −Δ

(cid:32) 𝜙𝑛+ 3

2 + 𝜙𝑛− 1

2

2

(cid:32) 𝜙𝑛+ 3

2 + 𝜙𝑛− 1

2

2

(cid:32) 𝜌𝑛+ 3

2 + 𝜌𝑛− 1

2

2

=

1
𝜖0

(cid:33)

(cid:33)

(cid:33)

.

−

+

1
𝑐2

(cid:32)

−

1
𝑐2

𝜙𝑛+ 3

2 − 𝜙𝑛+ 1
Δ𝑡

2

(cid:33)

(cid:32)

−

−

1
𝑐2

Δ𝑡

𝜙𝑛+ 1

2 − 𝜙𝑛− 1
Δ𝑡

2

𝜙𝑛+ 3

2 − 2𝜙𝑛+ 1
Δ𝑡2

2 + 𝜙𝑛− 1

2

(4.67)

(4.68)

(4.69)

(cid:33)

The last step comes from the fully implicit CDF-2 scalar wave equation (4.46). We thus see Gauss’s

Law is satisfied.

QED

Theorem 4.2.7. Under the assumption that 𝜖 𝑘

2 = 0, for 𝑘 ≤ 𝑛 + 1, both the explicit and semi-implicit

semi-discrete methods satisfy Gauss’s law

∇ · E𝑛+ 1

2 =

𝜌𝑛+ 1
2
𝜖0

.

(4.70)

Proof. We start by noting, under the assumption that 𝜖 𝑘

2 = 0, we have 𝜖 𝑘

1 = 0, for 𝑘 = 𝑛 + 1, from the

previous theorems. We carry out the proof for the explicit case and note that the semi-implicit case
is identical except for the fact that ∇𝜙𝑛+ 1

2 will be replaced by ∇

in the identity

2 + 𝜙𝑛− 1
2 )

(cid:17)

(cid:16) 1
2 (𝜙𝑛+ 3

E = −∇𝜙 − 𝜕𝑡A.

138

Evaluating this identity at time 𝑡𝑛+ 1

2 and making use of the semi-discrete potentials, we find

E𝑛+ 1

2 = −∇𝜙𝑛+ 1

2 −

A𝑛+1 − A𝑛
Δ𝑡

.

Again, taking the divergence gives

∇ · E𝑛+ 1

2 = −Δ𝜙𝑛+ 1

2 −

∇ · A𝑛+1 − ∇ · A𝑛
Δ𝑡

.

given that 𝜖 𝑛+1

1 = 0, then we can replace ∇ · A𝑛+1 with −

𝜙𝑛+ 3

2 −𝜙𝑛+ 1
𝑐2Δ𝑡

2

∇ · E𝑛+ 1

2 = −Δ𝜙𝑛+ 1

2 −

𝜙𝑛+ 3

2 − 2𝜙𝑛+ 1
𝑐2Δ𝑡2

2 + 𝜙𝑛− 1

2

𝜙𝑛+ 1

2 −𝜙𝑛− 1
𝑐2Δ𝑡

2

, so

and ∇ · A𝑛

with −

=

𝜌𝑛+ 1
2
𝜖0

.

Hence, we arrive at the conclusion of the theorem.

QED

Remark 4.2.3. The semi-discrete formulation method automatically satisfies ∇ · B = ∇ · (∇ × A) = 0.

Remark 4.2.4. All three time-consistent methods satisfy all of the involutions and the gauge condition,

as long as the method satisfies the semi-discrete continuity condition.

Remark 4.2.5. For all the methods we discuss, we incorporate the improved asymmetric Euler

method for our particle update method. A time-centered wave advance opens the door for time-

centering the particle pusher, which increases the accuracy of the solver and can be used to explore

aspects of symplecticity. This exploration is beyond the scope of this thesis but will be considered in

future work.

4.2.3 s-Stage Adams Bashforth

Consider the equation 𝑦(cid:48) = 𝑓 (𝑦, 𝑡). The 𝑠-stage Adams Bashforth method may be described as

follows:

For 1
𝑐2

𝑦𝑛 = 𝑦𝑛−1 + Δ𝑡

𝑛−1
(cid:213)

𝑖=𝑛−𝑠

(cid:0)𝑐𝑖 𝑓 (cid:0)𝑡𝑖, 𝑦𝑖(cid:1)(cid:1) .

(4.71)

𝜕2𝑢

𝜕𝑡2 − Δ𝑢 = 𝑆, defining 𝑣 := 𝜕𝑢

𝜕𝑡 , we rewrite it as the first order system

139

(cid:170)
(cid:174)
(cid:174)
(cid:172)
We then get the following update system

(cid:169)
(cid:173)
(cid:173)
𝑢
(cid:171)

𝑣

𝜕
𝜕𝑡

𝑐2 (𝑆 + Δ𝑢)

𝑣

.

(cid:170)
(cid:174)
(cid:174)
(cid:172)

= (cid:169)
(cid:173)
(cid:173)
(cid:171)

𝑣𝑛 = 𝑣𝑛−1 + 𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑖=𝑛−𝑠

𝑐𝑖 (cid:0)𝑆𝑖 + Δ𝑢𝑖(cid:1) ,

𝑢𝑛 = 𝑢𝑛−1 + Δ𝑡

𝑛−1
(cid:213)

𝑖=𝑛−𝑠

𝑐𝑖𝑣𝑖.

We apply this to our wave equations

𝜙 = 𝑣𝑛−1
𝑣𝑛

𝜙 + 𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑖=𝑛−𝑠

𝑐𝑖

(cid:18) 𝜌𝑖
𝜖0

(cid:19)

,

+ Δ𝜙𝑖

𝜙𝑛 = 𝜙𝑛−1 + Δ𝑡

𝑛−1
(cid:213)

𝑖=𝑛−𝑠

𝑐𝑖𝑣𝑖
𝜙.

A = v𝑛−1
v𝑛

A + 𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑐𝑖 (cid:0)𝜇0J𝑖 + ΔA𝑖(cid:1) ,

A𝑛 = A𝑛−1 + Δ𝑡

𝑖=𝑛−𝑠
𝑛−1
(cid:213)

𝑐𝑖v𝑖
A

.

We define the following residuals:

𝑖=𝑛−𝑠

𝐿 := −𝜙𝑛 + 𝜙𝑛−1 − 𝑐2Δ𝑡
𝜖 𝑛

𝑉 := −𝑣𝑛
𝜖 𝑛

𝜙 + 𝑣𝑛−1

𝜙 − 𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑖=𝑛−𝑠
𝑛−1
(cid:213)

𝑐𝑖∇ · A𝑖,

𝑐𝑖∇ · v𝑖
A

,

𝐶 := −𝜌𝑛 + 𝜌𝑛−1 − Δ𝑡
𝜖 𝑛

𝑐𝑖∇ · J𝑖.

𝑖=𝑛−𝑠
𝑛−1
(cid:213)

We now prove two lemmas.

𝑖=𝑛−𝑠

Lemma 4.2.6. 𝜖 𝑛

𝑉 is a linear combination of 𝜖𝐿, 𝜖𝐶, and 𝜖𝑉 at previous timesteps.

140

(4.72)

(4.73)

(4.74)

(4.75)

(4.76)

(4.77)

(4.78)

(4.79)

Proof. We take the definition of 𝜖𝑉 , (4.78), and plug in our update equations:

𝑉 = − 𝑣𝑛
𝜖 𝑛

𝜙 + 𝑣𝑛−1

𝜙 − 𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑖=𝑛−𝑠

∇ · v𝑖
A

(cid:19) (cid:33)

+ Δ𝜙𝑖

(cid:32)

= −

𝑣𝑛−1
𝜙 + 𝑐2Δ𝑡

(cid:32)

+

𝑣𝑛−2
𝜙 + 𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑐𝑖

𝑖=𝑛−𝑠
𝑛−2
(cid:213)

(cid:18) 𝜌𝑖
𝜖0
(cid:18) 𝜌𝑖
𝜖0

𝑐𝑖

(cid:19) (cid:33)

+ Δ𝜙𝑖

(4.80)

− 𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑖=𝑛−𝑠

We rearrange

𝑖=𝑛−𝑠−1
(cid:32)

𝑐𝑖∇ ·

v𝑖−1
A + 𝑐2Δ𝑡

𝑐 𝑗 (cid:0)𝜇0J 𝑗 + ΔA 𝑗 (cid:1)

(cid:33)

.

𝑖−1
(cid:213)

𝑗=𝑖−𝑠

− 𝑣𝑛−1

𝜙 + 𝑣𝑛−2

𝜙 − 𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑖=𝑛−𝑠
(cid:19)

+ Δ𝜙𝑖

𝑐𝑖∇ · v𝑖−1
A

+ 𝑐2Δ𝑡

𝑛−2
(cid:213)

𝑖=𝑛−𝑠−1

𝑐𝑖

(cid:18) 𝜌𝑖
𝜖0

(cid:19)

+ Δ𝜙𝑖

𝑖−1
(cid:213)

𝑗=𝑖−𝑠

𝑐 𝑗 (cid:0)𝜇0∇ · J 𝑗 + Δ∇ · A 𝑗 (cid:1)

− 𝑐2Δ𝑡

− 𝑐2Δ𝑡

𝑐𝑖

(cid:18) 𝜌𝑖
𝜖0

𝑐𝑖𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑖=𝑛−𝑠
𝑛−1
(cid:213)

𝑖=𝑛−𝑠

= − 𝑣𝑛−1

𝜙 + 𝑣𝑛−2

𝜙 − 𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑐𝑖∇ · v𝑖−1
A

𝑛−1
(cid:213)

(cid:32)

𝑐𝑖

−

𝑐2Δ𝑡

𝑖=𝑛−𝑠

𝜌𝑖
𝜖0

+

𝑖=𝑛−𝑠
𝜌𝑖−1
𝜖0

− 𝑐2Δ𝑡

(cid:32)

𝑐 𝑗 𝜇0∇ · J 𝑗 + Δ

𝑖−1
(cid:213)

𝑗=𝑖−𝑠

−𝜙𝑖 + 𝜙𝑖−1 − 𝑐2Δ𝑡

(cid:33)(cid:33)

𝑐 𝑗 ∇ · A 𝑗

𝑖−1
(cid:213)

𝑗=𝑖−𝑠

=𝜖 𝑛−1

𝑉 + 𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑖=𝑛−𝑠

𝑐𝑖

(cid:18) 1
𝜖0

(cid:19)

𝐶 + Δ𝜖 𝑖
𝜖 𝑖
𝐿

∴ 𝜖 𝑛

𝑉 is a linear combination of 𝜖𝐿, 𝜖𝐶, and 𝜖𝑉 at previous timesteps.

Lemma 4.2.7. 𝜖 𝑛

𝐿 is a linear combination of 𝜖𝐿 and 𝜖𝑉 at previous timesteps.

Proof. We write 𝜖 𝑛

𝐿 as follows

(4.81)

QED

141

𝐿 = −𝜙𝑛 + 𝜙𝑛−1 − 𝑐2Δ𝑡
𝜖 𝑛

𝑛−1
(cid:213)

𝑐𝑖∇ · A𝑖

(cid:32)

= −

𝜙𝑛−1 + Δ𝑡

𝑖=𝑛−𝑠
(cid:33)

𝑐𝑖𝑣𝑖
𝜙

(cid:32)

+

𝑛−1
(cid:213)

𝑖=𝑛−𝑠

𝑛−1
(cid:213)

𝑖=𝑛−𝑠

= −𝜙𝑛−1 + 𝜙𝑛−2 − 𝑐2Δ𝑡

= 𝜖 𝑛−1

𝐿 + Δ𝑡

𝑛−1
(cid:213)

𝑖=𝑛−𝑠

𝜖 𝑖
𝑉 .

𝜙𝑛−2 + Δ𝑡

𝑛−2
(cid:213)

(cid:33)

𝑐𝑖𝑣𝑖
𝜙

− 𝑐2Δ𝑡

(cid:32)

𝑐𝑖∇ ·

A𝑖 + Δ𝑡

𝑖−1
(cid:213)

𝑐𝑖∇ · A𝑖 + Δ𝑡

𝑖=𝑛−𝑠−1
𝑛−1
(cid:213)

𝑖=𝑛−𝑠

(cid:32)

𝑐𝑖

−𝑣𝑖

𝜙 + 𝑣𝑖−1

𝜙 − 𝑐2Δ𝑡

𝑖−1
(cid:213)

𝑗=𝑖−𝑠

𝑗=𝑖−𝑠
(cid:33)

𝑐 𝑗 ∇ · v

𝑗
A

𝑛−1
(cid:213)

𝑖=𝑛−𝑠

∴ 𝜖 𝑛

𝐿 is a linear combination of 𝜖𝐿 and 𝜖𝑣 at previous timesteps.

We take the above lemmas and prove the following theorem:

Theorem 4.2.8. If 𝜖 𝑖

𝑉 = 𝜖 𝑖

𝐿 = 𝜖 𝑖

𝐶 = 0 for 𝑖 ≤ 0, then 𝜖 𝑛

𝐿 = 𝜖 𝑛

𝑉 = 𝜖 𝑛

𝐶 = 0 ∀ 𝑛.

Proof. Assume 𝜖 𝑖

𝑉 = 𝜖 𝑖

𝐿 = 𝜖 𝑖

𝐶 = 0 for 𝑖 ≤ 0.

(cid:33)

𝑐 𝑗 v

𝑗
A

(4.82)

QED

We prove by induction. Using Lemma 4.2.6, we see

𝑉 = 𝜖 0
𝜖 1

𝑉 + 𝑐2Δ𝑡

0
(cid:213)

𝑖=1−𝑠

𝑐𝑖

(cid:18) 1
𝜖0

(cid:19)

𝐶 + Δ𝜖 𝑖
𝜖 𝑖
𝐿

= (0) + 𝑐2Δ𝑡

0
(cid:213)

𝑖=1−𝑠

𝑐𝑖 (

1
𝜖0

(0) + Δ(0)) = 0.

(4.83)

This may be clicked up to arbitrary 𝑛 using the exact same logic. So 𝜖 𝑛

𝑉 = 0 ∀ 𝑛. From Lemma

4.2.7, we know

𝐿 = 𝜖 0
𝜖 1

𝐿 + Δ𝑡

0
(cid:213)

𝜖 𝑖
𝑉 = (0) + Δ𝑡

𝑛−1
(cid:213)

(0) = 0.

So 𝜖 𝑛

𝐿 = 0 ∀ 𝑛. Knowing that 𝜖 𝑛

𝐿 = 𝜖 𝑛

𝑖=𝑛−𝑠
𝑖=1−𝑠
𝑉 = 0 ∀ 𝑛, we again use Lemma 4.2.6 to see

𝑉 = 𝜖 𝑛−1
𝜖 𝑛

𝑉 + 𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑐𝑖

𝐶 + Δ𝜖 𝑖
𝜖 𝑖
𝐿

(cid:19)

(cid:19)

(cid:18) 1
𝜖0
(cid:18) 1
𝜖0

𝜖 𝑖
𝐶 + Δ(0)

=⇒ (0) = (0) + 𝑐2Δ𝑡

𝑖=𝑛−𝑠
𝑛−1
(cid:213)

𝑐𝑖

𝑖=𝑛−𝑠

=⇒ 𝜖 𝑛−1

𝐶 = −

𝑛−2
(cid:213)

𝑛−𝑠

𝑐𝑖
𝑐𝑛−1

𝜖 𝑖
𝐶 .

142

(4.84)

(4.85)

Given this result, we use the above logic to show that 𝜖 𝑖

𝐶 = 0 ∀ 𝑖 ≤ 0 =⇒ 𝜖 1

𝐶 = 0 and so on to

arbitrary n.
∴ if 𝜖 𝑖

𝑉 = 𝜖 𝑖

𝐿 = 𝜖 𝑖

𝐶 = 0 for 𝑖 ≤ 0, then 𝜖 𝑛

𝐿 = 𝜖 𝑛

𝑉 = 𝜖 𝑛

𝐶 = 0 ∀ 𝑛.

QED

Finally, we connect satisfaction of the semi-discrete continuity equation and the semi-discrete

Lorenz gauge with satisfaction of Gauss’s Law, ∇ · E =

𝜌
𝜖0

.

Theorem 4.2.9. Assuming the involution ∇ · E =

𝜌
𝜖0

at 𝑡 = 0, if the residuals for the Lorenz gauge

condition (4.77) and the semi-discrete continuity equation (4.79) are zero for time level 𝑛, then
∇ · E𝑛 =

∀ 𝑛.

𝜌𝑛
𝜖0

Proof. We prove by induction. Assume 𝜖 𝑛

𝐶 = 𝜖 𝑛

𝐿 = 0 ∀ 𝑛.

By our assumption, ∇ · E0 =
Now we assume true for 𝑛, and consider the 𝑛 + 1 case. By definition we know E = −∇𝜙 − 𝜕A
𝜕𝑡 .

.

𝜌0
𝜖0

We recall vA − 𝜕A

𝜕𝑡 . So

E𝑛+1 = −∇𝜙𝑛+1 − v𝑛+1
A

,

=⇒ ∇ · E𝑛+1 = −Δ𝜙𝑛+1 − ∇ · v𝑛+1
A

(cid:32)

= −Δ𝜙𝑛+1 − ∇ ·

v𝑛
A + 𝑐2Δ𝑡

𝑛−1
(cid:213)

(cid:33)

𝑐𝑖 (cid:0)𝜇0J𝑖 + ΔA𝑖(cid:1)

= −Δ𝜙𝑛+1 − ∇ · v𝑛

A + 𝑐2Δ𝑡

𝑐𝑖 (cid:0)𝜇0∇ · J𝑖 + Δ∇ · A𝑖(cid:1)

= −Δ𝜙𝑛+1 − ∇ · v𝑛

A − 𝜇0𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑐𝑖∇ · J𝑖 − Δ

(cid:32)

𝑐2Δ𝑡

𝑛−1
(cid:213)

𝑖=𝑛−𝑠

(cid:33)

𝑐𝑖∇ · A𝑖

(4.86)

(cid:16)

𝜙𝑛+1 − 𝜙𝑛(cid:17)

+ Δ

𝑖=𝑛−𝑠
𝑛−1
(cid:213)

𝑖=𝑛−𝑠

(cid:16)

𝑖=𝑛−𝑠
𝜌𝑛+1 − 𝜌𝑛(cid:17)
𝜌𝑛
𝜖0

−

143

1
𝜖0
𝜌𝑛+1
𝜖0
𝜌𝑛
𝜖0

= −Δ𝜙𝑛+1 − ∇ · v𝑛

A +

= −Δ𝜙𝑛 − ∇ · v𝑛

A +

𝜌𝑛+1
𝜖0

−

= ∇ · E𝑛 +

=

𝜌𝑛+1
𝜖0

.

The sixth line is justified by our assumption, 𝜖 𝑛

𝐶 = 𝜖 𝑛

𝐿 = 0 ∀ 𝑛. The last step is justified by our

inductive hypothesis. This concludes the proof.

QED

Remark 4.2.6. The properties of this method were proven after the production of our third paper,

as such this remains theory and has yet to be confirmed by numerical results.

4.2.4 The Generalized s-stage Diagonal Implicit Runge-Kutta Method

In this section we will first establish an 𝑠-stage method for solving the wave equations and

continuity equation. We will then prove the properties this method, namely that conservation of

charge implies satisfaction of the Lorenz gauge condition. As with the CDF and BDF methods, we

will assume we are given J during the construction. In practice, J will come from a particle advance,

and there are several ways to go about constructing J at stage values.

4.2.4.1 The s-stage DIRK Method

Consider 1
𝑐2

𝜕2𝑢
𝜕𝑡 − Δ𝑢 = 𝑆, which we rewrite as a first order system:

= (cid:169)
(cid:173)
(cid:173)
(cid:171)
We apply an 𝑠-stage Runge-Kutta method, where defining 𝑆(𝑖) := 𝑆 (𝑥, 𝑡𝑛 + 𝑐𝑖 ℎ), we arrive at

(cid:169)
(cid:173)
(cid:173)
𝑢
(cid:171)

(cid:170)
(cid:174)
(cid:174)
(cid:172)

(cid:170)
(cid:174)
(cid:174)
(cid:172)

𝑣

.

𝑣

𝑐2 (𝑆 + Δ𝑢)

𝜕
𝜕𝑡

𝑘 𝑣
𝑖 = 𝑐2 (cid:169)
(cid:173)
(cid:171)

𝑖 = 𝑣𝑛 + ℎ
𝑘 𝑢

𝑢𝑛 + ℎ
𝑆(𝑖) + Δ (cid:169)
(cid:173)
(cid:171)
𝑎𝑖 𝑗 𝑘 𝑣
𝑗 ,

𝑖
(cid:213)

𝑖
(cid:213)

𝑗=1

,

𝑎𝑖 𝑗 𝑘 𝑢
𝑗 (cid:170)
(cid:174)
(cid:172)

(cid:170)
(cid:174)
(cid:172)

𝑢𝑛+1 = 𝑢𝑛 + ℎ

𝑣𝑛+1 = 𝑣𝑛 + ℎ

𝑗=1
𝑠
(cid:213)

𝑖=1
𝑠
(cid:213)

𝑖=1

𝑏𝑖 𝑘 𝑢
𝑖 ,

𝑏𝑖 𝑘 𝑣
𝑖 .

144

(4.87)

(4.88)

(4.89)

(4.90)

(4.91)

Here the 𝑎𝑖 𝑗 , 𝑏𝑖, and 𝑐𝑖 terms come from whatever Butcher tableau with which our Runge-Kutta
method is working. When solving for 𝑘 𝑣

𝑖 we use the definition of 𝑘 𝑢

𝑖 and vice versa to arrive at,

L𝑖 (cid:2)𝑘 𝑢
𝑖

(cid:3) = 𝑣𝑛 + ℎ

𝑖−1
(cid:213)

𝑗=1

𝑎𝑖 𝑗 𝑘 𝑣

L𝑖 (cid:2)𝑘 𝑣
𝑖

(cid:3) = 𝑐2 (cid:169)
(cid:173)
(cid:171)

𝑢𝑛 + ℎ
𝑆(𝑖) + Δ (cid:169)
(cid:173)
(cid:171)

𝑗 + ℎ𝑎𝑖𝑖 (cid:169)
𝑐2 (cid:169)
(cid:173)
(cid:173)
(cid:171)
(cid:171)
𝑖−1
(cid:213)
𝑎𝑖 𝑗 𝑘 𝑢
𝑗 (cid:170)
(cid:170)
(cid:174)
(cid:174)
(cid:172)
(cid:172)

𝑗=1

𝑢𝑛 + ℎ
𝑆(𝑖) + Δ (cid:169)
(cid:173)
(cid:171)

𝑖−1
(cid:213)

𝑗=1

𝑣𝑛 + ℎ

+ ℎ𝑎𝑖𝑖𝑐2Δ (cid:169)
(cid:173)
(cid:171)

,

𝑎𝑖 𝑗 𝑘 𝑢
𝑗 (cid:170)
(cid:170)
(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:172)
(cid:172)
(cid:172)
𝑖−1
(cid:213)
𝑎𝑖 𝑗 𝑘 𝑣
𝑗 (cid:170)
(cid:174)
(cid:172)

𝑗=1

(4.92)

.

(4.93)

Here we have defined L𝑖 :=

(cid:16)

(cid:17)

I − 1
𝛼2
𝑖

Δ

and 𝛼𝑖 := 1

ℎ𝑎𝑖𝑖𝑐 . Note the 𝑘 𝑗 values are known for 𝑗 < 𝑖.

We apply this to our system of wave equations, the Lorenz gauge, and the continuity equation.

To ease the comparison, we define 𝑣𝜙 :=

for 𝜙 and A are

𝜕𝜙

𝜕𝑡 and 𝑣A := 𝜕A

𝜕𝑡 . Doing so, we see the update statements

𝜙𝑛+1 = 𝜙𝑛 + ℎ

A𝑛+1 = A𝑛 + ℎ

𝜙 = 𝑣𝑛
𝑣𝑛+1

𝜙 + ℎ

A = v𝑛
v𝑛+1

A + ℎ

𝑠
(cid:213)

𝑖=1
𝑠
(cid:213)

𝑖=1
𝑠
(cid:213)

𝑖=1
𝑠
(cid:213)

𝑖=1

𝑏𝑖 𝑘 𝜙
𝑖 ,

𝑏𝑖 𝑘 A
𝑖 ,

𝑏𝑖 𝑘 𝑣 𝜙
𝑖

,

𝑏𝑖 𝑘 vA
𝑖

,

with the 𝑖 = 1, · · · , 𝑠 stage values computed according to

𝑘 𝜙
𝑖 = L−1
𝑖

𝑘 𝑣 𝜙
𝑖 = L−1
𝑖

𝑖 = L−1
kA

𝑖

𝑣A
𝑖 = L−1
k
𝑖


























𝑣𝑛
𝜙 + ℎ

𝑖−1
(cid:213)

𝑗=1

𝑎𝑖 𝑗 𝑘 𝑣 𝜙

𝑗 + ℎ𝑎𝑖𝑖𝑐2

𝜌(𝑖)
𝜖0

+ ℎ𝑎𝑖𝑖𝑐2Δ (cid:169)
(cid:173)
(cid:171)

𝜙𝑛 + ℎ

𝑖−1
(cid:213)

𝑗=1

𝜌(𝑖)
𝜖0

+ Δ (cid:169)
(cid:173)
(cid:171)
𝑖−1
(cid:213)

𝑗=1

v𝑛
A + ℎ

𝜙𝑛 + ℎ

𝑖−1
(cid:213)

𝑗=1

𝑎𝑖 𝑗 𝑘 𝜙
𝑗 (cid:170)
(cid:174)
(cid:172)

+ ℎ𝑎𝑖𝑖𝑐2Δ (cid:169)
(cid:173)
(cid:171)

𝑣𝑛
𝜙 + ℎ

𝑖−1
(cid:213)

𝑗=1

𝑎𝑖 𝑗 kvA

𝑗 + ℎ𝑎𝑖𝑖𝑐2𝜇0J(𝑖) + ℎ𝑎𝑖𝑖𝑐2Δ (cid:169)
(cid:173)
(cid:171)

A𝑛 + ℎ

A𝑛 + ℎ

𝜇0J(𝑖) + Δ (cid:169)
(cid:173)
(cid:171)

𝑖−1
(cid:213)

𝑗=1

𝑎𝑖 𝑗 kA
𝑗 (cid:170)
(cid:174)
(cid:172)

+ ℎ𝑎𝑖𝑖𝑐2Δ (cid:169)
(cid:173)
(cid:171)

v𝑛
A + ℎ

145

,








𝑎𝑖 𝑗 𝑘 𝜙
𝑗 (cid:170)
(cid:174)
(cid:172)







𝑎𝑖 𝑗 𝑘 𝑣 𝜙
𝑗 (cid:170)
(cid:174)
(cid:172)

,

𝑖−1
(cid:213)

𝑗=1

𝑎𝑖 𝑗 kA
𝑗 (cid:170)
(cid:174)
(cid:172)







𝑎𝑖 𝑗 kvA
𝑗 (cid:170)
(cid:174)
(cid:172)

.

𝑖−1
(cid:213)

𝑗=1

,








(4.94)

(4.95)

(4.96)

(4.97)

(4.98)

(4.99)

(4.100)

(4.101)

We do a much more straightforward update statement for the gauge and continuity update equations:

𝜙𝑛+1 = 𝜙𝑛 − 𝑐2ℎ

𝑠
(cid:213)

𝑏𝑖∇ · A(𝑖),

𝜌𝑛+1 = 𝜌𝑛 − ℎ

𝑖=1

𝑠
(cid:213)

𝑏𝑖∇ · J(𝑖).

(4.102)

(4.103)

In the above updates we have defined {A, J, 𝜌}(𝑖) as the 𝑖-th linear interpolation between time

levels 𝑛 and 𝑛 + 1 given by the Runge-Kutta method, i.e.,

𝑖=1

So it follows

𝑋 (𝑖) = (1 − 𝑐𝑖) 𝑋 𝑛 + 𝑐𝑖 𝑋 𝑛+1.

𝑟
(cid:213)

𝑖=1

𝑏𝑖 𝑋 (𝑖) =

𝑟
(cid:213)

𝑖=1

(cid:16)

(1 − 𝑐𝑖) 𝑋 𝑛 + 𝑐𝑖 𝑋 𝑛+1(cid:17)

𝑏𝑖

≡ 𝑅𝑟,𝑛 𝑋 𝑛 + 𝑅𝑟,𝑛+1𝑋 𝑛+1.

(4.104)

(4.105)

4.2.4.2 The Properties of the s-stage Method

We wish to link the satisfaction of the DIRK-𝑠 formulation of the Lorenz gauge condition (4.102)

with the DIRK-𝑠 continuity equation (4.103). To do so, we define the following five residuals:

𝜖 𝑛+1
1

𝜖 𝑛+1
2

𝜖 𝑛+1
3

𝜖 𝑛+1
4

𝜖 𝑛+1
5

:= −𝜙𝑛+1 + 𝜙𝑛 − 𝑐2ℎ

(cid:16)

:= −𝑣𝑛+1

𝜙 + 𝑣𝑛

𝜙 − 𝑐2ℎ
(cid:16)

:= −𝜌𝑛+1 + 𝜌𝑛 − ℎ

(cid:16)

𝑅𝑠,𝑛∇ · A𝑛 + 𝑅𝑠,𝑛+1∇ · A𝑛+1(cid:17)
(cid:17)

𝑅𝑠,𝑛∇ · v𝑛
A + 𝑅𝑠,𝑛+1∇ · v𝑛+1
𝑅𝑠,𝑛∇ · J𝑛 + 𝑅𝑠,𝑛+1∇ · J𝑛+1(cid:17)

A

:= −𝑘 𝜙,𝑛+1
𝑖

+ 𝑘 𝜙,𝑛

𝑖 − 𝑐2ℎ

(cid:16)

:= −𝑘 𝑣 𝜙,𝑛+1

𝑖

+ 𝑘 𝑣 𝜙,𝑛
𝑖

− 𝑐2ℎ

𝑅𝑠,𝑛∇ · kA,𝑛
(cid:16)

𝑖

𝑅𝑠,𝑛∇ · kvA,𝑛

𝑖

+ 𝑅𝑠,𝑛∇ · kA,𝑛+1

𝑖

(cid:17)

+ 𝑅𝑠,𝑛+1∇ · kvA,𝑛+1

𝑖

(cid:17)

(4.106)

(4.107)

(4.108)

(4.109)

(4.110)

We have a host of lemmas to prove that will assist with the main theorem. The first, Lemma 4.2.8,

is partially related to the residual of the continuity equation (4.108) and will assist in proving Lemmas

4.2.9 and 4.2.10, which are concerned with the residuals of the 𝑘 variables (4.109) and (4.110).

146

These lemmas assist in proving Lemmas 4.2.11 and 4.2.12, two additional lemmas concerned with

the residuals of the gauge condition (4.106) and its time derivative (4.107), respectively, and these

will finally assist in proving Theorem 4.2.10, that satisfaction of the semi-discrete continuity equation

implies satisfaction of the semi-discrete gauge condition.

Lemma 4.2.8. For any substep 𝑖, the following identity holds:

−

𝜌(𝑖),𝑛
𝜖0

+

𝜌(𝑖),𝑛−1
𝜖0

− 𝑐2ℎ𝜇0

(cid:16)

𝑅𝑠,𝑛∇ · J(𝑖),𝑛−1 + 𝑅𝑠,𝑛+1∇ · J(𝑖),𝑛(cid:17)

= 𝜇0𝑐2 (cid:16)

Proof. We use the identity 𝑐2 = 1
𝜇0𝜖0

and see

(1 − 𝑐𝑖) 𝜖 𝑛

3 + 𝑐𝑖𝜖 𝑛+1

3

(cid:17)

.

(4.111)

− 𝑐2ℎ𝜇0

(cid:16)

𝑅𝑠,𝑛∇ · J(𝑖),𝑛−1 + 𝑅𝑠,𝑛+1∇ · J(𝑖),𝑛(cid:17)

+

𝜌(𝑖),𝑛−1
𝜖0

−

𝜌(𝑖),𝑛
𝜖0
(cid:18)

−

= 𝜇0
= 𝜇0𝑐2 (cid:16)

+

𝜌(𝑖),𝑛−1
𝜌(𝑖),𝑛
𝜇0𝜖0
𝜇0𝜖0
−𝜌(𝑖),𝑛 + 𝜌(𝑖),𝑛−1 − ℎ

− 𝑐2ℎ𝜇0

(cid:16)

𝑅𝑠,𝑛∇ · J(𝑖),𝑛−1 + 𝑅𝑠,𝑛+1∇ · J(𝑖),𝑛(cid:17) (cid:19)

(4.112)

(cid:16)

𝑅𝑠,𝑛∇ · J(𝑖),𝑛−1 + 𝑅𝑠,𝑛+1∇ · J(𝑖),𝑛(cid:17)(cid:17)

.

Taking the interior portion, we further derive

(cid:16)

−

(1 − 𝑐𝑖) 𝜌𝑛 + 𝑐𝑖 𝜌𝑛+1(cid:17)
(cid:16)

+
(1 − 𝑐𝑖) J𝑛−1 + 𝑐𝑖J𝑛(cid:17)

𝑅𝑠,𝑛∇ ·

(cid:16)

(cid:16)

− ℎ

(1 − 𝑐𝑖) 𝜌𝑛−1 + 𝑐𝑖 𝜌𝑛(cid:17)

+ 𝑅𝑠,𝑛+1∇ ·

(cid:16)

(1 − 𝑐𝑖) J𝑛 + 𝑐𝑖J𝑛+1(cid:17)(cid:17)

= (1 − 𝑐𝑖)

(cid:16)

(cid:16)

+ 𝑐𝑖

−𝜌𝑛 + 𝜌𝑛−1(cid:17)
(cid:16)

−𝜌𝑛+1 + 𝜌𝑛(cid:17)
𝑅𝑠,𝑛∇ · J𝑛−1 + 𝑅𝑠,𝑛+1∇ · J𝑛(cid:17)

(cid:16)

− ℎ

(1 − 𝑐𝑖)

+ 𝑐𝑖

(cid:16)

𝑅𝑠,𝑛∇ · J𝑛 + 𝑅𝑠,𝑛+1∇ · J𝑛+1(cid:17)(cid:17)

(4.113)

= (1 − 𝑐𝑖)

(cid:16)

−𝜌𝑛 + 𝜌𝑛−1 − ℎ

(cid:16)

𝑅𝑠,𝑛∇ · J𝑛−1 + 𝑅𝑠,𝑛+1∇ · J𝑛(cid:17)(cid:17)

(cid:16)

+ 𝑐𝑖

−𝜌𝑛+1 + 𝜌𝑛 − ℎ

(cid:16)

𝑅𝑠,𝑛∇ · J𝑛 + 𝑅𝑠,𝑛+1∇ · J𝑛+1(cid:17)(cid:17)

= (1 − 𝑐𝑖) 𝜖 𝑛

3 + 𝑐𝑖𝜖 𝑛+1

3

.

We have thus demonstrated the identity.

QED

Lemma 4.2.9. The 𝜖4 residual (4.109) at time level 𝑛 is a linear combination of the residuals of the

Lorenz gauge (4.106) at time level 𝑛, the time derivative of the Lorenz gauge (4.107) at time level 𝑛,

and the continuity equation at time levels 𝑛 and 𝑛 + 1.

147

Proof. We prove by induction. For the base case, we see

4 = −𝑘 𝜙,𝑛
𝜖 𝑛

1

= −L−1
1

+ 𝑘 𝜙,𝑛−1
1
(cid:20)

− ℎ𝑐2 (cid:16)
(cid:18)

𝑣𝑛
𝜙 + ℎ𝑎11𝑐2

Δ𝜙𝑛 +

(cid:20)

+ L−1
1

𝑣𝑛−1
𝜙 + ℎ𝑎11𝑐2

(cid:18)

(cid:17)

1

+ 𝑅𝑛+1∇ · kA,𝑛

𝑅𝑛∇ · kA,𝑛−1
1
(cid:19)(cid:21)
𝜌(1),𝑛
𝜖0
𝜌(1),𝑛−1
𝜖0
ΔA𝑛−1 + 𝜇0J(1),𝑛−1(cid:17)(cid:105)
ΔA𝑛 + 𝜇0J(1),𝑛(cid:17)(cid:105)

(cid:19)(cid:21)

Δ𝜙𝑛−1 +

− ℎ𝑐2𝑅𝑛∇ · L−1
1

− ℎ𝑐2𝑅𝑛+1∇ · L−1
1

(cid:104)

= L−1
1

−𝑣𝑛

𝜙 + 𝑣𝑛−1
(cid:104)

+ ℎ𝑎11𝑐2L−1
1

(cid:20)

+ ℎ𝑎11𝑐2L−1
1

(cid:104)

A + ℎ𝑎11𝑐2 (cid:16)
v𝑛−1
A + ℎ𝑎11𝑐2 (cid:16)
(cid:104)
v𝑛
𝜙 − ℎ𝑐2 (cid:16)
𝑅𝑛∇ · v𝑛−1
−Δ𝜙𝑛 + Δ𝜙𝑛−1 − ℎ𝑐2 (cid:16)
𝜌(1),𝑛−1
𝜖0

𝜌(1),𝑛
𝜖0
1 + ℎ𝑎11𝑐4𝜇0

−

+

(cid:16)

(cid:104)

= L−1
1

2 + ℎ𝑎11𝑐2Δ𝜖 𝑛
𝜖 𝑛

(cid:17)(cid:105)

A

A + 𝑅𝑛+1∇ · v𝑛
∇ · A𝑛−1(cid:17)
(cid:16)
𝑅𝑛∇ · J(1),𝑛−1 + 𝑅𝑛+1∇ · J(1),𝑛(cid:17)(cid:21)
(cid:16)

+ 𝑅𝑛+1Δ (∇ · A𝑛)

𝑅𝑛Δ

(cid:17)(cid:105)

− ℎ𝑐2𝜇0

(1 − 𝑐1) 𝜖 𝑛

3 + 𝑐1𝜖 𝑛+1

3

(cid:17)(cid:105)

(4.114)

The last step is justified definitionally and by Lemma 4.2.8. We have shown the base case, now we
wish to show the inductive step. Assuming true for 𝑖 = 𝑙 − 1, we now consider 𝑖 = 𝑙:

𝑙

4 = −𝑘 𝜙,𝑛
𝜖 𝑛







= − L −1

𝑙








+ 𝑘 𝜙,𝑛−1
𝑙

− ℎ𝑐2 (cid:16)

𝑅𝑠,𝑛∇ · kA,𝑛−1

𝑙

+ 𝑅𝑠,𝑛+1∇ · kA,𝑛

𝑙

(cid:17)

𝑣𝑛−1
𝜙 + ℎ

𝑙−1
(cid:213)

𝑗=1

𝑎𝑙 𝑗 𝑘 𝑣𝜙 ,𝑛−1
𝑗

+ ℎ𝑎𝑙𝑙𝑐2

𝜌 (𝑙),𝑛−1
𝜖0

+ ℎ𝑎𝑙𝑙𝑐2Δ (cid:169)
(cid:173)
(cid:171)

𝜙𝑛−1 + ℎ

𝑙−1
(cid:213)

𝑗=1

𝑎𝑙 𝑗 𝑘 𝜙,𝑛−1
𝑗

𝑙−1
(cid:213)

𝑗=1








+L −1
𝑙

𝑣𝑛−2
𝜙 + ℎ

𝑎𝑙 𝑗 𝑘 𝑣𝜙 ,𝑛−2
𝑗

+ ℎ𝑎𝑙𝑙𝑐2

𝜌 (𝑙),𝑛−2
𝜖0

−ℎ𝑐2𝑅𝑠,𝑛∇ · (cid:169)
(cid:173)
(cid:171)

L −1
𝑙

v𝑛−2
A + ℎ

𝑙−1
(cid:213)

𝑗=1

𝑎𝑙 𝑗kvA,𝑛−2
𝑗

−ℎ𝑐2𝑅𝑠,𝑛+1∇ · (cid:169)
(cid:173)
(cid:171)

L −1
𝑙

v𝑛−1
A + ℎ

𝑙−1
(cid:213)

𝑗=1








𝜙𝑛−2 + ℎ

+ ℎ𝑎𝑙𝑙𝑐2Δ (cid:169)
(cid:173)
(cid:171)

𝑙−1
(cid:213)

𝑗=1

𝑎𝑙 𝑗 𝑘 𝜙,𝑛−2
𝑗

A𝑛−2 + ℎ
+ ℎ𝑎𝑙𝑙𝑐2𝜇0J(𝑙),𝑛−2 + ℎ𝑎𝑙𝑙𝑐2Δ (cid:169)
(cid:173)
(cid:171)

𝑎𝑙 𝑗kvA,𝑛−1
𝑗

A𝑛−1 + ℎ
+ ℎ𝑎𝑙𝑙𝑐2𝜇0J(𝑙) ,𝑛−1 + ℎ𝑎𝑙𝑙𝑐2Δ (cid:169)
(cid:173)
(cid:171)



(cid:170)

(cid:174)


(cid:172)




(cid:170)

(cid:174)


(cid:172)

𝑙−1
(cid:213)

𝑗=1

𝑎𝑙 𝑗kA,𝑛−2
𝑗



(cid:170)

(cid:174)


(cid:172)

𝑎𝑙 𝑗kA,𝑛−1
𝑗

𝑙−1
(cid:213)

𝑗=1

(cid:170)
(cid:174)
(cid:172)


(cid:170)
(cid:170)

(cid:174)
(cid:174)


(cid:172)
(cid:172)

(4.115)

.

