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

This paper develops structure-preserving, oscillation-eliminating discontinuous Galerkin (OEDG) schemes for ideal magnetohydrodynamics (MHD), as a sequel to our recent work [Peng, Sun, and Wu, OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws, 2023]. The schemes are based on a locally divergence-free (LDF) oscillation-eliminating (OE) procedure to suppress spurious oscillations while maintaining many of the good properties of original DG schemes, such as conservation, local compactness, and optimal convergence rates. The OE procedure is built on the solution operator of a novel damping equation – a simple linear ordinary differential equation (ODE) whose exact solution can be exactly formulated. Because this OE procedure does not interfere with DG spatial discretization and RK stage update, it can be easily incorporated to existing DG codes as an independent module. These features make the proposed LDF OEDG schemes highly efficient and easy to implement.In addition, we present a positivity-preserving (PP) analysis of the LDF OEDG schemes on Cartesian meshes via the optimal convex decomposition technique and the geometric quasi-linearization (GQL) approach. Efficient PP LDF OEDG schemes are obtained with the HLL flux under a condition accessible by the simple local scaling PP limiter.Several one- and two-dimensional MHD tests confirm the accuracy, effectiveness, and robustness of the proposed structure-preserving OEDG schemes.