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Recent development of structure-preserving geometric particle-in-cell (PIC)algorithms for Vlasov-Maxwell systems is summarized. With the arriving of 100petaflop and exaflop computing power, it is now possible to carry out directsimulations of multi-scale plasma dynamics based on first-principles. However,standard algorithms currently adopted by the plasma physics community do notpossess the long-term accuracy and fidelity required in these large-scalesimulations. This is because conventional simulation algorithms are based onnumerically solving the underpinning differential (or integro-differential)equations, and the algorithms used in general do not preserve the geometric andphysical structures of the systems, such as the local energy-momentumconservation law, the symplectic structure, and the gauge symmetry. As aconsequence, numerical errors accumulate coherently with time and long-termsimulation results are not reliable. To overcome this difficulty and tohardness the power of exascale computers, a new generation ofstructure-preserving geometric PIC algorithms have been developed. This newgeneration of algorithms utilizes modern mathematical techniques, such asdiscrete manifolds, interpolating differential forms, and non-canonicalsymplectic integrators, to ensure gauge symmetry, space-time symmetry and theconservation of charge, energy-momentum, and the symplectic structure. Thesehighly desired properties are difficult to achieve using the conventional PICalgorithms. In addition to summarizing the recent development and demonstratingpractical implementations, several new results are also presented, including astructure-preserving geometric relativistic PIC algorithm, the proof of thecorrespondence between discrete gauge symmetry and discrete charge conservationlaw, and a reformulation of the explicit non-canonical symplectic algorithm forthe discrete ...

Structure-Preserving Geometric Particle-in-Cell Methods for Vlasov-Maxwell Systems