Grouping all of these inside L −1

𝑙

, we can break all these apart, rearrange, and either using definitions or

148

Lemma 4.2.8 to see the argument passed to this operator takes the form:

− 𝑣𝑛−1

𝜙 + 𝑣𝑛−2
(cid:18)

+ ℎ𝑎𝑙𝑙𝑐2

−

𝜙 − ℎ𝑐2 (cid:16)
𝜌 (𝑙),𝑛−1
𝜖0

+

ℎ𝑐2𝑅𝑠,𝑛v𝑛−1
𝜌 (𝑙),𝑛−2
𝜖0

A + 𝑅𝑠,𝑛+1v𝑛−1
(cid:16)

A

− ℎ𝑐2𝜇0

(cid:17)

𝑅𝑠,𝑛∇ · J(𝑙),𝑛−2 + 𝑅𝑠,𝑛+1∇ · J(𝑙),𝑛−1(cid:17)(cid:19)

+ ℎ2𝑎𝑙𝑙𝑐2Δ

+ ℎ2𝑎𝑙𝑙𝑐2Δ

(cid:16)

−𝜙𝑛−1 + 𝜙𝑛−2 − ℎ𝑐2 (cid:16)
𝑙−1
(cid:213)

(cid:16)

𝑎𝑙 𝑗

−𝑘 𝜙,𝑛−1
𝑗

+ 𝑘 𝜙,𝑛−2
𝑗

𝑅𝑠,𝑛∇ · A𝑛−2 + 𝑅𝑠,𝑛+1∇ · A𝑛−1(cid:17)(cid:17)

(4.116)

− ℎ𝑐2 (cid:16)

𝑅𝑠,𝑛∇ · kA,𝑛−2 + 𝑅𝑠,𝑛+1∇ · kA,𝑛−1

𝑗

(cid:17)(cid:17)

𝑗=1

= 𝜖 𝑛−1
2

+ ℎ𝑎𝑙𝑙𝑐2𝜇0𝑐2 (cid:16)

(1 − 𝑐𝑙) 𝜖 𝑛−1

3

+ 𝑐𝑙𝜖 𝑛
3

(cid:17)

+ ℎ𝑎𝑙𝑙𝑐2Δ𝜖 𝑛−1

1

+ ℎ2𝑎11𝑐2Δ

𝑙−1
(cid:213)

𝑗=1

𝑎𝑙 𝑗 𝛼𝑛−1
𝑙

.

We define 𝛼𝑛−1

𝑙

to be some residual that is a linear combination of 𝜖 𝑛−1

1

, 𝜖 𝑛−1
2

, and 𝜖 𝑛−1

3

. This is justified by

our inductive hypothesis. We have demonstrated 𝜖 𝑛

4 is a linear combination of residuals of previous timesteps

for any number of substeps.

QED

Lemma 4.2.10. The 𝜖5 residual (4.110) at time level 𝑛 is a linear combination of the residuals of

the Lorenz gauge (4.106) at time level 𝑛, the time derivative of the Lorenz gauge (4.107) at time level

𝑛, the continuity equation at time levels 𝑛 and 𝑛 + 1, and the 𝜖4 residual (itself a linear combination

of the other three residuals) at time level 𝑛.

Proof. We prove by induction. For the base case, we see

5 = −𝑘 𝑣 𝜙,𝑛
𝜖 𝑛

1

= −L−1
1

+ 𝑘 𝑣 𝜙,𝑛−1
1
(cid:18)
(cid:20)

𝑐2

Δ𝜙𝑛 +

− ℎ𝑐2𝑅𝑛∇ · L−1
1

(cid:104)

𝑐2 (cid:16)

− ℎ𝑐2𝑅𝑛+1∇ · L−1
1

(cid:17)

1
(cid:18)

+ ℎ𝑎11Δ𝑣𝑛
𝜙

+ 𝑅𝑛+1∇ · kvA,𝑛
(cid:20)

𝑅𝑛∇ · kvA,𝑛−1
1
(cid:19)(cid:21)

− ℎ𝑐2 (cid:16)
𝜌(1),𝑛
𝜖0
ΔA𝑛−1 + 𝜇0J(1),𝑛−1 + ℎ𝑎11Δv𝑛−1
A
(cid:17)(cid:105)
𝑐2 (cid:16)

ΔA𝑛 + 𝜇0J(1),𝑛 + ℎ𝑎11Δv𝑛
A

+ L−1
1

𝑐2

(cid:104)

(cid:17)(cid:105)

Δ𝜙𝑛−1 +

𝜌(1),𝑛−1
𝜖0

(cid:19)(cid:21)

+ ℎ𝑎11Δ𝑣𝑛−1

𝜙

(4.117)

𝑅𝑛∇ · A𝑛−1 + 𝑅𝑛+1∇ · A𝑛(cid:17)(cid:17)(cid:105)

= 𝑐2L−1
1

+ 𝑐2L−1
1

(cid:104)

Δ

(cid:20)

−

+ ℎ𝑎11𝑐2L−1
1

(cid:16)

−𝜙𝑛 + 𝜙𝑛−1 − ℎ𝑐2 (cid:16)
𝜌(1),𝑛−1
𝜌(1),𝑛
𝜖0
𝜖0
(cid:104)
𝜙 + 𝑣𝑛−1

−𝑣𝑛

Δ

+

(cid:16)

= 𝑐2L−1
1

(cid:2)Δ𝜖 𝑛
1

(cid:3) + 𝑐4𝜇0L−1

1

− ℎ𝑐2𝜇0
𝜙 − ℎ𝑐2 (cid:16)
(cid:2)(1 − 𝑐1) 𝜖 𝑛

(cid:16)

𝑅𝑛∇ · J(1),𝑛−1 + 𝑅𝑛+1∇ · J(1),𝑛(cid:17) (cid:21)

𝑅𝑛∇ · A𝑛

(cid:17)(cid:17)(cid:105)

A + 𝑅𝑛+1∇ · v𝑛−1
(cid:3) + ℎ𝑎11𝑐2L−1

A

1

(cid:3)

(cid:2)𝜖 𝑛
2

3 + 𝑐1𝜖 𝑛+1

3

149

The last step is justified definitionally and by Lemma 4.2.8. We have proven the base case, now we

assume true for 𝑖 = 𝑙 − 1 substeps. We consider 𝑖 = 𝑙 and see

− 𝑘 𝑣 𝜙,𝑛
𝑙

+ 𝑘 𝑣 𝜙,𝑛−1
𝑙

− ℎ𝑐2 (cid:16)

𝑅𝑠,𝑛∇ · kvA,𝑛−1

𝑙

+ 𝑅𝑠,𝑛+1∇ · kvA,𝑛

𝑙

(cid:17)

𝑣𝑛−1
𝜙 + ℎ

𝑙−1
(cid:213)

𝑗=1

𝑎𝑙 𝑗 𝑘 𝑣 𝜙,𝑛−1
𝑗

𝑎𝑙 𝑗 𝑘 𝜙,𝑛−2
𝑗

𝑣𝑛−2
𝜙 + ℎ

𝑎𝑙 𝑗 𝑘 𝑣 𝜙,𝑛−2
𝑗

𝑙−1
(cid:213)

𝑗=1

𝜙𝑛−1 + ℎ

𝑙−1
(cid:213)

𝑗=1

𝑎𝑙 𝑗 𝑘 𝜙,𝑛−1
𝑗

+ Δ (cid:169)
(cid:173)
(cid:171)

= −L−1

𝑙

𝜌(𝑙),𝑛−1
𝜖0







𝜌(𝑙),𝑛−2
𝜖0

𝑙−1
(cid:213)

𝑗=1

𝜙𝑛−2 + ℎ

+ Δ (cid:169)
(cid:173)
(cid:171)
𝜇0J(𝑙),𝑛−2 + Δ (cid:169)
(cid:173)
(cid:171)

+ L−1
𝑙







− ℎ𝑐2𝑅𝑠,𝑛∇ · L−1







− ℎ𝑐2𝑅𝑠,𝑛+1∇ · L−1

𝑙

𝑙








𝜇0J(𝑙),𝑛−1 + Δ (cid:169)
(cid:173)
(cid:171)

A𝑛−1 + ℎ

A𝑛−2 + ℎ

(cid:170)
(cid:174)
(cid:172)

+ ℎ𝑎𝑙𝑙𝑐2Δ (cid:169)
(cid:173)
(cid:171)

(cid:170)
(cid:174)
(cid:172)
𝑙−1
(cid:213)

+ ℎ𝑎𝑙𝑙𝑐2Δ (cid:169)
(cid:173)
(cid:171)
𝑎𝑙 𝑗 kA,𝑛−2
𝑗

(cid:170)
(cid:174)
(cid:172)
𝑎𝑙 𝑗 kA,𝑛−1
𝑗

𝑗=1

𝑙−1
(cid:213)

𝑗=1








(cid:170)
(cid:174)
(cid:172)


(cid:170)

(cid:174)


(cid:172)

𝑙−1
(cid:213)

𝑗=1

v𝑛−2
A + ℎ

+ ℎ𝑎𝑙𝑙𝑐2Δ (cid:169)
(cid:173)
(cid:171)

(cid:170)
(cid:174)
(cid:172)

+ ℎ𝑎𝑙𝑙𝑐2Δ (cid:169)
(cid:173)
(cid:171)

v𝑛−1
A + ℎ

𝑙−1
(cid:213)

𝑗=1

𝑎𝑙 𝑗 kvA,𝑛−1
𝑗



(cid:170)

(cid:174)


(cid:172)

𝑎𝑙 𝑗 kvA,𝑛−1
𝑗







(4.118)

(cid:170)
(cid:174)
(cid:172)

We can group all of this within the L−1

𝑙 operator and, taking the entire argument, we rearrange and

see

−

+

𝜌(𝑙),𝑛−1
𝜌(𝑙), 𝑛 − 1
𝜖0
𝜖0
−𝜙𝑛−1 + 𝜙𝑛−2 − ℎ𝑐2 (cid:16)
(cid:16)
𝑙−1
(cid:213)

(cid:16)

+ Δ

+ ℎΔ

− ℎ𝑐2𝜇0

(cid:16)

𝑅𝑠,𝑛∇ · J(𝑙),𝑛−2 + 𝑅𝑠,𝑛+1∇ · J𝑛−1(cid:17)

𝑅𝑠,𝑛∇ · A𝑛−2 + 𝑅𝑠,𝑛+1∇ · A𝑛−1(cid:17)(cid:17)

𝑎𝑙 𝑗

−𝑘 𝜙,𝑛−1
𝑗

+ 𝑘 𝜙,𝑛−2
𝑗

− ℎ𝑐2 (cid:16)

𝑅𝑠,𝑛∇ · kA,𝑛−2

𝑗

+ 𝑅𝑠,𝑛+1∇ · kA,𝑛−1

𝑗

(cid:17)(cid:17)

𝑗=1

+ ℎ𝑎𝑙𝑙𝑐2Δ

(cid:16)

−𝑣𝑛−1

𝜙 + 𝑣𝑛−2

𝜙 − ℎ𝑐2 (cid:16)

𝑅𝑠,𝑛∇ · v𝑛−1

A + 𝑅𝑠,𝑛+1∇ · v𝑛−1

A

(cid:17)(cid:17)

(4.119)

+ ℎ2𝑎𝑙𝑙𝑐2Δ

𝑙−1
(cid:213)

𝑗=1

(cid:16)

𝑎𝑙 𝑗

−𝑘 𝑣 𝜙,𝑛−1
𝑗

+ 𝑘 𝑣 𝜙,𝑛−2
𝑗

− ℎ𝑐2 (cid:16)

𝑅𝑠,𝑛∇ · kvA,𝑛−2

𝑗

+ 𝑅𝑠,𝑛+1∇ · kvA,𝑛−1

𝑗

(cid:17)(cid:17)

= 𝜇0𝑐2 (cid:16)

(1 − 𝑐𝑙) 𝜖 𝑛−1

3

+ 𝑐𝑙𝜖 𝑛
3

(cid:17)

+ Δ𝜖1 + ℎΔ

𝑙−1
(cid:213)

𝑗=1

𝛼𝑛−1
𝑙

+ ℎ𝑎𝑙𝑙𝑐2Δ𝜖 𝑛−1

2

+ ℎ2𝑎𝑙𝑙𝑐2Δ

𝑙−1
(cid:213)

𝑗=1

𝛽𝑛−1
𝑙

.

Here we have set 𝛽𝑙 as some linear combination of 𝜖 𝑛−1

1

, 𝜖 𝑛−1
2

, and 𝜖 𝑛−1

3

, justified by our inductive

hypothesis. 𝛼𝑛−1
𝑙
We have demonstrated 𝜖 𝑛

is likewise a linear combination of these residuals and is justified by Lemma 4.2.9.

5 is a linear combination of residuals of previous timesteps for any amount

of substeps.

QED

150

With these two lemmas in hand, we will prove the following lemmas further relating the residuals.

Lemma 4.2.11. The residual of the Lorenz gauge condition (4.106) at time level 𝑛 + 1 is a linear

combination of the residuals of the Lorenz gauge (4.106), the time derivative of the Lorenz gauge

(4.107), and the continuity equation (4.108).

Proof. Consider the Lorenz gauge at time 𝑛 + 1:

1 = −𝜙𝑛+1 + 𝜙𝑛 − 𝑐2ℎ
𝜖 𝑛+1

(cid:16)

𝑅𝑠,𝑛∇ · A𝑛 + 𝑅𝑠,𝑛+1∇ · A𝑛+1(cid:17)

(4.120)

We plug in our update equations (4.94) and (4.95):

𝜖 𝑛+1
1 = −

(cid:32)

𝜙𝑛 + ℎ

𝑠
(cid:213)

(cid:33)

(cid:32)

+

𝑏𝑖 𝑘 𝜙,𝑛
𝑖

𝜙𝑛−1 + ℎ

𝑠
(cid:213)

(cid:33)

𝑏𝑖 𝑘 𝜙,𝑛−1
𝑖

(cid:32)

− 𝑐2ℎ

𝑖=1
(cid:32)

𝑅𝑠,𝑛∇ ·

A𝑛−1 + ℎ

𝑖=1
(cid:33)

𝑏𝑖kA,𝑛−1
𝑖

𝑠
(cid:213)

𝑖=1

+ 𝑅𝑠,𝑛+1∇ ·

(cid:32)

A𝑛 + ℎ

(cid:33)(cid:33)

𝑏𝑖kA,𝑛
𝑖

𝑠
(cid:213)

𝑖=1

(4.121)

= −𝜙𝑛 + 𝜙𝑛−1 − 𝑐2ℎ

(cid:16)

𝑅𝑠,𝑛∇ · A𝑛−1 + 𝑅𝑠,𝑛+1∇ · A𝑛(cid:17)

+ ℎ

(cid:16)

𝑏𝑖

𝑠
(cid:213)

𝑖=1

−𝑘 𝜙,𝑛

𝑖 + 𝑘 𝜙,𝑛−1

𝑖

(cid:16)

− 𝑐2ℎ

𝑅𝑠,𝑛kA,𝑛−1
𝑖

+ 𝑅𝑠,𝑛+1∇ · kA,𝑛

𝑖

(cid:17)(cid:17)

The first term is by definition 𝜖 𝑛

1 , the second term is a linear combination of 𝜖 𝑛

1 , 𝜖 𝑛

2 , 𝜖 𝑛

3 , and 𝜖 𝑛+1
QED

3

by Lemma 4.2.9.

Lemma 4.2.12. The residual of the time derivative of the Lorenz gauge condition (4.107) at time

level 𝑛 + 1 is a linear combination of the residuals of the Lorenz gauge (4.106) at time level 𝑛, the

time derivative of the Lorenz gauge (4.107) at time level 𝑛, the continuity equation (4.108) at time

levels 𝑛 and 𝑛 + 1.

Proof. Consider the residual of the time derivative of the Lorenz gauge at time 𝑛 + 1:

2 = −𝑣𝑛+1
𝜖 𝑛+1

𝜙 + 𝑣𝑛

𝜙 − 𝑐2ℎ

(cid:16)

𝑅𝑠,𝑛∇ · v𝑛

A + 𝑅𝑠,𝑛+1∇ · v𝑛+1

A

(cid:17)

(4.122)

151

We plug in our update equations (4.96) and (4.97):

𝜖 𝑛+1
2 = −

(cid:32)

𝑠
(cid:213)

(cid:33)

(cid:32)

+

𝑏𝑖 𝑘 𝑣 𝜙,𝑛
𝑖

𝑣𝑛
𝜙 + ℎ

𝑣𝑛−1
𝜙 + ℎ

𝑠
(cid:213)

𝑏𝑖 𝑘 𝑣 𝜙,𝑛−1
𝑖

(cid:33)

(cid:32)

− 𝑐2ℎ

𝑖=1
(cid:32)

𝑅𝑠,𝑛∇ ·

𝑠
(cid:213)

𝑖=1
(cid:33)

𝑏𝑖kvA,𝑛−1
𝑖

+ 𝑅𝑠,𝑛+1∇ ·

v𝑛−1
A + ℎ

𝑖=1
𝑅𝑠,𝑛∇ · v𝑛−1

(cid:16)

A + 𝑅𝑠,𝑛+1∇ · v𝑛

A

(cid:17)

𝜙 − 𝑐2ℎ

𝜙 + 𝑣𝑛−1
= −𝑣𝑛
𝑠
(cid:213)

(cid:16)

+ ℎ

𝑏𝑖

𝑖=1

−𝑘 𝑣 𝜙,𝑛
𝑖

+ 𝑘 𝑣 𝜙,𝑛−1
𝑖

− 𝑐2ℎ

(cid:16)

𝑅𝑠,𝑛kvA,𝑛−1
𝑖

+ 𝑅𝑠,𝑛+1∇ · kvA,𝑛

𝑖

(cid:17)(cid:17)

.

(cid:32)

v𝑛
A + ℎ

𝑠
(cid:213)

𝑖=1

(cid:33)(cid:33)

𝑏𝑖kvA,𝑛
𝑖

(4.123)

The first term is by definition 𝜖 𝑛
Lemma 4.2.10. The same lemma shows 𝜖 𝑛

2 , the second term is a linear combination of 𝜖 𝑛

1 , 𝜖 𝑛

2 , 𝜖 𝑛

3 , 𝜖 𝑛+1

3

, and 𝜖 𝑛

4 by

4 is a linear combination of the other four residuals. QED

With Lemmas 4.2.11 and 4.2.12 at hand, we are now ready to prove the following theorem.

Theorem 4.2.10. Assuming the residuals of the continuity and gauge condition (4.106)-(4.108) are

zero for timestep 𝑛 = 0, the residuals of the Lorenz gauge condition (4.106)-(4.107) are zero for all

time steps 𝑛 > 0 if the residual of the continuity equation (4.108) is zero for all time 𝑛 > 0.

Proof. Assume the residuals are zero for the starting conditions. Additionally, assume the residual for
the continuity equation is zero for all time. By Lemma 4.2.11, we know 𝜖 𝑛+1
of 𝜖 𝑛

is a linear combination

. ie

1 , 𝜖 𝑛

2 , 𝜖 𝑛

3 , and 𝜖 𝑛+1

3

1

𝜖 𝑛+1
1 = L−1

𝑖

By our assumption, 𝜖 𝑘

3 = 0 ∀ 𝑘. Thus

(cid:2)𝐴𝜖 𝑛

1 + 𝐵𝜖 𝑛

2 + 𝐶𝜖 𝑛

3 + 𝐷𝜖 𝑛+1

3

𝜖 𝑛+1
1 = L−1

𝑖

(cid:2)𝐴𝜖 𝑛

1 + 𝐵𝜖 𝑛

2

(cid:3) .

(cid:3) .

(4.124)

(4.125)

2 = 0, then by induction we have 𝜖 𝑛+1

1 = 𝜖 0
Clearly, if 𝜖 0
Lemma 4.2.12 gives that 𝜖 𝑛+1
show that our assumptions imply 𝜖 𝑘

2

is a linear combination of 𝜖 𝑛
2 = 0 ∀ 𝑘.

1 = 0. Thus 𝜖 𝑘
1 , 𝜖 𝑛

1 = 0 ∀ 𝑘.
3 , and 𝜖 𝑛+1

2 , 𝜖 𝑛

3

. Similar logic will

If the residual of the continuity equation (4.108) is zero for all time, then the residuals for the

Lorenz gauge (4.106)-(4.107) are zero for all time.

QED

152

Remark 4.2.7. In this generalization we have proven only one direction of the “if and only if”

relationship indicated by the above schemes. However, for our purposes we are only interested in

satisfaction of the continuity equation implying satisfaction of the Lorenz gauge, and so this suffices

for now. The other direction will be proven for DIRK-2, indicating generalization to 𝑠 stages is

possible, but this will require more work.

Remark 4.2.8. Given this unidirectional relation, we were able to prove the relationship using only

a generic linear combination of residuals, we did not need any specifics as to the coefficients. To

prove the other direction, these details will be necessary.

Remark 4.2.9. For neither DIRK-2 nor the generalized DIRK-𝑠 methods have we established that

Gauss’s law for electricity is satisfied if the semi-discrete Lorenz gauage condition is satisfied. Stages

beyond the first involve compounded inverse linear operators that render such a property intractable

to prove. This is not to suggest that Runge-Kutta methods exclude this property, for example the

implicit midpoint rule gives this relationship. Further work needs to be done to design Runge-Kutta

methods that demonstrably have this property.

4.2.4.3 The 2-stage Diagonal Implicit Runge-Kutta Method

We can prove that satisfaction of the semi-discrete Lorenz gauge implies satisfaction of the

semi-discrete continuity equation for the 2-stage DIRK method (DIRK-2), using some of the tools

provided above. The DIRK-2 method for the wave equation is as follows:

𝜙𝑛+1 = 𝜙𝑛 + ℎ

𝑛+1 = A

A

𝑛 + ℎ

(cid:16)

𝑏1𝑘 𝜙
(cid:16)

1

+ 𝑏2𝑘 𝜙

2

(cid:17)

,

𝑏1kA

1 + 𝑏2kA
2

(cid:17)

.

(4.126)

(4.127)

153

The substep variables are

(cid:20)

(cid:20)

(cid:104)

(cid:104)

𝑘 𝜙
1 = L−1

1

𝑘 𝜙
2 = L−1

2

1 = L−1
kA

1

2 = L−1
kA

2

𝑣𝑛
𝜙 + ℎ𝑎11𝑐2

(cid:18)

Δ𝜙𝑛 +

(cid:19)(cid:21)

,

𝜌(1)
𝜖0

𝜙 + ℎ𝑎21𝑐2𝑘 𝑣 𝜙
𝑣𝑛
+ ℎ𝑎11𝑐2 (cid:16)

1

v

𝑛
A

ΔA

𝑛
A

v

+ ℎ𝑎21𝑐2kvA
1

(1)(cid:17)(cid:105)

,

𝑛 + 𝜇0J
+ ℎ𝑎22𝑐2 (cid:16)

(cid:18)

+ ℎ𝑎22𝑐2

Δ𝜙𝑛 + ℎ𝑎21Δ𝑘 𝜙

1

+

(cid:19)(cid:21)

,

𝜌(2)
𝜖0

ΔA

𝑛 + ℎ𝑎21ΔkA

1 + 𝜇0J

(2)(cid:17)(cid:105)

.

As with DIRK-𝑠, we have defined L𝑖 :=

I − 1
𝛼2
𝑖
We see the update statements for 𝜙 and A are

Δ

(cid:16)

(cid:17)

, 𝛼𝑖 := 1

ℎ𝑎𝑖𝑖𝑐 , 𝑣𝜙 :=

𝜕𝜙

𝜕𝑡 , and 𝑣A := 𝜕A
𝜕𝑡 .

,

(cid:19)(cid:21)

Δ𝜙𝑛 +

𝜌(1)
𝜖0
+ ℎ𝑎11Δ𝑣𝑛
𝜙

(cid:19)(cid:21)

,

(cid:20)

(cid:20)

(cid:20)

(cid:20)

𝑘 𝜙
1 = L−1

1

𝑘 𝑣 𝜙
1 = L−1

1

𝑘 𝜙
2 = L−1

2

𝑘 𝑣 𝜙
2 = L−1

2

Likewise, for 𝑣𝜙 and vA:

(cid:18)

𝑣𝑛
𝜙 + ℎ𝑎11𝑐2

(cid:18)

𝑐2

Δ𝜙𝑛 +

𝜌(1)
𝜖0
𝜙 + ℎ𝑎21𝑐2𝑘 𝜙
𝑣𝑛
(cid:18) 𝜌(2)
𝜖0

𝑐2

1

(cid:18)

+ ℎ𝑎22𝑐2

Δ𝜙𝑛 + ℎ𝑎21Δ𝑘 𝜙

+ Δ𝜙𝑛 + ℎ𝑎22𝑐2Δ𝑘 𝜙

1

(cid:16)

+ ℎ𝑎22Δ

(cid:19)(cid:21)

,

1

+

𝜌(2)
𝜖0
𝜙 + ℎ𝑎21𝑘 𝑣 𝜙
𝑣𝑛

1

(cid:17)(cid:19)(cid:21)

.

𝜙 = 𝑣𝑛
𝑣𝑛+1

𝜙 + ℎ

𝑛+1
A

v

= v

𝑛
A

+ ℎ

(cid:16)

1

𝑏1𝑘 𝑣 𝜙
(cid:16)

𝑏1kvA
1

+ 𝑏2𝑘 𝑣 𝜙

2

𝑣
+ 𝑏2k
A
2

(cid:17)

(cid:17)

,

.

These have substep variables:

(cid:20)

(cid:20)

𝑘 𝑣 𝜙
1 = L−1

1

𝑘 𝑣 𝜙
2 = L−1

2

(cid:18)

𝑐2

Δ𝜙𝑛 +

𝜌(1)
𝜖0

+ ℎ𝑎11Δ𝑣𝑛
𝜙

(cid:19)(cid:21)

,

𝑐2

(cid:18) 𝜌(2)
𝜖0

+ Δ𝜙𝑛 + ℎ𝑎22𝑐2Δ𝑘 𝜙

1

+ ℎ𝑎22Δ

(cid:16)

𝜙 + ℎ𝑎21𝑘 𝑣 𝜙
𝑣𝑛

1

(cid:17)(cid:19)(cid:21)

,

1 = L−1
kvA

1

2 = L−1
kvA

2

(cid:104)

𝑐2 (cid:16)

(cid:104)

𝑐2 (cid:16)

ΔA

𝑛 + 𝜇0J

(1) + ℎ𝑎11Δv

(cid:17)(cid:105)

,

𝑛
A

𝜇0J

(2) + ΔA

𝑛 + ℎ𝑎22𝑐2ΔkA

1 + ℎ𝑎22Δ

(cid:16)

𝑛
A

v

+ ℎ𝑎21kvA
1

(cid:17)(cid:17)(cid:105)

.

154

(4.128)

(4.129)

(4.130)

(4.131)

(4.132)

(4.133)

(4.134)

(4.135)

(4.136)

(4.137)

(4.138)

(4.139)

(4.140)

(4.141)

Now, we can do a similar update statement for the gauge and continuity equations:

𝜙𝑛+1 = 𝜙𝑛 − 𝑐2ℎ

(cid:16)

𝑏1∇ · A

𝜌𝑛+1 = 𝜌𝑛 − ℎ

(cid:16)

𝑏1∇ · J

(2)(cid:17)

(1) + 𝑏2∇ · A
(2)(cid:17)

(1) + 𝑏2∇ · J

.

,

(4.142)

(4.143)

Here we have defined {𝐴, 𝐽}(𝑖) as the 𝑖th linear interpolation between 𝑛 and 𝑛 + 1 given by the

Runge-Kutta method, ie

So it follows

𝑋 (𝑖) = (1 − 𝑐𝑖) 𝑋 𝑛 + 𝑐𝑖 𝑋 𝑛+1.

𝑏1𝑋 (1) + 𝑏2𝑋 (2) = (𝑏1 (1 − 𝑐1) + 𝑏2 (1 − 𝑐1)) 𝑋 𝑛 + (𝑏1𝑐1 + 𝑏2𝑐2) 𝑋 𝑛+1

≡ 𝑅𝑛 𝑋 𝑛 + 𝑅𝑛+1𝑋 𝑛+1.

(4.144)

(4.145)

4.2.4.4 Properties of DIRK-2

Having described the update method, we now wish to link the satisfaction of the DIRK-2

formulation of the Lorenz gauge condition (4.142) with the DIRK-2 continuity equation (4.143). To

do so, we define the following three residuals:

𝜖 𝑛+1
1

𝜖 𝑛+1
2

𝜖 𝑛+1
3

:= −𝜙𝑛+1 + 𝜙𝑛 − 𝑐2ℎ

(cid:16)

𝑅𝑛∇ · A

𝑛 + 𝑅𝑛+1∇ · A

𝑛+1(cid:17)

,

:= −𝑣𝑛+1

𝜙 + 𝑣𝑛

(cid:16)

𝑅𝑛∇ · v

𝑛
A

𝜙 − 𝑐2ℎ
(cid:16)

(cid:17)

.

+ 𝑅𝑛+1∇ · v
𝑛+1
A
𝑛+1(cid:17)

,

:= −𝜌𝑛+1 + 𝜌𝑛 − ℎ

𝑅𝑛∇ · J

𝑛 + 𝑅𝑛+1∇ · J

(4.146)

(4.147)

(4.148)

Additionally, we have the following identities, the first two established by Lemma 4.2.8 and the

155

last two by the base cases of Lemmas 4.2.9 and 4.2.10, respectively.

−

𝜌(1),𝑛
𝜖0

+

𝜌(1),𝑛−1
𝜖0

−

𝜌(2),𝑛
𝜖0

+

𝜌(2),𝑛−1
𝜖0

−𝑘 𝜙,𝑛
1

+ 𝑘 𝜙,𝑛−1
1

−𝑘 𝑣 𝜙,𝑛
1

+ 𝑘 𝑣 𝜙,𝑛−1
1

− 𝑐2ℎ𝜇0
= 𝜇0𝑐2 (cid:16)

(cid:16)

𝑅𝑛∇ · J

(1),𝑛−1 + 𝑅𝑛+1∇ · J

(1),𝑛(cid:17)

(1 − 𝑐1) 𝜖 𝑛

3 + 𝑐1𝜖 𝑛+1

3

(cid:17)

,

(cid:16)

𝑅𝑛∇ · J

(1 − 𝑐2) 𝜖 𝑛

− 𝑐2ℎ𝜇0
= 𝜇0𝑐2 (cid:16)
− ℎ𝑐2 (cid:16)
𝑅𝑛∇ · kA,𝑛−1
(cid:104)

1

(2),𝑛−1 + 𝑅𝑛+1∇ · J

(2),𝑛(cid:17)

3 + 𝑐2𝜖 𝑛+1

3

(cid:17)

,

+ 𝑅𝑛+1∇ · kA,𝑛

1

(cid:17)

= L−1
1
− ℎ𝑐2 (cid:16)

2 + ℎ𝑎11𝑐2Δ𝜖 𝑛
𝜖 𝑛
𝑅𝑛∇ · kvA

,𝑛−1

1

(cid:104)

= 𝑐2L−1
1

Δ𝜖 𝑛

1 + 𝑐2𝜇0

1 + ℎ𝑎11𝑐4𝜇0
,𝑛

+ 𝑅𝑛+1∇ · kvA
(cid:16)

1
(1 − 𝑐1) 𝜖 𝑛

(4.149)

(4.150)

(4.151)

(cid:17)(cid:105)

,

(1 − 𝑐1) 𝜖 𝑛

3 + 𝑐1𝜖 𝑛+1

3

(cid:16)

(cid:17)

3 + 𝑐1𝜖 𝑛+1

3

(cid:17)

+ ℎ𝑎11𝜖 𝑛
2

(cid:105)

.

(4.152)

With these in mind, we wish to prove the following lemmas.

Lemma 4.2.13. The residual of the Lorenz gauge condition (4.146) at time level 𝑛 + 1 is a linear

combination of the residuals of the Lorenz gauge (4.146) at time level 𝑛, the time derivative of the

Lorenz gauge condition (4.147) at time level 𝑛, and the continuity equation (4.148) at time levels 𝑛

and 𝑛 + 1.

Proof. Consider the residual of the Lorenz gauge (4.146) at time 𝑛 + 1.

1 = −𝜙𝑛+1 + 𝜙𝑛 − 𝑐2ℎ
𝜖 𝑛+1

(cid:16)

𝑅𝑛∇ · A

𝑛 + 𝑅𝑛+1∇ · A

𝑛+1(cid:17)

.

We plug in our update equations (4.126)-(4.127) and see

𝜖 𝑛+1
1 = −

(cid:16)

𝜙𝑛 + ℎ

(cid:16)

− 𝑐2ℎ

(cid:16)

𝑅𝑛∇ ·

𝑏1𝑘 𝜙,𝑛
(cid:16)

1

+ 𝑏2𝑘 𝜙,𝑛
(cid:16)

2

𝑛−1 + ℎ

A

(cid:17)(cid:17)

(cid:16)

+

𝜙𝑛−1 + ℎ

(cid:16)

𝑏1kA,𝑛−1

1

+ 𝑏2kA,𝑛−1

2

(cid:17)(cid:17)

𝑏1𝑘 𝜙,𝑛−1
(cid:17)(cid:17)

1

+ 𝑏2𝑘 𝜙,𝑛−1
(cid:16)

2

+ 𝑅𝑛+1∇ ·

𝑛 + ℎ

A

(cid:16)

𝑏1kA,𝑛

1

+ 𝑏2kA,𝑛

2

(cid:17)(cid:17)(cid:17)

(4.153)

We note that the first variable in each parenthetical, if combined, forms the definition of residual

(4.146) at time level 𝑛. We do so and then use the definition of the 𝑘 variables (4.128)-(4.131) to

156

expand into the following:

1 = −𝜙𝑛 + 𝜙𝑛−1 − 𝑐2ℎ
𝜖 𝑛+1

(cid:16)

𝑅𝑛∇ · A
(cid:18)

𝑛−1 + 𝑅𝑛+1∇ · A
(cid:19)(cid:21)

𝑛(cid:17)

𝑣𝑛
𝜙 + ℎ𝑎11𝑐2

Δ𝜙𝑛 +

𝜌(1)
𝜖0

𝜙 + ℎ𝑎21𝑐2𝑘 𝑣 𝜙,𝑛
𝑣𝑛
(cid:18)

1

𝑣𝑛−1
𝜙 + ℎ𝑎11𝑐2

Δ𝜙𝑛−1 +

(cid:19)(cid:21)

𝜌(1),𝑛−1
𝜖0

(cid:18)

+ ℎ𝑎22𝑐2

Δ𝜙𝑛 + ℎ𝑎21Δ𝑘 𝜙,𝑛

1

+

(cid:19)(cid:21)

𝜌(2)
𝜖0

(cid:20)

(cid:20)

− ℎ𝑏1L−1
1

− ℎ𝑏2L−1
2

(cid:20)

(cid:20)

+ ℎ𝑏1L−1
1

+ ℎ𝑏2L−1
2

𝜙 + ℎ𝑎21𝑐2𝑘 𝑣 𝜙,𝑛−1
𝑣𝑛−1

1

+ ℎ𝑎22𝑐2

(cid:18)

Δ𝜙𝑛−1 + ℎ𝑎21Δ𝑘 𝜙,𝑛−1

1

+

(cid:19)(cid:21)

𝜌(2),𝑛−1
𝜖0

𝑛−1
∇ · v
A

+ ℎ𝑎11𝑐2 (cid:16)

Δ

(cid:16)

∇ · A

𝑛−1(cid:17)

+ 𝜇0∇ · J

(1),𝑛−1(cid:17)(cid:105)

(4.154)

− ℎ2𝑐2𝑅𝑛𝑏1L−1
1

− ℎ2𝑐2𝑅𝑛𝑏2L−1
2

(cid:104)

(cid:104)

,𝑛−1

(cid:105)

𝑛−1
∇ · v
A
(cid:104)

ℎ𝑎22𝑐2 (cid:16)

+ ℎ𝑎21𝑐2∇ · kvA
1
𝑛−1(cid:17)
(cid:16)

∇ · A

Δ
+ ℎ𝑎11𝑐2 (cid:16)

− ℎ2𝑐2𝑅𝑛𝑏2L−1
2
(cid:104)

− ℎ2𝑐2𝑅𝑛+1𝑏1L−1
1

𝑛
∇ · v
A

(cid:16)

+ ℎ𝑎21Δ

(cid:17)

+ 𝜇0∇ · J

(2),𝑛−1(cid:17)(cid:105)

∇ · kA,𝑛−1
1
(1),𝑛(cid:17)(cid:105)

Δ (∇ · A

𝑛) + 𝜇0∇ · J

− ℎ2𝑐2𝑅𝑛+1𝑏2L−1
2

(cid:2)∇ · v
𝑛
A
(cid:104)
− ℎ2𝑐2𝑅𝑛+1𝑏2L−1
2

+ ℎ𝑎21𝑐2∇ · kvA
1
ℎ𝑎22𝑐2 (cid:16)

Δ (∇ · A

,𝑛

(cid:3)

𝑛) + ℎ𝑎21Δ

(cid:16)

∇ · kA,𝑛

1

(cid:17)

+ 𝜇0∇ · J

(2),𝑛(cid:17)(cid:105)

We already noted the first row is the definition of the first residual at time level 𝑛. For the rest

of the rows, a pattern emerges in which every other line1 all have operators and coefficients that

allow their components to be grouped together. Doing so, a further pattern emerges such that the 𝑖th

1Consider indented lines as belonging to the line above.

