Explicit high-order non-canonical symplectic particle-in-cell algorithms for Vlasov-Maxwell systems

Abstract

Explicit high-order non-canonical symplectic particle-in-cell algorithms for classical particle-field systems governed by the Vlasov-Maxwell equations are developed. The algorithm conserves a discrete non-canonical symplectic structure derived from the Lagrangian of the particle-field system, which is naturally discrete in particles. The electromagnetic field is spatially-discretized using the method of discrete exterior calculus with high-order interpolating differential forms for a cubic grid. The resulting time-domain Lagrangian assumes a non-canonical symplectic structure. It is also gauge invariant and conserves charge. The system is then solved using a structure-preserving splitting method discovered by He et al., which produces five exactly-soluble sub-systems, and high-order structure-preserving algorithms follow by combinations. The explicit, high-order, and conservative nature of the algorithms is especially suitable for long-term simulations of particle-field systems with extremely large number of degrees of freedom on massively parallel supercomputers.The algorithms have been tested and verified by the two physics problems, i.e., the nonlinear Landau damping and the electron Bernstein wave.

Finite-dimensional collisionless kinetic theory

“…First, by using a field discretization as in Refs. 4,6, and 7, gauge invariance and charge conservation will be preserved exactly. Second, by using the macroparticle formalism from Refs.…”

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.…”

“…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.…”

“…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.…”