348 IGPM348_3.pdf        October 2012
TITLE An Asymptotic Preserving all Mach Number Scheme for the Euler Equations of Gas Dynamics - second revised version (October 2014)
AUTHORS Sebastian Noelle, Georgij Bispen, Koottungal Revi Arun, Maria Lukáčová-Medviďová, Claus-Dieter Munz
ABSTRACT We propose an asymptotic preserving (AP), all Mach number, Godunov-type finite volume scheme for the numerical solution of the compressible Euler equations of gas dynamics. The new scheme augments Klein’s non-stiff/stiff decomposition of the fluxes (J. Comput. Phys. 121:213- 237, 1995) with a higher order correction flux. We employ an explicit/implicit time discretisation (Cordier et al., J. Comput. Phys. 231:5685-5704, 2012) for the split fluxes, which we solve by an iterative approximation. This results in a second order partial differential equation (PDE) for the pressure. Due to our choice of the reference pressure, the stiff subsystem is hyperbolic, the correction flux is viscous and the second order PDE for the pressure is elliptic. In the new scheme only a single scalar equation needs to be solved in an iteration step; we prove that the AP property is guaranteed; the CFL number is only related to the non-stiff characteristic speeds, independent of the Mach number. The second order accuracy of the scheme is based on MUSCL-type reconstructions and an appropriate combination of Runge-Kutta and Crank-Nicolson time stepping strategies. The results of some benchmark problems are presented, which validate also the convergence, robustness and the efficiency of the scheme to capture the weakly compressible flow features accurately.
KEYWORDS Euler equations of gas dynamics, low Mach number limit, stiffness, semi-implicit time discretisation, flux decomposition, asymptotic preserving schemes.
DOI 10.1137/120895627
PUBLICATION SIAM Journal of Scientific Computing
36(6), B989–B1024. (36 pages)