157

component of each has the same coefficient and may be grouped together. If we do so, we see

1 = −𝜙𝑛 + 𝜙𝑛−1 − 𝑐2ℎ
𝜖 𝑛+1

(cid:16)

𝑛(cid:17)

𝑛−1 + 𝑅𝑛+1∇ · A
𝑅𝑛∇ · A
(cid:16)

𝜙 − 𝑐2ℎ

𝑅𝑛∇ · v

𝑛−1
A

(cid:17)(cid:105)

𝑛
A

+ 𝑅𝑛+1v
(cid:16)

(cid:16)

𝜙 + 𝑣𝑛−1
(cid:104)

−Δ𝜙𝑛 + Δ𝜙𝑛−1 − 𝑐2ℎ

𝑅𝑛Δ

∇ · A

𝑛−1(cid:17)

+ 𝑅𝑛+1Δ (∇ · A

+ ℎ𝑏1L−1
1

(cid:104)

−𝑣𝑛

+ ℎ2𝑎11𝑐2𝑏1L−1
1

+ ℎ2𝑎11𝑐2𝑏1L−1
1

+ ℎ2𝑏2𝑎21𝑐2L−1
2

+ ℎ2𝑏2𝑎22𝑐2L−1
2

+ ℎ2𝑏2𝑎22𝑐2L−1
2

+ ℎ2𝑎22𝑐2𝑏2L−1
2

(cid:20)

(cid:104)

(cid:104)

(cid:104)

(cid:20)

(cid:17)(cid:105)

𝑛)
(1),𝑛(cid:17) (cid:21)

+

−

𝜌(1),𝑛
𝜌(1),𝑛−1
𝜖0
𝜖0
− ℎ𝑐2 (cid:16)
−𝑘 𝑣 𝜙,𝑛
+ 𝑘 𝑣 𝜙,𝑛−1
1
1
−Δ𝜙𝑛 + Δ𝜙𝑛−1 − ℎ𝑐2 (cid:16)

−Δ𝑘 𝜙,𝑛
1
𝜌(2),𝑛
𝜖0

−

+ Δ𝑘 𝜙,𝑛−1
1
𝜌(2),𝑛−1
𝜖0

+

− 𝑐2ℎ𝜇0

(cid:16)

𝑅𝑛∇ · J

(1),𝑛−1 + 𝑅𝑛+1∇ · J

,𝑛−1

+ 𝑅𝑛+1∇ · kvA

1

(cid:17)(cid:105)

,𝑛

(4.155)

𝑅𝑛∇ · kvA
(cid:16)

1

∇ · A
(cid:16)

𝑅𝑛Δ
− ℎ𝑐2 (cid:16)

𝑛−1(cid:17)

(cid:17)(cid:105)

𝑛)

+ 𝑅𝑛+1 (∇ · A
(cid:16)

+ 𝑅𝑛+1Δ

(cid:17)

𝑅𝑛Δ

∇ · kA,𝑛−1

1

− ℎ𝑐2𝜇0

(cid:16)

𝑅𝑛∇ · J

(2),𝑛−1 + 𝑅𝑛+1∇ · J

1

∇ · kA,𝑛
(2),𝑛(cid:17) (cid:21)

(cid:17)(cid:17)(cid:105)

.

Utilizing the residual definitions (4.146)-(4.148) and the identities (4.149)-(4.152), we see finally

1 = 𝜖 𝑛
𝜖 𝑛+1

1

+ ℎ𝑏1L−1
1

(cid:3)

(cid:2)𝜖 𝑛
2

+ ℎ2𝑎11𝑐2𝑏1L−1
1

(cid:3)

(cid:2)Δ𝜖 𝑛
1

+ ℎ2𝑎11𝑐2𝑏1𝑐2𝜇0L−1
1

(cid:2)(1 − 𝑐1)𝜖 𝑛

+ ℎ2𝑏2𝑎21𝑐2L−1
2

(cid:104)

𝑐2L−1
1

(cid:104)

Δ𝜖 𝑛

1 + 𝑐2𝜇0

(cid:3)

3 + 𝑐1𝜖 𝑛+1
(cid:16)

3
(1 − 𝑐1) 𝜖 𝑛

3 + 𝑐1𝜖 𝑛+1

3

(cid:17)

(cid:105) (cid:105)

+ ℎ𝑎11𝜖 𝑛
2

(4.156)

+ ℎ2𝑏2𝑎22𝑐2L−1
2

+ ℎ2𝑏2𝑎22𝑐2L−1
2

(cid:3)

(cid:2)Δ𝜖 𝑛
1
(cid:104)

ΔL−1
1

(cid:104)

+ ℎ2𝑏2𝑎22𝑐2𝜇0𝑐2L−1
2

2 + ℎ𝑎11𝑐2Δ𝜖 𝑛
𝜖 𝑛
3 + 𝑐2𝜖 𝑛+1

(cid:2)(1 − 𝑐2) 𝜖 𝑛

3

(cid:3) .

1 + ℎ𝑎11𝑐4𝜇0

(cid:16)

(1 − 𝑐1) 𝜖 𝑛

3 + 𝑐1𝜖 𝑛+1

3

(cid:17)(cid:105) (cid:105)

Thus we see the residual for the gauge condition at 𝑡 = 𝑛 + 1, 𝜖 𝑛+1

1

, is a linear combination of the

three residuals at current or previous timesteps.

QED

We now prove a similar lemma concerning the time derivative values update.

Lemma 4.2.14. The residual of the time derivative of the Lorenz gauge condition (4.147) at time

level 𝑛 + 1 is a linear combination of the residuals of the Lorenz gauge (4.146) at time level 𝑛, the

158

time derivative of the Lorenz gauge condition (4.147) at time level 𝑛, and the continuity equation

(4.148) at time levels 𝑛 and 𝑛 + 1.

Proof. Consider the residual of time derivative of the Lorenz gauge at time 𝑛 + 1.

2 = −𝑣𝑛+1
𝜖 𝑛+1

𝜙 + 𝑣𝑛

𝜙 − 𝑐2ℎ

(cid:16)

𝑅𝑛∇ · v

𝑛
A

+ 𝑅𝑛+1∇ · v

(cid:17)

𝑛+1
A

The components of the righthand side may be expanded using the update equations (4.136) and

(4.137). We can further expand using the definitions of the 𝑘 variables, and then by the logic of

Lemma 4.2.13 we will find

(1 − 𝑐1) 𝜖 𝑛

3 + 𝑐1𝜖 𝑛+1

3

(cid:17)(cid:105)

2 = 𝜖 𝑛
𝜖 𝑛+1

2

+ ℎ𝑏1𝑐2L−1
1

+ ℎ𝑏1𝑐2L−1
1

(cid:3)

(cid:2)Δ𝜖 𝑛
1
𝜇0𝑐2 (cid:16)
(cid:104)
(cid:2)Δ𝜖 𝑛
2

(cid:3)

(cid:3)

+ ℎ𝑏2𝑐2L−1
2

+ ℎ2𝑏1𝑎11𝑐2L−1
1
(cid:2)𝜖 𝑛
1
𝜇0𝑐2 (cid:16)
(cid:104)

+ ℎ𝑏2𝑐2L−1
2

+ ℎ2𝑏2𝑎22𝑐2L−1
2

Δ

(cid:104)

+ ℎ2𝑏2𝑎22𝑐2L−1
2

(cid:3)

(cid:2)Δ𝜖 𝑛
2
(cid:104)

(cid:17)(cid:105)

(1 − 𝑐2) 𝜖 𝑛
(cid:104)
(cid:16)

3 + 𝑐2𝜖 𝑛+1
2 + ℎ𝑎11𝑐2Δ𝜖 𝑛
𝜖 𝑛

L−1
1

3

1 + ℎ𝑎11𝑐4𝜇0

(4.157)

(cid:17)(cid:105) (cid:17)(cid:105)

(cid:16)

(1 − 𝑐1) 𝜖 𝑛

3 + 𝑐1𝜖 𝑛+1

3

+ ℎ3𝑏2𝑎22𝑎21𝑐2L−1
2

(cid:16)

𝑐2L−1
1

(cid:104)

Δ𝜖 𝑛

1 + 𝑐2𝜇0

(cid:16)

(1 − 𝑐1) 𝜖 𝑛

3 + 𝑐1𝜖 𝑛+1

3

(cid:17)

+ ℎ𝑎11𝜖 𝑛
2

(cid:105) (cid:17)(cid:105)

.

Δ

Thus we see the residual for the time derivative of the gauge condition at 𝑡 = 𝑛 + 1, 𝜖 𝑛+1

2

, is a

linear combination of the three residuals at current or previous timesteps.

QED

Having proven the above lemmas, we wish to connect the satisfaction of the gauge condition

(4.146)-(4.147) with the continuity equation (4.148).

Theorem 4.2.11. Assuming the residuals of the continuity equation and gauge condition (4.146) -

(4.148) are zero for timesteps 𝑛 ∈ {−1, 0}, the residual of the continuity equation (4.148) is zero for

all time 𝑛 > 0 if and only if the residuals of the Lorenz gauge condition (4.146) - (4.147) are zero

for all time steps 𝑛 > 0.

159

Proof. Assume 𝜖 𝑘

3 = 0 ∀ 𝑘 ∈ Z.

Then by Theorem 4.2.10 we know 𝜖 𝑘
Now we prove the other direction. Suppose 𝜖 𝑘

1 = 𝜖 𝑘

2 = 0 for all time.
1 = 𝜖 𝑘

2 = 0 ∀ 𝑘 ∈ Z. By Lemma 4.2.13 𝜖 𝑛+1

1

’s

update equation becomes

0 = ℎ2𝑏1𝑎11𝑐4𝜇0L−1
1
(cid:104)

+ ℎ2𝑏2𝑎22𝑐2L−1
2

(cid:2)(1 − 𝑐1)𝜖 𝑛
(cid:104)

3 + 𝑐1𝜖 𝑛+1
(cid:16)

3

(cid:3) + ℎ2𝑏2𝑎21𝑐6𝜇0L−1
2
(cid:17)(cid:105) (cid:105)

ΔL−1
1

ℎ𝑎11𝑐4𝜇0

(1 − 𝑐1) 𝜖 𝑛

3 + 𝑐1𝜖 𝑛+1

3

(cid:104)(cid:16)

(cid:104)

L−1
1

(1 − 𝑐1) 𝜖 𝑛

3 + 𝑐1𝜖 𝑛+1

3

(cid:17)(cid:105) (cid:105)

+ ℎ2𝑏2𝑎22𝑐2𝜇0𝑐2L−1
2

(cid:2)(1 − 𝑐2) 𝜖 𝑛

3 + 𝑐2𝜖 𝑛+1

3

(cid:3)

Likewise, by Lemma 4.2.14, we see 𝜖 𝑛+1

2

’s update equation is

0 = ℎ𝑏1𝑐4𝜇0L−1
1

(cid:2)(1 − 𝑐1) 𝜖 𝑛

3 + 𝑐1𝜖 𝑛+1

+ ℎ3𝑏2𝑎22𝑎11𝑐6𝜇0L−1
2

+ ℎ3𝑏2𝑎22𝑎21𝑐6𝜇0L−1
2

(cid:2)ΔL−1
1
(cid:2)ΔL−1
1

2

3

(cid:3) + ℎ𝑏2𝑐4𝜇0L−1
(cid:3) (cid:3)
3 + 𝑐1𝜖 𝑛+1
3 + 𝑐1𝜖 𝑛+1

(cid:2)(1 − 𝑐1) 𝜖 𝑛
(cid:2)(1 − 𝑐1) 𝜖 𝑛

(cid:3) (cid:3)

3

3

(4.158)

(cid:3)

(4.159)

(cid:2)(1 − 𝑐2) 𝜖 𝑛

3 + 𝑐2𝜖 𝑛+1

3

We set 𝜖 𝑛
of 𝜖 𝑛+1
3

3 = 0 given our assumption that it is zero at the start. Thus we have two linear combinations
≡ 𝜖3. They are as follows

ℎ2𝑏1𝑎11𝑐4𝜇0𝑐1L−1
1

[𝜖3] + ℎ2𝑏2𝑎21𝑐6𝜇0𝑐1L−1
2

[𝜖3](cid:3)
[𝜖3](cid:3) + ℎ2𝑏2𝑎22𝑐4𝜇0𝑐2L−1

(cid:2)L−1
1

2

[𝜖3] = 0

+ℎ3𝑏2𝑎22𝑎21𝑐6𝜇0𝑐1L−1
2

(cid:2)ΔL−1
1

ℎ𝑏1𝑐4𝜇0𝑐1L−1
1

[𝜖3] + ℎ𝑏2𝑐4𝜇0𝑐2L−1
2

[𝜖3]

+ℎ3𝑏2𝑎22𝑎11𝑐6𝜇0𝑐1L−1
2

(cid:2)ΔL−1
1

[𝜖3](cid:3) + ℎ3𝑏2𝑎22𝑎21𝑐6𝜇0𝑐1L−1

2

(cid:2)ΔL−1
1

[𝜖3](cid:3) = 0

If we apply the L1 and L2 operators and then divide out common factors, these become

(4.160)

(4.161)

𝑏1𝑎11𝜇0𝑐1L2 [𝜖3] + 𝑏2𝑎21𝑐2𝜇0𝑐1𝜖3 + ℎ𝑏2𝑎22𝑎21𝑐2𝜇0𝑐1Δ𝜖3 + 𝑏2𝑎22𝜇0𝑐2L1 [𝜖3] = 0,

(4.162)

𝑏1𝜇0𝑐1L2 [𝜖3] + 𝑏2𝜇0𝑐2L1 [𝜖3] + ℎ2𝑏2𝑎22𝑎11𝑐2𝜇0𝑐1Δ𝜖3 + ℎ2𝑏2𝑎22𝑎21𝑐2𝜇0𝑐1Δ𝜖3 = 0.

(4.163)

We define some coefficients for convenience, 𝐴1 := 𝑏1𝑎11𝜇0𝑐1, 𝐵1 := 𝑏2𝑎21𝑐2𝜇0𝑐1, 𝐶1 :=

ℎ𝑏2𝑎22𝑎21𝑐2𝜇0𝑐1, 𝐷1 := 𝑏2𝑎22𝜇0𝑐2, 𝐴2 := 𝑏1𝜇0𝑐1, 𝐵2 := 𝑏2𝜇0𝑐2, 𝐶2 := ℎ2𝑏2𝑎22𝑎11𝑐2𝜇0𝑐1,

160

𝐷2 := ℎ2𝑏2𝑎22𝑎21𝑐2𝜇0𝑐1. Using these, along with the definition of the L{1,2} operators and grouping

like terms, we get

where we have defined

(I + Γ1Δ) 𝜖3 = 0,

(I + Γ2Δ) 𝜖3 = 0,

Γ1(ℎ) := ( 𝐴1 + 𝐵1 + 𝐷1)−1 (cid:16)
Γ2(ℎ) := ( 𝐴2 + 𝐵2)−1 (cid:16)

ℎ𝐶1 − (ℎ𝑎22𝑐)2 𝐴1 − (ℎ𝑎11𝑐)2 𝐷1

(cid:17)

,

ℎ2𝐶2 + ℎ2𝐷2 − (ℎ𝑎11𝑐)2 𝐴2 − (ℎ𝑎22𝑐)2 𝐵2

(4.164)

(4.165)

(4.166)

(4.167)

(cid:17)

.

Note we have assumed 𝐴1 + 𝐵1 + 𝐷1 ≠ 0 and 𝐴2 + 𝐵2 ≠ 0. Were either of these zero, we

would have Laplace’s equation, and this under periodic boundary conditions only permits 𝜖3 = 0.

Additionally, under periodic boundary conditions, these have nontrivial solutions if Γ1, Γ2 are

positive, otherwise 𝜖3 = 0. The nontrivial solution of a Helmholtz equation 𝑢 + 𝑘 2𝜕𝑥𝑥𝑢 = 0 has

the form 𝑢 = 𝐴 cos

(cid:17)

(cid:16) 1
𝑘 𝑥

+ 𝐵 sin

(cid:17)

(cid:16) 1
𝑘 𝑥

, if 𝑘 = 𝐿

𝑛𝜋 (assuming a domain [0, 𝐿)). The derivatives with

respect to whatever spatial dimensions are all associated with the same Γ value, so we know that

the dimensions must all be the same. With this in mind, we are operating in a domain of length

𝐿𝑥 = 𝐿 𝑦 = · · · = 𝐿. Now, say Γ1 ≠ Γ2. This would imply 𝜖3 would potentially satisfy one of the

equations, but certainly would not satisfy the other. Thus the solutions we are looking for exist if
Γ1(ℎ) = Γ2(ℎ) = (cid:0) 𝐿
𝑛𝜋
they both take this value and must find the ℎ values such that this is the case.

, where 𝑛 ∈ Z. If either Γ values are not this, then 𝜖3 = 0. We thus assume

(cid:1) 2

Γ1(ℎ) = ( 𝐴1 + 𝐵1 + 𝐷1)−1 𝐶1ℎ − 𝑐2 ( 𝐴1 + 𝐵1 + 𝐷1)−1 (cid:16)

𝑎2
22

𝐴1 + 𝑎2
11

𝐷1

(cid:17)

ℎ2 =

(cid:19) 2

,

(cid:18) 𝐿
𝑛𝜋

Γ2(ℎ) = ( 𝐴2 + 𝐵2)−1 (cid:16)

𝐶2 + 𝐷2 − 𝑎2
11

𝑐2 𝐴2 − 𝑎2
22

𝑐2𝐵2

(cid:17)

ℎ2 =

(cid:19) 2

.

(cid:18) 𝐿
𝑛𝜋

(4.168)

(4.169)

The above expressions give rise to the following two quadratic polynomials: (4.168) yields

A1ℎ2 + B1ℎ + C1 = 0 and (4.169) yields A2ℎ2 + B2ℎ + C2 = 0. For there to be a nontrivial solution
to 𝜖3, the coefficients of these two identities must be identical. Note C1 = C2 = − (cid:0) 𝐿
𝑛𝜋

. We next

(cid:1) 2

consider B1 = B2, ie given B2 = 0, we need B1 = 0, ie

𝐶1
𝐴1 + 𝐵1 + 𝐷1

= 0.

161

(4.170)

This can only happen if 𝐶1 = 0, ie 𝑏2𝑎22𝑎21𝑐2𝜇0𝑐1 = 0. Given this is a DIRK method, we know the

diagonal values are nonzero, so the only way this is achieved is for 𝑎21 = 0.

If we set A1 = A2, we would find that SDIRK methods (methods for which all 𝑎𝑖𝑖 are the same

value) do not permit this to be the case, but that DIRK methods for which 𝑎21 = 0 and 𝑐1 < 𝑐2 allow

for these values to be the same, in other words, there are DIRK methods for which 𝜖3 may have a

nontrivial solution. However, to rule this out as a possibility, consider (4.170) and the difference

(or lack thereof) between Γ1 and Γ2. We again know Γ1 = Γ2, thus Γ1 − Γ2 = 0, and so we move

everything into one quadratic equation to solve:

(A1 + A2) ℎ2 − B1ℎ = 0

(4.171)

From (4.170), we know B2 is zero. This implies the only possible timestep is ℎ = 0, hardly a timestep

in which we are interested. Thus, there does not exist a nontrivial solution for 𝜖3 for ℎ > 0. QED

4.2.5 Conclusion

In Section 3.3.1 we proved that satisfaction of the semi-discrete Lorenz gauge implies satisfaction

of the semi-discrete continuity equation and vice versa, as well as satisfaction of the semi-discrete

Lorenz gauge implies satisfaction of Gauss’s Law. In this section we extended this theory to

encompass not only arbitrary BDF-k methods, but also for a family of second-order time centered

methods, and Adams-Bashforth methods. Additionally we have proven that satisfaction of the

semi-discrete continuity equation implies satisfaction of the semi-discrete Lorenz gauge for arbitrary

DIRK-s methods, showing that the converse holds as well for DIRK-2, implying the possibility for

the property holding for higher order DIRK as well. We now have the theory, we move on to see if

this will yield numerical fruit.

4.3 Numerical Results

This section contains the numerical results for our PIC method using several wave solvers

described above. We consider two different test problems with dynamic and steady state qualities.

We first consider the Weibel instability [280], which is a streaming instability that occurs in a periodic

domain. The second example is a simulation of a non-relativistic drifting cloud of electrons in

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a periodic domain. For each example, we compare the performance of the different methods by

inspecting the fully discrete Lorenz gauge condition and tracking its behavior as a function of time.

Our exploration concentrates on the BDF-1, BDF-2, CDF-2, and DIRK-2 methods, typically with a

mesh refinement of 128 × 128. The DIRK-2 method’s weights are those explored by Qin and Zhang

[290] and has the following Butcher tableau:

1/4 1/4

0

3/4 1/2 1/4

1/2 1/2

All methods use a spectral solve to invert the L operator and compute gradients spectrally as well.

We make use of quadratic particle weighting as in [286, 287, 291, 292, 293, 294] (see Appendix F
for more details on interpolants), and we compute J𝑛+1 using x𝑛+1
point and v𝑛

𝑖 and 𝑞𝑖 as the value that is interpolated.

as the location of the interpolant

𝑖

We conclude the section by summarizing the key results of the experiments.

4.3.1 Weibel Instability

In the preceeding chapter, Section 3.4.1, we considered the Weibel instability for the BDF-1

wave solver coupled with the IAEM. The full description of the problem and the setup can be found

there. For convenience we repeat the table in in Table 4.2.

Parameter
Average number density ( ¯𝑛) [m−3]
Average electron temperature ( ¯𝑇) [K]

Electron angular plasma period (𝜔−1

𝑝𝑒) [s/rad]

Electron skin depth (𝑐/𝜔 𝑝𝑒) [m]
Electron drift velocity in 𝑥 (𝑣⊥) [m/s]
Maximum electron velocity in 𝑦 (𝑣 (cid:107)) [m/s]

Value
1.0 × 1010
1.0 × 104
1.772688 × 10−7
5.314386 × 101
𝑐/2
𝑐/100

Table 4.2 Plasma parameters used in the simulation of the Weibel instability. All simulation particles
are prescribed a drift velocity corresponding to 𝑣⊥ in the 𝑥 direction while the 𝑦 component of their
velocities are sampled from a uniform distribution scaled to the interval [−𝑣 (cid:107), 𝑣 (cid:107)).

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Weibel Instability Magnetic Magnitude vs Angular Plasma Period, Mesh Resolution:
128 × 128

(a) Naive Update, Quadratic Weighting

(b) Charge Conserving Update, Quadratic Weighting

Figure 4.1 The growth rate of the magnetic magnitude under the four methods under consideration.
We note that both the naive update (left) and charge conserving update (right) follow the theoretical
linear growth rate well, though the naive exhibits nonphysical oscillations and is clearly not as
smooth. The naive update, unlike the charge conserving update, does not use the continuity equation
to update 𝜌, but rather maps both J and 𝜌 to the mesh using quadratic particle weighting. When
we enforce continuity, the oscillations observed in not only the growth period, but also the final
state, are removed for all methods except DIRK-2, which triggers small two-stream instabilities (see
Remark 4.3.3).

Weibel Instability Gauge Error vs Angular Plasma Period, Mesh Resolution: 128 × 128

(a) Naive Update, Quadratic Weighting

(b) Charge Conserving Update, Quadratic Weighting

Figure 4.2 The error in the Lorenz gauge condition for the Webeil Instability on a 128 × 128 grid.
On the left we scatter charge and current density in a naive manner, on the right we have the setup in
which we scatter current and update charge from the continuity equation. We see the error in the
gauge is drastically reduced when we enforce continuity.

As noted earlier, we make use of higher order particle weighting as we are using a spectral solver.

We see not only good alignment with the linear growth rate predicted by theory (Figures 4.1), but

we also see good preservation of the Gauge condition (Figure 4.2) and Gauss’s Law (Figure 4.3) in

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all cases other than DIRK-2, which as we have previously described in Remark 4.2.9 does not have

the latter property. In Figure 4.1 we show the growth rate of the magnetic magnitude as a function

of angular plasma period. Figure 4.1a is the growth rate when using the ‘‘naive” weight where both

charge density and current density are mapped to a mesh using the same weighting scheme on the

collocated mesh. Figure 4.1b is the result of making use of the conservative method proposed in

this paper. This shows the physical benefits quite nicely, as we see the oscillations in the growth rate

eliminated and the change in the magnetic magnitude over time become much smoother. Likewise

we exhibit the significant improvement in the Gauge condition and Gauss’s law by comparing the

naive weighting (Figures 4.2a and 4.3a, respectively) with the conserving scheme (Figures 4.2b and

4.3b, respectively).

Remark 4.3.1. BDF-1, BDF-2, and CDF-2 with Crank-Nicolson all have diffusion, and as such this

removes high frequencies, whereas the high frequencies are resolved by the Green’s function solver

used in Chapter 2.

Remark 4.3.2. DIRK-2 behaves dispersively, and as such high frequencies will remain when we use

a spectral solver.

Remark 4.3.3. We note that DIRK-2 exhibits oscillations towards the end that eventually saturate

(Figure 4.4). Due to its dispersive nature (Remark 4.3.2) it perturbs the cold direction (𝑣⊥) and

triggers small two stream instabilities. Higher order particle weighting reduces the effect but does

not eliminate it.

All methods, BDF-1, BDF-2, CDF-2, and DIRK-2, capture the Weibel instability, which is a

more challenging problem due to the care needed in setting up the problem, and additionally, once

we enforce the semi-discrete continuity equation, they exhibit excellent gauge error and agreement

with Gauss’s Law (again, except DIRK-2 for Gauss). The DIRK-2 method gives results that merit

some comment. Figure 4.4 shows the gauge error increasing in an oscillatory fashion at around the

thirtieth plasma period, eventually stabilizing around the end of the run. DIRK-2 has no diffusion

and cannot resolve high frequencies, as such it triggers nonphysical two stream instabilities in the

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cold direction (Remark 4.3.3). Refining the interpolation scheme mitigates this effect, but does not

entirely remove it. We see the downstream effects of this in the magnetic magnitude growth in the

quadratic scheme (Figure 4.4b), though this matches the linear growth rate theory just as well as the

other methods, towards the end the magnetic magnitude begin oscillating slightly.

There are a variety of ways we can approach this problem, the simplest of which would be adding

numerical diffusion. Additionally, moving to a fully finite element basis based on 𝐶1 elements or

a new version of the Green’s function solvers (eg [295]) would mitigate the issue, and will be the

subject of future work. We are interested in developing Runge-Kutta methods that have the property

of satisfying Gauss’s Law if they satisfy the Gauge condition, and this too shall be the subject of

future inquiry.

Weibel Instability Error in Gauss’s Law vs Angular Plasma Period, Mesh Resolution:
128 × 128

(a) Naive Update, Quadratic Weighting

(b) Charge Conserving Update, Quadratic Weighting

Figure 4.3 The error in Gauss’s Law on a 128 × 128 grid. On the left we have scatter charge and
current density in a naive manner, on the right we have the setup in which we scatter current and
update charge from the continuity equation. We see the error in Gauss’s law is drastically reduced
with the exception of DIRK-2 (see Remark 4.2.9).

4.3.2 Drifting Cloud of Electrons

In a plasma that is macroscopically neutral, errors in Gauss’s law may be mitigated, as an error

in one direction caused by a negative particle may very well be cancelled by a likewise error from a

positive particle nearby. In Section 3.4.2 we introduced the 2𝐷 − 2𝑉 problem of a drifting cloud of

electrons. A stationary group of ions are distributed in a Gaussian about the center of the periodic

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Weibel Instability, Charge Conserving DIRK-2, Mesh Resolution: 128 × 128

(a) Gauge Error vs Angular Plasma Period

(b) Magnetic Magnitude vs Angular Plasma Period

Figure 4.4 We refine interpolation schemes by increasing the order of spline basis functions for the
DIRK-2 method. We see above excellent satisfaction for the gauge condition (left) until around the
thirtieth plasma period, after which oscillations begin forming for around twenty plasma periods,
eventually themselves stabilizing. This is unlike BDF-1, BDF-2, or CDF-2. Increasing the order
of interpolation reduces this phenomenon, but does not remove it. This is a two-stream instability
triggered by the DIRK-2 method’s dispersive nature (see Remarks 4.3.2 and 4.3.3).

domain,

(cid:104)

− 𝐿 𝑥
2

, −

(cid:17)

𝐿 𝑦
2

×

(cid:17)

(cid:104) 𝐿 𝑥
2

, 𝐿 𝑦
2

, with a mobile group of electrons distributed in the exact same

manner. This group of electrons is given a drift of 𝑐/100 in the 𝑥 and 𝑦 direction, which is enough

for some of them to escape the potential well of the ions, while the bulk of the electrons falls back

in the well (see Figure 4.7 for a visualization). Over time they move into the more neutral spaces,

spreading apart as they go, and as they do there is ample opportunity for violations in Gauss’ law and

the Lorenz gauge condition. In this section we modify the dimensions slightly from those in Section

3.4.2, increasing the domain from [−.5, .5]2 to [−8, 8]2, where the units are in Debye lengths, to

increase the amount of space that is traversed and make the problem more sensitive to errors in

Gauss’s Law. We increase the length of the run of the simulation to 100 angular plasma periods. A

summary of the plasma parameters may be found in Table 4.3.

In Figure 4.5 we observe significant improvement in the gauge error when we compare the naive

scatter method to the method enforcing charge conservation. Likewise when we compare satisfaction

of Gauss’s Law in Figure 4.6 between the naive scatter and charge conserving methods we find many

orders of magnitude of improvement, confirming our theory.

In this section we examined two numerical experiments. First we simulated the well known

Weibel Instability, showing our numerical simulations have good agreement with the known analytic

167

Parameter
Average number density ( ¯𝑛) [m−3]
Average electron temperature ( ¯𝑇) [K]

Electron angular plasma period (𝜔−1

𝑝𝑒) [s/rad]

Electron skin depth (𝑐/𝜔 𝑝𝑒) [m]
Electron drift velocity in 𝑥 and 𝑦 [m/s]

Value
1.0 × 1013
1.0 × 105
5.6057 × 10−9
1.6806
𝑐/100

Table 4.3 Plasma parameters used in the simulation of the Moving Cloud of Electrons problem. All
simulation particles are prescribed a drift velocity corresponding to 𝑣𝑥 = 𝑣 𝑦 = 𝑐
100 with a thermal
velocity corresponding to a Maxwellian.

Moving Cloud Gauge Error vs Angular Plasma Period, Mesh Resolution: 128 × 128

(a) Naive Conserving Update, Quadratic Weighting

(b) Charge Conserving Update, Quadratic Weighting

Figure 4.5 The error in the Lorenz gauge for the Moving Cloud of Electrons problem. We see on the
left side relatively high errors in Gauss, whereas the right side shows significantly less.

solutions. We then simulated a non-relativistic cloud of electrons, an experiment particularly

sensitive to gauge errors. Both problems showed negligible variation in the semi-discrete Lorenz

gauge and significant improvement in Gauss’s Law. The results of both of these experiments are in

line with the theoretical results we presented in section 4.2.

4.4 Conclusion

In Chapter 3 we noted a number of ways of preserving the Lorenz gauge under the BDF-1

scheme. In particular, Section 3.2.1 we noted that if the time and spatial derivative was applied

consistently to the continuity equation and the wave equations, we not only satisfied the semi-discrete

Lorenz gauge condition if and only if we satisfied the semi-discrete continuity equation, but we

additionally satisfied Gauss’s Law if we satisfied the semi-discrete Lorenz gauge. In this chapter

we have extended this property to a variety of wave solvers, generalizing to BDF-𝑘 and extending

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Moving Cloud Error in Gauss’s Law vs Angular Plasma Period, Mesh Resolution: 128 × 128

(a) Naive Conserving Update, Quadratic Weighting

(b) Charge Conserving Update, Quadratic Weighting

Figure 4.6 The error in Gauss’s law for the Moving Cloud of Electrons problem. We see on the left
side relatively high errors in Gauss, whereas the right side shows significantly less. As previously
discussed, because the Runge-Kutta method is not a nested operator in time in a similar manner to
the CDF or BDF methods, so there is no straightforward way to enable DIRK methods to satisfy
Gauss’s Law, as we would need to apply a DIRK time difference to a time difference to get the
second time derivative. See Remark 4.2.9.

this property to a family of second order time-centered methods as well as arbitrary stage DIRK-𝑠

methods, though the DIRK-𝑠 methods lacked the desireable property of satisfying Gauss’s Law if

the semi-discrete continuity equation was satisfied. We then gave numerical evidence to confirm the

theory.

This is what was to be shown.

We now turn to future inquiries.

169

(a) 𝑡 = 0

(b) 𝑡 = 0.5

(c) 𝑡 = 1.0

(d) 𝑡 = 2.0

Figure 4.7 A Gaussian distribution of electrons and ions are placed on a periodic domain at 𝑡 = 0.0.
The electrons are given a drift velocity (see Table 4.3) in addition to their standard thermal Maxwellian
velocity. Unlike the previous work’s moving cloud of electrons problem in Section 3.4.2, only some
electrons escape the well (𝑡 = .5), the rest are pulled back in (𝑡 = 1.0), causing a unique pattern to
emerge as electrons are now travelling in opposing directions (𝑡 = 2.0).

170

CHAPTER 5

FUTURE WORK

The above work has seen the development of a PIC method that displays a number of properties.

Chapter 2 showcases solving the modified Helmholtz equation using a Green’s function, exhibiting

the ability to retain physical characteristics, a resilience to numerical heating, close adherence to

theory, and flexibility with regards to boundary conditions and geometries (though this component

remains underexplored). Chapters 3 and 4 take this method and show the circumstances under which

the Lorenz gauge and Gauss’s law will be satisfied, solving the modified Helmholtz equation using

either the FFT or sixth order finite difference method. This method, too, exhibits close adherence

to theory. This PIC method, though still in its infancy, has much potential, and in this chapter

we will briefly consider some future lines of inquiry to develop this promising method. First, in

Section 5.1, we will consider the addition of particle source and sink terms, observing how this

may change the theory linking the continuity equation with the Lorenz gauge. In Section 5.3 We

will then consider developing kernel based derivatives and how that may enable us to return to our

use of one dimensional Green’s functions against a dimensionally split spatial operator. In Section

5.4 we will explore extending this theory to a Finite Element framework. Finally, in Section 5.5,

we will consider adding a collision operator to our method. It should be emphasized that these are

thumbnail sketches attempting to show the feasibility of what is being proposed.

5.1 Source Terms

In Section 2.4 one of the numerical experiments considered was an expanding beam, in which

particles were injected into the domain on the left side of the box and removed upon contacting a

boundary. Similar to this, a sheath problem was also considered, which had no particles injected

but still removed particles upon contact with a boundary. Continuity is obviously violated here, as

charge is showing up out of nowhere. In other words, our continuity equation now has a source term

𝜕 𝜌
𝜕𝑡

+ ∇ · J = 𝜎.

(5.1)

How this impacts the theory developed by Chapters 3 and 4 remains to be explored. Under the

171

BDF-1 scheme, we see the discretization of the continuity equation becomes

𝜌𝑛+1 − 𝜌𝑛
Δ𝑡

+ ∇ · J𝑛+1 = 𝜎𝑛+1.

So the residuals would form something akin to

𝜖 𝑛+1
1 =

𝜖 𝑛+1
2 =

𝜙𝑛+1 − 𝜙𝑛
1
Δ𝑡
𝑐2
𝜌𝑛+1 − 𝜌𝑛
Δ𝑡

+ ∇ · A𝑛+1,

+ ∇ · J𝑛+1 − 𝜎𝑛+1.

We have the standard update equations

(cid:20)

𝜙𝑛+1 = L−1

∇ · A𝑛+1 = L−1 (cid:104)

1
𝛼2

2𝜙𝑛 − 𝜙𝑛−1 +

𝜌𝑛+1
𝜖0
2∇ · A𝑛 − ∇ · A𝑛−1 +

(cid:21)

,

𝜇0
𝛼2

(cid:105)

.

∇ · J

Considering the Lorenz gauge with this unknown source term

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

1
𝑐2

𝜙𝑛+1 − 𝜙𝑛
Δ𝑡

+ ∇ · A𝑛+1 =

1
𝛼2

(cid:21)

(cid:21)

𝜌𝑛+1
𝜖0
𝜌𝑛
1
𝜖0
𝛼2
𝜇0
∇ · J𝑛+1(cid:105)
𝛼2
(cid:21)

(cid:20)

2𝜙𝑛 − 𝜙𝑛−1 +

L−1

(cid:20)

2𝜙𝑛−1 − 𝜙𝑛−2 +

L−1

1
𝑐2Δ𝑡
1
𝑐2Δ𝑡
+ L−1 (cid:104)

−

2∇ · A𝑛 − ∇ · A𝑛−1 +
(cid:20) 1
𝜙𝑛 − 𝜙𝑛−1
Δ𝑡
𝑐2
(cid:20) 𝜌𝑛 − 𝜌𝑛−1
Δ𝑡
1 − 𝜖 𝑛−1

2𝜖 𝑛

+

+ ∇ · J𝑛

1

𝜇0
𝛼2

(cid:21)

= 2L−1

L−1

+

𝜇0
𝛼2
= L−1 (cid:104)

2 + 𝜎𝑛+1(cid:105)
𝜖 𝑛+1

+ ∇ · A𝑛

− L−1

(cid:20) 1
𝑐2

𝜙𝑛−1 − 𝜙𝑛−2
Δ𝑡

(cid:21)

+ ∇ · A𝑛−1

.

(5.7)

In other words, the residual of the Lorenz gauge at 𝑛 + 1 is equal to a linear combination of the

Lorenz gauge residuals from previous timesteps, the residual of the continuity equation at 𝑛 + 1,

and the source term of the continuity equation at 𝑛 + 1. Said differently, it appears as though the

Lorenz gauge itself now has a source term identical to that of the continuity equation. Modifying

the residual term, we arrive at

˜𝜖 𝑛+1
1 =

1
𝑐2

𝜙𝑛+1 − 𝜙𝑛
Δ𝑡

+ ∇ · A𝑛+1 + L−1 (cid:2)𝜎𝑛+1(cid:3) = 𝜖 𝑛+1

1 + L−1 (cid:2)𝜎𝑛+1(cid:3) .

(5.8)

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We now consider how the Lorenz gauge itself behaves with a source. Assuming 𝜖 0

1 = 0 (or

1 = L−1 (cid:2)𝜎0(cid:3)) and 𝜖 𝑛
˜𝜖 0

2 = 0 ∀ 𝑛, we consider
1 = L−1 (cid:104)
˜𝜖 𝑛+1
= L−1 (cid:104)

2𝜖 𝑛

1 − 𝜖 𝑛−1

1

2𝜖 𝑛

1 − 𝜖 𝑛−1

1

= L−1 (cid:2)2𝜖 𝑛

1 − 𝜖 𝑛−1

1

2 + 𝜎𝑛+1(cid:105)
𝜖 𝑛+1
(0) + 𝜎𝑛+1(cid:105)

+

𝜇0
𝛼2
𝜇0
𝛼2
+ 𝜎𝑛+1(cid:3) .

+

By assumption we know 𝜖 𝑛

1 = 𝜖 𝑛−1

1

= 0. So we see ˜𝜖 𝑛+1

1 = L−1 (cid:2)𝜎𝑛+1

1

(5.9)

(cid:3) =⇒ 𝜖 𝑛+1

1 = 0.

In short, there is good reason to suggest that the theory can absorb a source term in the continuity

equation. There are several practical next steps that follow. First, this theory itself is somewhat

speculative. What does it mean for the Lorenz gauge to have a source term, especially in argument of

the L−1 operator? The physical explanation of a source term for the Lorenz gauge is also something

of a vexed question and will need to be explored. This has not been experimentally tested, and

numerical experiments need to follow theory.

5.2 Rotated Grid

In Section 1.4.4, the difficulties of rotating a Yee grid while retaining a cartesian set of coordinates

were discussed. This difficulty is primarily due to the staggered nature of the mesh, and while

work has been done to mitigate these errors (eg [174, 179]), the amount of effort to work around

the Yee mesh while retaining a finite difference formulation is nontrivial. Yee’s requirement to

maintain a uniformity in its mesh makes it difficult to simulate a domain that is not in line with

the coordinates of the simulation reference frame (Figure 5.1). The MOLT method’s nonstaggered

nature, combined with its ability to have nonuniform lines of integration, enables it to handle these

sort of domains with greater ease (Figure 5.2). The MOLT method has already been demonstrated

to have remarkable geometric flexibility, showcased in elliptical domains in [265, 266, 267, 268],

and extended even to simulation of an A6 magnetron in [296].

Initial investigations into the ability of MOLT to simulate electromagnetic waves compared to

the Yee scheme are promising. A simple square domain, [𝐿, 𝐿] × [𝐿, 𝐿] with 𝐿 = 4 with PEC

boundaries is set up. A Gaussian source is injected into the magnetic field centered at

(cid:17)

(cid:16)

− 1
2

, 1
2

. The

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𝐸𝑥
𝐸𝑦

𝐵𝑧 ⊗

𝑦

𝑥

Figure 5.1 A 2D Yee grid in Transverse Electric (TE) mode. The domain itself is noted by the
inscribed square.

