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 wave speed 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 high-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.

翻译：我们提出一种新颖的结构保持方法，用于逼近可压缩理想磁流体动力学（MHD）方程。该方法采用非散度形式的MHD方程，将磁场对动量和总机械能的贡献处理为源项。我们的方法基于马丘克-斯特朗分裂技术，包含三个独立模块：可压缩欧拉求解器、源项系统求解器以及总机械能更新过程。该方案允许自由选择欧拉方程求解器，同时采用旋度相容有限元空间离散磁场，从而精确保持对偶约束。我们证明该方法能保持不变域特性，包括密度正性、内能正性以及比熵最小值原理。若用于求解欧拉方程的格式可保持总能量守恒，则所得的MHD格式也可证明保持总能量守恒；同理，若欧拉方程求解格式满足熵稳定，则所得MHD格式亦满足熵稳定。本方法的CFL条件不依赖于磁声波速度，仅取决于欧拉系统的常规最大波速。通过求解多个理想MHD问题验证方法有效性，结果表明该方法在光滑问题中能实现空间高阶精度，同时在激波流体动力学区域展现出无条件鲁棒性。