Hamiltonian splitting for the Vlasov–Maxwell equations

Abstract

Abstract. -A new splitting is proposed for solving the Vlasov-Maxwell system. This splitting is based on a decomposition of the Hamiltonian of the Vlasov-Maxwell system and allows for the construction of arbitrary high order methods by composition (independent of the specific deterministic method used for the discretization of the phase space). Moreover, we show that for a spectral method in space this scheme satisfies Poisson's equation without explicitly solving it. Finally, we present some examples in the context of the time evolution of an electromagnetic plasma instability which emphasizes the excellent behavior of the new splitting compared to methods from the literature.

Explicit high-order noncanonical symplectic algorithms for ideal two-fluid systems

“…The development of geometric algorithms for these systems can be challenging. However, recently significant advances have been achieved in the development of structure preserving geometric algorithms for charged particle dynamics [37][38][39][40][41][42][43][44][45][46][47][48][49][50], the Vlasov-Maxwell systems [3,4,[51][52][53][54][55][56][57][58][59][60][61], compressible ideal MHD [62,63], and incompressible fluids [64,65]. All of these methods have demonstrated unparalleled long-term numerical accuracy and fidelity compared with conventional methods.…”

Hamiltonian time integrators for Vlasov-Maxwell equations

“…In modern plasma physics and accelerator physics, numerical integration of the Vlasov-Maxwell equations is an important tool for theoretical studies, and varieties of numerical algorithms have been developed. Recently, geometric integration methods [1][2][3][4], which are designed in the spirit of preserving the intrinsic structures of a dynamical system, have been developed for plasma physics applications [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. By preserving properties such as the Poisson structure of a Hamiltonian system and the invariant volume form of a source-free system, geometric integration methods usually generate numerical results with superior long-term behavior compared to other methods [4,20], and are thus more suitable for large-scale, long-term simulations.…”

“…It is known that the Vlasov-Maxwell system is a Hamiltonian system with respect to a Poisson bracket [21][22][23]. In a recent paper [18], Crouseilles, Einkemmer, and Faou proposed an innovative Hamiltonian splitting method for the Vlasov-Maxwell equations based on a bracket first suggested in Ref. [21].…”

“…[24] that the bracket adopted in Ref. [18] is not Poisson, because it does not satisfy the Jacobi identity [22,23]. Very disappointedly, if the Hamiltonian splitting method proposed in Ref.…”

“…Very disappointedly, if the Hamiltonian splitting method proposed in Ref. [18] is applied with the correct Poisson bracket, a.k.a. the MorrisonMarsden-Weinstein (MMW) bracket [22][23][24][25], one of the subsystems cannot be solved exactly in general.…”