Structure-preserving geometric particle-in-cell methods for Vlasov-Maxwell systems

Abstract

Recent development of structure-preserving geometric particle-in-cell (PIC) algorithms for Vlasov-Maxwell systems is summarized. With the arriving of 100 petaflop and exaflop computing power, it is now possible to carry out direct simulations of multi-scale plasma dynamics based on first-principles. However, standard algorithms currently adopted by the plasma physics community do not possess the long-term accuracy and fidelity required in these large-scale simulations. This is because conventional simulation algorithms are based on numerically solving the underpinning differential (or integro-differential) equations, and the algorithms used in general do not preserve the geometric and physical structures of the systems, such as the local energy-momentum conservation law, the symplectic structure, and the gauge symmetry. As a consequence, numerical errors accumulate coherently with time and long-term simulation results are not reliable. To overcome this difficulty and to hardness the power of exascale computers, a new generation of structure-preserving geometric PIC algorithms have been developed. This new generation of algorithms utilizes modern mathematical techniques, such as discrete manifolds, interpolating differential forms, and non-canonical symplectic integrators, to ensure gauge symmetry, space-time symmetry and the conservation of charge, energy-momentum, and the symplectic structure. These highly desired properties are difficult to achieve using the conventional PIC algorithms. In addition to summarizing the recent development and demonstrating practical implementations, several new results are also presented, including a structure-preserving geometric relativistic PIC algorithm, the proof of the correspondence between discrete gauge symmetry and discrete charge conservation law, and a reformulation of the explicit non-canonical symplectic algorithm for the discrete Poisson bracket using the variational approach. Numerical examples are given to verify the advantages of the structure-preserving geometric PIC algorithms in comparison with the conventional PIC methods. * hongqin@ustc.edu.cn i.e., J div d * I,J J J,l + Dt (ρ I,l ) = 0 ,where div d * is the transpose of −∇ d defined in Eq. (A11).If Eq. (149) is satisfied initially, according to Eq. (148),we can see that Eq. (149) will be automatically satisfied for all time steps and there is no need to solve it.Similarly, we can prove that the discrete Lagrangians of the particle-field systems defined by Eqs. (159) and (167) also admit discrete charge conservation law due to the fact that they are gauge symmetric. Numeric examplesUsing the C programming language and the Message Passing Interface (MPI), we have implemented the 2nd-order explicit Hamiltonian splitting PIC (EHSPIC) algorithm, the 1st-order variational symplectic charge-conservative relativistic PIC (VSCRPIC) algorithm, and the conventional Boris-Yee PIC (BYPIC) scheme for comparison study. Among these PIC schemes, the VSCRPIC algorithm is relativistic, and both EHSPIC and VSCRPIC algorithms a...

Machine learning and serving of discrete field theories

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