In this work, we present a positivity-preserving adaptive filtering approach for discontinuous spectral element approximations of the ideal magnetohydrodynamics equations. This approach combines the entropy filtering method (Dzanic and Witherden, J. Comput. Phys., 468, 2022) for shock capturing in gas dynamics along with the eight-wave method for enforcing a divergence-free magnetic field. Due to the inclusion of non-conservative source terms, an operator-splitting approach is introduced to ensure that the positivity and entropy constraints remain satisfied by the discrete solution. Furthermore, a computationally efficient algorithm for solving the optimization process for this nonlinear filtering approach is presented. The resulting scheme can robustly resolve strong discontinuities on general unstructured grids without tunable parameters while recovering high-order accuracy for smooth solutions. The efficacy of the scheme is shown in numerical experiments on various problems including extremely magnetized blast waves and three-dimensional magnetohydrodynamic instabilities.

翻译：本文提出了一种针对非连续谱元近似求解理想磁流体力学方程的正性保持自适应滤波方法。该方法结合了气体动力学激波捕捉的熵滤波方法（Dzanic and Witherden, J. Comput. Phys., 468, 2022）与实现无散磁场的八波方法。由于包含非守恒源项，本文引入了算子分裂策略以确保离散解满足正性和熵约束条件。此外，针对该非线性滤波方法的优化求解过程，提出了计算高效的算法。所构建的格式能够在无需可调参数的情况下，稳健处理非结构化网格上的强间断，同时恢复光滑解的高阶精度。数值实验展示了该方法在极端磁化激波与三维磁流体动力学不稳定性等多种问题中的有效性。