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

TL;DR

The paper develops a structure-preserving oscillation-eliminating discontinuous Galerkin (OEDG) method for the ideal MHD equations that simultaneously enforces a locally divergence-free magnetic field and positivity of density and pressure. It introduces an oscillation-eliminating (OE) damping step, solved exactly, after each Runge–Kutta stage, integrated as a non-intrusive module that preserves conservation and optimal convergence. A rigorous positivity-preserving (PP) analysis is carried out using geometric quasi-linearization (GQL) and optimal convex decomposition, with an upwind discretization of the Godunov–Powell source term ensuring PP for cell averages. Numerical tests in 1D and 2D demonstrate high-order accuracy, suppression of nonphysical oscillations near shocks, and robustness across challenging low-density, low-pressure, and low-plasma-beta regimes, highlighting the method’s potential for scalable MHD simulations on structured and, prospectively, unstructured meshes.

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.

Figures (15)

• Figure 1: The sketch near the interface $x=x_{i+\frac{1}{2}}$: the left/right part of the gray area represents the local region of cell $I_{ij}/I_{i+1,j}$ near the interface with a thickness of $\varepsilon\to0^{+}$.
• Figure 2: The three HLL wave patterns at the interface $x=x_{i+\frac{1}{2}}$ with the (estimated) minimum wave speed $\mathscr{V}_{1,l}\left(x_{i+\frac{1}{2}}, y_{j}^{(\mu)}\right)$ and the (estimated) maximum wave speed $\mathscr{V}_{1,r}\left(x_{i+\frac{1}{2}}, y_{j}^{(\mu)}\right)$.
• Figure 3: Numerical solutions of the first MHD shock tube problem at time $t = 0.2$ with $800$ cells (symbols "$\circ$") and $4000$ cells (solid lines). Left column from top to bottom: $\rho$, $u_{1}$, $u_{3}$, $B_{3}$. Right column from top to bottom: $p$, $u_{2}$, $B_{2}$, $E$.
• Figure 4: Numerical solutions of the second MHD shock tube problem at time $t = 0.16$ with $800$ cells (symbols "$\circ$") and $4000$ cells (solid lines). Left column from top to bottom: $\rho$, $u_{1}$, $u_{3}$, $B_{3}$. Right column from top to bottom: $p$, $u_{2}$, $B_{2}$, $E$.
• Figure 5: Numerical solutions of the third MHD shock tube problem at time $t = 0.1$ computed by the third-order OEDG scheme with $800$ cells (symbols "$\circ$") and $4000$ cells (solid lines). Left column from top to bottom: $\rho$, $u_{1}$, $B_{2}$. Right column from top to bottom: $p$, $u_{2}$, $E$.

Theorems & Definitions (10)

• Remark 1
• Remark 2
• Lemma 1: GQL representation Wu2018Positivity
• Lemma 2: Wu2018A
• Lemma 3
• Lemma 4
• Theorem 1
• Corollary 1: PP via Zhang--Shu convex decomposition
• Corollary 2: PP via optimal convex decomposition for $\mathbb P^2$ and $\mathbb P^3$
• Remark 3