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.
翻译:在本文中,对压缩流体动力学和磁力动力(MHD)方程式采用了一种符合能量的有限差异办法。对于压缩磁力动力学和磁力动力学等式,则使用产品规则得出一种符合能量的有限差异配方,以空间差异为标准。内电、动能和磁能方程式的保存特性可以在不明确解决总能量方程式的情况下在离散一级得到满足。数字溶液中的冲击波和不连续性通过非线性过滤办法稳定下来。过滤法的能量一致分解办法也可通过引入粘力和耐热速率产生。由此产生的能源一致性配方程式可以在各种中央差异、非线性过滤和时间整合办法下实施。第二级和第五级方案是在拟议的配方程式的基础上实施的。保护特性和现有办法的稳健性通过一二维数字试验得到证明。拟议办法成功地处理了极高马赫数量和低贝塔条件中最严格的问题。