A Locally Divergence-Free Oscillation-Eliminating Discontinuous Galerkin Method for Ideal Magnetohydrodynamic Equations

TL;DR

This paper introduces a high-order discontinuous Galerkin method for ideal compressible MHD equations that effectively eliminates oscillations near discontinuities and maintains a divergence-free magnetic field, improving simulation stability and accuracy.

Contribution

The paper develops a novel locally divergence-free oscillation-eliminating DG method that is easily integrated into existing codes and ensures stability and accuracy in MHD simulations.

Findings

High-order accuracy demonstrated on benchmark cases.

Effective suppression of spurious oscillations near discontinuities.

Stable under normal CFL conditions.

Abstract

Numerical simulations of ideal compressible magnetohydrodynamic (MHD) equations are challenging, as the solutions are required to be magnetic divergence-free for general cases as well as oscillation-free for cases involving discontinuities. To overcome these difficulties, we develop a locally divergence-free oscillation-eliminating discontinuous Galerkin (LDF-OEDG) method for ideal compressible MHD equations. In the LDF-OEDG method, the numerical solution is advanced in time by using a strong stability preserving Runge-Kutta scheme. Following the solution update in each Runge-Kutta stage, an oscillation-eliminating (OE) procedure is performed to suppress spurious oscillations near discontinuities by damping the modal coefficients of the numerical solution. Subsequently, on each element, the magnetic filed of the oscillation-free DG solution is projected onto a local divergence-free space, to satisfy the divergence-free condition. The OE procedure and the LDF projection are fully decoupled from the Runge-Kutta stage update, and can be non-intrusively integrated into existing DG codes as independent modules. The damping equation of the OE procedure can be solved exactly, making the LDF-OEDG method remain stable under normal CFL conditions. These features enable a straightforward implementation of a high-order LDF-OEDG solver, which can be used to efficiently simulate the ideal compressible MHD equations. Numerical results for benchmark cases demonstrate the high-order accuracy, strong shock capturing capability and robustness of the LDF-OEDG method.