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

TL;DR

The paper tackles the dual challenges of preserving the magnetic divergence-free constraint $\nabla\cdot\mathbf{B}=0$ and suppressing spurious oscillations near discontinuities in ideal MHD simulations. It develops the Locally Divergence-Free Oscillation-Eliminating Discontinuous Galerkin (LDF-OEDG) method, which combines a non-intrusive OE post-processing that damps high-order modal coefficients with a local divergence-free projection of the magnetic field, all within a SSP Runge-Kutta time-stepping framework. The LDF-OEDG scheme ensures exact divergence-free satisfaction inside each element and stable, high-order performance under normal CFL restrictions, while remaining easy to integrate into existing RKDG codes. Numerical tests on multiple 2D MHD benchmarks demonstrate third-order accuracy for $k=2$, robust shock capturing, and superior resolution compared with traditional DG approaches, highlighting the method’s practicality and effectiveness for ideal MHD simulations.

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.