Quadratic conservative scheme for relativistic Vlasov--Maxwell system

TL;DR

This paper introduces a quadratic conservative numerical scheme for the relativistic Vlasov--Maxwell system that strictly preserves charge, momentum, and energy conservation laws, addressing a long-standing issue in plasma simulations.

Contribution

The paper presents a novel quadratic conservative discretization method that ensures exact conservation laws for the relativistic Vlasov--Maxwell system, improving the accuracy of plasma simulations.

Findings

Successfully conserves charge, momentum, and energy in simulations

Validates the scheme with relativistic two-stream and Weibel instabilities

Enables first-principles studies of mesoscopic and macroscopic plasma physics

Abstract

For more than half a century, most of the plasma scientists have encountered a violation of the conservation laws of charge, momentum, and energy whenever they have numerically solve the first-principle equations of kinetic plasmas, such as the relativistic Vlasov--Maxwell system. This fatal problem is brought by the fact that both the Vlasov and Maxwell equations are indirectly associated with the conservation laws by means of some mathematical manipulations. Here we propose a quadratic conservative scheme, which can strictly maintain the conservation laws by discretizing the relativistic Vlasov--Maxwell system. A discrete product rule and summation-by-parts are the key players in the construction of the quadratic conservative scheme. Numerical experiments of the relativistic two-stream instability and relativistic Weibel instability prove the validity of our computational theory, and the proposed strategy will open the doors to the first-principle studies of mesoscopic and macroscopic plasma physics.