Superconvergence and accuracy enhancement of discontinuous Galerkin solutions for Vlasov-Maxwell equations

TL;DR

This paper demonstrates that a post-processing technique can significantly enhance the accuracy and resolution of discontinuous Galerkin methods for solving the Vlasov-Maxwell system, achieving superconvergence and better capturing plasma phenomena.

Contribution

It proves superconvergence of DG solutions for the Vlasov-Maxwell system and shows how post-processing improves accuracy and resolution in plasma simulations.

Findings

Superconvergence order of (2k+1/2) in negative norm for DG solutions.

Post-processing enhances accuracy and resolution of electromagnetic fields and distribution functions.

Numerical tests confirm improved performance in plasma instability simulations.

Abstract

This paper considers the discontinuous Galerkin (DG) methods for solving the Vlasov-Maxwell (VM) system, a fundamental model for collisionless magnetized plasma. The DG methods provide accurate numerical description with conservation and stability properties. However, to resolve the high dimensional probability distribution function, the computational cost is the main bottleneck even for modern-day supercomputers. This work studies the applicability of a post-processing technique to the DG solution to enhance its accuracy and resolution for the VM system. In particular, we prove the superconvergence of order $(2k+\frac{1}{2})$ in the negative order norm for the probability distribution function and the electromagnetic fields when piecewise polynomial degree $k$ is used. Numerical tests including Landau damping, two-stream instability and streaming Weibel instabilities are considered showing the performance of the post-processor.