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.

翻译：数值模拟磁流体动力学（MHD）面临显著挑战，包括抑制间断（如激波）附近的伪振荡以及保持关键物理结构（如磁场无散约束和密度、压力的正性）。本文为理想MHD方程开发了保持结构特性的消振荡间断Galerkin（OEDG）格式。该格式利用局部无散（LDF）消振荡（OE）过程来抑制伪振荡，同时保留磁场的LDF特性以及原始DG格式的诸多优良属性，如守恒性、局部紧致性和最优收敛阶。OE过程基于一种新型阻尼方程的解算子——该阻尼方程为一组可精确求解的线性常微分方程组，无需任何离散化。OE过程在每个龙格-库塔阶段后执行，且不影响DG空间离散，便于作为独立模块集成至现有DG代码中。此外，本文利用最优凸分解技术与几何拟线性化（GQL）方法，在笛卡尔网格上对LDF OEDG格式进行了严格的保正性（PP）分析。通过仅在单元平均值离散方程中引入Godunov-Powell源项的适当离散化，并在可通过简单PP限制器实现的条件约束下，推导出高效的PP LDF OEDG格式。一系列一维和二维MHD测试验证了所提出保持结构特性的OEDG格式的精度、有效性和鲁棒性。