The computation of compressible flows at all Mach numbers is a very challenging problem. An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime, while it can deal with stiffness and accuracy in the low Mach number regime. This paper designs a high order semi-implicit weighted compact nonlinear scheme (WCNS) for the all-Mach isentropic Euler system of compressible gas dynamics. To avoid severe Courant-Friedrichs-Levy (CFL) restrictions for low Mach flows, the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components. A third-order implicit-explicit (IMEX) method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives. The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit. One- and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.

中文翻译：

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All-Mach等熵Euler系统的高阶半隐含加权紧致非线性格式

所有马赫数下的可压缩流的计算是一个非常具有挑战性的问题。解决此问题的有效数值方法需要在高马赫数状态下具有冲击捕捉能力，而在低马赫数状态下可以处理刚度和精度。本文针对可压缩气体动力学的全马氏等熵欧拉系统，设计了一个高阶半隐式加权紧致非线性方案（WCNS）。为了避免对低马赫数流量施加严格的Courant-Friedrichs-Levy（CFL）限制，将Euler方程中的非线性通量分为刚性和非刚性分量。三阶隐式显式（IMEX）方法用于拆分分量的时间离散化，五阶WCNS用于通量导数的空间离散化。高阶IMEX方法是渐近保留的，并且在零马赫数限制内渐近精确。给出了可压缩和不可压缩状态下的一维和二维数值示例，以证明所设计的IMEX WCNS的优势。