We present a divergence-free semi-implicit finite volume scheme for the simulation of the ideal magnetohydrodynamics (MHD) equations which is stable for large time steps controlled by the local transport speed at all Mach and Alfv\'en numbers. An operator splitting technique allows to treat the convective terms explicitly while the hydrodynamic pressure and the magnetic field contributions are integrated implicitly, yielding two decoupled linear implicit systems. The linearity of the implicit part is achieved by means of a semi-implicit time linearization. This structure is favorable as second-order accuracy in time can be achieved relying on the class of semi-implicit IMplicit-EXplicit Runge-Kutta (IMEX-RK) methods. In space, implicit cell-centered finite difference operators are designed to discretely preserve the divergence-free property of the magnetic field on three-dimensional Cartesian meshes. The new scheme is also particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, since no explicit numerical dissipation is added to the implicit contribution and the time step is scale independent. Likewise, highly magnetized flows can benefit from the implicit treatment of the magnetic fluxes, hence improving the computational efficiency of the novel method. The convective terms undergo a shock-capturing second order finite volume discretization to guarantee the effectiveness of the proposed method even for high Mach number flows. The new scheme is benchmarked against a series of test cases for the ideal MHD equations addressing different acoustic and Alfv\'en Mach number regimes where the performance and the stability of the new scheme is assessed.

翻译：我们提出了一种无散度的半隐式有限体积格式，用于模拟理想磁流体力学（MHD）方程。该格式在任意马赫数和阿尔文数下，由局部输运速度控制的大时间步长下保持稳定。通过算子分裂技术，对流项被显式处理，而流体动力学压力和磁场贡献则被隐式整合，从而得到两个解耦的线性隐式系统。隐式部分的线性化通过半隐式时间线性化实现。这种结构有利于使用半隐式IMplicit-EXplicit龙格-库塔（IMEX-RK）方法达到时间方向上的二阶精度。在空间上，我们设计了隐式单元中心有限差分算子，以离散方式在三维笛卡尔网格上保持磁场的无散度性质。由于隐式部分未添加显式数值耗散且时间步长与尺度无关，新格式特别适用于低马赫数流动以及MHD方程不可压缩极限情形。同样地，强磁化流动可受益于磁通量的隐式处理，从而提升新方法的计算效率。对流项采用激波捕捉的二阶有限体积离散化，确保所提方法在高马赫数流动中依然有效。我们通过一系列针对理想MHD方程在不同声学和阿尔文马赫数区域内的算例测试，评估了新方案的性能与稳定性。