Large Time Step and Asymptotic Preserving Numerical Schemes for the Gas Dynamics Equations with Source Terms

Abstract

We propose a large time step and asymptotic preserving scheme for the gas dynamics equations with external forces and friction terms. By asymptotic preserving, we mean that the numerical scheme is able to reproduce at the discrete level the parabolic-type asymptotic behaviour satisfied by the continuous equations. By large time-step, we mean that the scheme is stable under a CFL stability condition driven by the (slow) material waves, and not by the (fast) acoustic waves as it is customary in Godunov-type schemes. Numerical evidences are proposed and show a gain of several orders of magnitude in both accuracy and efficiency.

High‐order asymptotic‐preserving schemes for linear systems: Application to the Goldstein–Taylor equations

“…We once again use (6)- (9) to observe that the leading effect of the second order extension is to replace the term in (1 + t) −(3/2 + p) by a term in (1 + t) −(2 + p) since the residual in 2…”

“…The aim of asymptotic-preserving schemes is to preserve this asymptotic behavior at the numerical level. There is a huge literature on this topic and the design of first-order asymptotic-preserving scheme is now well understood, see for instance the recent book [1] and the references therein, and without any attempt to be exhaustive the following works [2][3][4][5][6][7][8][9][10][11][12][13][14]. In the present work and in order to get this asymptotic-preserving property, we follow a very simple and original approach first proposed in [15] which consists in modifying the numerical diffusion associated with the usual Godunov-type schemes in such a way that the consistency error of the scheme gets uniform with respect to the stiff parameters.…”

An all-speed relaxation scheme for gases and compressible materials

“…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]). …”

Study of a New Asymptotic Preserving Scheme for the Euler System in the Low Mach Number Limit

“…The method derived and studied in this work belongs to the class of schemes called asymptotic preserving (AP). Such type of schemes have been developed in [4] and in [6,8] for the specific problem related to the compressible-incompressible passage. However, in [4] the method is based on a Lagrange projection method and on a splitting procedure that allows to decouple the acoustic and the transport phenomenons.…”

“…Such type of schemes have been developed in [4] and in [6,8] for the specific problem related to the compressible-incompressible passage. However, in [4] the method is based on a Lagrange projection method and on a splitting procedure that allows to decouple the acoustic and the transport phenomenons. While in [8,6], the authors split the pressure through the introduction of a numerical parameter which must be tuned depending on the problem in order for the scheme to work.…”