The space charge effect is one of the most important collective effects in beam dynamic studies. In many cases, numerical simulations are inevitable in order to get a clear under-standing of this effect. The particle-particle interaction algorithms and the article-in-cell algorithms are widely used in space charge effect simulations. But they both have difficulties in dealing with highly correlated beams with abnormal distributions or complicated geometries. We developed a new algorithm to calculate the three dimensional self-field between charged particles by combining the differential algebra (DA) techniques with the fast multipole method (FMM). The FMM hierarchically decomposes the whole charged domain into many small regions. For each region it uses multipole expansions to represent the potential/field contributions from the particles far away from the region and then converts the multipole expansions into a local expansion inside the region. The potential/field due to the far away particles is calculated from the expansions and the potential/field due to the nearbyparticles is calculated from the Coulomb force law. The DA techniques are used in the calculation, translation and converting of the expansions. The new algorithm scales linearly with the total number of particles and it is suitable for any arbitrary charge distribution. Using the DA techniques, we can calculate both the potential/field and its high order derivatives, which will be useful for the purpose of including the space charge effect into transfer mapsin the future.We first present the single level FMM, which decomposes the whole domain into boxes of the same size. It works best for charge distributions that are not overly non-uniform. Then we present the multilevel fast multipole algorithm (MLFMA), which decomposes the whole domain into different sized boxes according to the charge density. Finer boxes are generated where the higher charge density exists; thus the algorithm works for any arbitrary charge distribution. A Message Passing Interface (MPI) based parallel version of the MLFMA is developed, so that we can take advantage of cluster machines and enhance our simulation ability. The algorithms are described in details and the numerical experimental results about the efficiency and accuracy of the algorithm are presented and discussed. In the end, we give an example of using this algorithm in the photo emission process simulation. Some simulation related topics are discussed, such as: how to choose the proper units for the variables in the beam dynamic equations, how to transform the space charge fields from the bunch frame to the laboratory frame, and how to avoid artificial collisions between the charged particles.
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