Parameter Value

𝜏
𝜎𝑥
𝜎𝑦
𝑡0
𝑥0
𝑦0
𝜃

1
1
20
1
20
6𝜏
− 1
2
− 1
2
𝜋
6

Table 5.1 Table of parameter values for the source function.

root source, S, is modified for both the Yee grid and the MOLT method to ensure the sources are

physically the same:

(cid:32)

(cid:32)

S(𝑥, 𝑦, 𝑡) = exp

−

(cid:16) 𝑡 − 𝑡0
𝜏

(cid:17) 2

+

(cid:19) 2

(cid:18) 𝑥 − 𝑥0
𝜎𝑥

+

(cid:18) 𝑦 − 𝑦0
𝜎𝑦

(cid:19) 2(cid:33)(cid:33)

( ˆx + ˆy + ˆz) ,

S𝑦𝑒𝑒 (𝑥, 𝑦, 𝑡) = ∇ × S,
𝜕S
𝜕𝑡

S𝑀𝑂 𝐿𝑇 (𝑥, 𝑦, 𝑡) =

.

174

(5.10)

(5.11)

(5.12)

𝜙, 𝐴1, 𝐴2
𝑥 boundaries
𝑦 boundaries

𝑦

𝑥

Figure 5.2 A 2D MOLT grid. The domain itself is noted by the inscribed square. The collocation of
the nodes allow for easier coding as well as more consistent boundaries applied. Additionally, with
a little extra coding, the staircases may be filled in (shown in the colored nodes, blue for 𝑥 lines, red
for 𝑦 lines.

The parameters for S may be found in Table 5.1. The values of the magnetic field and the vector

potentials at (0, 0) are recorded. This was done with the standard Yee grid as well as the first and

second order MOLT BDF methods.1

The aliasing associated with Yee [173] causes the electromagnetic structure of the rotated box to

quickly deviate from that of the unrotated,2 unlike MOLT, whose rotated box’s values closely track

1It was observed that the amplitudes of the waves from BDF1 and BDF2, especially BDF1, were not matching that
of Yee, given their dissipative nature. An alternative method discussed in [267, 268] that is purely dispersive (though
artificial dissipation may be added if one wishes) was also tested. This is the basic MOLT method with an averaging
parameter 𝛽 ∈ (0, 2]. The semidiscretized form of the wave equation takes the following form

1
𝑐2

𝑢𝑛+1 − 2𝑢𝑛 + 𝑢𝑛−1
Δ𝑡2

(cid:18)

𝑢𝑛 +

− Δ

𝑢𝑛+1 − 2𝑢𝑛 + 𝑢𝑛−1
𝛽2

(cid:19)

= 𝑆.

(5.13)

This is rearranged and solved using the standard MOLT wave solver with a Green’s function discussed in Section 2.2.
The 𝛽 parameter makes this solver purely dispersive, and this non-dissipative method matched the amplitude of Yee
at the start. As will be shown, the dissipative methods hold excellent agreement between their rotated and nonrotated
components, unlike the Yee grid. Not shown is the dispersive 𝛽 method holding equally well.

2This is a very naive implementation of the Yee mesh. If a node, that is, the bottom left corner of the Yee cell, was
detected as being within the domain, then the full cell was included, even if the individual components were not within
the domain. [173] does show some slight improvement in matching the modal content of the unroated mesh if a more

175

with that of the unrotated. Even 𝐵𝑧, whose values have to computed with numerical derivatives and

interpolation of the rotated grid’s values, has consistent values between rotated and unrotated grids.

It should be emphasized, this is for a naively staircased grid, the edges have not been filled out yet,

and we still see very close adherence to the nonrotated domain (Figures 5.6, 5.7, and 5.8). Shown

also are the fields themselves at 𝑡 = 25. The Yee grid (Figure 5.3) shows a completely different

field once the domain is rotated, while the MOLT BDF1 and BDF2 fields (Figures 5.4 and 5.5,

respectively) retain a close similarity between rotated and unrotated domains.

Yee computed 𝐵𝑧 field at 𝑡 = 25, Mesh Resolution: 256 × 256

(a) Unrotated Domain

(b) Rotated Domain

Figure 5.3 𝐵𝑧 field at 𝑡 = 25 as computed by the Yee method. The rotated domain is of a completely
different nature than that of the unrotated. The red dot indicates the location of the listener node,
whose values are recorded over time and displayed in Figure 5.6.

The next step for this process will be to incorporate macroparticles. The first stage is simple,

noting the success of the naively staircased structure above, we can do a staircased domain with

particles and observe the results. Once this is completed, the staircases will need to be filled

in. From the wave’s perspective this is simple and has already been accomplished by Causley,

careful selection of electric field points is made, but this improvement is rather small, around 2%. There are ways of
improving this significantly, such as shown in [180] (see Section 1.4.4 for more discussion), but the main point here is
that a naive implementation of both MOLT and Yee has very different results, the former retaining the structure of the
waves, the latter losing it.

176

MOLT BDF1 computed 𝐵𝑧 field at 𝑡 = 25, Mesh Resolution: 256 × 256

(a) Unrotated Domain

(b) Rotated Domain

Figure 5.4 𝐵𝑧 field at 𝑡 = 25 as computed by the MOLT BDF1 method. The rotated domain is
closely related to that of the unrotated. The red dot indicates the location of the listener node, whose
values are recorded over time and displayed in Figure 5.6.

Thavappiragasam, and others. The particles will require more thought. Given the unstructured

nature of the mesh near the edges, care will need to be made to interpolate the particles in such a

way that will preserve charge.

5.3 Kernel Based Derivatives

The ability to handle complex geometries is certainly attractive. However, if the work of this

thesis has hammered one point home, it is that preserving the involutions is important, and that the

Green’s function method, the one that is able to handle the nontrivial geometry described, does not

preserve the involutions. This is due to the nature of the derivative of the potentials, the derivative

of the continuity equation, and the Laplacian embedded in the modified Helmholtz operator, L.

Ultimately, if L is inverted using a Green’s function, then we must use a similar method to compute

the ∇ · J term in the continuity equation.

[295] has begun exploration into a kernel based method for computing derivatives of arbitrary

fields. In considering L = I − 1
𝛼2

𝜕2
𝜕𝑥2 , they define an additional operator, D := I − L−1, expanding

177

MOLT BDF2 computed 𝐵𝑧 field at 𝑡 = 25, Mesh Resolution: 256 × 256

(a) Unrotated Domain

(b) Rotated Domain

Figure 5.5 𝐵𝑧 field at 𝑡 = 25 as computed by the MOLT BDF2 method. The rotated domain is
closely related to that of the unrotated. The red dot indicates the location of the listener node, whose
values are recorded over time and displayed in Figure 5.6.

it into a Neumann series:3

1
𝛼2

𝜕2
𝜕𝑥2

= I − L = L

(cid:16)

L−1 − I

(cid:17)

= −L

(cid:16)

I − L−1(cid:17)

= −LD = − (I − D)−1 D = −

∞
(cid:213)

𝑝=1

D 𝑝.

(5.14)

We can do similar work to obtain a first order derivative operator, though it must be noted that these

will be directional in nature. We consider the operators L𝐿 := I − 1
𝛼

𝜕
𝜕𝑥 and L𝑅 := I + 1
𝛼

𝜕
𝜕𝑥 . Defining

D𝐿 := I − L−1

𝐿 and D𝑅 := I − R−1

𝑅 , we similarly find that the first order derivatives may be written

in terms of Neumann series:

1
𝛼

𝜕+
𝜕𝑥

1
𝛼

𝜕−
𝜕𝑥

= I − L𝐿 = L𝐿

= L𝑅 − I = L𝑅

(cid:16)

(cid:16)

L−1

𝐿 − I

I − L−1
𝐿

(cid:17)

(cid:17)

= −D𝐿 (I − D𝐿)−1 = −

∞
(cid:213)

𝑝=1

D

𝑝
𝐿 ,

= D𝑅 (I − D𝑅)−1 =

∞
(cid:213)

𝑝=1

D

𝑝
𝑅,

(5.15)

(5.16)

3A Neumann series considers a bounded linear operator T : 𝑋 → 𝑋, where 𝑋 is a Banach space. Assuming
holds. This
for |𝑥| < 1. For more details, see Chapters 1 and

(cid:107)T (cid:107) < 1 (with (cid:107)T (cid:107) = inf 𝛽 ∈ 𝐵, where 𝐵 = {𝛽|(cid:107)T [𝑥] ≤ 𝛽(cid:107)𝑥(cid:107) ∀ 𝑥 ∈ 𝑋 }), the identity (I − T ) −1 = (cid:205)∞
is the operator equivalent to the geometric series, (1 − 𝑥) −1 = (cid:205)∞
𝑛=0
2 of [297].

𝑘=0 T 𝑘

𝑥𝑛

178

Value of 𝐵𝑧 at Center of Domain, Mesh Resolution: 256 × 256

(a) Yee

(b) MOLT BDF1
Figure 5.6 Value of 𝐵𝑧 (0, 0) along time for Yee, MOLT BDF1, and MOLT BDF2. Both unrotated
and rotated grids values are tracked. Both should be the same value, and MOLT’s rotated grid is
much closer to that of its unrotated grid. This is despite the fact that the magnetic field must be
numerically derived, and that for the rotated grid this requires two interpolation steps resulting in a
much more inaccurate magnetic field than the Yee grid would produce. It is also worth mentioning
that the rotated Yee values are the order of magnitude of the waves themselves, unlike that of the
MOLT BDF1 and BDF2 methods.

(c) MOLT BDF2

These differential operators, D∗, are all defined in terms of identity operators and L−1

∗ operators.

The L−1

∗ operators are known.

L−1 [𝑣] (𝑥) =

𝛼

2

L−1

𝐿 [𝑣] (𝑥) = 𝛼

L−1

𝑅 [𝑣] (𝑥) = 𝛼

∫ 𝑏

𝑎
∫ 𝑏

𝑥
∫ 𝑥

𝑎

𝑒−𝛼|𝑥−𝑠|𝑣(𝑠)𝑑𝑠 + 𝐴𝑒−𝛼(𝑥−𝑎) + 𝐵𝑒−𝛼(𝑏−𝑥),

𝑒−𝛼(𝑠−𝑥)𝑣(𝑠)𝑑𝑠 + 𝐵𝐿𝑒−𝛼(𝑏−𝑥),

𝑒−𝛼(𝑥−𝑠)𝑣(𝑠)𝑑𝑠 + 𝐴𝑅𝑒−𝛼(𝑥−𝑎).

(5.17)

(5.18)

(5.19)

See Appendix G for derivation of L−1;4 L−1

𝐿 and L−1

𝑅 are similarly derived.

[295] has already had success in simulating the 1D wave equation, 1D heat equation, and others

using this method to approximate spatial derivative component of these equations to arbitrary

accuracy. They were also able to do so for nonuniform meshes. Given the nature of this derivative

and its direct relationship to the L operator, there is the promise of achieving the Gauge and Gauss’s

law accuracy displayed in chapters 3 and 4 while keeping the Green’s function solution paradigm.

Given the success of [295], the next step will be applying this operator to compute the ∇ · J

4It is worth noting that L = 1

2 (L𝐿 + L𝑅).

179

MOLT BDF1 𝐴1 and 𝐴2 Values at Center of Domain, Mesh Resolution: 256 × 256

(a) MOLT BDF1 𝐴1 Values

(b) MOLT BDF1 𝐴2 Values

Figure 5.7 Value of 𝐴1(0, 0) and 𝐴2(0, 0) along time for MOLT. The adherence of the rotated grid’s
values to those of the unrotated is much closer than the magnetic field, indicating that the errors in the
magnetic field between MOLT’s rotated and unrotated likely stems from the numerical derivatives
and two interpolation steps.

term in the continuity equation whilst in the context of the dimensionally split Green’s function

wave solver used in Chapter 2. If numerical experiments indicate success, theory will need to be

developed in order to understand how the relationship between these two operators is formed.

5.4 Finite Element Method Formulation

There are three main approaches to mesh based numerical solutions of PDEs, Finite Difference,

Finite Element, and Finite Volume. This thesis has been primarily interested in the first paradigm.

Finite Element, though more involved, offers more rigor when it comes to error bounds, and gives

even more flexibility with regards to geometry. The basic idea of the Finite Element Method (FEM) is

to decompose the domain into a set of nonuniform elements. A common element shape is a triangle

(2D) or tetrahedral (3D), though other shapes certainly exist. We can construct a finite element

method for our wave equations as follows. First, we discretize in space using a triangularization Tℎ

180

MOLT BDF2 𝐴1 and 𝐴2 Values at Center of Domain, Mesh Resolution: 256 × 256

(a) MOLT BDF2 𝐴1 Values

(b) MOLT BDF2 𝐴2 Values

Figure 5.8 Value of 𝐴1(0, 0) and 𝐴2(0, 0) along time for MOLT BDF2. The adherence of the rotated
grid’s values to those of the unrotated is much closer than the magnetic field, indicating that the
errors in the magnetic field between MOLT’s rotated and unrotated likely stems from the numerical
derivatives and two interpolation steps.

of the domain Ω such that Ω = (cid:205)𝑇 ∈Tℎ

𝑇. We declare the following function spaces:

ℎ := (cid:8)𝑣 ∈ 𝐶0(Ω) | 𝑣|𝑇 ∈ 𝑝𝑟 (𝑇)(cid:9) ⊆ 𝐻1(Ω),
𝑉 0
ℎ := (cid:8)𝑣 ∈ 𝐶1(Ω) | 𝑣|𝑇 ∈ 𝑝𝑟 (𝑇)(cid:9) ⊆ 𝐻2(Ω),
𝑉 1
ℎ := (cid:8)𝑣 ∈ 𝐶2(Ω) | 𝑣|𝑇 ∈ 𝑝𝑟 (𝑇)(cid:9) ⊆ 𝐻3(Ω).
𝑉 2

All are equipped with the following inner product

(𝑢, 𝑣)Tℎ :=

(cid:213)

𝑇 ∈Tℎ

(𝑢, 𝑣)𝑇 :=

∫

(cid:213)

𝑇 ∈Tℎ

𝑇

𝑢𝑣.

We wish to find 𝜙ℎ, 𝐴𝑖

ℎ ∈ 𝑉 0

ℎ , where Aℎ =

(cid:16)

ℎ, 𝐴𝑦
𝐴𝑥

ℎ, 𝐴𝑧

ℎ

(cid:17)𝑇

such that

1
𝑐2

(cid:18) 𝜕2𝜙ℎ
𝜕𝑡2
(cid:32) 𝜕2 𝐴𝑖
ℎ
𝜕𝑡2

1
𝑐2

Tℎ

(cid:33)

Tℎ

, 𝑍𝑖
ℎ

(cid:19)

, 𝜂ℎ

+ (∇𝜙ℎ, ∇𝜂ℎ)Tℎ =

(cid:19)

, 𝜂ℎ

(cid:18) 𝜌
𝜖0

Tℎ

ℎ ,
, ∀ 𝜂ℎ ∈ 𝑉 0

+ (cid:0)∇𝐴𝑖

ℎ, ∇𝑍𝑖
ℎ

(cid:1)

= (cid:0)𝜇0𝐽𝑖, 𝑍𝑖
ℎ

(cid:1)

Tℎ

Tℎ

, ∀ 𝑍𝑖

ℎ .
ℎ ∈ 𝑉 0

181

(5.20)

(5.21)

(5.22)

(5.23)

(5.24)

(5.25)

If we consider the Lorenz gauge condition, 𝐹ℎ := 1
𝑐2

𝜕𝜙ℎ
𝜕𝑡 + ∇ · Aℎ = 0 as being satisfied at the

start, we find that plugging it into the FEM formulation with test function 𝐺 ℎ ∈ 𝑉 0

ℎ of the wave

equation yields the following:

(cid:18) 1
𝑐2

𝜕2𝐹ℎ
𝜕𝑡2

(cid:19)

, 𝐺 ℎ

Tℎ

+ (∇𝐹ℎ, ∇𝐺 ℎ)Tℎ = · · ·

(cid:18) 1
𝑐2
(cid:18) 1
𝑐2
(cid:18) 1
𝑐2

+

+

𝜕
𝜕𝑡

𝜕
𝜕𝑡

𝜕2
𝜕𝑡2

(cid:20) 𝜕2
𝜕𝑡2

(cid:21)

(cid:19)

, 𝐺 ℎ

[𝜙ℎ]

Tℎ

(cid:19)

[∇𝜙ℎ] , ∇𝐺 ℎ

Tℎ
(cid:19)

[∇ · Aℎ] , 𝐺 ℎ

Tℎ

+ (∇∇ · Aℎ, ∇𝐺 ℎ)Tℎ

(5.26a)

(5.26b)

(5.26c)

(5.26d)

(5.26e)

(5.26b) combines with (5.26c) to acquire 1
𝑐2

(cid:16) 𝜕
𝜕𝑡

(cid:105)

(cid:104) 𝜌
𝜖0

(cid:17)

, 𝐺 ℎ

Tℎ

by our wave equation setup. This is

accomplished by simply pulling the time derivative out, which we can do given the inner product is

an integral over space, not time. We would like to do the same with (5.26d) and (5.26e), however,

(5.26e) is going to cause issues. We take (5.26e), decompose it, and using integration by parts we

observe:

(∇∇ · Aℎ, ∇𝐺 ℎ)Tℎ =

=

=

=

(∇∇ · Aℎ, ∇𝐺 ℎ)𝑇

[− (Δ∇ · Aℎ, 𝐺 ℎ)𝑇 + (cid:104)(∇∇ · Aℎ) · n, 𝐺 ℎ(cid:105)𝜕𝑇 ]

[− (∇ · ΔAℎ, 𝐺 ℎ)𝑇 + (cid:104)(∇∇ · Aℎ) · n, 𝐺 ℎ(cid:105)𝜕𝑇 ]

(5.27)

[(ΔAℎ, ∇𝐺 ℎ)𝑇 + (cid:104)(∇∇ · Aℎ) · n, 𝐺 ℎ(cid:105)𝜕𝑇 − (cid:104)ΔAℎ · n, 𝐺 ℎ(cid:105)𝜕𝑇 ] .

(cid:213)

𝑇 ∈Tℎ
(cid:213)

𝑇 ∈Tℎ
(cid:213)

𝑇 ∈Tℎ
(cid:213)

𝑇 ∈Tℎ

Were 𝐴ℎ ∈ 𝑉 2

ℎ , then we have continuity along the second spatial derivative. Summing over all

triangles would cancel each of these terms out. Low orders of smoothness prevent this from

happening. Therefore, as the Lorenz gauge evolves over time under this framing, we observe the

182

two highlighted terms emerge.

(cid:18) 1
𝑐2

𝜕2𝐹ℎ
𝜕𝑡2

(cid:19)

, 𝐺 ℎ

+ (∇𝐹ℎ, ∇𝐺 ℎ)Tℎ = · · ·

=

1
𝑐2

(cid:18)

∇ ·

(cid:32)(cid:18) 1
𝜕
𝜕𝑡
𝑐2
(cid:18) 𝜕2Aℎ
𝜕𝑡2

Tℎ
𝜕2𝜙ℎ
𝜕𝑡2

(cid:19)

, 𝐺 ℎ

Tℎ

(cid:33)

+ (∇𝜙ℎ, ∇𝐺 ℎ)Tℎ

(cid:19)

(cid:19)

, 𝐺 ℎ

Tℎ

+ (∇ · (∇Aℎ) , ∇𝐺 ℎ)𝑇

[(cid:104)(∇∇ · Aℎ) · n, 𝐺 ℎ(cid:105)𝜕𝑇 − (cid:104)ΔAℎ · n, 𝐺 ℎ(cid:105)𝜕𝑇 ] .

(cid:213)

𝑇 ∈Tℎ

+

+

(5.28a)

(5.28b)

(5.28c)

(5.28d)

The two lines, (5.28b) and (5.28c), reduce to

1
𝑐2

(cid:18) 𝜕
𝜕𝑡

(cid:21)

(cid:20) 𝜌
𝜖0

(cid:19)

, 𝐺 ℎ

Tℎ

+ (𝜇0∇ · J, 𝐺 ℎ)Tℎ = 𝜇0

(cid:18) 𝜕 𝜌
𝜕𝑡

+ ∇ · J, 𝐺 ℎ

(cid:19)

Tℎ

= 0.

(5.29)

From (5.28d) there is possibly a nonzero remainder in the case of 𝐴ℎ ∉ 𝑉 2

ℎ . In other words, if the

elements are not at least continuous up to the second derivative, the gauge will not be preserved.

What is to be done with this? The next steps are relatively straightforward. First, we could simply

implement 𝑉 2

ℎ elements. This is certainly tractable if we use a very simple mesh and geometry. Under

this assumption we can implement 𝑉 2

ℎ elements just to prove the theory plays out numerically. But

𝑉 2

ℎ elements are notoriously difficult to use in anything other than a very simple mesh, so assuming

this works, for this method to gain any practical use, we would need to consider 𝑉 1

ℎ elements. A few

approaches are considered.

We could consider constraints to the elements. It is important to note that it is not continuity per

se that is eliminating those remainder terms, if the sums themselves cancel without continuity that

works just as well. Additionally, There has also been good research done on boosting the smoothness

of the elements to higher orders. SIAC filtering [298, 299, 300] or the lifting method recently

introduced in [301]. If the elements are raised to higher levels of smoothness, then their derivatives

will match on the boundaries, eliminating the offending remainder.

Dr. Sining Gong has suggested an alternative approach. Instead of smoothing the elements,

what if we modified the J source term, projecting it onto 𝑉 2

ℎ in a particular way? This, too, will be

explored.

183

Clearly 𝑉 0

ℎ = 𝑉 2

ℎ + 𝑉 2,⊥

ℎ

, where 𝑉 2,⊥

ℎ

, (𝑣, 𝑢) = 0. We consider the bases of 𝑉 2

𝑉 2,⊥
ℎ and 𝑉 2,⊥
𝑖=𝑀+1. We consider the
ℎ
following matrices, 𝐵𝑁×𝑁 = (cid:0)𝜙𝑖, 𝜙 𝑗 (cid:1) and 𝐶𝑁×𝑁 = (cid:0)∇𝜙𝑖, ∇𝜙 𝑗 (cid:1). We decompose these into block

is the orthogonal complement to 𝑉 2
𝑖=1 and {𝜙𝑖}𝑁
as {𝜙𝑖}𝑀

ℎ

ℎ , that is, ∀ 𝑣 ∈ 𝑉 2

ℎ , 𝑢 ∈

diagonal matrices as follows

𝐵 = (cid:169)
(cid:173)
(cid:173)
(cid:171)

𝐵11 𝐵12

𝐵21 𝐵22

(cid:170)
(cid:174)
(cid:174)
(cid:172)

𝐶11 𝐶12
𝐶 = (cid:169)
(cid:173)
(cid:173)
𝐶21 𝐶22
(cid:171)

(cid:170)
(cid:174)
(cid:174)
(cid:172)

.

From now on we will drop the boldface for J and A given that we will be solving each of the

vector components individually. Having defined these, we consider a projection of 𝐽 onto 𝑉 2
𝐽 = (cid:205)𝑀
𝐿𝐽
𝑖 𝜙𝑖. Call the vector of coefficients L𝐽
𝑖=1

define a projection operator onto 𝑉 2 as P𝑉 2

ℎ . We

. We

ℎ

further define

˜𝐽 := P𝑉 2

ℎ

𝐽 +

(cid:32) (cid:32)

𝐶21

(cid:18) 1
𝑐2

𝐵11 + 𝐶11

(cid:19) −1(cid:33)

(cid:33)𝑇

L𝐽

(𝜙𝑖) 𝑁

𝑖=𝑀+1

.

(5.30)

It is worth noting that P𝑉 2

˜𝐽. It is also worth noting that we can annotate

which is equal to (cid:0) ˜𝐽, 𝑣(cid:1)

𝑖=1, where each index 𝑗 in L𝐽

ℎ

𝐽 = P𝑉 2
Tℎ, as 𝐵11L𝐽 (𝜙𝑖) 𝑀

ℎ

(cid:16)P𝑉 2

ℎ

(cid:17)

𝐽, 𝑣

,

Tℎ
corresponds to 𝑣 = 𝜙 𝑗 for

𝑣 ∈ 𝑉 2

ℎ . So if we set 𝑣 = 𝜙 𝑗 , then this value is the 𝑗th row of 𝐵11 multiplied against L𝐽
We are now interested in solving the following variational problem for 𝐴𝑖
ℎ ∈ 𝑉 2
ℎ :

ℎ, 𝐽𝑖

.

(cid:32) 𝜕2 𝐴𝑖
ℎ
𝜕𝑡2

1
𝑐2

(cid:33)

, 𝑤ℎ

Tℎ

+ (cid:0)∇𝐴𝑖

ℎ, ∇𝑤ℎ(cid:1)

= 𝜇0

(cid:0) ˜𝐽𝑖, 𝑤ℎ(cid:1)

Tℎ

Tℎ

ℎ .
, ∀ 𝑤ℎ ∈ 𝑉 0

(5.31)

From the above we know 𝑤 = 𝑣 + 𝑢, where 𝑣 ∈ 𝑉 2

ℎ and 𝑢 ∈ 𝑉 2,⊥

ℎ

. So we are really looking to satisfy

the following system:

(cid:32) 𝜕2 𝐴𝑖
ℎ
𝜕𝑡2

(cid:32) 𝜕2 𝐴𝑖
ℎ
𝜕𝑡2

1
𝑐2

1
𝑐2

(cid:33)

, 𝑣ℎ

Tℎ

(cid:33)

Tℎ

, 𝑢ℎ

𝑢ℎ + 𝑣ℎ = 𝑤ℎ.

+ (cid:0)∇𝐴𝑖

ℎ, ∇𝑣ℎ(cid:1)

= 𝜇0

(cid:0) ˜𝐽𝑖

ℎ, 𝑣ℎ(cid:1)

Tℎ

Tℎ

ℎ .
, ∀ 𝑣ℎ ∈ 𝑉 2

+ (cid:0)∇𝐴𝑖

ℎ, ∇𝑢ℎ(cid:1)

= 𝜇0

(cid:0) ˜𝐽𝑖

ℎ, 𝑢ℎ(cid:1)

Tℎ

, ∀ 𝑢ℎ ∈ 𝑉 2,⊥

ℎ

.

Tℎ

(5.32)

(5.33)

(5.34)

To be more specific, we are looking to solve (5.32) while satisfying (5.33). When we do so,

we by definition satisfy (5.31). First, as usual we semi-discretize this system in time. We are now

184

considering a BVP at some time level 𝑛. Given the plethora of indices already being annotated, we

do not notate the time level, assuming they are all at the same time level. (We can also without loss

of generality assume the source term ˜𝐽 contains the previous timesteps of 𝐴. With that bookkeeping

out of the way, we see

( 𝐴ℎ, 𝑢ℎ) =

(cid:32) 𝑀
(cid:213)

𝑖=0

(cid:33)

𝑎𝑖𝜙𝑖, 𝑢ℎ

,

(∇𝐴ℎ, ∇𝑢ℎ) =

(cid:32) 𝑀
(cid:213)

𝑖=0

Tℎ

(cid:33)

𝑎𝑖∇𝜙𝑖, ∇𝑢ℎ

.

Tℎ

(5.35)

(5.36)

Now, 𝑢ℎ may take any form. What if we had it take the form 𝜙1? Then clearly the results of

these two inner products would be (cid:205)𝑀

𝑖 𝜙𝑖𝑎𝑖𝜙1 or (cid:205)𝑀

𝑖 ∇𝜙𝑖𝑎𝑖∇𝜙1, respectively. And of course this

logic works for arbitary 𝜙 𝑗 . We can effectively anotate this as 𝐵11a and 𝐶11a, respectively, with the

𝑗th index corresponding to 𝑢ℎ taking the value of 𝜙 𝑗 . In other words, (5.32), when fully discretized,

takes the form

This, in turn, implies that a = 𝜇0

(cid:16) 1
𝑐2

𝐵11a + 𝐶11a = 𝜇0L𝐽 .

1
𝑐2
𝐵11 + 𝐶11(cid:17) −1

L𝐽

.

Now, let us consider (5.33). First, it is clear, given 𝐴𝑖

ℎ ∈ 𝑉 2

ℎ , that

consider

(cid:16)

∇𝐴𝑖

ℎ, ∇𝑢ℎ

(cid:17)

.

Tℎ

(5.37)

(cid:16)

(cid:17)

𝐴𝑖
ℎ, 𝑢ℎ

Tℎ

= 0. Now we must

Similar to the logic used above, we know (5.33) holds for all 𝑢ℎ, including 𝑢ℎ = 𝜙 𝑗 , though it is

worth noting here that this 𝜙 𝑗 is a member of 𝑉 2,⊥

ℎ

. The corresponding value of

(cid:16)

∇𝐴𝑖

ℎ, ∇𝜙 𝑗

(cid:17)

is then

(cid:16)(cid:205)𝑀
𝑖=1

∇

(cid:17)

𝑎𝑖𝜙𝑖

∇𝜙 𝑗 . We can effectively annotate this 𝐶21a. So it follows

(cid:0)∇𝐴𝑖

ℎ, ∇𝑣ℎ(cid:1)

Tℎ

(cid:32)

=

∇

(cid:32) 𝑀
(cid:213)

𝑖=0

(cid:33)

(cid:33)

𝑎𝑖𝜙𝑖

, ∇𝑣ℎ

(cid:32)

(cid:32)

=

∇

𝜇0

(cid:18) 1
𝑐2

𝐵11 + 𝐶11

Tℎ
(cid:19) −1

(cid:33)

(cid:33)

, ∇𝑣ℎ

(𝜙𝑖) 𝑀
𝑖=1

Tℎ

(5.38)

= 𝜇0𝐶21

(cid:19) −1

𝐵11 + 𝐶11

(cid:18) 1
𝑐2

L𝐽 (𝜙𝑖) 𝑁

𝑖=𝑀+1

.

As we can see this is exactly what our source term ˜𝐽 is when we only consider the 𝑉 2,⊥

ℎ

test

functions. This proves that, with a little tweaking of the source term, we can satisfy the wave equation

185

using 𝑉 0

ℎ elements, ie

(cid:0)𝐴𝑖

ℎ, 𝑣ℎ(cid:1)

Tℎ

+ (cid:0)∇𝐴𝑖

ℎ, ∇𝑣ℎ(cid:1)

Tℎ

= 𝜇0

(cid:0) ˜𝐽𝑖

ℎ, 𝑣ℎ(cid:1)

Tℎ

ℎ .
, ∀ 𝑣ℎ ∈ 𝑉 0

(5.39)

This eliminates the spurious terms, (5.28d).

This has yet to be implemented in any code form, but provides a promising alternative to the

above listed approaches to the problem.

5.5 Collisions

Collisions were briefly mentioned towards the end of Section 1.4.2. Our PIC method has so far

only been interested in collisionless plasmas, as these are more straightforward, not to mentioned

computationally cheaper, than collisional plasmas. There are a number of ways to approach this

problem. First, it is worth noting how we can model collisions. [302] gives a good overview of the

problem. We consider this by first adding a collision operator to the right hand side of the Vlasov

equation:

𝜕 𝑓𝛼
𝜕𝑡

+ v · ∇𝑥 𝑓𝛼 + F · ∇𝑣 𝑓𝛼 =

𝑎𝛼,𝛽Q (cid:0) 𝑓𝛼, 𝑓𝛽(cid:1)

(cid:213)

𝛽

(5.40)

Here 𝑎𝛼,𝛽 = 1

𝜖 represents the collision frequency between two species of elections, with 𝜖 being
the Knudsen number, the ratio of the mean free path and the domain length. Sending 𝑎𝛼,𝛽 → 0 ∀ 𝛼, 𝛽

approaches the Vlasov equation. Q is the collision operator, in this case the Landau collision operator

(see Section 1.4.2 for more details).

Q ( 𝑓𝛼, 𝑓𝛽) = ∇𝑣 ·

∫

R3

𝐴(v − v∗) 𝑓𝛽 (v∗)∇𝑣 𝑓𝛼 (v) − 𝑓𝛼 (v)∇𝑣∗

𝑓𝛽 (v∗)𝑑v∗.

(5.41)

[7] derives both integral operators in Chapter 21. [302] also gives a good discussion on both

from a more numerical perspective. Standard practice is to define 𝑓 := 𝑓 (v) and 𝑓∗ := 𝑓 (v∗), so we

will use this from here on out. 𝐴 is the collision kernel, defined by

𝐴(z) := 𝐶 (cid:107)z(cid:107)𝛾 (cid:16)

(cid:107)z(cid:107)2𝐼𝑑 − z ⊗ z

(cid:17)

≡ (cid:107)z(cid:107)𝛾+2Π(z).

(5.42)

𝐶 is a constant (typically unity). Π(z) is the projection matrix onto {z}⊥. 𝐼𝑑 is the 𝑑 dimensional

identity matrix. −𝑑 ≤ 𝛾 ≤ 1 dictates the type of potential governing the system, hard potentials

186

for 𝛾 > 0, Maxwellian molecules for 𝛾 = 0, and 𝛾 < 0 corresponding to soft potentials. 𝛾 = −3

corresponds to Coulomb interactions.

Carrillo et al [303] developed a collision kernel for the Landau-Fokker-Plank equation, first for

the homogenous Landau equation, then generalizing this to the Vlasov-Landau equation in [304].

These works conserve mass, momentum, and energy as well, with the added bonus of increasing a

regularized entropy. The former’s results have been replicated (Figure 5.9), the latter’s will be as

well, which will provide a good platform from which to launch.

In the former, they consider the spatially homogenous system, meaning only the time derivative

on the lefthand side remains (homogeneity implies uniform spatial distribution, so ∇𝑥 𝑓 = 0, and

homogeneity also implies no internal forces, so F · ∇𝑣 𝑓 = 0). The Landau collision operator is

added to the righthand side:

𝜕 𝑓
𝜕𝑡

= Q ( 𝑓 , 𝑓 ) = ∇𝑣 ·

∫

R𝑑

𝐴(v − v∗) 𝑓 𝑓∗

(cid:0)∇𝑣 log ( 𝑓 ) − ∇𝑣∗ log ( 𝑓∗)(cid:1) 𝑑v∗.

(5.43)

The Q term may be rewritten in terms of the variation of the entropy functional:

Q ( 𝑓 , 𝑓 ) = ∇𝑣 ·

(cid:20)∫

(cid:18)

R𝑑

𝐴(v − v∗)

(cid:18)

∇𝑣

𝛿𝐸
𝛿 𝑓

− ∇𝑣∗

(cid:19)

𝛿𝐸∗
𝛿 𝑓∗

(cid:19)

(cid:21)

𝑓

.

𝑓∗𝑑v∗

This 𝐸 and

𝛿𝐸
𝛿 𝑓 correspond to the entropy functional and its variational derivative:

𝐸 :=

∫

R𝑑

𝑓 log( 𝑓 )𝑑v,

=⇒

𝛿𝐸
𝛿 𝑓

= log( 𝑓 ).

(5.44)

(5.45)

(5.46)

They regularize the entropy functional and its variation by defining a mollifier function, 𝜓𝜀:5

𝐸𝜀 ( 𝑓 ) :=

∫

R𝑑

( 𝑓 ∗ 𝜓𝜀) log ( 𝑓 ∗ 𝜓𝜀)𝑑v,

=⇒

𝛿𝐸𝜀
𝛿 𝑓

= 𝜓𝜀 ∗ log ( 𝑓 ∗ 𝜓𝜀) ,

𝐹𝜀 ( 𝑓 ) := ∇

𝛿𝐸𝜀
𝛿 𝑓

,

𝜓𝜀 (v) :=

1
(2𝜋𝜀)

𝑑
2

exp

(cid:18)

−

(cid:107)v(cid:107)2
2𝜀

(cid:19)

.

(5.47)

(5.48)

(5.49)

(5.50)

5The 𝐹𝜀 term is a novelty introduced here to make a future step clearer.

187

This leaves us with the homogenous Landau equation defined by the regularized entropy functional

𝜕 𝑓
𝜕𝑡

= Q𝜀 ( 𝑓 , 𝑓 ) := ∇𝑣 ·

(cid:20)∫

(cid:18)

R𝑑

𝐴(v − v∗)

(cid:18)

∇𝑣

𝛿𝐸𝜀
𝛿 𝑓

− ∇𝑣∗

(cid:19)

𝛿𝐸𝜀,∗
𝛿 𝑓∗

(cid:19)

(cid:21)

𝑓

𝑓∗𝑑v∗

≡ −∇𝑣 · (𝑈𝜀 ( 𝑓 ) 𝑓 ) .

(5.51)

When we take the gradient of

𝛿𝐸 𝜀
𝛿 𝑓 , we can immediately see the advantage of convolving 𝑓 with 𝜓𝜀.

Exploiting the convolution property

𝑑
𝑑𝑥 ( 𝑓 (𝑥) ∗ 𝑔(𝑥)) =

𝑑𝑓
𝑑𝑥 ∗ 𝑔, we see 𝐹𝜀 = ∇𝜓𝜀 ∗ log( 𝑓 ∗ 𝜓𝜀).

Recalling that we are assuming homogeneity in space, we let the particle distribution function

for 𝑁 particles each with associated velocities at time 𝑡, 𝑣𝑖 and associated weights 𝑤𝑖 assume the

following form

𝑓 𝑁 (𝑡, v) =

𝑁
(cid:213)

𝑖=1

𝑤𝑖𝛿(v − v𝑖 (𝑡)).

If we assume this form, we get the following

𝜕 𝑓 𝑁
𝜕𝑡

=

=

=

𝑁
(cid:213)

𝑖=0
𝑁
(cid:213)

𝑖=0
𝑁
(cid:213)

𝑖=0

𝑤𝑖

𝜕
𝜕𝑡

𝛿 (v − v𝑖 (𝑡))

(cid:18)

𝑤𝑖

∇v𝑖 𝛿 (v − v𝑖 (𝑡)) ·

(cid:19)

𝑑v𝑖
𝑑𝑡

(cid:18)

𝑤𝑖 (−∇v𝛿 (v − v𝑖 (𝑡))) ·

(cid:19)

𝑑v𝑖
𝑑𝑡

= −∇v ·

𝑁
(cid:213)

(cid:18)

𝑖=0

𝑤𝑖𝛿 (v − v𝑖 (𝑡))

(cid:19)

𝑑v𝑖
𝑑𝑡

= −∇v ·

(cid:16)
𝑈𝜀

(cid:16)

𝑓 𝑁 (cid:17)

𝑓 𝑁 (cid:17)

(5.52)

(5.53)

The second step is justified from a property of the Dirac delta function,

𝜕𝑏 𝛿(𝑎 − 𝑏).
The last step is from (5.51), where we define 𝑈𝜀. Now, before we take another step, it is important

𝜕𝑎 𝛿(𝑎 − 𝑏) = − 𝜕

𝜕

to note another property, that 𝑓 (𝛿(𝑥 − 𝑦1))𝛿(𝑥 − 𝑦2) = 0 for 𝑦1 ≠ 𝑦2, assuming 𝑓 (0) = 0. The

reasoning is simple, if 𝑥 = 𝑦1, then 𝑓 (𝑦1) may be nonzero, but 𝛿(𝑥 − 𝑦2) = 0. Similarly, for 𝑥 = 𝑦2,

the 𝑓 (0) resolves to 0. We see 𝑈𝜀 ( 𝑓 ) (𝑡, v𝑖) resolves to zero if v𝑖 = 0 due to the collision kernel 𝐴,

188

and so passing in 𝛿(v − v𝑖) will have a similar effect. Now, by this logic,

(cid:16)

𝑓 𝑁 (cid:17)

𝑈𝜀

𝑓 = 𝑈𝜀

(cid:32) 𝑁
(cid:213)

𝑖=0

𝑤𝑖𝛿(v − v𝑖 (𝑡))

(cid:33) 𝑁
(cid:213)

𝑤𝑖𝛿 (v − v𝑖 (𝑡))

= 𝛿𝑖 𝑗𝑈𝜀

(cid:32) 𝑁
(cid:213)

𝑖=0

𝑗=0
(cid:33) 𝑁
(cid:213)

𝑤𝑖𝛿(v − v𝑖 (𝑡))

𝑗=0

𝑤𝑖𝛿 (v − v𝑖 (𝑡))

(5.54)

=

𝑁
(cid:213)

𝑖=0

𝑈𝜀 (𝑤𝑖𝛿 (v − v𝑖)) 𝑤𝑖𝛿 (v − v𝑖) .

