In this thesis we will analyze and enhance two schemes for kinetic equations. Namely the discontinous Galerkin (DG) methods for solving the the Vlasov-Maxwell (VM) system and a hybrid method for solving the time-dependent radiation transport equation (RTE). In Chapter 2 we will consider the DG methods for solving the VM system, a fundamental model for collisionless magnetized plasma. The DG methods provide accurate numerical description with conservation and stability properties. However, to resolve the high dimensional probability distribution function, the computational cost is the main bottleneck even for modern-day supercomputers. The first part of this thesis studies the applicability of a post-processing technique to the DG solution to enhance its accuracy and resolution for the VM system. This postprocessor is applied at the final time of the simulation, and its cost is negligible, it succeeds by producing a high-resolution solution with the same cost of computing a low-resolution one, thus saving computational time in the process. In particular, we prove the superconvergence of order $(2k+\frac{1}{2})$ in the negative order norm for the probability distribution function and the electromagnetic fields when piecewise polynomial degree $k$ is used. Numerical tests including Landau damping, two-stream instability and streaming Weibel instabilities are considered showing the performance of the post-processor. This is based on joint work with Yingda Cheng, Juntao Huang and Jennyfer Ryan [1] In Chapter 3, we prove rigorous error estimates for a hybrid method introduced in [2] for solving the time-dependent RTE. The method relies on a splitting of the kinetic distribution function for the radiation into uncollided and collided components. A high-resolution method (in angle) is used to approximate the uncollided components and a low-resolution method is used to approximate the the collided component. After each time step, the kinetic distribution is reinitialized to be entirely uncollided. For this analysis, we consider a mono-energetic problem on a periodic domains, with constant material cross-sections of arbitrary size. We assume the uncollided equation is solved exactly and the collided part is approximated in angle via a spherical harmonic expansion ($\Peqn$ method). Using a non-standard set of semi-norms, we obtain estimates of the form $C(\e,\sigma,\dt)N^{-s}$ where $s\geq 1$ denotes the regularity of the solution in angle, $\e$ and $\sigma$ are scattering parameters, $\dt$ is the time-step before reinitialization, and $C$ is a complicated function of $\e$, $\sigma$, and $\dt$. These estimates involve analysis of the multiscale RTE that includes, but necessarily goes beyond, usual spectral analysis. We also compute error estimates for the monolithic $\Peqn$ method with the same resolution as the collided part in the hybrid. Our results highlight the benefits of the hybrid approach over the monolithic discretization in both highly scattering and streaming regimes. This is based in a joint work with Cory D. Hauck and Victor Decaria [3]
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