Structure-Preserving Oscillation-Eliminating Discontinuous Galerkin Schemes for Ideal MHD Equations: Locally Divergence-Free and Positivity-Preserving

TL;DR

This paper introduces novel structure-preserving discontinuous Galerkin schemes for ideal MHD that eliminate oscillations, preserve divergence-free magnetic fields, and ensure positivity of physical quantities, with rigorous analysis and verified numerical tests.

Contribution

The paper develops a locally divergence-free oscillation-eliminating DG scheme with a novel damping-based OE procedure that preserves physical structures and positivity, easily integrable into existing codes.

Findings

Effective suppression of spurious oscillations near discontinuities.

Preservation of divergence-free magnetic fields and positivity.

Verified accuracy and robustness through numerical tests.

Abstract

Numerically simulating magnetohydrodynamics (MHD) poses notable challenges, including the suppression of spurious oscillations near discontinuities (e.g., shocks) and preservation of essential physical structures (e.g., the divergence-free constraint of magnetic field and the positivity of density and pressure). This paper develops structure-preserving oscillation-eliminating discontinuous Galerkin (OEDG) schemes for ideal MHD. The schemes leverage a locally divergence-free (LDF) oscillation-eliminating (OE) procedure to suppress spurious oscillations while retaining the LDF property of magnetic field and many desirable attributes of original DG schemes, such as conservation, local compactness, and optimal convergence rates. The OE procedure is based on the solution operator of a novel damping equation, a linear system of ordinary differential equations that are exactly solvable without any discretization. The OE procedure is performed after each Runge-Kutta stage and does not impact DG spatial discretization, facilitating its easy integration into existing DG codes as an independent module. Moreover, this paper presents a rigorous positivity-preserving (PP) analysis of the LDF OEDG schemes on Cartesian meshes, utilizing the optimal convex decomposition technique and the geometric quasi-linearization (GQL) approach. Efficient PP LDF OEDG schemes are derived by incorporating appropriate discretization of Godunov-Powell source terms into only the discrete equations of cell averages, under a condition achievable through a simple PP limiter. Several one- and two-dimensional MHD tests verify the accuracy, effectiveness, and robustness of the proposed structure-preserving OEDG schemes.