Combining (5.53) and (5.54), we observe

−∇v ·

𝑁
(cid:213)

(cid:18)

𝑖=0

𝑤𝑖𝛿 (v − v𝑖 (𝑡))

(cid:19)

𝑑v𝑖
𝑑𝑡

= −∇v ·

𝑁
(cid:213)

𝑖=0

𝑈𝜀 (𝑤𝑖𝛿 (v − v𝑖)) 𝑤𝑖𝛿 (v − v𝑖) .

(5.55)

This implies

𝑑v𝑖
𝑑𝑡

= 𝑈𝜀

(cid:16)

𝑓 𝑁 (cid:17)

(𝑡, v𝑖).

−∇v ·

(cid:16)
𝑈𝜀

(cid:16)

𝑓 𝑁 (cid:17)

𝑓 𝑁 (cid:17)

= −∇v ·

(cid:32)
𝑈𝜀

(cid:16)

𝑓 𝑁 (cid:17)

𝑁
(cid:213)

𝑖=0

(cid:33)

𝑤𝑖𝛿 (v − v𝑖 (𝑡))

Setting (5.53) equal to (5.57), we see

𝑁
(cid:213)

(cid:18)

− ∇v ·

𝑤𝑖𝛿(v − v𝑖 (𝑡))

𝑖=0
(cid:16)

= 𝑈𝜀

𝑓 𝑁 (cid:17)

=⇒

𝑑v𝑖
𝑑𝑡

(𝑡, v𝑖 (𝑡)).

(cid:19)

𝑑v𝑖
𝑑𝑡

= −∇v ·

(cid:32)
𝑈𝜀

(cid:16)

𝑓 𝑁 (cid:17)

𝑁
(cid:213)

𝑖=0

(cid:33)

𝑤𝑖𝛿(v − v𝑖 (𝑡))

,

This gives us our velocity update equation

𝑑v𝑖
𝑑𝑡

= 𝑈𝜀 ( 𝑓 𝑁 ) (𝑡, v𝑖 (𝑡)) = −

𝑁
(cid:213)

𝑗

𝑤 𝑗 𝐴(v𝑖 − v 𝑗 ) (cid:2)𝐹 𝑁

𝜀 (v𝑖) − 𝐹 𝑁

𝜀 (v 𝑗 )(cid:3)

The convolution of 𝑓 (v) and 𝜓𝜀 (v) is (cid:205)𝑘 𝑤 𝑘 𝜓𝜀 (v − v𝑘 ). Knowing this, we write

𝐹 𝑁
𝜀 (v) =

∫

R𝑑

∇𝜓𝜀 (v − v∗) log

(cid:33)

𝑤 𝑘 𝜓𝜀 (v∗ − v𝑘 )

𝑑v∗

(cid:32) 𝑁
(cid:213)

𝑘

We can convert all of this to a discrete version. We see

(5.56)

(5.57)

(5.58)

(5.59)

(5.60)

𝑑 ¯v𝑖
𝑑𝑡

= ¯𝑈𝜀 ( ¯𝑓 𝑁 ) (𝑡, ¯𝑣𝑖) := −

𝑁
(cid:213)

𝑗

𝑤 𝑗 𝐴 (cid:0) ¯v𝑖 − ¯v 𝑗 (cid:1) (cid:2) ¯𝐹 𝑁

𝜀 ( ¯v𝑖) − ¯𝐹 𝑁
𝜀

(cid:0) ¯v 𝑗 (cid:1)(cid:3) .

(5.61)

189

The discrete versions of 𝐸 𝑁

𝜀 and 𝐹 𝑁

𝜀 are shown to be

¯𝐹𝜀 ( 𝑓 ) (v𝑖) :=

¯𝐸𝜀 ( 𝑓 ) (v𝑖) :=

(cid:34)

(cid:34)

(cid:213)

𝑙

(cid:213)

𝑙

ℎ𝑑∇𝜓𝜀 (cid:0) ¯v𝑖 − v𝑐
𝑙

(cid:1) log

(cid:32)

(cid:213)

(cid:33)(cid:35)

𝑤 𝑘 𝜓𝜀 (cid:0)v𝑐

𝑙 − ¯v𝑘 (cid:1)

(cid:32)

(cid:213)

ℎ𝑑

𝑖

𝑤𝑖𝜓𝜀 (cid:0)v𝑐

𝑙 − ¯v𝑖(cid:1)

𝑘
(cid:33)

log

(cid:32)

(cid:213)

𝑘

𝑤 𝑘 𝜓𝜀 (cid:0)v𝑐

𝑙 − ¯v𝑘 (cid:1)

(cid:33)(cid:35)

.

(5.62)

(5.63)

Here, we have defined v𝑐

𝑙 to be the velocity cell centers and ℎ to be the spacing between the centers.

The summation comes from computing the integral according to the trapezoid rule.

𝑑v𝑖
𝑑𝑡 may be

computed using any difference scheme, this paper uses first order backward difference.

(a) Fractional Error, (cid:107) 𝑓𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 − 𝑓𝑒𝑥𝑎𝑐𝑡 (cid:107)2/(cid:107) 𝑓𝑒𝑥𝑎𝑐𝑡 (cid:107)2

(b) Total Energy Over Time

Figure 5.9 The fractional error between 𝑓𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 and 𝑓𝑒𝑥𝑎𝑐𝑡 refined over velocity space (left) and
the total energy over time refined over time (right). This matches the results of [303], indicating our
foundation is secure and the real work may begin.

A number of desirable properties are proven for the continuous case, with some of these properties

transferring to the discrete. Of particular note is the preservation of the physical properties such as

positivity, conservation of mass, momentum, and energy, and entropy decay.

Work has been done to replicate the results of this paper, which includes four sample problems,

two of which have analytic solutions to compare to. We chose the first one, which gives the following

190

collision kernel and exact solution:

𝐴 (z) =

𝑓𝑒𝑥𝑎𝑐𝑡 (𝑡, v) =

(cid:17)

,

(cid:16)

1
16
1
2𝜋𝐾 exp

(cid:107)z(cid:107)2𝐼𝑑 − z ⊗ z
(cid:18) −(cid:107)v(cid:107)2
2𝐾

(cid:19) (cid:18) 2𝐾 − 1
𝐾

+

1 − 𝐾
2𝐾 2

(cid:19)

(cid:107)v(cid:107)2

, 𝐾 := 1 −

1
2

exp

(cid:16)

−

𝑡

(cid:17)

8

.

(5.64)

(5.65)

Having obtained our numerical update to our velocities, we obtain the numerical solution

𝑓𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 (𝑡, v) =

(cid:16)

𝜓𝜀 ∗ ¯𝑓 𝑁 (𝑡, v)

(cid:17)

=

𝑁
(cid:213)

𝑖=1

𝑤𝑖𝜓𝜀 (𝑣 − ¯𝑣𝑖 (𝑡)).

(5.66)

The comparisons of the numerical distribution with the exact are displayed in Figure 5.9. We

notice that the norm is bound, though certainly not nontrivial. More importantly is the observation

that the total energy hardly varies throughout the simulation. This is in direct agreement with the

observations obtained by [303] (a comparison between the plots will show they are identical).

The next steps will be to extend this work to relativistic settings, as the current operator is

accurate only for nonrelativistic settings. The ability to handle relativity is one of the benefits of our

method brings to the table, so we need to develop a collision operator that can match. Discussions

with Dr. Hu are already underway as of April 2025 to begin this work.

The first step will be to replace the standard nonrelativistic kernel with the relativistic kernel

described in [305, 306, 307].6 Once this is done, we will apply the same exact initial conditions

given by [303] in their test problems, ensuring it reduces to the nonrelativistic. Given there are fewer

extant relativistic collision test problems, we may need to extend one or more of these test problems

to the relativistic setting.

Once this is done, we will use the more recent work of [304] where they extended to heterogeneous

settings. Throughout all of this, it is worth noting that runtime becomes nontrivial. The computation

of the macroparticles already composes the lion’s share of the compute resources necessary in a
PIC code. Implementing a collision operator installs an O (cid:0)𝑁 2(cid:1) operation, where 𝑁 is the number

of particles. This is extremely expensive. Work has been done to bring collision operators down

to O (𝑁 log (𝑁)), with Carrillo et al drawing special attention to the Fourier spectral methods

6All have a different way of formulating this kernel. Presumably they all reduce to the same form, but I have not

tested this.

191

in particular given they can exploit the Landau operator’s convolution components. Pareschi et

al [308] considered a spectral solve, which they then combined with a splitting technique [309]

which Zhang and Gamba [310, 311] built upon. They split the Fokker-Planck-Landau equation into

two subproblems, the Vlasov-Poisson (They simplify Vlasov-Maxwell to Vlasov-Poisson) and the

homogenous Landau problem via time-splitting. This, too, may be an avenue of advance.

5.6 Conclusion

Einstein once said, “As our circle of knowledge expands, so does the circumference of darkness

surrounding it.” And indeed, we see that having established a new PIC method that preserves

involutions has only led to more questions surrounding it. There are a number of exciting lines of

inquiry that this method has unlocked, only a few of which I have discussed in this chapter. We may

consider how our theory extends to include source and sink terms, develop kernel based methods

of taking derivatives that would enable the Green’s function method to satisfy the involutions,

extend our theory to a Finite Element formulation to push the geometric capabilities even further, or

implement collision operators to better capture the physical phenomena of a system. Suffice to say,

there is much to explore.

192

CHAPTER 6

CONCLUSION

In this dissertation we have covered a short sequence of projects concerning the development of a

new Particle-in-Cell method. First, in Chapter 2 we hammered together Causley et al’s the MOLT

method of solving waves together with the Asymmetrical Euler Method particle pusher of Gibbon

et al, modifying the pusher in order to better preserve physical properties. This forging resulted in a

new PIC method that was shown to have a number of desirable properties. This method features

an unconditionally stable wave solver, a particle pusher that displays volume preserving properties,

features a mesh is collocated, is geometrically flexible, is able to handle a naive bilinear interpolation

scheme while preserving physical properties of the system, is resilient to numerical heating, and

refines at a rather fast rate.

In Chapter 3 we noted a lacuna within this method, that the Lorenz gauge and Gauss’s laws

had nontrivial errors that, while bound, were still concerning. We considered how to solve this

problem, first developing theory to show that satisfaction of the continuity equation implied not

only satisfaction of the Lorenz gauge (and vice versa), but that satisfaction of the Lorenz gauge in

turn implied satisfaction of Gauss’s laws. We additionally developed a gauge correction method

that would manually enforce the Lorenz gauge. To do this we ceased solving the boundary value

problems resulting from the method of lines transpose using a Green’s function and instead employed

either the Fast Fourier Transform or sixth order Finite Difference method.

In Chapter 4 we generalized the theory developed in Chapter 3 from the first order Backward

Difference Formula to all Backward Difference Formulas, a family of time centered second order

methods, and all Adams-Bashforth methods. All 𝑠-stage diagonally implicit Runge-Kutta methods

were shown to satisfy the Lorenz gauge if they satisfied the continuity equation, with the two stage

method being proven to hold the if and only if relationship, suggesting the possibility of extending

this to arbitrary 𝑠 stage methods.

Finally, in Chapter 5 we considered a number of possible lines of inquiry this work could lead

to. These inquiries included adding source terms to the continuity equation while maintaining the

193

Lorenz gauge and Gauss’s laws, developing kernel based methods for computing derivatives in order

to implement the Green’s function method described in Chapter 2 while keeping the theory explored

in Chapters 3 and 4, extending from a Finite Difference formulation to a Finite Element formulation,

and adding collision operators.

There is a quote attributed to the painter John Ruskin, “If you can paint one leaf you can paint

the world.” Certainly the contributions of this thesis compared with the ever widening domain of

science are roughly proportional to a leaf compared with the vast complexity of the nature it inhabits.

But it is my hope that this study tells us something not only about the simulation of charged particles

in electromagnetic waves, but about this truly wonderful cosmos we inhabit.

Sit finis libri non finis quaerendi.

194

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Boltzmann and Landau Transport Equations”. PhD thesis. ”University of Texas at Austin”,
2014.

[311] Chenglong Zhang and Irene M. Gamba. “A conservative scheme for Vlasov Poisson Landau
modeling collisional plasmas”. In: Journal of Computational Physics 340 (2017), pp. 470–
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[312] Allen Taflove and Morris E. Brodwin. “Numerical Solution of Steady-State Electromag-
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387-95366-3.

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Electricity and Magnetism. Nottingham: Printed for the author by T. Wheelhouse, 1828.

221

APPENDIX A

NONDIMENSIONALIZATION OF EQUATIONS

We nondimensionalize four sets of equations. The second may be found in Chapter 1 of [34], the

last two may be found in the appendix of [2] as well as [34], and the first is no less in debt to Bill

Sand’s patient instruction in nondimensionalization.1

The setup for the non-dimensionalization used in all the following sections considers the following

substitutions:

x → 𝐿 ˜x,

v → 𝑉 ˜v ≡

𝑡 → 𝑇 ˜𝑡,
𝐿
𝑇 ˜v, P → 𝑃 ˜P ≡

𝑀 𝐿
𝑇

˜P,

𝜙 → 𝜙0 ˜𝜙, A → 𝐴0 ˜A,

𝑛 → ¯𝑛 ˜𝑛,

𝑞𝑖 → 𝑄 ˜𝑞𝑖, 𝑚𝑖 → 𝑀 ˜𝑚𝑖,

𝜌 → 𝑄 ¯𝑛 ˜𝜌,

J → 𝑄 ¯𝑛𝑉 ˜J ≡

𝑄 ¯𝑛𝐿
𝑇

˜J,

E → 𝐸0 ˜E, B → 𝐵0 ˜B.






Here, we use ¯𝑛 to denote a reference number density [m−3], 𝑄 is the scale for charge in [C],

and we also introduce 𝑀, which represents the scale for mass [kg]. The values for 𝑄 and 𝑀 are

set according to the electrons, so that 𝑄 = |𝑞𝑒 | and 𝑀 = 𝑚𝑒. For reasons that will become clear in

Section A.2, we choose the scales for the potentials 𝜙0 and 𝐴0 to be

𝜙0 =

𝐿2𝑀
𝑇 2𝑄

,

𝐴0 =

𝐿𝑀
𝑇𝑄

.

A natural choice of the scales for 𝐿 and 𝑇 are the Debye length and angular plasma period, which

are defined, respectively, by

𝐿 = 𝜆𝐷 =

(cid:115)

𝜖0𝑘 𝐵 ¯𝑇
¯𝑛𝑞2
𝑒

[m],

𝑇 = 𝜔−1

𝑝𝑒 =

(cid:114) 𝑚𝑒𝜖0
¯𝑛𝑞2
𝑒

[s/rad],

where 𝑘 𝐵 is the Boltzmann constant, 𝑚𝑒 is the electron mass, 𝑞𝑒 is the electron charge, and ¯𝑇 is

an average macroscopic temperature for the plasma. We choose to select these scales for all test

1And honesty compels me to admit probably copied from a whiteboard tutorial by him.

222

problems considered in section 2.4.2 with the exception of the beam problems, in which the length

scale 𝐿 corresponds to the longest side of the simulation domain and 𝑇 is the crossing time for a

particle that is injected into the domain. In most cases, the user will need to provide a macroscopic

temperature ¯𝑇 [K] in addition to the reference number density ¯𝑛 to compute 𝜆𝐷 and 𝜔−1

𝑝𝑒. Note that

the plasma period can be obtained from the angular plasma period 𝑇 after multiplying the latter by

2𝜋.

A.1 Maxwell’s Equations

We have Maxwell’s equations:

,

∇ · E =

𝜌
𝜖0
∇ · B = 0,

∇ × E = −

𝜕B
𝜕𝑡

,

∇ × B = 𝜇0J +

1
𝑐2

𝜕E
𝜕𝑡

.

(A.1a)

(A.1b)

(A.1c)

(A.1d)

Where E is the electric field, B is the magnetic field, J is the current density, 𝜌 is the charge

density, 𝜖0 and 𝜇0 are the vacuum permittivity and vacuum permeability, respectively.

We have the position vector x, the velocity vector v, time 𝑡, number density 𝑛, and charge 𝑞.

Now we get

𝐸0
𝐿
𝐵0
𝐿
𝐸0
𝐿
𝐵0
𝐿

˜𝜌,

˜∇ · ˜E =

𝑄 ¯𝑛
𝜖0
˜∇ · ˜B = 0,

˜∇ × ˜E = −

𝐵0
𝑇

,

𝜕 ˜B
𝜕 ˜𝑡
𝐿
𝑇

˜∇ × ˜B = 𝜇0𝑄 ¯𝑛

˜J +

(A.2a)

(A.2b)

(A.2c)

(A.2d)

1
𝑐2

𝐸0
𝑇

𝜕 ˜E
𝜕 ˜𝑡

.

We set

𝐸0 =

𝑀𝑉
𝑄𝑇

=

𝑀 𝐿
𝑄𝑇 2

, 𝐵0 =

𝑀
𝑄𝑇

223

which yields

˜∇ · ˜E =

𝐿𝑄 ¯𝑛
𝐸0𝜖0

˜𝜌,

˜∇ · ˜B = 0,

˜∇ × ˜E = −

𝐿
𝑇

𝐵0
𝐸0

𝜕 ˜B
𝜕 ˜𝑡

= −

𝐿
𝑇

˜∇ × ˜B = 𝜇0𝑄 ¯𝑛

𝐿2
𝑇

1
𝐵0

˜J +

For Gauss’s Law:

𝑀
𝑄𝑇
𝑀𝑉
𝑄𝑇
𝐸0
𝐵0

1
𝑐2

𝜕 ˜B
𝜕 ˜𝑡

= −

𝐿
𝑇

1
𝑉

𝜕 ˜B
𝜕 ˜𝑡

= −

𝜕 ˜B
𝜕 ˜𝑡

,

𝐿
𝑇

𝜕 ˜E
𝜕 ˜𝑡

.

𝐿𝑄 ¯𝑛
𝐸0𝜖0

=

𝐿𝑄 ¯𝑛
𝑀𝑉
𝑄𝑇 𝜖0

=

𝐿𝑄2𝑇 ¯𝑛
𝑀𝑉𝜖0

=

𝑄2𝑇 2 ¯𝑛
𝑀𝜖0

.

Call this 𝜎−1
1

:=

𝑄2𝑇 2 ¯𝑛
𝑀𝜖0

. For Faraday’s Law (A.1c):

𝐿
𝑇

𝐵0
𝐸0

𝐿
𝑇

=

𝑀
𝑄𝑇
𝑀𝑉
𝑄𝑇

𝐿
𝑇

1
𝑉

=

= 1

For Ampere’s Law, the first term (A.1d):

𝜇0𝑄 ¯𝑛𝐿2
𝑇

1
𝐵0

=

𝜇0𝑄 ¯𝑛𝐿2
𝑇

1
𝑀
𝑄𝑇

=

𝜇0𝑄 ¯𝑛𝐿2
𝑇

𝑄𝑇
𝑀

=

𝜇0𝑄2𝐿2 ¯𝑛
𝑀

Call this 𝜎2 :=

𝜇0𝑄2 𝐿2 ¯𝑛

𝑀 . For the second term:

1
𝑐2

𝐸0
𝐵0

𝐿
𝑇

=

1
𝑐2

𝑀𝑉
𝑄𝑇
𝑀
𝑄𝑇

𝐿
𝑇

=

1
𝑐2

𝑉 𝐿
𝑇

=

𝑉 2
𝑐2

.

Call this 𝜅 := 𝑐
𝑉 .

Thus we have

,

˜∇ · ˜E =

˜𝜌
𝜎1
˜∇ · ˜B = 0,
𝜕 ˜B
𝜕 ˜𝑡
1
𝜅2

˜∇ × ˜E = −

˜∇ × ˜B = 𝜎2 ˜J +

,

𝜕 ˜E
𝜕 ˜𝑡

.

We can now drop the tildes for simplicity. This completes the nondimensionalization.

224

(A.3a)

(A.3b)

(A.3c)

(A.3d)

(A.4a)

(A.4b)

(A.4c)

(A.4d)

A.2 Equations of Motion in E-B form

We have the Newton-Lorenz equations

𝑑x𝑖
𝑑𝑡
𝑑v𝑖
𝑑𝑡

= v𝑖,
𝑞𝑖
𝑚𝑖

=

(cid:16)

E(x𝑖) + v𝑖 × B(x𝑖)

(cid:17)

.

(A.5)

(A.6)

We insert the scales into the position equation (A.5), we obtain the following non-dimensional

form

𝐿
𝑇

𝑑 ˜x𝑖
𝑑 ˜𝑡

𝐿
𝑇 ˜v𝑖 =⇒

=

𝑑 ˜x𝑖
𝑑 ˜𝑡

= ˜v𝑖.

Following the same process for the velocity equation, after some rearrangement, we obtain
˜𝑞𝑖
𝑄
𝑀
˜𝑚𝑖
(cid:18) 𝑇𝑄
𝑉 𝑀

(cid:0)𝐸0E + 𝐵0𝑉 ˜v × ˜B(cid:1)
𝑇𝑄
𝑀

𝐵0 ˜v𝑖 × ˜B

𝐸0E +

𝑑 ˜v
𝑑 ˜𝑡

=⇒

𝑉
𝑇
𝑑 ˜v
𝑑 ˜𝑡

˜𝑞𝑖
𝑟𝑖

=

=

(cid:19)

.

Here we have introduced the non-dimensional electric and magnetic fields ˜E and ˜B, which are

normalized by 𝐸0 and 𝐵0, respectively, and 𝑟𝑖 = 𝑚𝑖/𝑀 is a mass ratio . From the definitions
B = ∇ × A and E = −∇𝜙 − 𝜕A

𝜕𝑡 , we can express these scales in terms of 𝜙0 and 𝐴0 as
𝐴0
𝐿

, 𝐵0 =

𝐸0 =

𝜙0
𝐿

.

Therefore,2 the non-dimensionalized equation for the velocity can be expressed in terms of these

scales as

𝑑 ˜v𝑖
𝑑 ˜𝑡

=

=

=

≡

˜𝑞𝑖
𝑟𝑖
˜𝑞𝑖
𝑟𝑖
˜𝑞𝑖
𝑟𝑖
˜𝑞𝑖
𝑟𝑖

˜E +

𝜙0
𝐿

(cid:16)

(cid:16) 𝑇𝑄
𝑉 𝑀

𝜙0
𝐿
𝑇𝑄
(𝐿/𝑇) 𝑀
(cid:16)𝑇 2𝑄𝜙0
𝐿2𝑀

˜E +

,

(cid:17)

𝑇𝑄
𝑀

𝐴0
𝐿 ˜v𝑖 × ˜B
𝐴0
𝑇𝑄
𝐿 ˜v𝑖 × ˜B
𝑀
𝑇𝑄 𝐴0
(cid:17)
𝑀 𝐿 ˜v𝑖 × ˜B

˜E +

,

(cid:17)

,

(cid:16)

𝛼1 ˜E + 𝛼2 ˜v𝑖 × ˜B

(cid:17)

,

where we have introduced the parameters

𝛼1 =

𝑇 2𝑄𝜙0
𝐿2𝑀

, 𝛼2 =

𝑇𝑄 𝐴0
𝑀 𝐿

.

2The E0 =

𝜙0
L initially seems off, until you see 𝐸0 ˜E = −

𝜙0
𝐿 ∇ ˜𝜙 − 𝐴0

𝑇 ˜A inherently requires

𝜙0
𝐿 = 𝐴0
𝑇 .

225

We then select 𝜙0 and 𝐴0 so that 𝛼1 = 𝛼2 = 1, i.e.,

𝜙0 =

𝐿2𝑀
𝑇 2𝑄

,

𝐴0 =

𝑀 𝐿
𝑇𝑄

,

which, happily, results in an the non-dimensional system

𝑑x𝑖
𝑑𝑡
𝑑v𝑖
𝑑𝑡

= v𝑖,
𝑞𝑖
𝑟𝑖

=

(cid:16)

E + v𝑖 × B

(cid:17)

,

(A.7)

(A.8)

(A.9)

where we have dropped the tildes for convenience of notation.

A.3 Maxwell’s Equations under the Lorenz Gauge

We now consider the vector and scalar wave equations formed by converting the E and B fields

of Maxwell’s equations to vector and scalar potentials under the Lorenz gauge.

− Δ𝜙 =

,

𝜌
𝜖0

− ΔA = 𝜇0J,

+ ∇ · A = 0.

1
𝑐2

1
𝑐2
1
𝑐2

𝜕2𝜙
𝜕𝑡2
𝜕2A
𝜕𝑡2
𝜕𝜙
𝜕𝑡






(A.10a)

(A.10b)

(A.10c)

We non-dimensionalize the field equations by substituting scales introduced at the beginning of the

section into the equations (A.10a) - (A.10c), which gives

1
𝑐2

1
𝑐2
1
𝑐2

𝜙0
𝑇 2
𝐴0
𝑇 2
𝜙0
𝑇






𝜕2 ˜𝜙
𝜕 ˜𝑡2
𝜕2 ˜A
𝜕 ˜𝑡2
𝜕 ˜𝜙
𝜕 ˜𝑡

+

−

−

𝜙0
𝐿2
𝐴0
𝐿2

˜Δ ˜𝜙 =

˜Δ ˜A =

˜𝜌,

𝑄 ¯𝑛
𝜖0
𝜇0𝑄 ¯𝑛𝐿
𝑇

˜J,

𝐴0
𝐿

˜∇ · ˜A = 0.

The first equation can be rearranged to obtain

𝐿2
𝑐2𝑇 2

𝜕2 ˜𝜙
𝜕 ˜𝑡2

− ˜Δ ˜𝜙 =

𝐿2𝑄 ¯𝑛
𝜖0𝜙0

˜𝜌.

Similarly, with the second equation we obtain

𝐿2
𝑐2𝑇 2

𝜕2 ˜A
𝜕 ˜𝑡2

− ˜Δ ˜A =

𝑄 ¯𝑛𝑉 𝐿2
𝑐2𝜖0 𝐴0

˜J,

226

where we have used 𝑉 = 𝐿𝑇 −1 as well as the fact that 𝑐2 = (𝜇0𝜖0)−1 . Finally, the gauge condition

becomes

𝜙0𝑉
𝑐2 𝐴0

𝜕 ˜𝜙
𝜕 ˜𝑡

+ ˜∇ · ˜A = 0.

Introducing the normalized speed of light 𝜅 = 𝑐/𝑉, and selecting 𝜙0 and 𝐴0 according to (A.7), we

find that the above equations simplify to (dropping the tildes)

𝜕𝑡𝑡 𝜙 − Δ𝜙 =

𝜌,

1
𝜎1

𝜕𝑡𝑡A − ΔA = 𝜎2J,

𝜕𝑡 𝜙 + ∇ · A = 0,

1
𝜅2
1
𝜅2
1
𝜅2






where we have introduced the new parameters

𝜎1 =

𝑀𝜖0
𝑄2𝑇 2 ¯𝑛

, 𝜎2 =

𝑄2𝐿2 ¯𝑛𝜇0
𝑀

.

(A.11)

(A.12)

(A.13)

(A.14)

These are nothing more than normalized versions of the permittivity and permeability constants in

the original equations (matching exactly the normalized constants in Section A.1).

A.4 Particle Equations of Motion

We now wish to non-dimensionalize the particle equations of motion under the Lorenz gauge.

The generalized momentum equations are as follows:

=

(cid:113)

𝑐2 (P𝑖 − 𝑞𝑖A)
𝑐2 (P𝑖 − 𝑞𝑖A)2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2

,

= −𝑞𝑖∇𝜙 +

𝑞𝑖𝑐2 (∇A) · (P𝑖 − 𝑞𝑖A)
𝑐2 (P𝑖 − 𝑞𝑖A)2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2

(cid:113)

.

𝑑x𝑖
𝑑𝑡

𝑑P𝑖
𝑑𝑡






(A.15)

(A.16)

Starting with the position equation (A.15), we substitute the scales introduced at the beginning of

this section and obtain

𝐿
𝑇

𝑑 ˜x𝑖
𝑑 ˜𝑡

=

(cid:113)

𝑐2 (cid:0)𝑃 ˜P𝑖 − 𝑄 𝐴0 ˜𝑞𝑖 ˜A(cid:1)
𝑐2 (cid:0)𝑃 ˜P𝑖 − 𝑄 𝐴0 ˜A(cid:1) 2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2

.

This equation can be simplified further by using the definition of 𝐴0 from (A.7) and noting that

the scale for momentum is 𝑃 = 𝑀 𝐿𝑇 −1. Defining the normalized mass ˜𝑚𝑖 = 𝑚𝑖/𝑀 and charge

227

˜𝑞𝑖 = 𝑞𝑖/𝑄, we obtain the non-dimensionalized position equation

𝑑 ˜x𝑖
𝑑 ˜𝑡

=

(cid:113)

𝜅2 (cid:0) ˜P𝑖 − ˜𝑞𝑖 ˜A(cid:1)

,

𝜅2 (cid:0) ˜P𝑖 − ˜𝑞𝑖 ˜A(cid:1) 2 + (cid:0) ˜𝑚𝑖𝜅2(cid:1) 2

where 𝜅 = 𝑐/𝑉 is, again, the normalized speed of light.

Following an identical treatment for the generalized momentum equation (A.16) and appealing

to the definition of the scales (A.7), after some simplifications, we obtain

𝑑 ˜P𝑖
𝑑 ˜𝑡

= − ˜𝑞𝑖 ˜∇ ˜𝜙 +

(cid:113)

˜𝑞𝑖𝜅2 (cid:0) ˜∇ ˜A(cid:1) · (cid:0) ˜P𝑖 − ˜𝑞𝑖 ˜A(cid:1)

.

𝜅2 (cid:0) ˜P𝑖 − ˜𝑞𝑖 ˜A(cid:1) 2 + (cid:0) ˜𝑚𝑖𝜅2(cid:1) 2

Therefore, the non-dimensional form of the relativistic equations of motion is given by (dropping

tildes)

𝑑x𝑖
𝑑𝑡

𝑑P𝑖
𝑑𝑡

=

(cid:113)

𝜅2 (P𝑖 − 𝑞𝑖A)
𝜅2 (P𝑖 − 𝑞𝑖A)2 + (cid:0)𝑚𝑖𝜅2(cid:1) 2

,

= −𝑞𝑖∇𝜙 +

𝑞𝑖𝜅2 (∇A) · (P𝑖 − 𝑞𝑖A)
𝜅2 (P𝑖 − 𝑞𝑖A)2 + (cid:0)𝑚𝑖𝜅2(cid:1) 2

.

(cid:113)






Performing analogous manipulations in the non-relativistic limit leads to the system (again

dropping tildes)

𝑑x𝑖
𝑑𝑡
𝑑P𝑖
𝑑𝑡





=

1
𝑚𝑖

(P𝑖 − 𝑞𝑖A) ,

= −𝑞𝑖∇𝜙 +

𝑞𝑖
𝑚𝑖

(∇A) · (P𝑖 − 𝑞𝑖A) .

228

APPENDIX B

DERIVING THE PARTICLE EQUATIONS OF MOTION FROM FIRST PRINCIPLES

We wish to derive the equations of motion in terms of the vector and scalar potentials. To do so,

we begin with the relativistic Lagrangians for a free particle and a particle interacting in a field,

obtained from Chapter 12 of [32]1 (see also [34]):

L 𝑓 𝑟𝑒𝑒 = −

𝑚𝑐2
𝛾

,

L𝑖𝑛𝑡 = −𝑞𝜙 + 𝑞v · A.

We combine these to acquire the Lagrangian:

L = −

𝑚𝑐2
𝛾

+ 𝑞v · A − 𝑞𝜙.

We define the canonical momentum as

P := ∇vL.

Knowing

𝑑
𝑑𝑣𝑖

𝛾−1 = −

𝛾𝑣𝑖
𝑐2 , ie ∇v𝛾−1 = −

𝛾v
𝑐2 yields

P = 𝛾𝑚v + 𝑞A.

(B.1)

(B.2)

(B.3)

(B.4)

(B.5)

What is 𝛾𝑚v other than momentum? Thus we define p := 𝛾𝑚v as the ordinary kinetic momentum.

The Hamiltonian is written in terms of the Lagrangian as

1Jackson uses Gaussian units, which we convert here to SI.

H := P · v − L.

(B.6)

229

We note

(cid:18)

1 −

𝛾 =

(cid:19) − 1

2

v2
𝑐2

− 1
2

(cid:17) 2

(cid:16) 1
𝛾𝑚 (P − 𝑞A)
𝑐2

(cid:170)
(cid:174)
(cid:174)
(cid:172)

1 −

= (cid:169)
(cid:173)
(cid:173)
(cid:171)

=⇒ 𝛾−2 = 1 −

(P − 𝑞A)2

=⇒ 1 = 𝛾2 −

(P − 𝑞A)2

1
(𝛾𝑚𝑐)2
1
(𝛾𝑚𝑐)2
1
(𝑚𝑐)2

=⇒ − 𝛾2 = −1 −

(P − 𝑞A)2

(B.7)

(cid:115)

=⇒ 𝛾 =

1 +

1
(𝑚𝑐)2

(P − 𝑞A)2

We may derive the following identity from (B.5) and (B.7):

P − 𝑞A

v =

𝑚(cid:113) 1

(𝑚𝑐)2 (P − 𝑞A)2 + 1
𝑐2 (P − 𝑞A)
𝑐2 (P − 𝑞A)2 + (cid:0)𝑚𝑐2(cid:1) 2

=

(cid:113)

.

(B.8)

We substitute this into the Lagrangian to acquire

H = P · v − L

=

=

=

=

(cid:113)

= P · (cid:169)
(cid:173)
(cid:173)
(cid:171)

(cid:18)

𝑐2 (P − 𝑞A)
𝑐2 (P − 𝑞A)2 + (cid:0)𝑚𝑐2(cid:1) 2
𝑚𝑐2(cid:17) 2(cid:19) − 1

𝑚𝑐2
𝛾

− (cid:169)
−
(cid:173)
(cid:173)
(cid:171)

(cid:170)
(cid:174)
(cid:174)
(cid:172)
(P − 𝑞A) · 𝑐2 (P − 𝑞A) + 𝑚𝑐2

𝑐2 (P − 𝑞A)
𝑐2 (P − 𝑞A)2 + (cid:0)𝑚𝑐2(cid:1) 2
(cid:18)

+ 𝑞 (cid:169)
(cid:173)
(cid:173)
(cid:171)

1 +

(cid:113)

(cid:16)

2

· A − 𝑞𝜙(cid:170)
(cid:174)
(cid:174)
(cid:172)
(cid:19) − 1

(cid:170)
(cid:174)
(cid:174)
(cid:172)
(P − 𝑞A)2

2

𝑐2 (P − 𝑞A)2 +

1
(𝑚𝑐)2

(cid:18)

(cid:18)

𝑐2 (P − 𝑞A)2 +

(cid:16)

𝑚𝑐2(cid:17) 2(cid:19) − 1

2

𝑐2 (P − 𝑞A)2 +

(cid:16)

𝑚𝑐2(cid:17) 2 (cid:18) (cid:16)

𝑚𝑐2(cid:17) 2

+ 𝑐2 (P − 𝑞A)2

(cid:19) − 1

2

+ 𝑞𝜙

𝑐2 (P − 𝑞A)2 +

(cid:16)

𝑚𝑐2(cid:17) 2(cid:19) − 1
2 (cid:18)

𝑐2 (P − 𝑞A)2 +

(cid:16)

𝑚𝑐2(cid:17) 2(cid:19)

+ 𝑞𝜙

(cid:113)

𝑐2 (P − 𝑞A)2 + (cid:0)𝑚𝑐2(cid:1) 2 + 𝑞𝜙.

230

+ 𝑞𝜙

(B.9)

We compute the Hamiltonian for a system of 𝑁 particles by summing over the Hamiltonian of

each particle. Given a particle 𝑖, and evaluating A and 𝜙 at x𝑖, we know

𝑑x𝑖
𝑑𝑡

=

𝜕H
𝜕P𝑖

=

(cid:113)

𝑐2(P𝑖 − 𝑞𝑖A)
𝑐2 (P𝑖 − 𝑞𝑖A)2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2

𝑑P𝑖
𝑑𝑡

= −

𝜕H
𝜕x𝑖

= −𝑞𝑖∇𝜙 +

𝑞𝑖𝑐2(∇A) · (P𝑖 − 𝑞𝑖A)
𝑐2 (P𝑖 − 𝑞𝑖A)2 + (cid:0)𝑚𝑖𝑐2(cid:1) 2

(cid:113)

We can resimplify in terms of 𝛾𝑖

𝑑x𝑖
𝑑𝑡
𝑑P𝑖
𝑑𝑡

=

1
𝛾𝑖𝑚𝑖

(P𝑖 − 𝑞𝑖A) ,

= −𝑞𝑖∇𝜙 +

𝑞𝑖
𝛾𝑖𝑚𝑖

(∇A) · (P𝑖 − 𝑞𝑖A) .

If we let assume a nonrelativistic setting, ie 𝛾 ≈ 1 then we see the system simplifies to

𝑑x𝑖
𝑑𝑡
𝑑P𝑖
𝑑𝑡

=

=

1
𝑚𝑖
𝑞𝑖
𝑚𝑖

(P𝑖 − 𝑞𝑖A) ,

(∇A) · (P𝑖 − 𝑞𝑖A) − 𝑞𝑖∇𝜙

(B.10)

(B.11)

(B.12)

(B.13)

(B.14)

(B.15)

Thus we have the particle equations of motion in terms of the vector and scalar potentials.

231

APPENDIX C

DERIVING THE BOLTZMANN EQUATION FROM FIRST PRINCIPLES

Matthew W. Kunz has a set of lecture notes [67] upon which this is based, and it was Dr. Shiping

Zhou who walked the Christlieb group through them. The main difference between these lecture

notes and this (other than the addition some steps to make things clearer) will be the units, this uses

SI whereas the lecture notes uses Gaussian.

We first begin by considering the evolution of a system of particles of species 𝛼 in phase space.

This system may be described by the following function

𝐹𝛼 (𝑡, r, v) =

𝑁 𝛼(cid:213)

𝑖=1

𝛿 (r − R𝛼𝑖 (𝑡)) 𝛿 (v − V𝛼𝑖 (𝑡)) .

(C.1)

Here 𝑁𝛼 is the number of particles of species 𝛼, r is physical space and v is velocity space. This

model views the collection of particles as a summation of delta functions over phase space. Newton’s

law and the Lorentz force give us

𝑑R𝛼𝑖
𝑑𝑡

= V𝛼𝑖,

1
𝑚𝛼

F𝛼𝑖 = −

𝑑V𝛼𝑖
𝑑𝑡

=

𝑞𝛼
𝑚𝛼

(E𝑚 + V𝛼𝑖 × B𝑚)

(C.2)

The 𝑚 indicates “microphysical,” meaning the electric and magnetic fields are the result of

the particles themselves, nothing external. These are functions of r and 𝑡. These of course satisfy

Maxwell’s equations, and given the linearity, we can incorporate external fields without issue. Now,

taking a time derivative will show how the system evolves over time. Before we do so, the following

properties will be useful:

𝑎𝛿 (𝑎 − 𝑏) = 𝑏𝛿 (𝑏 − 𝑎) ,
𝜕
𝜕𝑎

𝑓 (𝑎 − 𝑏) = −

𝜕
𝜕𝑏

𝑓 (𝑎 − 𝑏).

