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.

翻译：本文旨在发展用于理想磁流体力学（MHD）的保持结构、消振荡间断伽辽金（OEDG）格式，作为我们近期工作[彭、孙、吴，OEDG：双曲守恒律的消振荡间断伽辽金方法，2023]的后续。该格式基于局部无散（LDF）的消振荡（OE）过程，以抑制非物理振荡，同时保留原始间断伽辽金格式的诸多优良性质，如守恒性、局部紧致性和最优收敛速度。OE过程构建于一种新型阻尼方程的解算子之上——该方程为一阶线性常微分方程（ODE），其精确解可显式给出。由于此OE过程不干扰DG空间离散与龙格-库塔（RK）阶段更新，因此可作为独立模块轻松嵌入现有DG代码。这些特性使所提LDF OEDG格式兼具高效率与易实现性。此外，我们通过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格式进行了保正性（PP）分析。结合HLL通量与简单局部缩放保正限制器所得条件，我们获得了高效的保正LDF OEDG格式。多个一维与二维MHD数值测试验证了所提保持结构的OEDG格式的精度、有效性与鲁棒性。