Numerical simulations of ideal compressible magnetohydrodynamic (MHD) equations are challenging, as the solutions are required to be magnetic divergence-free for general cases as well as oscillation-free for cases involving discontinuities. To overcome these difficulties, we develop a locally divergence-free oscillation-eliminating discontinuous Galerkin (LDF-OEDG) method for ideal compressible MHD equations. In the LDF-OEDG method, the numerical solution is advanced in time by using a strong stability preserving Runge-Kutta scheme. Following the solution update in each Runge-Kutta stage, an oscillation-eliminating (OE) procedure is performed to suppress spurious oscillations near discontinuities by damping the modal coefficients of the numerical solution. Subsequently, on each element, the magnetic filed of the oscillation-free DG solution is projected onto a local divergence-free space, to satisfy the divergence-free condition. The OE procedure and the LDF projection are fully decoupled from the Runge-Kutta stage update, and can be non-intrusively integrated into existing DG codes as independent modules. The damping equation of the OE procedure can be solved exactly, making the LDF-OEDG method remain stable under normal CFL conditions. These features enable a straightforward implementation of a high-order LDF-OEDG solver, which can be used to efficiently simulate the ideal compressible MHD equations. Numerical results for benchmark cases demonstrate the high-order accuracy, strong shock capturing capability and robustness of the LDF-OEDG method.

翻译：理想可压缩磁流体力学（MHD）方程的数值模拟具有挑战性，因为其解需在一般情况下满足磁场无散条件，且在涉及间断的情况下需无振荡。为克服这些困难，本文针对理想可压缩MHD方程发展了一种局部无散振荡消除型间断伽辽金（LDF-OEDG）方法。在LDF-OEDG方法中，数值解通过使用强稳定性保持龙格-库塔格式进行时间推进。在每个龙格-库塔阶段解更新之后，执行振荡消除（OE）程序，通过阻尼数值解的模态系数来抑制间断附近的虚假振荡。随后，在每个单元上，将无振荡DG解的磁场投影到局部无散空间，以满足无散条件。OE程序和LDF投影与龙格-库塔阶段更新完全解耦，可作为独立模块非侵入式集成到现有DG代码中。OE程序的阻尼方程可精确求解，使得LDF-OEDG方法能在正常CFL条件下保持稳定性。这些特性使得高阶LDF-OEDG求解器易于实现，可用于高效模拟理想可压缩MHD方程。基准测试案例的数值结果证明了LDF-OEDG方法的高阶精度、强激波捕捉能力及鲁棒性。