(C.3a)

(C.3b)

The reasoning for these is straightforward enough, if 𝑎 ≠ 𝑏, then 𝛿 (𝑎 − 𝑏) = 0, if 𝑎 = 𝑏, then

𝛿 (𝑎 − 𝑏) = ∞, but more importantly the integral of 𝛿 (𝑎 − 𝑏) will be 𝑎 = 𝑏. In either case, the

first identity holds. The second immediately follows from the chain rule. If 𝜉 := 𝑎 − 𝑏, then
𝜕 𝑓
𝜕𝑎 =

𝜕 𝑓
𝜕𝜉 . Now we take a derivation of the particle

𝜕 𝑓
𝜕𝜉 . Similar reasoning yields

𝜕 𝑓
𝜕𝑏 = −

𝜕𝜉
𝜕𝑎 =

𝜕 𝑓
𝜕𝜉

232

𝑑V𝛼𝑖
𝑑𝑡

𝑑V𝛼𝑖
𝑑𝑡

· ∇V𝛼𝑖 𝛿 (r − R𝛼𝑖 (𝑡)) 𝛿 (v − V𝛼𝑖 (𝑡))

(cid:21)

· ∇r𝛿 (r − R𝛼𝑖 (𝑡)) 𝛿 (v − V𝛼𝑖 (𝑡))

(cid:21)

𝛿 (r − R𝛼𝑖 (𝑡)) 𝛿 (v − V𝛼𝑖 (𝑡))

· ∇R𝛼𝑖 𝛿 (r − R𝛼𝑖 (𝑡)) 𝛿 (v − V𝛼𝑖 (𝑡)) +

(cid:20) 𝑑R𝛼𝑖
𝑑𝑡

· ∇r𝛿 (r − R𝛼𝑖 (𝑡)) 𝛿 (v − V𝛼𝑖 (𝑡)) +

[V𝛼𝑖 · ∇r𝛿 (r − R𝛼𝑖 (𝑡)) 𝛿 (v − V𝛼𝑖 (𝑡))]

distribution function:

𝜕𝐹𝛼
𝜕𝑡

=

=

𝜕
𝜕𝑡
𝑁 𝛼(cid:213)

𝑁 𝛼(cid:213)

𝑖=1
(cid:20) 𝑑R𝛼𝑖
𝑑𝑡

𝑖=1
𝑁 𝛼(cid:213)

= −

𝑖=1
𝑁 𝛼(cid:213)

= −

𝑖=1
𝑁 𝛼(cid:213)
(cid:20) (cid:18) 𝑞𝛼
𝑚𝛼

−

𝑖=1
𝑁 𝛼(cid:213)

(E𝑚 (R𝛼𝑖, 𝑡) + V𝛼𝑖 × B𝑚 (R𝛼𝑖, 𝑡))

(cid:19)

· ∇r𝛿 (r − R𝛼𝑖 (𝑡)) 𝛿 (v − V𝛼𝑖 (𝑡))

(cid:21)

= −

[v · ∇r𝛿 (r − R𝛼𝑖 (𝑡)) 𝛿 (v − V𝛼𝑖 (𝑡))]

𝑖=1
𝑁 𝛼(cid:213)
(cid:20) (cid:18) 𝑞𝛼
𝑚𝛼

𝑖=1

−

(E𝑚 (r, 𝑡) + v × B𝑚 (r, 𝑡))

(cid:19)

· ∇r𝛿 (r − R𝛼𝑖 (𝑡)) 𝛿 (v − V𝛼𝑖 (𝑡))

(cid:21)

= −v · ∇r𝐹𝛼 −

𝑞𝛼
𝑚𝛼

(E𝑚 (r, 𝑡) + v × B𝑚 (r, 𝑡)) · ∇v𝐹𝛼

(C.4)

The move from the fourth and fifth line to the sixth and seventh is justified by the first identity,

(C.3a), the move from the second to the third line is justified by the second identity (C.3b). Given

this equality, it is clear

(cid:20) 𝜕
𝜕𝑡

+ v · ∇r +

𝑞𝛼
𝑚𝛼

(E𝑚 + v × B𝑚) · ∇v

(cid:21)

𝐹𝛼 = 0.

(C.5)

This is the Klimontovich equation. From this we can derive the Vlasov equation, which is useful

given that simulating the Klimontovich equation is computationally intractable.1 We define the

ensemble average (basically a smoothing function)

(cid:10)𝐺 (cid:0)𝐹𝛼, 𝐹𝛽, ..., 𝐹𝛾(cid:1) (cid:11) :=

∫

𝑑X𝑎𝑙𝑙 𝑃𝑁𝐺 (cid:0)𝐹𝛼, 𝐹𝛽, ..., 𝐹𝛾(cid:1)

(C.6)

1Rigorously speaking, simulation of the Klimontovich equation requires the simulation of every particle in the

system, which is not PIC.

233

Here the 𝑃𝑁 term is the probability of a particle having a certain set of initial conditions. We then

define

𝑓𝛼 (x, 𝑡) := (cid:104)𝐹𝛼 (x, 𝑡)(cid:105).

(C.7)

We consider this the number of particles of species 𝛼 per unit phase space. We can then derive

the time evolution for 𝑓𝛼. We first define a perturbation of the fields and 𝐹𝛼 function by

𝐹𝛼 = (cid:104)𝐹𝛼(cid:105) + 𝛿𝐹𝛼, E𝑚 = (cid:104)E𝑚(cid:105) + 𝛿E, B𝑚 = (cid:104)B𝑚(cid:105) + 𝛿B,

𝑓𝛼 := (cid:104)𝐹𝛼(cid:105), E := (cid:104)E𝑚(cid:105), B := (cid:104)B𝑚(cid:105),

(cid:104)𝛿𝐹𝛼(cid:105) = (cid:104)𝛿E(cid:105) = (cid:104)𝛿B(cid:105) = 0.

(C.8)

(C.9)

(C.10)

The E, B, and 𝐹𝛼 terms represent the smoother portions of the fields and distribution function.

The fields in particular are now “macroscopic” fields, averaging out all of the plasma particles. the

𝛿 terms are the remainder, they represent the spiky and discrete nature of the particles. The last line

is justified by definition. If we apply the ensemble average to an ensemble average, nothing happens.

As such, applying the ensemble average to any of the terms in (C.8) necessitates the ensemble

average of the 𝛿 term becoming zero. ie for some value 𝑇:

𝑇 = (cid:104)𝑇(cid:105) + 𝛿𝑇

=⇒ (cid:104)𝑇(cid:105) = (cid:104)(cid:104)𝑇(cid:105) + 𝛿𝑇(cid:105)

=⇒ (cid:104)𝑇(cid:105) = (cid:104)(cid:104)𝑇(cid:105)(cid:105) + (cid:104)𝛿𝑇(cid:105)

=⇒ (cid:104)𝑇(cid:105) = (cid:104)𝑇(cid:105) + (cid:104)𝛿𝑇(cid:105)

=⇒ (cid:104)𝛿𝑇(cid:105) = 0.

We run the average over (C.5) and decompose its fields and distribution function as described

above. When we do this, it becomes

𝜕
𝜕𝑡

(cid:104) 𝑓𝛼 + 𝛿𝐹𝛼(cid:105) + v · ∇r(cid:104) 𝑓𝛼 + 𝛿𝐹𝛼(cid:105) +

(cid:28) 𝑞𝛼
𝑚𝛼

((E + 𝛿E) + v × (B + 𝛿B)) · ∇v ( 𝑓𝛼 + 𝛿𝐹𝛼)

(cid:29)

.

(C.11)

The third term we must be careful, the ensemble average falls upon the whole term, whereas the first

two terms only 𝐹𝛼 is a function of r. We take a closer look at the third term, which upon multiplying,

234

we see becomes the sum of the following four terms:

(cid:28) 𝑞𝛼
𝑚𝛼
(cid:28) 𝑞𝛼
𝑚𝛼
(cid:28) 𝑞𝛼
𝑚𝛼
(cid:28) 𝑞𝛼
𝑚𝛼

(E + v × B) · ∇v 𝑓𝛼

(cid:29)

(E + v × B) · ∇v𝛿𝐹𝛼

,

(cid:29)

,

(𝛿E + v × 𝛿B) · ∇v 𝑓𝛼

(cid:29)

(𝛿E + v × 𝛿B) · ∇v𝛿𝐹𝛼

,

(cid:29)

.

(C.12a)

(C.12b)

(C.12c)

(C.12d)

Now, by definition (cid:104)E(cid:105) = E, (cid:104)B(cid:105) = B, and (cid:104) 𝑓𝛼(cid:105) = 𝑓𝛼. Additionally, given v is not a function of

space, we see (cid:104)v(cid:105) = v. As such we may pull them out of the ensemble average, and, recalling the

ensemble average of all of the delta terms is zero, we see these terms become

𝑞𝛼
𝑚𝛼
𝑞𝛼
𝑚𝛼
𝑞𝛼
𝑚𝛼
(cid:28) 𝑞𝛼
𝑚𝛼

(E + v × B) · ∇v 𝑓𝛼,

(E + v × B) · ∇v (cid:104)𝛿𝐹𝛼(cid:105) = 0,

((cid:104)𝛿E(cid:105) + v × (cid:104)𝛿B(cid:105)) · ∇v 𝑓𝛼 = 0,

(𝛿E + v × 𝛿B) · ∇v𝛿𝐹𝛼

(cid:29)

.

(C.13a)

(C.13b)

(C.13c)

(C.13d)

With these facts in mind, we see the Klimontovich equation under the ensemble average (C.11)

becomes

𝜕 𝑓𝛼
𝜕𝑡

+ v · ∇r 𝑓𝛼 +

𝑞𝛼
𝑚𝛼

(E + v × B) · ∇v 𝑓𝛼 = −

(cid:28) 𝑞𝛼
𝑚𝛼

(𝛿E + v × 𝛿B) · ∇v𝛿𝐹𝛼

(cid:29)

(C.14)

This is the Boltzmann equation, which reduces to the Vlasov equation if the 𝛿 terms are negligible.

235

APPENDIX D

THE YEE AND BORIS METHOD

D.1 The Yee Grid

Named for the applied mathematician, Kane Yee (born 1934), the Yee scheme [86] is a clever

numerical technique to model electromagnetic waves via the finite difference method. The brilliance

of this technique lies in its staggering of the electric and magnetic fields (see figures D.1 and

D.2). Where most solvers will collocate both the fields and their constituent components, the Yee

grid staggers them, and in so doing enforces the involutions (Gauss’s Law for both electricity and

magnetism). It is derived as follows:1

We have Maxwell’s equations, (1.7a) - (1.7d). We take the curl equations and view them in

component form, first we will approach (1.7c).

𝜕𝐸𝑧
𝜕𝑦

𝜕𝐸𝑦
𝜕𝑧

−

= −

𝜕𝐵𝑥
𝜕𝑡

𝜕𝐸𝑥
𝜕𝑧

−

𝜕𝐸𝑧
𝜕𝑥

= −

𝜕𝐵𝑦
𝜕𝑡

𝜕𝐸𝑦
𝜕𝑥

−

𝜕𝐸𝑥
𝜕𝑦

= −

𝜕𝐵𝑧
𝜕𝑡

,

,

,

(D.1a)

(D.1b)

(D.1c)

First, we stagger the electric and magnetic fields in time, with the magnetic field being on integer

timesteps and the electric field being on half steps. We also stagger the fields in space as show in the

diagram. We do so using the standard finite difference approach, Taylor expanding in both space

and time:

2

𝐸 𝑛+ 1
𝑧,𝑖, 𝑗+1,𝑘+ 1
2

− 𝐸 𝑛+ 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

Δ𝑦

𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘+1

− 𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

Δ𝑧

−

−

𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘+1

− 𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

Δ𝑧

2

𝐸 𝑛+ 1
𝑧,𝑖+1, 𝑗,𝑘+ 1
2

− 𝐸 𝑛+ 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

Δ𝑥

= −

= −

𝐵𝑛+1

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

− 𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

Δ𝑡

,𝑘+ 1
2

,

(D.2a)

𝐵𝑛+1
𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

− 𝐵𝑛

𝑦,𝑖+ 1
2

Δ𝑡

, 𝑗,𝑘+ 1
2

,

(D.2b)

1I am indebted to Dr. Raymond Rumpf at University of Texas at El Paso, who has an excellent Youtube series

walking through the Yee grid and how it is derived.

236

𝐵𝑦

𝐸𝑧

𝐸𝑥

𝐵𝑥

𝐵𝑧

𝐸𝑦

𝑧

𝑥

𝑦

Figure D.1 A 3D Yee grid cell. Values are located on the arrows.

𝐸 𝑛+ 1

2
𝑦,𝑖+1, 𝑗+ 1
2

− 𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘
Δ𝑥

,𝑘

−

𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗+1,𝑘

− 𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

Δ𝑦

𝐵𝑛+1
𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖+ 1
2

Δ𝑡

= −

, 𝑗+ 1
2

,𝑘

.

(D.2c)

We can then rearrange these to form update equations:

𝐵𝑛+1

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

= 𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

𝐵𝑛+1
𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

= 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

− Δ𝑡 (cid:169)
(cid:173)
(cid:173)
(cid:171)

− Δ𝑡 (cid:169)
(cid:173)
(cid:173)
(cid:171)

2

𝐸 𝑛+ 1
𝑧,𝑖, 𝑗+1,𝑘+ 1
2

− 𝐸 𝑛+ 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

Δ𝑦

𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘+1

− 𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

Δ𝑧

237

𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘+1

− 𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

Δ𝑧

2

𝐸 𝑛+ 1
𝑧,𝑖+1, 𝑗,𝑘+ 1
2

− 𝐸 𝑛+ 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

Δ𝑥

−

−

,

(cid:170)
(cid:174)
(cid:174)
(cid:172)

,

(cid:170)
(cid:174)
(cid:174)
(cid:172)

(D.3a)

(D.3b)

⊗

⊗

⊗

⊗

⊗

⊗

⊗

⊗

⊗

𝐸𝑥

𝐸𝑦

𝐵𝑧 ⊗

𝑦

𝑥

Figure D.2 A 2D Yee grid cell in Transverse Electric (TE) mode. The magnetic field is pointed out
of the page.

𝐵𝑛+1
𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘 = 𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘 − Δ𝑡 (cid:169)
(cid:173)
(cid:173)
(cid:171)

𝐸 𝑛+ 1

2
𝑦,𝑖+1, 𝑗+ 1
2

− 𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘
Δ𝑥

,𝑘

−

𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗+1,𝑘

− 𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

Δ𝑦

We may now repeat the process with (1.7d). Taking it in component form:

𝜕𝐵𝑧
𝜕𝑦

𝜕𝐵𝑦
𝜕𝑧

−

𝜕𝐵𝑥
𝜕𝑧

𝜕𝐵𝑦
𝜕𝑥

−

−

𝜕𝐵𝑧
𝜕𝑥

𝜕𝐵𝑥
𝜕𝑦

= 𝜇0𝐽𝑥 +

= 𝜇0𝐽𝑦 +

= 𝜇0𝐽𝑥 +

1
𝑐2

𝜕𝐸𝑥
𝜕𝑡

1
𝑐2

𝜕𝐸𝑦
𝜕𝑡

1
𝑐2

𝜕𝐸𝑧
𝜕𝑡

,

,

,

(D.3c)

.

(cid:170)
(cid:174)
(cid:174)
(cid:172)

(D.4a)

(D.4b)

(D.4c)

We then Taylor expand:

𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖+ 1
2

Δ𝑦

, 𝑗− 1
2

,𝑘

−

𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

− 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘− 1
2

Δ𝑧

= 𝜇0𝐽𝑛− 1
2
𝑥,𝑖+ 1
2

+

, 𝑗,𝑘

1
𝑐2

𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

− 𝐸 𝑛− 1
2
𝑥,𝑖+ 1
2

Δ𝑡

, 𝑗,𝑘

,

(D.5a)

238

𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

− 𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

Δ𝑧

,𝑘− 1
2

−

𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖− 1
2

, 𝑗+ 1
2

,𝑘

Δ𝑥

= 𝜇0𝐽𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

+

,𝑘

1
𝑐2

𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

− 𝐸 𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

Δ𝑡

,𝑘

,

(D.5b)

𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

− 𝐵𝑛

𝑦,𝑖− 1
2

Δ𝑥

, 𝑗,𝑘+ 1
2

−

𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

− 𝐵𝑛

𝑥,𝑖, 𝑗− 1
2

,𝑘+ 1
2

Δ𝑦

= 𝜇0𝐽𝑛− 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

+

1
𝑐2

𝐸 𝑛+ 1
2
𝑧,𝑖, 𝑗,𝑘+ 1
2

− 𝐸 𝑛− 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

Δ𝑡

.

(D.5c)

And once more we rearrange these to form update equations

𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘 = 𝐸 𝑛− 1

2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

+ 𝑐2Δ𝑡

(cid:32) 𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖+ 1
2

Δ𝑦

, 𝑗 − 1
2

,𝑘

−

𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

− 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘− 1
2

Δ𝑧

− 𝜇0𝐽 𝑛− 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

(cid:33)

,

(D.6a)

𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘 = 𝐸 𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

+ 𝑐2Δ𝑡

(cid:32) 𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

− 𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

Δ𝑧

,𝑘− 1
2

−

𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖− 1
2

, 𝑗+ 1
2

,𝑘

Δ𝑥

− 𝜇0𝐽 𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

(cid:33)

,

(D.6b)

𝐸 𝑛+ 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

= 𝐸 𝑛− 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

+ 𝑐2Δ𝑡

(cid:32) 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

− 𝐵𝑛

𝑦,𝑖− 1
2

Δ𝑥

, 𝑗,𝑘+ 1
2

−

𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

− 𝐵𝑛

𝑥,𝑖, 𝑗 − 1
2

,𝑘+ 1
2

Δ𝑦

− 𝜇0𝐽 𝑛− 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

(cid:33)

.

(D.6c)

From here we can begin reducing our dimensions. For example, let us say there is no change in

the 𝑧 direction. This reduces the above equations to

𝐵𝑛+1

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

= 𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

2

𝐸 𝑛+ 1
𝑧,𝑖, 𝑗+1,𝑘+ 1
2

− 𝐸 𝑛+ 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

Δ𝑦

,

(cid:170)
(cid:174)
(cid:174)
(cid:172)

− Δ𝑡 (cid:169)
(cid:173)
(cid:173)
(cid:171)

𝐵𝑛+1
𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

= 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

2

𝐸 𝑛+ 1
𝑧,𝑖+1, 𝑗,𝑘+ 1
2

− 𝐸 𝑛+ 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

Δ𝑥

,

(cid:170)
(cid:174)
(cid:174)
(cid:172)

−

− Δ𝑡 (cid:169)
(cid:173)
(cid:173)
(cid:171)

239

(D.7a)

(D.7b)

𝐵𝑛+1
𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘 = 𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

𝐸 𝑛+ 1

2
𝑦,𝑖+1, 𝑗+ 1
2

,𝑘 − Δ𝑡 (cid:169)
(cid:173)
(cid:173)
(cid:171)

− 𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘
Δ𝑥

,𝑘

−

𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗+1,𝑘

− 𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

Δ𝑦

.

(cid:170)
(cid:174)
(cid:174)
(cid:172)

𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘 = 𝐸 𝑛− 1

2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

+ 𝑐2Δ𝑡

(cid:32) 𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗− 1
2

,𝑘

Δ𝑦

− 𝜇0𝐽𝑛− 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

(cid:33)

,

𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘 = 𝐸 𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

(cid:32)

+ 𝑐2Δ𝑡

−

𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖− 1
2

, 𝑗+ 1
2

,𝑘

Δ𝑥

− 𝜇0𝐽𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

(cid:33)

,

(D.7c)

(D.8a)

(D.8b)

𝐸 𝑛+ 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

= 𝐸 𝑛− 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

+ 𝑐2Δ𝑡

(cid:32) 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

− 𝐵𝑛

𝑦,𝑖− 1
2

Δ𝑥

, 𝑗,𝑘+ 1
2

−

𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

− 𝐵𝑛

𝑥,𝑖, 𝑗 − 1
2

,𝑘+ 1
2

Δ𝑦

− 𝜇0𝐽 𝑛− 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

(cid:33)

.

(D.8c)

Notice now that this set of equations can be decoupled into what is known as Tranverse Magnetic

(TM), or 𝐸𝑧 and Transverse Electric (TE) 𝐵𝑧 mode.

𝐵𝑛+1

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

= 𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

𝐵𝑛+1
𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

= 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

2

𝐸 𝑛+ 1
𝑧,𝑖, 𝑗+1,𝑘+ 1
2

− 𝐸 𝑛+ 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

Δ𝑦

,

(cid:170)
(cid:174)
(cid:174)
(cid:172)

− Δ𝑡 (cid:169)
(cid:173)
(cid:173)
(cid:171)

2

𝐸 𝑛+ 1
𝑧,𝑖+1, 𝑗,𝑘+ 1
2

− 𝐸 𝑛+ 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

Δ𝑥

,

(cid:170)
(cid:174)
(cid:174)
(cid:172)

−

− Δ𝑡 (cid:169)
(cid:173)
(cid:173)
(cid:171)

(D.9a)

(D.9b)

𝐸 𝑛+ 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

= 𝐸 𝑛− 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

+ 𝑐2Δ𝑡

(cid:32) 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

− 𝐵𝑛

𝑦,𝑖− 1
2

Δ𝑥

, 𝑗,𝑘+ 1
2

−

𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

− 𝐵𝑛

𝑥,𝑖, 𝑗 − 1
2

,𝑘+ 1
2

Δ𝑦

− 𝜇0𝐽 𝑛− 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

(cid:33)

.

𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘 = 𝐸 𝑛− 1

2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

+ 𝑐2Δ𝑡

(cid:32) 𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗− 1
2

,𝑘

Δ𝑦

− 𝜇0𝐽𝑛− 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

(cid:33)

,

𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘 = 𝐸 𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

(cid:32)

+ 𝑐2Δ𝑡

−

𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖− 1
2

, 𝑗+ 1
2

,𝑘

Δ𝑥

− 𝜇0𝐽𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

(cid:33)

,

240

(D.9c)

(D.10a)

(D.10b)

𝐵𝑛+1
𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘 = 𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

𝐸 𝑛+ 1

2
𝑦,𝑖+1, 𝑗+ 1
2

,𝑘 − Δ𝑡 (cid:169)
(cid:173)
(cid:173)
(cid:171)

− 𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘
Δ𝑥

,𝑘

−

𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗+1,𝑘

− 𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

Δ𝑦

(D.10c)

.

(cid:170)
(cid:174)
(cid:174)
(cid:172)

We can reduce down to one dimension by eliminating changes in the 𝑦 direction. This will

eliminate equations (D.9a) and (D.10a) and simplify equations (D.9c) and (D.10c).

𝐵𝑛+1
𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

= 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

2

𝐸 𝑛+ 1
𝑧,𝑖+1, 𝑗,𝑘+ 1
2

− 𝐸 𝑛+ 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

Δ𝑥

,

(cid:170)
(cid:174)
(cid:174)
(cid:172)

−

− Δ𝑡 (cid:169)
(cid:173)
(cid:173)
(cid:171)

𝐵𝑛+1
𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘 = 𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

𝐸 𝑛+ 1

2
𝑦,𝑖+1, 𝑗+ 1
2

,𝑘 − Δ𝑡 (cid:169)
(cid:173)
(cid:173)
(cid:171)

− 𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘
Δ𝑥

,𝑘

.

(cid:170)
(cid:174)
(cid:174)
(cid:172)

𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘 = 𝐸 𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

(cid:32)

+ 𝑐2Δ𝑡

−

𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖− 1
2

, 𝑗+ 1
2

,𝑘

Δ𝑥

− 𝜇0𝐽𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

(cid:33)

,

𝐸 𝑛+ 1
2
𝑧,𝑖, 𝑗,𝑘+ 1
2

= 𝐸 𝑛− 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

+ 𝑐2Δ𝑡

(cid:32) 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

− 𝐵𝑛

𝑦,𝑖− 1
2

, 𝑗,𝑘+ 1
2

Δ𝑥

− 𝜇0𝐽𝑛− 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

(cid:33)

.

This is again decoupled into two sets of update equations.

𝐸 𝑛+ 1
2
𝑧,𝑖, 𝑗,𝑘+ 1
2

= 𝐸 𝑛− 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

+ 𝑐2Δ𝑡

(cid:32) 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

− 𝐵𝑛

𝑦,𝑖− 1
2

, 𝑗,𝑘+ 1
2

Δ𝑥

− 𝜇0𝐽𝑛− 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

(cid:33)

.

𝐵𝑛+1
𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

= 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

2

𝐸 𝑛+ 1
𝑧,𝑖+1, 𝑗,𝑘+ 1
2

− 𝐸 𝑛+ 1

2
𝑧,𝑖, 𝑗,𝑘+ 1
2

Δ𝑥

,

(cid:170)
(cid:174)
(cid:174)
(cid:172)

−

− Δ𝑡 (cid:169)
(cid:173)
(cid:173)
(cid:171)

𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘 = 𝐸 𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

(cid:32)

+ 𝑐2Δ𝑡

−

𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖− 1
2

, 𝑗+ 1
2

,𝑘

Δ𝑥

− 𝜇0𝐽𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

(cid:33)

,

241

(D.11a)

(D.11b)

(D.12a)

(D.12b)

(D.13a)

(D.13b)

(D.14a)

𝐵𝑛+1
𝑧,𝑖+ 1
2

,𝑘 − Δ𝑡 (cid:169)
(cid:173)
(cid:173)
(cid:171)
Thus we have our Yee grid for 1, 2, and 3 dimensions. This is an explicit update equation and is

,𝑘 = 𝐵𝑛

(D.14b)

𝑧,𝑖+ 1
2

(cid:170)
(cid:174)
(cid:174)
(cid:172)

, 𝑗+ 1
2

, 𝑗+ 1
2

.

,𝑘
Δ𝑥

𝐸 𝑛+ 1

2
𝑦,𝑖+1, 𝑗+ 1
2

− 𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

limited to timesteps of size [86, 312]:

Δ𝑡 ≤

1
Δ𝑦2 + 1
Δ𝑥2 + 1
Δ𝑧2

𝑐(cid:113) 1

.

(D.15)

We now turn to derive this value.

D.1.1 CFL Condition

The following is based on [312].2 To ensure stability we will derive the Courant–Friedrichs–Lewy

(CFL) number for the Yee scheme.

Consider the two curl equations in vacuum without current:

∇ × E = −

1
𝑐

𝜕B
𝜕𝑡

,

∇ × B =

1
𝑐

𝜕E
𝜕𝑡

.

(D.16)

(D.17)

The 1

𝑐2 term typically on the RHS of (D.17) has been spread over to (D.16) for reasons that will

soon become apparent. This is justified by simply rescaling our system, or can be thought of as

switching over to Gaussian units.

Multiplying (D.17) by the imaginary unit 𝑖 and subtracting (D.16) from it yields:

(cid:18)
𝑖

𝜕E
𝜕𝑡

+

(cid:19)

𝜕B
𝜕𝑡

(𝑖E + B)

1
𝑐
𝜕
1
𝜕𝑡
𝑐
𝜕
1
𝜕𝑡
𝑐

𝑖∇ × B − ∇ × E =

=⇒ ∇ × (𝑖B − E) =

=⇒ 𝑖∇ × (B + 𝑖E) =
𝜕V
𝜕𝑡

=⇒ 𝑖∇ × V =

1
𝑐

(𝑖E + B)

(D.18)

2There are other ways of going about it in perhaps a more simple way, but Taflove and Brodwin have a really clever
move where they separate the spatial and temporal components of the scheme and solve an eigenvalue problem for both.
They do simplify things by setting 𝑐 = 1 which I have generalized.

242

where V := B + 𝑖E. To assert stability we solve the following eigenvalue problem for the numerical

derivatives:

𝑖∇ × V = 𝜆V.

1
𝑐

𝜕V
𝜕𝑡

= 𝜆V,

(D.19)

(D.20)

First, we consider (D.19). Now, we assume V𝑙,𝑚,𝑛 = 𝑉0𝑒𝑖(𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧) as the “arbitrary

lattice spatial mode.” Taking the central difference scheme, we know

𝜕V𝑙,𝑚,𝑛
𝜕𝑥

≈

V𝑙+ 1

2

,𝑚,𝑛 − V𝑙− 1
Δ𝑥

2

,𝑚,𝑛

with similar results for 𝑦 and 𝑧. Let us consider the partial differential approximation on this particular

vector field:

V𝑙+ 1

2

,𝑚,𝑛 − V𝑙− 1
Δ𝑥

2

,𝑚,𝑛

=

=

=

=

=

(cid:16)

(cid:16)

𝑘 𝑥

(cid:17)

𝑙+ 1
2

𝑉0𝑒𝑖

Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧

(cid:17)

(cid:16)

(cid:16)

(cid:17)

𝑘 𝑥

𝑙− 1
2

Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧

− 𝑉0𝑒𝑖
Δ𝑥
𝑖𝑘 𝑥Δ𝑥 − 𝑉0𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1) 𝑒− 1

2

(cid:17)

𝑖𝑘 𝑥Δ𝑥

𝑉0𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1) 𝑒 1

2

Δ𝑥

𝑒 1

2

𝑖𝑘 𝑥Δ𝑥 − 𝑒− 1

2

𝑖𝑘 𝑥Δ𝑥(cid:17)

2𝑖 sin

(cid:19)(cid:19)

𝑘𝑥Δ𝑥

(cid:18) 1
2

𝑉0𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1)
Δ𝑥
𝑉0𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1)
Δ𝑥

(cid:16)

(cid:18)

2𝑖 sin

(cid:17)

𝑘𝑥Δ𝑥

(cid:16) 1
2
Δ𝑥

V𝑙,𝑚,𝑛

Therefore, we can simplify (D.19) to

sin

(cid:17)

𝑘𝑥Δ𝑥

sin

(cid:16) 1
2
Δ𝑥

−2 (cid:169)
(cid:173)
(cid:173)
(cid:171)
Breaking into components yields

,

𝑘 𝑦Δ𝑦

(cid:16) 1
2
Δ𝑦

(cid:17)

,

sin

(cid:17)

𝑘 𝑧Δ𝑧

(cid:16) 1
2
Δ𝑧

(cid:170)
(cid:174)
(cid:174)
(cid:172)

× V𝑙,𝑚,𝑛 = 𝜆V𝑙,𝑚,𝑛

(D.21)

sin

(cid:17)

𝑘 𝑦Δ𝑦

(cid:16) 1
2
Δ𝑦

𝑉𝑧,𝑙,𝑚,𝑛 −

sin

(cid:17)

𝑘 𝑧Δ𝑧

(cid:16) 1
2
Δ𝑧

𝑉𝑦,𝑙,𝑚,𝑛

(cid:170)
(cid:174)
(cid:174)
(cid:172)

−2 (cid:169)
(cid:173)
(cid:173)
(cid:171)

= 𝜆𝑉𝑥,𝑙,𝑚,𝑛,

(D.22a)

243

(cid:17)

(cid:17)

sin

𝑘 𝑧Δ𝑧

(cid:16) 1
2
Δ𝑧

sin

𝑘𝑥Δ𝑥

(cid:16) 1
2
Δ𝑥

−2 (cid:169)
(cid:173)
(cid:173)
(cid:171)

−2 (cid:169)
(cid:173)
(cid:173)
(cid:171)

𝑉𝑥,𝑙,𝑚,𝑛 −

sin

(cid:17)

𝑘𝑥Δ𝑥

(cid:16) 1
2
Δ𝑥

𝑉𝑧,𝑙,𝑚,𝑛

(cid:170)
(cid:174)
(cid:174)
(cid:172)

= 𝜆𝑉𝑦,𝑙,𝑚,𝑛,

(D.22b)

𝑉𝑦,𝑙,𝑚,𝑛 −

sin

(cid:17)

𝑘 𝑦Δ𝑦

(cid:16) 1
2
Δ𝑦

𝑉𝑥,𝑙,𝑚,𝑛

= 𝜆𝑉𝑧,𝑙,𝑚,𝑛.

(D.22c)

(cid:170)
(cid:174)
(cid:174)
(cid:172)

Now, we know the form V𝑙,𝑚,𝑛 = V0𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1)

, so it follows the individual components

will take this form. We substitute this into the (D.22) equations:

sin

(cid:17)

𝑘 𝑦Δ𝑦

(cid:16) 1
2
Δ𝑦

sin

(cid:17)

𝑘 𝑧Δ𝑧

(cid:16) 1
2
Δ𝑧

−2 (cid:169)
(cid:173)
(cid:173)
(cid:171)

−2 (cid:169)
(cid:173)
(cid:173)
(cid:171)

𝑉𝑧,𝑙,𝑚,𝑛𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1)

−

sin

(cid:17)

𝑘 𝑧Δ𝑧

(cid:16) 1
2
Δ𝑧

𝑉𝑦,𝑙,𝑚,𝑛𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1) (cid:170)
(cid:174)
(cid:174)
(cid:172)

= 𝜆𝑉𝑥,𝑙,𝑚,𝑛𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1) , (D.23a)

𝑉𝑥,𝑙,𝑚,𝑛𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1)

−

sin

(cid:17)

𝑘𝑥Δ𝑥

(cid:16) 1
2
Δ𝑥

𝑉𝑧,𝑙,𝑚,𝑛𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1) (cid:170)
(cid:174)
(cid:174)
(cid:172)

= 𝜆𝑉𝑦,𝑙,𝑚,𝑛𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1) , (D.23b)

sin

(cid:17)

𝑘𝑥Δ𝑥

(cid:16) 1
2
Δ𝑥

−2 (cid:169)
(cid:173)
(cid:173)
(cid:171)

𝑉𝑦,𝑙,𝑚,𝑛𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1)

−

sin

Dividing by 𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1)

yields

(cid:17)

𝑘 𝑦Δ𝑦

(cid:16) 1
2
Δ𝑦

𝑉𝑥,𝑙,𝑚,𝑛𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1) (cid:170)
(cid:174)
(cid:174)
(cid:172)
= 𝜆𝑉𝑧,𝑙,𝑚,𝑛𝑒𝑖 (cid:0)𝑘 𝑥𝑙Δ𝑥+𝑘 𝑦𝑚Δ𝑦+𝑘 𝑧𝑛Δ𝑧(cid:1) .

(D.23c)

sin

(cid:17)

𝑘 𝑦Δ𝑦

(cid:16) 1
2
Δ𝑦

𝑉𝑧,𝑙,𝑚,𝑛 −

sin

𝑘 𝑧Δ𝑧

(cid:16) 1
2
Δ𝑧

sin

(cid:17)

𝑘 𝑧Δ𝑧

(cid:16) 1
2
Δ𝑧

𝑉𝑥,𝑙,𝑚,𝑛 −

sin

𝑘𝑥Δ𝑥

(cid:16) 1
2
Δ𝑥

(cid:17)

(cid:17)

𝑉𝑦,𝑙,𝑚,𝑛

𝑉𝑧,𝑙,𝑚,𝑛

(cid:170)
(cid:174)
(cid:174)
(cid:172)

(cid:170)
(cid:174)
(cid:174)
(cid:172)

−2 (cid:169)
(cid:173)
(cid:173)
(cid:171)

−2 (cid:169)
(cid:173)
(cid:173)
(cid:171)

244

= 𝜆𝑉𝑥,𝑙,𝑚,𝑛,

(D.24a)

= 𝜆𝑉𝑦,𝑙,𝑚,𝑛,

(D.24b)

sin

(cid:17)

𝑘𝑥Δ𝑥

(cid:16) 1
2
Δ𝑥

𝑉𝑦,𝑙,𝑚,𝑛 −

sin

(cid:17)

𝑘 𝑦Δ𝑦

(cid:16) 1
2
Δ𝑦

𝑉𝑥,𝑙,𝑚,𝑛

(cid:170)
(cid:174)
(cid:174)
(cid:172)

−2 (cid:169)
(cid:173)
(cid:173)
(cid:171)

= 𝜆𝑉𝑧,𝑙,𝑚,𝑛.

(D.24c)

This can be rearranged

𝜆𝑉𝑥,𝑙,𝑚,𝑛 − 2

sin

(cid:17)

𝑘 𝑧Δ𝑧

(cid:16) 1
2
Δ𝑧

𝑉𝑦,𝑙,𝑚,𝑛 + 2

sin

2

(cid:17)

𝑘 𝑧Δ𝑧

(cid:16) 1
2
Δ𝑧

𝑉𝑥,𝑙,𝑚,𝑛 + 𝜆𝑉𝑦,𝑙,𝑚,𝑛 − 2

(cid:17)

(cid:17)

sin

𝑘 𝑦Δ𝑦

(cid:16) 1
2
Δ𝑦

sin

𝑘𝑥Δ𝑥

(cid:16) 1
2
Δ𝑥

𝑉𝑧,𝑙,𝑚,𝑛 = 0,

(D.25a)

𝑉𝑧,𝑙,𝑚,𝑛 = 0,

(D.25b)

sin

−2

(cid:17)

𝑘 𝑦Δ𝑦

(cid:16) 1
2
Δ𝑦

𝑉𝑥,𝑙,𝑚,𝑛 + 2

sin

(cid:17)

𝑘𝑥Δ𝑥

(cid:16) 1
2
Δ𝑥

𝑉𝑦,𝑙,𝑚,𝑛 + 𝜆𝑉𝑧,𝑙,𝑚,𝑛 = 0.

(D.25c)

This takes the form

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

𝜆

−2

𝑘 𝑧Δ𝑧)
sin ( 1
2
Δ𝑧

2

2

−2

𝑘 𝑧Δ𝑧)
sin ( 1
2
Δ𝑧
𝑘 𝑦Δ𝑦)
sin ( 1
2
Δ𝑦

𝜆

𝑘 𝑥Δ𝑥)
sin ( 1
2
Δ𝑥

2

𝑘 𝑦Δ𝑦)
sin ( 1
2
Δ𝑦
𝑘 𝑥Δ𝑥)
sin ( 1
2
Δ𝑥

−2

𝜆

𝑉𝑥,𝑙,𝑚,𝑛
(cid:169)
(cid:173)
(cid:173)
𝑉𝑦,𝑙,𝑚,𝑛
(cid:173)
(cid:173)
(cid:173)
𝑉𝑧,𝑙,𝑚,𝑛
(cid:171)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

=

0
(cid:170)
(cid:174)
(cid:174)
0
(cid:174)
(cid:174)
(cid:174)
0
(cid:172)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(D.26)

We want the determinant to be zero to allow a nontrivial solution. Using software to compute the

determinant yields:

𝜆3 + 4𝜆

(cid:18)

sin2

(cid:19)

(cid:18) 𝑘𝑥Δ𝑥
2

+ sin2

(cid:19)

(cid:18) 𝑘 𝑦Δ𝑦
2

+ sin2

(cid:19)(cid:19)

(cid:18) 𝑘 𝑧Δ𝑧
2

= 0.

One zero is obviously 𝜆 = 0, but the nontrivial zero is
(cid:32)

𝜆2 = −4

𝑘𝑥Δ𝑥)

sin2( 1
2
Δ𝑥

𝑘 𝑦Δ𝑦)

+

sin2( 1
2
Δ𝑦

+

𝑘 𝑧Δ𝑧)

sin2( 1
2
Δ𝑧

(cid:33)

.

(D.27)

(D.28)

In other words, 𝜆 is purely imaginary. Furthermore, the largest magnitude it can have is for values

of 𝑘𝑥, 𝑘 𝑦, and 𝑘 𝑧 such that the numerator is 1. ie:

(cid:115)

|𝜆| ≤ 2

1
Δ𝑥

+

1
Δ𝑦

+

1
Δ𝑧

.

245

(D.29)

Having established this, we next consider (D.20). Discretizing with respect to time yields

1
𝑐

V𝑛+ 1

2 − V𝑛− 1
Δ𝑡

2

= 𝜆V𝑛.

(D.30)

Let 𝑞 := V

V𝑛 be the solution growth factor for all timesteps 𝑛. This yields the following:

𝑛+ 1
2

V𝑛+ 1

2

2 − V𝑛− 1
V𝑛
1
= 𝑐𝜆Δ𝑡
𝑞

=⇒ 𝑞 −

= 𝜆Δ𝑡

(D.31)

=⇒ 𝑞2 − 𝑐𝜆Δ𝑡𝑞 − 1 = 0.

(a) Stability region over the complex plane

(b) Stability region over the imaginary axis

Figure D.3 The stability region over the complex plane with Δ𝑡 = 𝑐 = 1. The maximum absolute
value between 𝑞− and 𝑞+ (solutions for polynomial’s ± term) is plotted. We see that only values
along the imaginary axis can satisfy |𝑞| ≤ 1, and these are for I [𝜆] ≤ 2𝑐
Δ𝑡 .

Plugging this into the quadratic formula yields the result 𝑞 = 𝑐𝜆Δ𝑡

2 ±

(cid:114)

1 +

(cid:17) 2

(cid:16) 𝑐𝜆Δ𝑡
2

. Note we need

|𝑞| ≤ 1 for this to be stable. From (D.28) we know R [𝜆] = 0.3 So we know 𝜆 is purely imaginary.
3Taflove and Brodwin actually start out with this equation and just assert R [𝜆] = 0 from only (D.31). This makes
some intuitive sense. Say the imaginary component were zero. Any real component would immediately push |𝑞| > 1.
How this would shake out if 𝜆 were complex is somewhat more rickety, though plugging it into matlab (Figure D.3)
does certainly indicate that the real component must be zero in order to satisfy |𝑞| ≤ 1. We get a much clearer view of
this constraint from (D.29).

