High order semi-implicit weighted compact nonlinear scheme for the all-Mach isentropic Euler system

Yanqun Jiang,Xun Chen,Shuguang Zhou,Tao Xiong,Xu Zhang

doi:10.1186/s42774-020-00052-9

Yanqun Jiang, Xun Chen + Show 3 more

Open Access

https://doi.org/10.1186/s42774-020-00052-9

Copy DOI

Abstract

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.

Highlights

Low-Mach number flows in fluid dynamics that are slow compared with the speed of sound can be described by the following scaled isentropic Euler equations.ρt + ∇ · = 0,t +∇ · + 1 ε2 ∇ p = (1)where ρ(x, t) > 0, u(x, t) = (u(x, t), v(x, t)) and q = ρu(x, t) are the density, velocity and momentum of the fluid, respectively, and ε > 0 is the scaled Mach number that is a measure of compressibility of the fluid
3 IMEX method we briefly describe the framework of IMEX time discretization designed for the numerical integration of the stiff ordinary differential equation (ODE) as follows
7 Conclusions In this work, we have designed a high order IMEX weighted compact nonlinear scheme (WCNS) for the compressibleincompressible limit problems described as scaled isentropic Euler equations

Summary

Introduction

In view of the good performance of the IMEX time discretization method and the WCNS’s, we design a high order IMEX WCNS for the all-Mach isentropic Euler system based on the acoustic/advection splitting strategy [10]. Remark 3 To update the solutions from tn to tn+1 by the pth-order IMEX-ARSp WCNS, we first solve the elliptic equation (33) to obtain ρ(k) and calculate explicitly q(k) using (36). The IMEX-ARSp scheme is asymptotically stable with the time-step restrictions (38) or (39) independent of the Mach number ε.

Results

Conclusion