In this paper, an energy-consistent finite difference scheme for the compressible hydrodynamic and magnetohydrodynamic (MHD) equations is introduced. For the compressible magnetohydrodynamics, an energy-consistent finite difference formulation is derived using the product rule for the spatial difference. The conservation properties of the internal, kinetic, and magnetic energy equations can be satisfied in the discrete level without explicitly solving the total energy equation. The shock waves and discontinuities in the numerical solution are stabilized by nonlinear filtering schemes. An energy-consistent discretization of the filtering schemes is also derived by introducing the viscous and resistive heating rates. The resulting energy-consistent formulation can be implemented with the various kinds of central difference, nonlinear filtering, and time integration schemes. The second- and fifth-order schemes are implemented based on the proposed formulation. The conservation properties and the robustness of the present schemes are demonstrated via one- and two-dimensional numerical tests. The proposed schemes successfully handle the most stringent problems in extremely high Mach number and low beta conditions.

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基于非线性滤波的可压缩流体动力学和磁流体动力学的能量一致有限差分格式

本文介绍了可压缩流体动力学和磁流体动力学（MHD）方程的能量一致的有限差分格式。对于可压缩的磁流体动力学，使用空间差的乘积规则推导能量一致的有限差分公式。内部，动能和磁能方程的守恒性质可以在离散级别得到满足，而无需明确求解总能量方程。数值解中的冲击波和不连续性通过非线性滤波方案得以稳定。通过引入粘性和电阻加热速率，还可以得出过滤方案的能量一致性离散化。由此产生的能量一致性公式可通过各种中心差，非线性滤波，和时间积分方案。二阶和五阶方案是基于所提出的公式来实现的。通过一维和二维数值测试证明了本方案的保存特性和鲁棒性。拟议的方案成功地解决了在极高的马赫数和低贝塔条件下最严格的问题。