246

This means, letting 𝜆 = ¯𝜆𝑖:

𝑞𝑞∗ = (cid:169)
(cid:173)
(cid:171)

¯𝜆𝑖Δ𝑡
2

¯𝜆𝑖Δ𝑡
2

(cid:115)

±

1 +

(cid:115)

±

1 −

(cid:19) 2

(cid:18) ¯𝜆𝑖Δ𝑡
2

(cid:19) 2

(cid:18) ¯𝜆Δ𝑡
2

𝑞 =

=

(cid:115)

𝑐

¯𝜆𝑖Δ𝑡
2

±

1 −

+ 1 −

(cid:19) 2

(cid:18) 𝑐 ¯𝜆Δ𝑡
2
1
(cid:0)𝑐 ¯𝜆Δ𝑡(cid:1) 2 ,
2

= −

= 1 −
(cid:114)

(cid:0)𝑐 ¯𝜆Δ𝑡(cid:1) 2.

1
2

=⇒ |𝑞| =

1 −

(cid:19) 2

(cid:18) 𝑐 ¯𝜆Δ𝑡
2

(cid:18) 𝑐 ¯𝜆Δ𝑡
2

(cid:170)
(cid:174)
(cid:172)
(cid:19) 2

−

𝑐 ¯𝜆𝑖Δ𝑡
2

(cid:169)
(cid:173)
(cid:171)

(cid:115)

±

1 −

(cid:19) 2

(cid:18) 𝑐 ¯𝜆Δ𝑡
2

(cid:170)
(cid:174)
(cid:172)

(D.32)

This implies ¯𝜆 ≤ 2𝑐

Δ𝑡 . We can therefore set up the following inequality, knowing |𝜆| ≤ 2

Δ𝑡 by stability

requirement and |𝜆| ≤ 2

Δ𝑥 + 1

Δ𝑦 + 1

Δ𝑧 by the nature of the grid:

(cid:113) 1

(cid:115)

2

1
Δ𝑥

+

1
Δ𝑦

+

1
Δ𝑧

≤

2𝑐
Δ𝑡

=⇒ Δ𝑡 ≤

1
Δ𝑦 + 1
Δ𝑥 + 1
Δ𝑧

(cid:113) 1

𝑐

.

(D.33)

D.1.2 Satisfaction of the Involutions

With the inevitable coding pains that come with bookkeeping a staggered mesh, inevitably we

ask why the grid is staggered. This is due to the satisfaction of the involutions. Assuming a lack of

charge, we wish to enforce ∇ · E = ∇ · B = 0, and the staggered nature of the Yee grid does exactly

this.

Proof. Assume the involutions are satisfied at 𝑡 = 𝑛 and 𝑡 = 𝑛 + 1
now consider ∇ · E𝑛+ 1

2 under the Yee grid:

2, ie ∇ · B𝑛 = ∇ · E𝑛− 1

2 = 0. We

𝐸 𝑛+ 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

− 𝐸 𝑛+ 1
2
𝑥,𝑖− 1
2

, 𝑗,𝑘

Δ𝑥

+

𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

− 𝐸 𝑛+ 1

2
𝑦,𝑖, 𝑗− 1
2

,𝑘

Δ𝑦

247

𝐸 𝑛+ 1
2
𝑧,𝑖, 𝑗,𝑘+ 1
2

− 𝐸 𝑛+ 1

2
𝑥,𝑖, 𝑗,𝑘− 1
2

Δ𝑧

+

.

(D.34)

Applying the update equations (D.3) and (D.6), we see the following:

(cid:32)

𝐸 𝑛− 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

+ 𝑐2Δ𝑡

(cid:32) 𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

−𝐵𝑛

,𝑘
Δ𝑦

𝑧,𝑖+ 1
2

, 𝑗 − 1
2

,𝑘

−

𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

−𝐵𝑛

(cid:33)(cid:33)

, 𝑗,𝑘− 1
2

𝑦,𝑖+ 1
2

Δ𝑧

Δ𝑥

−𝐵𝑛

𝑧,𝑖− 1
2

, 𝑗+ 1
2

,𝑘
Δ𝑦

, 𝑗 − 1
2

,𝑘

−

𝐵𝑛

𝑦,𝑖− 1
2

, 𝑗,𝑘+ 1
2

−𝐵𝑛

(cid:33)(cid:33)

, 𝑗,𝑘− 1
2

𝑦,𝑖− 1
2

Δ𝑧

Δ𝑥

,𝑘− 1
2

−

𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

−𝐵𝑛

,𝑘
Δ𝑥

𝑧,𝑖− 1
2

(cid:33)(cid:33)

, 𝑗+ 1
2

,𝑘

−𝐵𝑛

Δ𝑧

𝑥,𝑖, 𝑗 − 1
2

𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗 − 1
2

,𝑘− 1
2

−

−𝐵𝑛

,𝑘
Δ𝑥

𝑧,𝑖− 1
2

(cid:33)(cid:33)

, 𝑗 − 1
2

,𝑘

Δ𝑦

, 𝑗,𝑘+ 1
2

−

𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

−𝐵𝑛

,𝑘+ 1
2
Δ𝑦

𝑥,𝑖, 𝑗 − 1
2

(cid:33)(cid:33)

,𝑘+ 1
2

−𝐵𝑛

𝑦,𝑖− 1
2

Δ𝑥

(cid:32)

𝐸 𝑛− 1
2
𝑥,𝑖− 1
2

, 𝑗,𝑘

+ 𝑐2Δ𝑡

(cid:32) 𝐵𝑛

𝑧,𝑖− 1
2

−

𝐸 𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

+ 𝑐2Δ𝑡

(cid:32) 𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

(cid:32)

+

−𝐵𝑛

Δ𝑧

𝑥,𝑖, 𝑗+ 1
2

Δ𝑦

(cid:32)

𝐸 𝑛− 1

2
𝑦,𝑖, 𝑗− 1
2

,𝑘

+ 𝑐2Δ𝑡

(cid:32) 𝐵𝑛

𝑥,𝑖, 𝑗 − 1
2

,𝑘+ 1
2

−

(cid:32) 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

+ 𝑐2Δ𝑡

(cid:32)

𝐸 𝑛− 1
2
𝑧,𝑖, 𝑗,𝑘+ 1
2

+

(cid:32)

𝐸 𝑛− 1
2
𝑧,𝑖, 𝑗,𝑘+ 1
2

(cid:32) 𝐵𝑛

𝑦,𝑖+ 1
2

+ 𝑐2Δ𝑡

−

Δ𝑧

, 𝑗,𝑘− 1
2

𝑦,𝑖− 1
2

−𝐵𝑛

Δ𝑥

, 𝑗,𝑘− 1
2

−

𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

Δ𝑧

−𝐵𝑛

,𝑘− 1
2
Δ𝑦

𝑥,𝑖, 𝑗 − 1
2

(cid:33)(cid:33)

,𝑘− 1
2

.

(D.35f)

(D.35a)

(D.35b)

(D.35c)

(D.35d)

(D.35e)

This may be rearranged into two components. The first:

𝐸 𝑛− 1
2
𝑥,𝑖+ 1
2

, 𝑗,𝑘

− 𝐸 𝑛− 1
2
𝑥,𝑖− 1
2

, 𝑗,𝑘

Δ𝑥

+

𝐸 𝑛− 1

2
𝑦,𝑖, 𝑗+ 1
2

,𝑘

− 𝐸 𝑛− 1

2
𝑦,𝑖, 𝑗− 1
2

,𝑘

Δ𝑦

𝐸 𝑛− 1
2
𝑧,𝑖, 𝑗,𝑘+ 1
2

− 𝐸 𝑛− 1

2
𝑥,𝑖, 𝑗,𝑘− 1
2

Δ𝑧

+

,

(D.36)

and the second (with the common 𝑐2Δ𝑡 left out for clarity):

𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

−𝐵𝑛

,𝑘
Δ𝑦

𝑧,𝑖+ 1
2

, 𝑗 − 1
2

,𝑘

𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

−𝐵𝑛

, 𝑗,𝑘− 1
2

𝐵𝑛

𝑧,𝑖− 1
2

, 𝑗+ 1
2

𝑦,𝑖+ 1
2

Δ𝑧

−𝐵𝑛

,𝑘
Δ𝑦

𝑧,𝑖− 1
2

, 𝑗 − 1
2

,𝑘

𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

−𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘− 1
2

Δ𝑧

+

+

𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

−𝐵𝑛

Δ𝑥

𝑦,𝑖− 1
2

, 𝑗,𝑘+ 1
2

𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗+ 1
2

−𝐵𝑛

,𝑘
Δ𝑥

𝑧,𝑖− 1
2

, 𝑗+ 1
2

,𝑘

𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

−𝐵𝑛

,𝑘+ 1
2
Δ𝑦

𝑥,𝑖, 𝑗 − 1
2

,𝑘+ 1
2

−

−

−

−
Δ𝑥

−
Δ𝑦

−
Δ𝑧

𝐵𝑛

𝑥,𝑖, 𝑗 − 1
2

,𝑘+ 1
2

−𝐵𝑛

𝑥,𝑖, 𝑗 − 1
2

,𝑘− 1
2

Δ𝑧

𝐵𝑛

𝑦,𝑖− 1
2

, 𝑗,𝑘+ 1
2

−𝐵𝑛

𝑦,𝑖− 1
2

Δ𝑧

𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗 − 1
2

−𝐵𝑛

,𝑘
Δ𝑥

𝑧,𝑖− 1
2

, 𝑗,𝑘− 1
2

, 𝑗 − 1
2

,𝑘

−
Δ𝑥

−
Δ𝑦

𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘− 1
2

−𝐵𝑛

𝑦,𝑖− 1
2

Δ𝑥

𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

, 𝑗,𝑘− 1
2

−
Δ𝑧

−𝐵𝑛

,𝑘− 1
2
Δ𝑦

𝑥,𝑖, 𝑗 − 1
2

,𝑘− 1
2

.

(D.37)

248

The first term goes to zero by our assumption. The second may be further rearranged to show all

terms that have like components:

(cid:18)

(cid:18)

(cid:18)

(cid:18)

𝐵𝑛

𝑧,𝑖+ 1
2

𝐵𝑛

𝑧,𝑖− 1
2

𝐵𝑛

𝑧,𝑖+ 1
2

− 𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗− 1
2

,𝑘

, 𝑗+ 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖− 1
2

, 𝑗− 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖− 1
2

, 𝑗+ 1
2

,𝑘

, 𝑗+ 1
2

,𝑘

, 𝑗+ 1
2

,𝑘

𝐵𝑛

𝑧,𝑖+ 1
2

, 𝑗− 1
2

,𝑘

− 𝐵𝑛

𝑧,𝑖− 1
2

, 𝑗− 1
2

,𝑘

1
Δ𝑥Δ𝑦

+

−

−

+

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(cid:18)

(cid:18)

(cid:18)

(cid:18)

(cid:18)

(cid:18)

(cid:18)

(cid:18)

(cid:19)

(cid:19)

(cid:19)

(cid:19)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

+

1
Δ𝑥Δ𝑧

+

1
Δ𝑦Δ𝑧

−

+

+

−

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

+

−

−

+

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

− 𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘− 1
2

𝐵𝑛

𝑦,𝑖− 1
2

𝐵𝑛

𝑦,𝑖+ 1
2

𝐵𝑛

𝑦,𝑖+ 1
2

, 𝑗,𝑘+ 1
2

, 𝑗,𝑘+ 1
2

, 𝑗,𝑘− 1
2

− 𝐵𝑛

𝑦,𝑖− 1
2

, 𝑗,𝑘− 1
2

− 𝐵𝑛

𝑦,𝑖− 1
2

, 𝑗,𝑘+ 1
2

− 𝐵𝑛

𝑦,𝑖− 1
2

, 𝑗,𝑘− 1
2

𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

𝐵𝑛

𝑥,𝑖, 𝑗− 1
2

𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘+ 1
2

,𝑘+ 1
2

− 𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

− 𝐵𝑛

𝑥,𝑖, 𝑗− 1
2

− 𝐵𝑛

𝑥,𝑖, 𝑗− 1
2

𝐵𝑛

𝑥,𝑖, 𝑗+ 1
2

,𝑘− 1
2

− 𝐵𝑛

𝑥,𝑖, 𝑗− 1
2

,𝑘− 1
2

,𝑘− 1
2

,𝑘+ 1
2

,𝑘− 1
2

(cid:19)

(cid:19)

(cid:19)

(cid:19)

(cid:19)

(cid:19)

(cid:19)

(cid:19)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(D.38)

The colors indicate which terms cancel out with which. Thus this term reduces to zero as well,
leaving us with the conclusion that ∇ · En+ 1
same for ∇ · B𝑛+1.

2 = 0 under the Yee grid. A similar process shows the

QED

D.2 The Boris Push

Named after Jay Boris, who developed the method in the early 1970s [91], it is still one of the

most popular particle methods owing to its volume preserving property, bound on energy error,

and its simplicity [313]. We will consider both the nonrelativistic and slightly more complicated

relativistic setup.4

D.2.1 Nonrelativistic Boris

We split our updates into two discrete times, Δ𝑡 and Δ𝑡

2 . We have velocity at the half timesteps

and position at the full timesteps, and we will use the Newton-Lorentz equations of motion to do our

updates

4Significantly, the Boris push handles relativity just fine, unlike other structure-preserving methods.

249

𝜃

v+

v−

v(cid:48) × s

v(cid:48)

v− × t

Figure D.4 The Boris Rotation

𝑑
𝑑𝑡

𝛾𝑚v = F = 𝑞(E + v × B),

𝑑
𝑑𝑡 x = v.

(D.39a)

(D.39b)

Here 𝛾 is the relativistic factor 𝛾 :=

(cid:19) − 1

2

(cid:18)(cid:113)

1 − 𝑣2
𝑐2

. For simplicity we let 𝑣 << 𝑐, and so 𝛾 ≈ 1.

Using the second-order accurate center difference scheme for the LHS of (D.39a) and (D.39b) and

averaging the velocity value over two timesteps for the RHS of (D.39a) yields

v𝑛+ 1

2 − v𝑛− 1
Δ𝑡

2

(cid:32)

E𝑛 +

𝑞
𝑚

=

v𝑛+ 1

2

2 + v𝑛− 1
Δ𝑡

x𝑛+1 − x𝑛
Δ𝑡

= v𝑛+ 1
2 .

(cid:33)

,

× B𝑛

(D.40a)

(D.40b)

We can solve this efficiently by the so-called Boris Push, first published in [91] but also described

with simplicity in [80], [35] and many others.5

We rewrite the velocity vector v𝑛± 1

2 = v± ±

𝑞E
𝑚

Δ𝑡
2 . We put this back into (D.40a), which gives

5Also well described in https://www.particleincell.com/2011/vxb-rotation/

250

(cid:0)v+ + v−(cid:1) × B

=

𝑞
v+ − v−
2𝑚
Δ𝑡
𝑞
Δ𝑡
=⇒ v+ − v− =
𝑚
2
𝑞
𝑚 v+ × B = v− +
=⇒ v+ − v+ × t = v− + v− × t

=⇒ v+ −

Δ𝑡
2

(cid:0)v+ × B + v− × B(cid:1)
Δ𝑡
2

𝑞
𝑚 v− × B

(D.41)

Here we let t := Δ𝑡
2

𝑞
𝑚 B. We know v−, to get from v− to v+ is simply the vector v+ − v−. To
begin doing so, we will construct a vector v(cid:48) that is perpendicular to v+ − v− and B. To keep things

perpendicular to B is straightforward, given B is pointing out of the page (in the 𝑧 direction), we

leave v(cid:48) lying on the plane of the page (in the 𝑥, 𝑦 plane). From geometry it is clear v(cid:48) will bisect v+

and v−. We do this by incrementing v− as such:

v(cid:48) := v− + v− × t.

(D.42)

From D.41 we may safely conclude that v(cid:48) bisects v+ and v−. ie this action is to rotate v− halfway

towards v+, though its magnitude is not guaranteed to match that of v+. Thus, we simply need to

rotate it again by some vector s.

v+ = v− + v(cid:48) × s.

(D.43)

We know s will point in the same direction as t, but we need to pin down its magnitude. Recall

that we require (cid:107)v−(cid:107) = (cid:107)v+(cid:107). We can derive the magnitude of s as follows

(cid:107)v+(cid:107) = (cid:107)v− + v(cid:48) × s(cid:107)

= (cid:107)v− + (v− + v− × t) × s(cid:107)

= (cid:107)v− + v− × s + (v− × t) × s(cid:107).

(D.44)

Using the vector identity (A × B) × C = (A · C) B − (B · C) A gives us (v− × t) × s = (v− · s) t −

(t · s) v−. This gives

(cid:107)v+(cid:107) = (cid:107)v− + v− × s + (v− · s) t − (t · s) v−(cid:107).

(D.45)

251

Noting that v− and s are orthogonal and s and t are parallel, and for clarity the letting 𝑡 := (cid:107)t(cid:107),

etc. this yields

(cid:107)v+(cid:107) = (cid:107)v− + v− × s − 𝑡𝑠v−(cid:107)

= (cid:107) (1 − 𝑡𝑠) v− + v− × s(cid:107).

(D.46)

Noting again that v− and s are orthogonal, we know (cid:107)v− × s(cid:107) = (cid:107)s(cid:107)(cid:107)v−(cid:107). We also know, if a

and b are orthogonal vectors, then (cid:107)a − b(cid:107)2 = (cid:107)a + b(cid:107)2 = (cid:107)a(cid:107) + (cid:107)b(cid:107)2. Therefore

(cid:107)v+(cid:107)2 = (cid:107) (1 − 𝑡𝑠) v− + v− × s(cid:107)2

= (cid:107) (1 − 𝑡𝑠) v−(cid:107)2 + (cid:107)v− × s(cid:107)2

= (1 − 𝑡𝑠)2 (cid:107)v−(cid:107)2 + (cid:107)v−(cid:107)2(cid:107)s(cid:107)2.

Remembering our requirement that (cid:107)v+(cid:107) = (cid:107)v−(cid:107), we may conclude

(1 − 𝑡𝑠)2 + 𝑠2 = 1

− 2𝑡𝑠 = 0

=⇒ 1 − 2𝑡𝑠 + 𝑡2𝑠2 + 𝑠2 = 1
1 + 𝑡2(cid:17)
=⇒ 𝑠2 (cid:16)
1 + 𝑡2(cid:17)
=⇒ 𝑠2 (cid:16)
2𝑡𝑠
1 + 𝑡2
2𝑡
.
1 + 𝑡2

=⇒ 𝑠2 =

=⇒ 𝑠 =

= 2𝑡𝑠

(D.47)

(D.48)

D.2.2 The Relativistic Boris Push

What if 𝛾 ≠ 1? Consider time 𝑡𝑛 = 𝑛Δ𝑡. Let 𝑚 be the resting mass of a particle, and let u := 𝛾v,

where 𝛾 =

(cid:113)

1 + 𝑢2

𝑐2 . For convenience we again let boldface denote a vector, nonboldface denote the

magnitude. The relativistic Lorentz force can be written

u𝑛+ 1

2 − u𝑛− 1
Δ𝑡

2

𝑞
𝑚

=

(cid:34)

E𝑛 +

1
𝑐

u𝑛+ 1

2

2 + u𝑛− 1
2𝛾𝑛

Similar to the nonrelativistic case, we let

u− := u𝑛− 1

2 +

𝑞Δ𝑡
2𝑚 E𝑛,

252

(cid:35)

.

× B𝑛

(D.49)

(D.50)

u+ := u𝑛+ 1

2 −

𝑞Δ𝑡
2𝑚 E𝑛.

This is plugged into D.49, which yields

u+ − u−
Δ𝑡

=

𝑞
2𝛾𝑛𝑚𝑐

(cid:0)(u+ + u−) × B𝑛(cid:1) .

Similar to the above, we define

And using similar logic, this yields

u(cid:48) := u− + u− × t.

u+ = u− + u(cid:48) × s.

Here we’ve defined

t :=

s :=

,

𝑞Δ𝑡
2𝛾𝑛𝑚𝑐
2𝑡
1 + 𝑡2

.

(D.51)

(D.52)

(D.53)

(D.54)

(D.55)

(D.56)

We’ve averaged 𝛾𝑛 :=

(cid:113)

1 + (cid:0) 𝑢−
𝑐

(cid:1) 2 =

(cid:113)

1 + (cid:0) 𝑢+
𝑐

(cid:1) 2

After acquiring the next velocity using D.51,

we step the location using Newton’s law

𝑑
𝑑𝑡 x = v,

which yields

x𝑛+1 = x𝑛 + Δ𝑡v𝑛+ 1

2 = x𝑛 + Δ𝑡 u𝑛+ 1
2
𝛾𝑛+ 1
2

.

(D.57)

(D.58)

253

APPENDIX E

PROVING VILLASENOR AND BUNEMAN’S SCHEME CONSERVES CHARGE

We wish to prove that Villasenor and Buneman’s scheme [37] is charge conserving, that is

𝜕 𝜌
𝜕𝑡

+ ∇ · J = 0.

(E.1)

Proof. Consider a charge that has moved from one place to another, as displayed in Figure E.2.

Without loss of generality, we only take the case in which the particle remains in the same cells. If it

moves cells, simply take the subpaths (see figure E.3) that remain within their respective cells and

repeat the following procedure. Defining the following:

w := x − X 𝑗,𝑘 ,

Δw := w
(cid:0)w

¯w :=

𝑛,

𝑛+1 − w
𝑛(cid:1)
𝑛+1 + w
2

,

It will be useful to note the following identity:

Δ𝑤1 ¯𝑤2 + Δ𝑤2 ¯𝑤1 = 𝑤𝑛+1

1

2 − 𝑤𝑛
𝑤𝑛+1

2

𝑤𝑛
1

.

So we have 𝜌 at two points, 𝜌𝑛

and 𝜌𝑛+1, and we can approximate the time derivative as

𝜕 𝜌
𝜕𝑡

=

𝑗,𝑘 − 𝜌𝑛
𝜌𝑛+1
𝑗,𝑘
Δ𝑡

.

(E.2)

(E.3)

(E.4)

We scatter 𝜌 linearly, that is, for a particle of charge 𝑞, we have 𝜌𝑖, 𝑗 = 1

Δ𝑥Δ𝑦 𝑞(1 − 𝑤1)(1 − 𝑤2),

where 𝑤1 and 𝑤2 are the weights from our shape functions, and

1

Δ𝑥Δ𝑦 is our volume. This is shown

in figure E.1. So it follows

254

𝑤1

𝜌𝑖, 𝑗+1

𝜌𝑖+1, 𝑗+1

(1 − 𝑤1 ) 𝑤2

𝑤1𝑤2

Δ𝑦

𝑝

(1 − 𝑤1 ) (1 − 𝑤2 )

𝑤1 (1 − 𝑤2 )

𝜌𝑖, 𝑗

Δ𝑥

𝑤2

𝜌𝑖+1, 𝑗

Figure E.1 A square with an interior point connected to the sides. The distances between the point
and the square’s sides are labeled.

𝑗,𝑘 − 𝜌𝑛
𝜌𝑛+1
𝑗,𝑘
Δ𝑡

=

=

=

=

1
Δ𝑥Δ𝑦
1
Δ𝑥Δ𝑦
1
Δ𝑥Δ𝑦
1
Δ𝑥Δ𝑦

(cid:16)

(cid:16)

𝑞
Δ𝑡
𝑞
Δ𝑡
𝑞
Δ𝑡

𝑞 (cid:0)1 − 𝑤𝑛+1

1

2

(cid:1) − 𝑞 (cid:0)1 − 𝑤𝑛
(cid:1) (cid:0)1 − 𝑤𝑛+1
1
Δ𝑡
2 + 𝑤𝑛+1

1 − 𝑤𝑛+1

1

1 − 𝑤𝑛+1

(cid:1) (cid:0)1 − 𝑤𝑛
2

(cid:1)

2 − 1 + 𝑤𝑛
𝑤𝑛+1

1 + 𝑤𝑛

2 − 𝑤𝑛

1

(cid:17)

𝑤𝑛
2

(E.5)

−Δ𝑤1 − Δ𝑤2 + 𝑤𝑛+1

1

2 − 𝑤𝑛
𝑤𝑛+1

1

𝑤𝑛
2

(cid:17)

(−Δ𝑤1 − Δ𝑤2 + Δ𝑤1 ¯𝑤2 + Δ𝑤2 ¯𝑤1)

Now, considering ∇ · J, we expand using finite difference

𝐽𝑛+ 1
2
1, 𝑗+ 1
2

,𝑘

− 𝐽𝑛+ 1
2
1, 𝑗− 1
2

,𝑘

Δ𝑥

𝐽𝑛+ 1
2
1, 𝑗,𝑘+ 1
2

− 𝐽𝑛+ 1

2
1, 𝑗,𝑘− 1
2

Δ𝑦

+

∇ · J =

(E.6)

Noting 𝐽

1, 𝑗− 1
2

,𝑘 = 𝐽

2, 𝑗,𝑘− 1
2

= 0, given the shape functions we are using, we use Verbonceour’s

description of Villasenor and Buneman’s scheme to scatter the particle [80]:

255

𝑖 + 1, 𝑗 + 1

I1,𝑖+ 1

2

, 𝑗+1

𝑖, 𝑗 + 1

𝑤𝑛+1
1

𝑝𝑛+1

Δ𝑦

I2,𝑖, 𝑗+ 1

2

I2,𝑖+1, 𝑗+ 1

2

𝑤𝑛
1

𝑖, 𝑗

𝑝𝑛
𝑤𝑛
2

I1,𝑖+ 1

2

, 𝑗

Δ𝑥

𝑤𝑛+1
2

𝑖 + 1, 𝑗

Figure E.2 A particle traversing within a single cell. Figure inspired by [80].

𝑖, 𝑗 + 1

I1,𝑖+ 1

2

, 𝑗+1

I2,𝑖, 𝑗+ 1

2

Δ𝑦

𝛿𝑥1

𝛿𝑦1

𝑝𝑛

I1,𝑖+ 3

2

, 𝑗+1

𝑝𝑛+1

𝑖 + 1, 𝑗 + 1

𝛿𝑦2

𝛿𝑥2
I2,𝑖+1, 𝑗+ 1

2

𝑖 + 2, 𝑗 + 1

I2,𝑖+2, 𝑗+ 1

2

𝑖, 𝑗

I1,𝑖+ 1

2

, 𝑗

Δ𝑥

𝑖 + 1, 𝑗

I1,𝑖+ 3

2

, 𝑗

𝑖 + 2, 𝑗

Figure E.3 A particle traversing across a cell boundary. Figure inspired by [80].

256

𝐽
1, 𝑗+ 1
2

,𝑘 =

𝐽
1, 𝑗,𝑘+ 1
2

=

1
Δ𝑦
1
Δ𝑥

𝐼

𝐼

,𝑘 =

1, 𝑗+ 1
2

2, 𝑗,𝑘+ 1
2

=

1
Δ𝑦
1
Δ𝑥

𝑞
Δ𝑡
𝑞
Δ𝑡

(Δ𝑤1 (1 − ¯𝑤2)) ,

(Δ𝑤2 (1 − ¯𝑤1)) .

Plugging these into E.6, we get

∇ · J =

𝑞
Δ𝑡

(cid:32) 1
Δ𝑦 Δ𝑤1 (1 − ¯𝑤2)
Δ𝑥

+

Δ𝑥 Δ𝑤2 (1 − ¯𝑤1)
1
Δ𝑦

(cid:33)

=

=

=

1
Δ𝑥Δ𝑦
1
Δ𝑥Δ𝑦
1
Δ𝑥Δ𝑦

𝑞
Δ𝑡
𝑞
Δ𝑡
𝑞
Δ𝑡

(Δ𝑤1 (1 − ¯𝑤2) + Δ𝑤2 (1 − ¯𝑤1))

(Δ𝑤1 − Δ𝑤1 ¯𝑤2 + Δ𝑤2 − Δ𝑤2 ¯𝑤1)

(Δ𝑤1 + Δ𝑤2 − Δ𝑤1 ¯𝑤2 − Δ𝑤2 ¯𝑤1) .

Thus we see

𝜕 𝜌
𝜕𝑡

= −∇ · J.

(E.7)

(E.8)

QED

257

APPENDIX F

SPLINES

The purpose of this is to make clearer how we derive multidimensional higher order particle weighting.

The linear and quadratic schemes can be found in [35], with [80] giving a brief overview of linear

and good visualizations of higher order schemes as well. A generalization using B-Splines, or Basis

Splines, is discussed in detail by de Boor [314].1

F.1 Constructing Splines

Here we will go over some of the theoretical underpinnings of particle weighting. I am working

primarily with the above footnoted pdf document and [289].

First, it’s important to emphasize that, for our purposes, we are considering a node at the center

of whatever spline we are considering. The particle we are interpolating to or from the node is

relative to this center.

The zeroth order basis function is defined as follows

𝑗, 𝑗+1(𝑥) =
𝐵0

1,

0,

if 𝐾 𝑗 < 𝑥 < 𝐾 𝑗+1,

otherwise.





The higher order basis functions are defined as

𝐵𝑘
𝑗, 𝑗+𝑘+1(𝑥) =

𝑥 − 𝐾 𝑗
𝐾 𝑗+𝑘 − 𝐾 𝑗

𝐵𝑘−1

𝑗, 𝑗+𝑘 (𝑥) +

𝐾 𝑗+𝑘+1 − 𝑥
𝐾 𝑗+𝑘+1 − 𝐾 𝑗+1

𝐵𝑘−1
𝑗+1, 𝑗+𝑘+1(𝑥).

(F.1)

(F.2)

This is all one dimensional, but by taking a tensor product 𝑛 times we can get 𝑛 + 1 dimensions.

We can get a better idea of what this all means by building the first few layers. Consider the

zeroth order basis which may be geometrically represented in as below:2

1This gitlab site is also very useful: https://gitlab.com/Makogan/mathproofs/blob/master/B-splines/b-splines.pdf
2Though this triangle representation is somewhat common, I’m particularly grateful for the gitlab site for making it

more interpretable.

258

𝐵0

𝑗, 𝑗+1

𝐾 𝑗

𝑁 𝑗

𝐾 𝑗+1

Assuming a uniform grid, this width of this spline is Δ𝑥, so if a particle is within a distance
2 , ie − Δ𝑥
2 , then 𝑁 𝑗 takes a value of 1. We can build a grid as follows, with a

of Δ𝑥

2 ≤ 𝑥 𝑝 − 𝐾 𝑗 < Δ𝑥
particle 𝑝 falling between 𝐾 𝑗 and 𝐾 𝑗+1:

𝐵0

𝑗−2, 𝑗−1

𝐵0

𝑗−1, 𝑗

𝐵0

𝑗, 𝑗+1

𝐵0

𝑗+1, 𝑗+2

𝐾 𝑗−2

𝑁 𝑗−2

𝐾 𝑗−1

𝑁 𝑗−1

𝑝

𝐾 𝑗

𝑁 𝑗

𝐾 𝑗+1

𝑁 𝑗+1

𝐾 𝑗+2

Clearly 𝑁 𝑗 will be assigned a value of 1 and the rest will be assigned a value of zero. This may

be normalized by considering




Let’s click things up to linear order. Now, instead of the node being between knots, we see the

𝑥 − 𝐾 𝑗
Δ𝑥

if 0 < ¯𝑥 < 1,

otherwise.

¯𝑥 ≡

(F.3)

𝑗, 𝑗+1

𝐵0

0,

1,

=

(cid:18)

(cid:19)

nodes as lining up with the knots (this will become a pattern, even splines have nodes between knots,

odd splines line up). Our first order spline takes the following form:

𝐵1

𝑗, 𝑗+2 =

𝑥 − 𝐾 𝑗
𝐾 𝑗+1 − 𝐾 𝑗

𝐵0

𝑗, 𝑗+1 +

𝐾𝑖+2 − 𝑥
𝐾 𝑗+2 − 𝐾 𝑗+1

𝐵0

𝑗+1, 𝑗+2

(F.4)

The triangle diagram takes the following appearance, with a particle 𝑝 falling between 𝑁 𝑗−1 and

𝑁 𝑗 again:

𝐵1

𝑗−2, 𝑗

𝐵1

𝑗−1, 𝑗+1

𝐵1

𝑗, 𝑗+2

𝐵0

𝑗−2, 𝑗−1

𝐵0

𝑗−1, 𝑗

𝐵0

𝑗, 𝑗+1

𝐵0

𝑗+1, 𝑗+2

{𝐾, 𝑁 } 𝑗−2

{𝐾, 𝑁 } 𝑗−1

{𝐾, 𝑁 } 𝑗

𝑝

{𝐾, 𝑁 } 𝑗+1

{𝐾, 𝑁 } 𝑗+2

259

We therefore see that the value mapped to 𝑁 𝑗 is

𝑗−1, 𝑗+1( 𝑝) =
𝐵1

=

=

𝑝 − 𝐾 𝑗−1
𝐾 𝑗 − 𝐾 𝑗−1
𝑝 − 𝐾 𝑗−1
𝐾 𝑗 − 𝐾 𝑗−1
𝐾 𝑗+1 − 𝑝
Δ𝑥

.

𝑗−1, 𝑗 ( 𝑝) +
𝐵0

𝐾 𝑗+1 − 𝑝
𝐾 𝑗+1 − 𝐾 𝑗

(0) +

𝐾 𝑗+1 − 𝑝
𝐾 𝑗+1 − 𝐾 𝑗

(1)

𝑗, 𝑗+1( 𝑝)
𝐵0

(F.5)

(F.6)

(F.7)

This corresponds to how far away the particle is from 𝑁 𝑗 . Note it gets larger the closer 𝑝 comes to

𝑁 𝑗 . Correspondingly, we see the value mapped to 𝑁 𝑗 is

𝑗, 𝑗+2( 𝑝) =
𝐵1

=

=

𝑝 − 𝐾 𝑗
𝐾 𝑗+1 − 𝐾 𝑗
𝑝 − 𝐾 𝑗
𝐾 𝑗+1 − 𝐾 𝑗
𝑝 − 𝐾 𝑗
Δ𝑥

.

𝑗, 𝑗+1( 𝑝) +
𝐵0

𝐾 𝑗+2 − 𝑝
𝐾 𝑗+2 − 𝐾 𝑗+1

(1) +

𝐾 𝑗+2 − 𝑝
𝐾 𝑗+2 − 𝐾 𝑗+1

(0)

𝑗+1, 𝑗+2( 𝑝)
𝐵0

(F.8)

(F.9)

(F.10)

This corresponds to how far away 𝑝 is from 𝑁 𝑗+1. It too gets larger the closer 𝑝 comes to 𝑁 𝑗 . We

can also rewrite it centered around the node it is to the right of (which has programmatic benefits).

If 𝐾 𝑗 is the location of the node immediately to the left of 𝑝, then we write

𝑗−1, 𝑗+1 ( 𝑝) =
𝐵1

(cid:0)𝐾 𝑗 + Δ𝑥(cid:1) − 𝑝
Δ𝑥

= 1 −

𝑝 − 𝐾 𝑗
Δ𝑥

(F.11)

All other 𝑁 values will be found to have nothing mapped to them due to the zeroth splines. We

see adding the two nonzero values results in unity.

We again click up to second order splines.

𝐵2

𝑗−2, 𝑗+1

𝐵2

𝑗−1, 𝑗+2

𝐵2

𝑗, 𝑗+3

𝐵1

𝑗−2, 𝑗

𝐵1

𝑗−1, 𝑗+1

𝐵1

𝑗, 𝑗+2

𝐵1

𝑗+1, 𝑗+3

𝐵0

𝑗−2, 𝑗−1
𝑁 𝑗−2

𝐾 𝑗−1

𝐵0

𝑗−1, 𝑗
𝑁 𝑗−1

𝐵0

𝑗, 𝑗+1
𝑁 𝑗

𝑝

𝐾 𝑗

𝐾 𝑗−2

𝐵0

𝑗+1, 𝑗+2
𝑁 𝑗+1

𝐾 𝑗+2

𝐵0

𝑗+2, 𝑗+3
𝑁 𝑗+2

𝐾 𝑗+3

𝐾 𝑗+1

We see nodes 𝑁 𝑗−1, 𝑁 𝑗 , and 𝑁 𝑗+1 will all receive values. The value of 𝑁 𝑗−1 is

260

𝑗−2, 𝑗+1( 𝑝) =
𝐵2

=

+

=

+

=

=

𝑝 − 𝐾 𝑗−2
𝐾 𝑗 − 𝐾 𝑗−2
𝑝 − 𝐾 𝑗−2
𝐾 𝑗−1 − 𝐾 𝑗−2
𝐾 𝑗+1 − 𝑝
𝐾 𝑗+1 − 𝐾 𝑗−1
𝑝 − 𝐾 𝑗−2
𝐾 𝑗−1 − 𝐾 𝑗−2
𝐾 𝑗+1 − 𝑝
𝐾 𝑗+1 − 𝐾 𝑗−1
𝐾 𝑗+1 − 𝑝
𝐾 𝑗+1 − 𝐾 𝑗−1
(cid:18) 𝐾 𝑗+1 − 𝑝
1
Δ𝑥
2

𝐾 𝑗 − 𝑝
𝐾 𝑗+1 − 𝐾 𝑗−1

𝑗−1, 𝑗+1( 𝑝)
𝐵1

𝑗−2, 𝑗−1( 𝑝) +
𝐵0

𝐾 𝑗 − 𝑝
𝐾 𝑗 − 𝐾 𝑗−1

𝑗−1, 𝑗 ( 𝑝) +
𝐵0

𝐾 𝑗+1 − 𝑝
𝐾 𝑗+1 − 𝐾 𝑗

𝑗, 𝑗+1( 𝑝)
𝐵0

(cid:19)

𝑗−1, 𝑗 ( 𝑝)
𝐵0
(cid:19)

𝑗−2, 𝑗 ( 𝑝) +
𝐵1
(cid:18) 𝑝 − 𝐾 𝑗−2
𝐾 𝑗−1 − 𝐾 𝑗−2

(cid:18) 𝑝 − 𝐾 𝑗−1
𝐾 𝑗 − 𝐾 𝑗−1
(cid:18) 𝑝 − 𝐾 𝑗−2
𝐾 𝑗−1 − 𝐾 𝑗−2

(0) +

𝐾 𝑗 − 𝑝
𝐾 𝑗 − 𝐾 𝑗−1

(0) +

𝐾 𝑗+1 − 𝑝
𝐾 𝑗+1 − 𝐾 𝑗

(cid:19)

(1)

=

𝐾 𝑗+1 − 𝑝
2Δ𝑥
(cid:18) 𝐾 𝑗 + Δ𝑥 − 𝑝
Δ𝑥

(cid:19) 2

𝐾 𝑗+1 − 𝑝
Δ𝑥
(cid:18)

=

1
2

1 −

(cid:18) 𝑝 − 𝐾 𝑗−1
𝐾 𝑗 − 𝐾 𝑗−1
𝐾 𝑗+1 − 𝑝
𝐾 𝑗+1 − 𝐾 𝑗
(cid:19) 2
1
2

=

𝑝 − 𝐾 𝑗
Δ𝑥

(cid:19) 2

.

