We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wavespeed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering second-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.

中文翻译：

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理想可压缩磁流体力学系统的保结构数值方法

我们引入了一种新颖的结构保持方法来近似可压缩理想磁流体动力学（MHD）方程。该技术使用非发散公式来解决 MHD 方程，其中磁场对动量和总机械能的贡献被视为源项。我们的方法使用 Marchuk-Strang 分裂技术，涉及三个不同的组件：可压缩欧拉求解器、源系统求解器和总机械能的更新程序。该方案允许在选择欧拉方程求解器时具有很大的自由度，同时使用符合旋度的有限元空间对磁场进行离散化，从而精确保留对合约束。我们证明该方法保留了不变的域属性，包括密度正性、内能正性和比熵最小原理。如果用于求解欧拉方程的方案使总能量守恒，则可以证明所得 MHD 方案能够使总能量守恒。类似地，如果用于求解欧拉方程的方案是熵稳定的，则所得 MHD 方案也是熵稳定的。在我们的方法中，CFL 条件不依赖于磁声波速度，而仅依赖于欧拉系统的通常最大波速。为了验证我们方法的有效性，我们解决了各种理想的 MHD 问题，表明该方法能够为平滑问题提供空间二阶精度，同时还在冲击流体动力学领域提供无条件鲁棒性。