Hamiltonian time integrators for Vlasov-Maxwell equations

Abstract

Hamiltonian integration methods for the Vlasov-Maxwell equations are developed by a Hamiltonian splitting technique. The Hamiltonian functional is split into five parts, i.e., the electrical energy, the magnetic energy, and the kinetic energy in three Cartesian components.Each of the subsystems is a Hamiltonian system with respect to the Morrison-MarsdenWeinstein Poisson bracket and can be solved exactly. Compositions of the exact solutions yield Poisson structure preserving, or Hamiltonian, integration methods for the VlasovMaxwell equations, which have superior long-term fidelity and accuracy.

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

“…The whole system is solved using the Hamiltonian splitting method discovered by He et al [4], which was been successfully adopted in constructing symplectic particle-in-cell schemes [3]. Because of its structure preserving and explicit nature, this algorithm is especially suitable for large-scale simulations for physics problems that are multi-scale and require long-term fidelity and accuracy.…”

“…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.…”

“…High-order Whitney interpolating forms [3] are used to ensure the gauge symmetry of Maxwell's equations. The discrete Poisson bracket for the ideal two-fluid system is obtained by the similar technique that is used in obtaining the discrete Vlasov-Maxwell bracket [3], and the final numerical scheme is constructed by the powerful Hamiltonian split method [3,4,47]. We note that for the existing structure preserving method for the compressible fluid [62], all fields are discretized over a moving mesh, which does not apply to cases where the mesh deforms significantly during the evolution, such as in a rotating fluid.…”

“…With the assistance of Whitney interpolating forms [1][2][3], this scheme preserves the gauge symmetry of the electromagnetic field, and the pressure field is naturally derived from the discrete internal energy.The whole system is solved using the Hamiltonian splitting method discovered by He et al [4], which was been successfully adopted in constructing symplectic particle-in-cell schemes [3]. Because of its structure preserving and explicit nature, this algorithm is especially suitable for large-scale simulations for physics problems that are multi-scale and require long-term fidelity and accuracy.…”

Multi-scale full-orbit analysis on phase-space behavior of runaway electrons in tokamak fields with synchrotron radiation

“…[26] and use a relativistic volume-preserving algorithm (VPA) [32]. As a geometric algorithm, the relativistic VPA possesses long-term numerical accuracy and stability [24,25,[31][32][33][34][35][36][37][38][39][40][41]. The secular full-orbit dynamics of runaway electrons is obtained through directly solving the Lorentz force equations.…”