A Weakly Asymptotic Preserving Low Mach Number Scheme for the Euler Equations of Gas Dynamics

Abstract

We propose a low Mach number, Godunov-type finite volume scheme for the numerical solution of the compressible Euler equations of gas dynamics. The scheme combines Klein's non-stiff/stiff decomposition of the fluxes [J. Comput. Phys., 121 (1995), pp. 213-237] with an explicit/implicit time discretization [F. Cordier, P. Degond, and A. Kumbaro, J. Comput. Phys., 231 (2012), pp. 5685-5704] for the split fluxes. This results in a scalar second order partial differential equation (PDE) for the pressure, which we solve by an iterative approximation. Due to our choice of a crucial reference pressure, the stiff subsystem is hyperbolic, and the second order PDE for the pressure is elliptic. The scheme is also uniformly asymptotically consistent. Numerical experiments show that the scheme needs to be stabilized for low Mach numbers. Unfortunately, this affects the asymptotic consistency, which becomes nonuniform in the Mach number, and requires an unduly fine grid in the small Mach number limit. On the other hand, the CFL number is only related to the nonstiff characteristic speeds, independently of the Mach number. Our analytical and numerical results stress the importance of further studies of asymptotic stability in the development of asymptotic preserving schemes.

An all-speed relaxation scheme for gases and compressible materials

“…where D (∂ x u) upw is the upwind spatial discretization (24) and D (∂ x u) cent is the centered one. M loc is the local Mach number, which can be computed on the numerical solution at the previous time step.…”

“…In order to do this, pressure gradient-type terms are introduced inside the momentum equation, allowing the splitting of the fast and the slow scales. These schemes have a formal proof of the Asymptotic Preserving (AP) property, namely their lower order expansion is a consistent and stable discretization of the incompressible limit (see also [24,25,26,27]). …”

Modelling of phase transitions in granular flows

“…Several ideas have been proposed to tackle this issue among which preconditioning methods (see [28]) which consist in multiplying the time derivatives by an appropriate matrix in order to alter the stiff eigenvalues of the system, and pressure correction methods, which extend to the compressible setting the projection techniques introduced by Chorin [8], Temam [27] in the incompressible framework (see for instance the works of Harlow, Amsden [14] or more recently the works of Herbin, Kheriji, Latché [16]). We are interested in this last category and more precisely in the recent techniques of implicit/explicit (IMEX) discretizations proposed for instance by Klein [18], Degond et al [10], Degond and Tang [13], Cordier, Degond, Kumbaro [9], Noelle et al [23]. These methods consist in splitting the pressure into a stiff and a non-stiff part, the first one being treated implicitly in time whereas the non-stiff part is treated explicitly.…”

Adaptive discontinuous evolution Galerkin method for dry atmospheric flow

“…Another common approach is based on semi-implicit time discretization; here the fast waves that are of less interest are approximated implicitly, whereas slow advection is treated explicitly. Several methods following this idea can be found, e.g., in [2,10,16,23,24,35,34,39,47] to name just a few.…”