Local Energy Conservation Law for Spatially-Discretized Hamiltonian Vlasov-Maxwell System

TL;DR

This paper proves that spatially-discretized Hamiltonian Vlasov-Maxwell systems inherently preserve local energy conservation laws in space-time, extending the understanding of structure-preserving algorithms in plasma physics.

Contribution

It establishes a general theoretical link between global and local conservation laws in Hamiltonian discretizations of the Vlasov-Maxwell system.

Findings

Spatial discretization preserves local energy conservation.

Global conservation implies local conservation in local algorithms.

Supports structure-preserving geometric algorithms.

Abstract

Structure-preserving geometric algorithm for the Vlasov-Maxwell (VM) equations is currently an active research topic. We show that spatially-discretized Hamiltonian systems for the VM equations admit a local energy conservation law in space-time. This is accomplished by proving that for a general spatially-discretized system, a global conservation law always implies a discrete local conservation law in space-time when the algorithm is local. This general result demonstrates that Hamiltonian discretizations can preserve local conservation laws, in addition to the symplectic structure, both of which are the intrinsic physical properties of infinite dimensional Hamiltonian systems in physics.