(cid:19)

(F.12)

(0)

Similar, we consider the 𝐵2

𝑗, 𝑗+3 spline:

𝐾 𝑗+3 − 𝑝
𝐾 𝑗+3 − 𝐾 𝑗

𝑗, 𝑗+1( 𝑝) +
𝐵0

𝑗+1, 𝑗+3( 𝑝)
𝐵1
𝐾 𝑗+1 − 𝑝
𝐾 𝑗+2 − 𝐾 𝑗+1

(cid:19)

𝑗+1, 𝑗+2( 𝑝)
𝐵0

𝐾 𝑗+3 − 𝑝
𝐾 𝑗+3 − 𝐾 𝑗+2

𝑗+1, 𝑗+2( 𝑝) +
𝐵0
𝐾 𝑗+1 − 𝑝
𝐾 𝑗+2 − 𝐾 𝑗+1

(1) +

(cid:19)

(0)

(cid:19)

𝑗+2, 𝑗+3( 𝑝)
𝐵0

(F.13)

𝑗, 𝑗+2( 𝑝) +
𝐵1
(cid:18) 𝑝 − 𝐾 𝑗
𝐾 𝑗+1 − 𝐾 𝑗
(cid:18) 𝑝 − 𝐾 𝑗−1
𝐾 𝑗+2 − 𝐾 𝑗+1

(cid:18) 𝑝 − 𝐾 𝑗
𝐾 𝑗+1 − 𝐾 𝑗
(cid:18) 𝑝 − 𝐾 𝑗−1
𝐾 𝑗+2 − 𝐾 𝑗+1

𝑗, 𝑗+3( 𝑝) =
𝐵2

=

+

=

+

=

=

𝑝 − 𝐾 𝑗
𝐾 𝑗+2 − 𝐾 𝑗
𝑝 − 𝐾 𝑗
𝐾 𝑗+2 − 𝐾 𝑗
𝐾 𝑗+3 − 𝑝
𝐾 𝑗+3 − 𝐾 𝑗+1
𝑝 − 𝐾 𝑗
𝐾 𝑗+2 − 𝐾 𝑗
𝐾 𝑗+3 − 𝑝
𝐾 𝑗+3 − 𝐾 𝑗+1
𝑝 − 𝐾 𝑗+1
𝐾 𝑗+2 − 𝐾 𝑗
1
2

(cid:18) 𝑝 − 𝐾 𝑗+1
Δ𝑥

(0) +

(cid:19)

(0)

𝐾 𝑗+3 − 𝑝
𝐾 𝑗+3 − 𝐾 𝑗+2
𝑝 − 𝐾 𝑗+1
Δ𝑥
(cid:19) 2

(cid:18)

=

𝑝 − 𝐾 𝑗+1
𝐾 𝑗+1 − 𝐾 𝑗
1
2

𝑝 − 𝐾 𝑗+1
2Δ𝑥
(cid:18) 𝑝 − (𝐾 𝑗 + Δ𝑥)
Δ𝑥

(cid:19) 2

=

=

1
2

−1 +

𝑝 − 𝐾 𝑗
Δ𝑥

(cid:19) 2

.

261

Lastly, we consider 𝐵 𝑗−1, 𝑗+2:

𝑗−1, 𝑗+2( 𝑝) =
𝐵2

=

+

=

+

=

=

=

=

=

=

=

𝑗−1, 𝑗+1( 𝑝) +
𝐵1
(cid:18) 𝑝 − 𝐾 𝑗−1
𝐾 𝑗 − 𝐾 𝑗−1

(cid:18) 𝑝 − 𝐾 𝑗
𝐾 𝑗+1 − 𝐾 𝑗
(cid:18) 𝑝 − 𝐾 𝑗−1
𝐾 𝑗 − 𝐾 𝑗−1

(cid:18) 𝑝 − 𝐾 𝑗
𝐾 𝑗+1 − 𝐾 𝑗
𝐾 𝑗+1 − 𝑝
𝐾 𝑗+1 − 𝐾 𝑗

𝑝 − 𝐾 𝑗−1
𝐾 𝑗+1 − 𝐾 𝑗−1
𝑝 − 𝐾 𝑗−1
𝐾 𝑗+1 − 𝐾 𝑗−1
𝐾 𝑗+2 − 𝑝
𝐾 𝑗+2 − 𝐾 𝑗
𝑝 − 𝐾 𝑗−1
𝐾 𝑗+1 − 𝐾 𝑗−1
𝐾 𝑗+2 − 𝑝
𝐾 𝑗+2 − 𝐾 𝑗
𝑝 − 𝐾 𝑗−1
𝐾 𝑗+1 − 𝐾 𝑗−1
𝑝 − 𝐾 𝑗−1
2Δ𝑥

𝐾 𝑗+2 − 𝑝
𝐾 𝑗+2 − 𝐾 𝑗−1

𝑗, 𝑗+2( 𝑝)
𝐵1

𝑗−1, 𝑗 ( 𝑝) +
𝐵0

𝑗, 𝑗+1( 𝑝) +
𝐵0

𝐾 𝑗+1 − 𝑝
𝐾 𝑗+1 − 𝐾 𝑗
𝐾 𝑗+2 − 𝑝
𝐾 𝑗+2 − 𝐾 𝑗+1
(cid:19)

(cid:19)

𝑗, 𝑗+1( 𝑝)
𝐵0

𝑗+1, 𝑗+2( 𝑝)
𝐵0

(0) +

𝐾 𝑗+1 − 𝑝
𝐾 𝑗+1 − 𝐾 𝑗
𝐾 𝑗+2 − 𝑝
𝐾 𝑗+2 − 𝐾 𝑗+1

(1)

(cid:19)

(0)

(1) +

+

𝐾 𝑗+2 − 𝑝
𝐾 𝑗+2 − 𝐾 𝑗

𝑝 − 𝐾 𝑗
𝐾 𝑗+1 − 𝐾 𝑗

+

𝐾 𝑗+1 − 𝑝
Δ𝑥
𝑝 − (𝐾 𝑗 − Δ𝑥)
2Δ𝑥
𝑝 − 𝐾 𝑗 + Δ𝑥
2Δ𝑥
(cid:18) 𝑝 − 𝐾 𝑗
2Δ𝑥
(cid:18) 𝑝 − 𝐾 𝑗
2Δ𝑥
(cid:18) 𝑝 − 𝐾 𝑗
Δ𝑥
(cid:18) 𝑝 − 𝐾 𝑗
1
Δ𝑥
2
(cid:18) 𝑝 − 𝐾 𝑗
Δ𝑥

1
2
1
2

+ 1

1
2

(cid:19) 2

(cid:19) 2

+

+

+

= −

= −

+

𝐾 𝑗+2 − 𝑝
2Δ𝑥
(𝐾 𝑗 + Δ𝑥) − 𝑝
Δ𝑥

+

(cid:19)

(cid:19)

𝐾 𝑗 + Δ𝑥 − 𝑝
Δ𝑥
(cid:19) (cid:18) 𝐾 𝑗 − 𝑝
Δ𝑥
(cid:19) (cid:18) 𝐾 𝑗 − 𝑝
Δ𝑥
𝑝 − 𝐾 𝑗
Δ𝑥

(cid:19) (cid:18)

−

+ 1

+ 1

𝑝 − 𝐾 𝑗
Δ𝑥

+

𝑝 − 𝐾 𝑗
Δ𝑥
(𝐾 𝑗 + 2Δ𝑥) − 𝑝
2Δ𝑥
𝐾 𝑗 + 2Δ𝑥 − 𝑝
2Δ𝑥
(cid:18) 𝐾 𝑗 − 𝑝
2Δ𝑥
(cid:18) 𝐾 𝑗 − 𝑝
2Δ𝑥
(cid:18)

+ 1

+ 1

+

+

(cid:19)

𝑝 − 𝐾 𝑗
Δ𝑥
(cid:19) 𝑝 − 𝐾 𝑗
Δ𝑥
(cid:19) 𝑝 − 𝐾 𝑗
Δ𝑥
(cid:19) 𝑝 − 𝐾 𝑗
Δ𝑥

+ 1

+ 1

+

−

𝑝 − 𝐾 𝑗
2Δ𝑥
𝑝 − 𝐾 𝑗
Δ𝑥
(cid:18) 𝑝 − 𝐾 𝑗
Δ𝑥

+

−

(cid:19) 2

.

1
2

−

1
1
2
2
𝑝 − 𝐾 𝑗
Δ𝑥

(cid:19) 2

(cid:18) 𝑝 − 𝐾 𝑗
Δ𝑥

+

1
2

=

3
4

−

This is slightly different than the formula given in Chapter 8 of [35], which is

𝑆2

(cid:0)𝑋 𝑗 − 𝑥(cid:1) =

(cid:34)

1
Δ𝑥

𝑆2

(cid:0)𝑋 𝑗±1 − 𝑥(cid:1) =

(cid:19) 2(cid:35)

(cid:18) 𝑥 − 𝑋 𝑗
Δ𝑥

(cid:19) 2

±

𝑥 − 𝑋 𝑗
Δ𝑥

,

.

−

3
4
(cid:18) 1
2

1
2Δ𝑥

(cid:19)

(F.14)

(F.15)

(F.16)

The 1

Δ𝑥 coefficient is no problem, this is simply the authors considering the density rather than the
value. A few pages prior they have the requirement Δ𝑥 (cid:205) 𝑗 𝑆(𝐾 𝑗 − 𝑥) = 1. In interests of generality

262

(for example if we want to interpolate a field to a particle, we are interested in its value, not its

density), we have neglected this coefficient.

The second difference is more subtle. The location 𝑋 𝑗 of a node 𝑁 𝑗 is the halfway point between

𝐾 𝑗 and 𝐾 𝑗+1, ie 𝑋 𝑗 =

𝐾 𝑗 +𝐾 𝑗+1
2

= 𝐾 𝑗 + Δ𝑥

2 . Let us consider what our formulas look like under this

coordinate frame:

(cid:18)

1 −

(cid:19) 2

𝑝 − 𝐾 𝑗
Δ𝑥

1
2

3
4

−

(cid:18) 𝑝 − 𝐾 𝑗
Δ𝑥

−

1
2

(cid:19) 2

(cid:18)

1
2

−1 +

(cid:19) 2

𝑝 − 𝐾 𝑗
Δ𝑥

=

=

=

=

=

=

=

1
2

1
2

3
4

3
4

1
2

1
2

1
2

(cid:16)

𝑝 −

𝑋 𝑗 − Δ𝑥
2
Δ𝑥

(cid:17)

2

(cid:170)
(cid:174)
(cid:174)
(cid:172)

1 −

(cid:169)
(cid:173)
(cid:173)
(cid:171)
(cid:18) 1
2

−

(cid:19) 2

,

𝑝 − 𝑋 𝑗
Δ𝑥
(cid:16)

𝑋 𝑗 − Δ𝑥
2
Δ𝑥

𝑝 −

− (cid:169)
(cid:173)
(cid:173)
(cid:171)
(cid:18) 𝑝 − 𝑋 𝑗
Δ𝑥

−

(cid:19) 2

,

(cid:17)

2

−

1
2

(cid:170)
(cid:174)
(cid:174)
(cid:172)

(cid:16)

𝑝 −

𝑋 𝑗 − Δ𝑥
2
Δ𝑥

−1 +

(cid:17)

2

(cid:170)
(cid:174)
(cid:174)
(cid:172)

(cid:169)
(cid:173)
(cid:173)
(cid:171)
(cid:18)

−

1
2

+

(cid:18) 1
2

−

(cid:19) 2

𝑝 − 𝑋 𝑗
Δ𝑥
𝑝 − 𝑋 𝑗
Δ𝑥

(cid:19) 2

.

(F.17)

(F.18)

(F.19)

This is, without the 1

Δ𝑥 prefactor, precisely the formulation of Birdsall and Langdon.

This covers the splines we have used in this paper. Of course, with sufficient tedium higher order

splines may be constructed. All of these may be made 𝑑 dimensional by taking the tensor product of

the splines 𝑑 − 1 times. It is also worth noting that a spline of order 𝑛 is equal to the convolution of

the (𝑛 − 1)th spline with the 0th order spline. As 𝑛 → ∞, the spline shape approaches a Gaussian

[35] (see Figure F.1).

Birdsall and Langdon do not provide the variance, so we will show that it is

𝑛+1
12 .

Proof. We do so by induction. First, consider the zeroth order spline 𝐵0(𝑥). We see, from the

263

Figure F.1 The first 10 basis splines. 𝐵𝑛 is generated by convolving 𝐵0 against itself 𝑛 times, eg
𝐵1 = 𝐵0 ∗ 𝐵0. As 𝑛 → ∞, 𝐵𝑛 approaches a Gaussian with variance 𝜎2

𝑛 = 𝑛+1
12 .

standard definition of variance:

∫ ∞

−∞

𝑥2𝐵0(𝑥)𝑑𝑥 =

𝑥3

1
3

(cid:12)
(cid:12)
(cid:12)
(cid:12)

1
2

− 1
2

=

1
24

− −

1
24

=

.

1
12

(F.20)

This shows our base case. We move on to the induction step. Assume true for 𝑛 − 1, we now

consider 𝑛.

∫ ∞

−∞

𝑥2𝐵𝑛 (𝑥)𝑑𝑥 =

=

=

∫ ∞

−∞
∫ ∞

−∞
∫ ∞

−∞

𝑥2 (𝐵𝑛−1 ∗ 𝐵0) (𝑥)𝑑𝑥

(cid:18)∫ ∞

𝑥2

𝐵𝑛−1(𝑠 − 𝑥)𝐵0(𝑠)𝑑𝑠

𝐵0(𝑠)

𝑥2𝐵𝑛−1(𝑠 − 𝑥)𝑑𝑥

−∞
(cid:18)∫ ∞

−∞

264

(cid:19)

(cid:19)

𝑑𝑥

𝑑𝑠

(F.21)

We consider the innermost integral. We will do a change of variables, 𝑡 = 𝑠 − 𝑥, 𝑑𝑡 = 𝑑𝑠. The

integration bounds are still infinity. It is also worth noting, as it will soon be relevant, that all basis

spline functions are even, that their volume is always unity, and that an even function multiplied by

an odd function is odd, so the integral over it is zero. Having said all this, we continue:

∫ ∞

−∞

(𝑠 − 𝑡)2 𝐵𝑛−1(𝑡)𝑑𝑡 =

=

∫ ∞

−∞
∫ ∞

−∞

(cid:16)

𝑠2 − 2𝑠𝑡 + 𝑡2(cid:17)

𝑠2𝐵𝑛−1(𝑡)𝑑𝑡 −

= 𝑠2 − 0 +

𝑛

12

.

𝐵𝑛−1(𝑡)𝑑𝑡
∫ ∞

2𝑠𝑡𝐵𝑛−1(𝑡)𝑑𝑡 +

−∞

∫ ∞

−∞

𝑡2𝐵𝑛−1(𝑡)𝑑𝑡

(F.22)

The last step is justified by the aforementioned properties and the inductive hypothesis. We now see

(F.21) becomes

∫ ∞

−∞

𝐵0(𝑠)

(cid:18)∫ ∞

−∞

𝑥2𝐵𝑛−1(𝑠 − 𝑥)𝑑𝑥

(cid:19)

𝑑𝑠 =

=

=

∫ ∞

−∞
∫ ∞

−∞
1
2

+

𝑛

(cid:17)

12
𝑛

𝑑𝑠

∫ ∞

12

−∞

𝐵0(𝑠)

(cid:16)

𝑠2 +

𝑠2𝐵0(𝑠)𝑑𝑠 +
𝑛

𝑛 + 1
12

.

=

12

𝐵0(𝑠)𝑑𝑠

(F.23)

The last step is simply by definition of the variance, which was already proven, and the volume

of 𝐵0 being unity. We have shown the variance of the 𝑛th order B-spline, 𝜎2

𝑛 , is

proof.

𝑛+1
12 , finishing the

QED

265

APPENDIX G

GREEN’S FUNCTIONS

The following is adapted and expanded out of Duffy’s excellent textbook [275].

The PIC method laid out in chapter 2 relies on the concept of Green’s functions, named after

the mathematician George Green (1793-1841), who laid down the theoretical framework for them

[315], motivated by finding a generic solution to Poisson’s equation. Given how much Chapter 2

relies on them, it is a worthwhile exercise to consider what these functions are and how they work.

Consider the linear differential equation L [𝑢] = 𝑓 (𝑥), where L is a linear differential operator, 𝑓

is a known source function, and 𝑢 is the unknown solution. We let 𝑎 and 𝑏 be the left and right

boundary conditions, respectively. Were we able to find an inverse operator, L−1, we would find the

solution, 𝑢(𝑥) = L−1 [ 𝑓 (𝑥)]. The discrete analog of this is finding the matrix L−1 for L such that

L−1L = 𝐼, where 𝐼 is the identity matrix. The logic is as follows:

Note

∫ 𝑏
𝑎

𝛿(𝑥 − 𝑠) 𝑓 (𝑠)𝑑𝑠 = 𝑓 (𝑥). Now, let 𝐺 be a function such that L [𝐺 (𝑥; 𝑠)] = 𝛿(𝑥 − 𝑠). We

multiply both sides by 𝑓 (𝑠) and integrate, observing

∫ 𝑏

𝑎

L [𝐺 (𝑥; 𝑠)] 𝑓 (𝑠)𝑑𝑠 =

∫ 𝑏

𝑎

𝛿(𝑥 − 𝑠) 𝑓 (𝑠)𝑑𝑠 = 𝑓 (𝑥).

(G.1)

Noting that L is an operator on 𝑥, not 𝑠, we can pull it out of the integrand. L [∫ 𝑏
𝑓 (𝑥). Now, given L [𝑢] = 𝑓 (𝑥), it immediately follows L [𝑢] = L [∫ 𝑏

𝑎

𝐺 (𝑥; 𝑠) 𝑓 (𝑠)𝑑𝑠]. Therefore,

𝑎

𝐺 (𝑥; 𝑠) 𝑓 (𝑠)𝑑𝑠] =

so long as L is invertible, we know

𝑢 =

∫ 𝑏

𝑎

𝐺 (𝑥; 𝑠) 𝑓 (𝑠)𝑑𝑠.

(G.2)

Correspondingly, we will find a function 𝐺 such that L [𝐺] = 𝛿(𝑥), where 𝛿(𝑥) is the Dirac delta

function.

Morse and Feshbach gave a list of properties of Green’s functions 𝐺 (𝑥; 𝜉) applied to Sturm-

Liousville problems:

which are

(cid:20)

𝑑
𝑑𝑥

𝑓 (𝑥)

(cid:21)

𝑑𝑦
𝑑𝑥

+ 𝑝(𝑥)𝑦 = −𝑞(𝑥),

(G.3)

266

• Satisfaction of homogenous equation, ie 𝑞(𝑥) = 0, when 𝑥 ≠ 𝜉.

• Satisfies homogenous boundary conditions, ie 𝐺 (𝑎; 𝜉) = 𝐺 (𝑏; 𝜉) = 0 where 𝑎 and 𝑏 are the

left and rightmost boundaries, respectively, or lim𝑥→±∞ 𝐺 (𝑥; 𝜉) = 0.

• Symmetry in 𝑥 and 𝜉.

• Satisfaction of the following conditions

𝑑𝐺
𝑑𝑥

(cid:12)
(cid:12)
(cid:12)
(cid:12)𝑥=𝜉+

− 𝑑𝐺
𝑑𝑥

(cid:12)
(cid:12)
(cid:12)
(cid:12)𝑥=𝜉 −

= − 1

𝑓 (𝜉) .

G.1 A Simple Example, Poisson’s Equation in 1D

Duffy gives a good example in chapter 2 which will elucidate things. Consider the simple

boundary value problem describing an elastic string along [0, 𝐿] with uniform tensile force 𝑇:

𝑇

𝑑2𝑢
𝑑𝑥2

= 𝑓 (𝑥),

𝑢(0) = 𝑢(𝐿) = 0.

(G.4a)

(G.4b)

Note this corresponds to a Sturm-Liousville problem with 𝑓𝑆𝐿 (𝑥) = −𝑇, 𝑝(𝑥) = 0, and 𝑞(𝑥) =

𝑓𝑝 (𝑥) ( 𝑓𝑝 being the forcing function for the Poisson equation, 𝑓𝑆𝐿 being the 𝑓 function in the

canonical Sturm-Liousville equation (keeping notation as consistent with Duffy as possible)). We

are interested in finding the displacement for a given point 𝜉 and no more. In other words, we

consider the force function 𝛿(𝑥 − 𝜉), solving for the corresponding function 𝐺 (𝑥, 𝜉). This yields

If 𝑥 ≠ 𝜉, then we get

Which yields

𝑇

𝑑2𝐺
𝑑𝑥2

= 𝛿(𝑥 − 𝜉),

𝐺 (0; 𝜉) = 𝐺 (𝐿; 𝜉) = 0.

𝑑2𝐺
𝑑𝑥2

= 0,

𝐺 (𝑥; 𝜉) =

𝑎𝑥 + 𝑏

if x < 𝜉

𝑐𝑥 + 𝑑

if x > 𝜉





267

(G.5a)

(G.5b)

(G.6)

(G.7)

We can use the boundaries to get some of these values. If 𝐺 (0; 𝜉) = 0, then surely 𝑏 = 0.

Likewise, if 𝐺 (𝐿; 𝜉) = 0, then surely 𝑑 = −𝑐𝐿. Thus we get

𝐺 (𝑥; 𝜉) =

𝑎𝑥

if x < 𝜉

𝑐(𝑥 − 𝐿)

if x > 𝜉

(G.8)





But what of the case 𝑥 = 𝜉? We know that the function must be continuous (Morse and Feshbach’s

second requirement, also intuitively makes sense, a broken string is unphyisical). So we know

𝑎𝜉 = 𝑐(𝜉 − 𝐿).

(G.9)

Now, setting 𝑥 = 𝜉, consider

𝑑2𝐺
𝑑𝑥2 (𝜉; 𝜉) = 𝛿(0). Using the property of the delta function, we

integrate over an increasingly small boundary, [𝜉 − 𝜖, 𝜉 + 𝜖]1

𝑇

lim
𝜖→0

𝑑𝐺
𝑑𝑥

(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑥=𝜉+𝜖

𝑥=𝜉−𝜖

= 1.

This implies

(cid:20) 𝑑𝐺
𝑑𝑥

lim
𝜖→0

(𝜉 + 𝜖; 𝜉) −

𝑑𝐺
𝑑𝑥

(𝜉 − 𝜖; 𝜉)

(cid:21)

=

1
𝑇

.

(G.10)

(G.11)

Denote lim𝜖→0 𝜉 ± 𝜖 as 𝜉± Now, we know what the derivatives of 𝐺 are, above and below the 𝜉

point.

𝑑𝐺
𝑑𝑥

𝑑𝐺
𝑑𝑥

(𝜉−; 𝜉) = 𝑎

(𝜉+; 𝜉) = 𝑐

Noting 𝑐 =

𝑎𝜉
𝜉−𝐿 , we get

=⇒

=⇒

1
𝑇

− 𝑎 =

𝑐 − 𝑎 =
𝑎𝜉
𝜉 − 𝐿
𝑎𝐿
𝜉 − 𝐿

=

1
𝑇
𝜉 − 𝐿
𝐿𝑇

=⇒ 𝑎 =

(G.12a)

(G.12b)

(G.13)

1
𝑇

1This gives another justification for the fourth requirement Morse and Feshbach give, which results in the same

thing.

268

Defining 𝑥> := max(𝑥, 𝜉) and 𝑥< := min(𝑥, 𝜉), this finally gives us

𝐺 (𝑥; 𝜉) =

1
𝑇 𝐿

(𝑥> − 𝐿)𝑥<

Thus we know

∫ 𝐿

𝑢(𝑥) =

𝑓 (𝜉)𝐺 (𝑥; 𝜉)𝑑𝜉

0
𝑥 − 𝐿
𝑇 𝐿

=

∫ 𝑥

0

𝑓 (𝜉)𝜉𝑑𝜉 +

𝑥
𝑇 𝐿

∫ 𝐿

𝑥

𝑓 (𝜉)(𝜉 − 𝐿)𝑑𝜉.

(G.14)

(G.15)

This is the solution for any source function 𝑓 . How to compute the integral is of course another

question, and most source functions will not give an analytic solution to these integrals, but we can

apply any quadrature method we wish to arbitrary accuracy.

G.2 The Modified Helmholtz Equation in 1D

Duffy goes over the Helmholtz equation, Δ𝑢 + 𝜆𝑢 = − 𝑓 (𝑥) in chapter 6 of [275]. We are

interested in the modified Helmholtz function, Δ𝑢 − 𝜆𝑢 = −𝑆(𝑥), which in the above work we notate

as

(cid:18)

I −

(cid:19)

Δ

1
𝛼2

𝑢 = 𝑆(x).

Given we dimensionally split the operator to one dimension, this reduces to

(cid:18)

I −

(cid:19)

1
𝛼2

𝜕2
𝜕𝑥2

𝑢 = 𝑓 (𝑥).

(G.16)

(G.17)

This corresponds to a Sturm-Liousville problem with 𝑓 (𝑥) = − 1

𝛼2 , 𝑝(𝑥) = 1, and 𝑞(𝑥) = −𝑆(𝑥).

We now want to find the Green’s function. We can use the roadmap Duffy gives in solving the

Helmholtz equation to do so. Similar to solving for the Green’s function for Poisson’s equation,2 we

consider the following boundary value problem

𝐺 − 1
𝛼2

𝜕2𝐺
𝜕𝑥2 = 𝛿(𝑥 − 𝜉),

−∞ < 𝑥, 𝜉 < ∞.





(G.18)

2Which is just a special case of the Helmholtz equation, modified or otherwise, where 𝜆 = 0.

269

This is slightly different than what we did with Poisson’s equation, in that our boundaries here

are infinite. This is a free space Green’s function. The solutions are as follows

𝐺 (𝑥; 𝜉) = 𝐴𝑒−𝛼(𝑥−𝜉) + 𝐵𝑒𝛼(𝑥−𝜉),

𝑥 < 𝜉,

𝐺 (𝑥; 𝜉) = 𝐶𝑒−𝛼(𝑥−𝜉) + 𝐷𝑒𝛼(𝑥−𝜉),

𝑥 > 𝜉.





(G.19)

The solution must be zero at infinity (Morse and Freshbach’s second requirement). This requires us

to set 𝐴 = 𝐷 = 0. We now consider the continuity requirements. The continuity of the function 𝐺

itself requires 𝐵 = 𝐶. The last requirement listed by Morse and Feschbach yields

𝑑𝐺
𝑑𝑥

(cid:12)
(cid:12)
(cid:12)
(cid:12)𝑥=𝜉+

−

𝑑𝐺
𝑑𝑥

(cid:12)
(cid:12)
(cid:12)
(cid:12)𝑥=𝜉 −

= 𝛼2.

Using these requirements we see

𝑑
𝑑𝑥

(cid:2)𝐶𝑒−𝛼(𝑥−𝜉)(cid:3)

𝑥=𝜉+ −

(cid:2)𝐵𝑒𝛼(𝑥−𝜉)(cid:3)

𝑥=𝜉 − = −𝛼2

𝑑
𝑑𝑥

=⇒ − 𝛼𝐶𝑒−𝛼(𝑥−𝜉)

− 𝛼𝐵𝑒𝛼(𝑥−𝜉)

=⇒ 𝐶𝑒−𝛼(𝑥−𝜉)

+ 𝐵𝑒𝛼(𝑥−𝜉)

= 𝛼

(cid:12)
(cid:12)
(cid:12)
(cid:12)𝑥=𝜉 −

(cid:12)
(cid:12)
(cid:12)
(cid:12)𝑥=𝜉+

(cid:12)
(cid:12)
(cid:12)
(cid:12)𝑥=𝜉+

(cid:12)
(cid:12)
(cid:12)
(cid:12)𝑥=𝜉 −

= −𝛼2

(G.20)

(G.21)

If we consider 𝜉+ = 𝜉 + 𝜖 and 𝜉− = 𝜉 − 𝜖 and let 𝜖 → 0, we see this results in 𝐶 + 𝐵 = 𝛼. W already
know 𝐶 = 𝐵, so it follows 𝐶 = 𝐵 = 𝛼

2 . Thus our Green’s function is

𝐺 (𝑥; 𝜉) =

𝑒−𝛼|𝑥−𝜉 |.

𝛼

2

(G.22)

270

APPENDIX H

DERIVING THE QUADRATURE WEIGHTS

We want to derive the quadrature weights originally established in [268].

We are looking to compute the integral
∫ 𝑥𝑖

𝛼

𝑥𝑖−1

𝑒−𝛼(𝑥𝑖−𝜉)𝑣(𝜉)𝑑𝜉.

(H.1)

To do so, we project 𝑣 to a polynomial 𝑝. Given the set of polynomials is complete, we know

any function 𝑓 (𝑥) = (cid:205)∞
𝑗=0

𝑐 𝑗 𝑥 𝑗

. So we want to use this to get an approximation of 𝑣. First, we let

𝜉 = Δ𝑥𝜏 + 𝑥𝑖−1. So the above integral (H.1), is equal to

𝛼Δ𝑥𝑖

∫ 1

0

𝑒−𝛼Δ𝑥𝑖 (1−𝜏)𝑣(Δ𝑥𝑖𝜏 + 𝑥𝑖−1)𝑑𝜏.

(H.2)

Here we have defined Δ𝑥𝑖

:= 𝑥𝑖 − 𝑥𝑖−1. We want the polynomial 𝑝 to interpolate through the

points 𝑣𝑖−1, 𝑣𝑖, and 𝑣𝑖+1, where 𝑣 𝑗 := 𝑣(𝑥 𝑗 ). For simplicity of integration, we want to construct

𝑝 (𝑥) = 𝑐0 + 𝑐1(𝑥 − 𝑥𝑖−1) + 𝑐2(𝑥 − 𝑥𝑖−1)2, as this gives 𝑝 (Δ𝑥𝑖𝜏 + 𝑥𝑖−1) = 𝑐0 + 𝑐1(Δ𝑥𝑖𝜏) + 𝑐2(Δ𝑥𝑖𝜏)2.

To do so, we need 𝑝(𝑥 𝑗 ) = 𝑣 𝑗∀ 𝑗 ∈ {𝑖 − 1, 𝑖, 𝑖 + 1}. If we are centering around a particular point 𝑐,

we acquire the linear system

1 𝑥𝑖−1 − 𝑐

(𝑥𝑖−1 − 𝑐)2

1

𝑥𝑖 − 𝑐

(𝑥𝑖 − 𝑐)2

1 𝑥𝑖+1 − 𝑐

(𝑥𝑖+1 − 𝑐)2

𝑐0

𝑐1

𝑐2

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

𝑣𝑖−1

𝑣𝑖

𝑣𝑖+1

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

=

.

(H.3)

From our requirements listed above, we want to center this around 𝑐 = 𝑥𝑖−1. For now we will

assume a uniformly staggered grid. Ie 𝑥𝑖 − 𝑥𝑖−1 = Δ𝑥 ∀ 𝑖. This gives the system

Inverting yields

1

0

0

1 Δ𝑥

Δ𝑥2

1 2Δ𝑥 4Δ𝑥2

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

𝑐0

𝑐1

𝑐2

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

=

.

𝑣𝑖−1

𝑣𝑖

𝑣𝑖+1

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

𝑐0

𝑐1

𝑐2

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

=

1
2

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

2

0

0

− 3
4
Δ𝑥
Δ𝑥
Δ𝑥2 − 2
Δ𝑥2

1

− 1
Δ𝑥

1
Δ𝑥2

.

𝑣𝑖−1

𝑣𝑖

𝑣𝑖+1

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

271

(H.4)

(H.5)

So, approximating 𝑣(𝑥) as 𝑝(𝑥) = 𝑐0 + 𝑐1(𝑥 − 𝑥𝑖−1) + 𝑐2(𝑥 − 𝑥𝑖−1)2, where the coefficients are

defined as above, we note

𝑝(Δ𝑥𝜏 + 𝑥𝑖−1) = 𝑐0 + 𝑐1(Δ𝑥𝜏 + 𝑥𝑖−1 − 𝑥𝑖−1) + 𝑐2(Δ𝑥𝜏 + 𝑥𝑖−1 − 𝑥𝑖−1)2

= 𝑐0 + 𝑐1Δ𝑥𝜏 + 𝑐2Δ𝑥2𝜏2.

As such, (H.2) becomes

𝛼Δ𝑥

∫ 1

0

𝑒−𝛼Δ𝑥(1−𝜏) (cid:16)

𝑐0 + 𝑐1Δ𝑥𝜏 + 𝑐2Δ𝑥2𝜏2(cid:17)

𝑑𝜏

(H.6)

Defining 𝜈 := 𝛼Δ𝑥 and 𝑑 := 𝑒−𝜈

, (H.6) will be evaluated

𝑒−𝜈(1−𝜏) 𝑑𝜏 + 𝜈Δ𝑥𝑐1

∫ 1

0

𝜏𝑒−𝜈(1−𝜏) 𝑑𝜏 + 𝜈Δ𝑥2𝑐2

∫ 1

0

𝜏2𝑒−𝜈(1−𝜏) 𝑑𝜏

∫ 1

𝜈

𝑒−𝜈(1−𝜏) (cid:16)

𝑐0 + 𝑐1Δ𝑥𝜏 + 𝑐2Δ𝑥2𝜏2(cid:17)

𝑑𝜏

0
∫ 1

=𝜈𝑐0

=𝜈𝑐0

0
(cid:20) 1
𝜈

+𝑐2𝜈Δ𝑥2

+𝑐1

= − 𝑐0
Δ𝑥
𝜈
Δ𝑥2
𝜈2

+𝑐2

𝑒−𝜈(1−𝜏)

(cid:21) 1

0

(cid:20) 1
𝜈2
(cid:20) 1
𝜈3

+𝑐1𝜈Δ𝑥

(𝜈𝜏 − 1)𝑒−𝜈(1−𝜏)

(cid:21) 1

0

(𝜈2𝜏2 − 2𝜈𝜏 + 2)𝑒−𝜈(1−𝜏)

(cid:21) 1

0

(cid:2)𝑒−𝜈(1−𝜏)(cid:3) 1
0
[(𝜈𝜏 − 1)𝑒−𝜈(1−𝜏)]1
0

(cid:2)(𝜈2𝜏2 − 2𝜈𝜏 + 2)𝑒−𝜈(1−𝜏)(cid:3) 1
0

.

This becomes

𝑐0 [1 − 𝑑] + 𝑐1

=𝑐0 [1 − 𝑑] + 𝑐1

[(𝜈 − 1) − −𝑑] + 𝑐2
Δ𝑥2
𝜈2

[𝜈 − 1 + 𝑑] + 𝑐2

=𝑣𝑖−1 [1 − 𝑑] +

(3𝑣𝑖−1 + 4𝑣𝑖 − 𝑣𝑖+1)

Δ𝑥
𝜈
Δ𝑥
𝜈
(cid:18) 1
2Δ𝑥

Δ𝑥2
𝜈2

[(𝜈2 − 2𝜈 + 2) − 2𝑑]

[𝜈2 − 2𝜈 + 2 − 2𝑑]
(cid:19) Δ𝑥
𝜈

[𝜈 − 1 + 𝑑]

(cid:18)

+

1
2Δ𝑥2

𝑣𝑖−1 − 2𝑣𝑖 +

1
2

𝑣𝑖+1

(cid:19) Δ𝑥2
𝜈2

[𝜈2 − 2𝜈 + 2 − 2𝑑]

=𝑣𝑖−1 [1 − 𝑑] + ((3𝑣𝑖−1 + 4𝑣𝑖 − 𝑣𝑖+1))

1
2𝜈

[𝜈 − 1 + 𝑑]

+ (𝑣𝑖−1 − 2𝑣𝑖 + 𝑣𝑖+1)

1
2𝜈2

[𝜈2 − 2𝜈 + 2 − 2𝑑]

272

(H.7)

(H.8)

We will sort these out by components. First, the 𝑣𝑖−1 component:

(cid:18)

𝑣𝑖−1

1 − 𝑑 − 3

𝜈 − 1 + 𝑑
2𝜈

+

𝜈2 − 2𝜈 + 2 − 2𝑑
2𝜈2

(cid:19)

(cid:19)

=𝑣𝑖−1

=𝑣𝑖−1

=𝑣𝑖−1

=𝑣𝑖−1

=𝑣𝑖−1

=𝑣𝑖−1

=𝑣𝑖−1

=𝑣𝑖−1

(cid:18) 2𝜈2 − 2𝑑𝜈2
2𝜈2

− 3

𝜈2 − 𝜈 + 𝑑𝜈
2𝜈2

+

𝜈2 − 2𝜈 + 2 − 2𝑑
2𝜈2

(cid:19)

(cid:18)

(cid:18)

(cid:18)

(cid:19)

(cid:19)

+

+

+

−

−𝑑 +

2𝑑𝜈2
2𝜈2

(cid:18) 2𝜈2 − 2𝑑𝜈2 − 3𝜈2 + 3𝜈 − 3𝑑𝜈 + 𝜈2 − 2𝜈 + 2 − 2𝑑
2𝜈2
2 − 2𝑑
2𝜈2
1 − 𝑑
𝜈2
2𝜈 − 2𝑑𝜈
2𝜈2
1 − 𝑑
𝜈
1 − 𝑑
𝜈
1 − 𝑑
𝜈2

𝜈 − 3𝑑𝜈
2𝜈2
2𝜈 − 𝜈 − 3𝑑𝜈
2𝜈2
−𝜈 − 𝑑𝜈
2𝜈2
−𝜈 − 𝑑𝜈
2𝜈2
1 + 𝑑
2𝜈
1 − 𝑑
𝜈

1 − 𝑑
𝜈2
(cid:19)
1 − 𝑑
𝜈2
1 + 𝑑
2𝜈

1 − 𝑑
𝜈2
(cid:19)

−𝑑 −

−𝑑 +

−𝑑 +

−𝑑 +

−

+

+

+

+

+

+

+

(cid:18)

(cid:18)

(cid:18)

(cid:19)

(cid:19)

.

We define 𝑄 := −𝑑 + 1−𝑑

𝜈2 − 1+𝑑
us the final form of the 𝑣𝑖−1 coefficient, 𝑄 + 𝑅.

𝜈 and 𝑅 := 1−𝑑

2𝜈 for reasons that will become apparent, which gives

Taking the 𝑣𝑖 components:

[𝜈 − 1 + 𝑑] − 2

4

1
2𝜈
2𝜈2 − 2𝜈 + 2𝑑𝜈
𝜈2

1
2𝜈2
𝜈2 − 2𝜈 + 2 − 2𝑑
𝜈2

−

[𝜈2 − 2𝜈 + 2 − 2𝑑]

=

=

=

2𝜈2 − 2𝜈 + 2𝑑𝜈 − 𝜈2 + 2𝜈 − 2 + 2𝑑
𝜈2
2𝑑𝜈
𝜈2

𝜈2
𝜈2

+

+

2𝑑 − 2
𝜈2
1 − 𝑑
𝜈2
1 − 𝑑
𝜈2
1 − 𝑑
𝜈

+

+

− 2

=1 − 2

=1 − 2

=1 −

2𝑑
𝜈
𝑑 − 1
+
𝜈
(cid:18) 1 − 𝑑
𝜈2

𝑑 + 1
𝜈
1 + 𝑑
2𝜈

−

(cid:19)

We recognize the second component, we’ll defined 𝑃 := 1 − 1−𝑑

𝜈 . So the 𝑣𝑖 component is 𝑃 − 2𝑅.

273

Taking the 𝑣𝑖+1 components:

−

1
2𝜈

[𝜈 − 1 + 𝑑] +

1
2𝜈2

[𝜈2 − 2𝜈 + 2 − 2𝑑]

=

=

=

=

=

=

−𝜈2 + 𝜈 − 𝑑𝜈
2𝜈2

+

𝜈2 − 2𝜈 + 2 − 2𝑑
2𝜈2

−𝜈2 + 𝜈 − 𝑑𝜈 + 𝜈2 − 2𝜈 + 2 − 2𝑑
2𝜈
−𝜈 − 𝑑𝜈 + 2 − 2𝑑
2𝜈2

−𝜈 − 𝑑𝜈
2𝜈2
−1 − 𝑑
2𝜈
1 − 𝑑
𝜈2

−

+

2 − 2𝑑
2𝜈2
1 − 𝑑
𝜈2
1 + 𝑑
2𝜈

.

+

This is exactly 𝑅. Thus we have our quadrature weights for J :

𝑗 [𝑣] = 𝑃𝑣(𝑥 𝑗 ) + 𝑄𝑣(𝑥 𝑗−1) + 𝑅 (cid:0)𝑣(𝑥 𝑗+1) − 2𝑣(𝑥 𝑗 ) + 𝑣(𝑥 𝑗−1)(cid:1) .
J 𝐿

(H.9)

The quadrature weights for J 𝑅

may be similarly derived.

274