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All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations

Published online by Cambridge University Press:  20 August 2015

Pierre Degond  and
Min Tang
Pierre Degond*
Affiliation:
Institute of Mathematics of Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France
Min Tang*
Affiliation:
Institute of Mathematics of Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France
*
Email:degond@mip.ups-tlse.fr
Corresponding author.Email:tangmin1002@gmail.com
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Abstract

An all speed scheme for the Isentropic Euler equations is presented in this paper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficulty in the low Mach regime. The key idea of our all speed scheme is the special semi-implicit time discretization, in which the low Mach number stiff term is divided into two parts, one being treated explicitly and the other one implicitly. Moreover, the flux of the density equation is also treated implicitly and an elliptic type equation is derived to obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared with previous semi-implicit methods, firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs to be solved implicitly which reduces much computational cost. We develop this semi-implicit time discretization in the framework of a first order Local Lax-Friedrichs (or Rusanov) scheme and numerical tests are displayed to demonstrate its performances.

Keywords

65M0665Z0576N9976L05Low Mach numberIsentropic Euler equationscompressible flowincompressible limitasymptotic preservingRusanov scheme
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 1 , July 2011 , pp. 1 - 31
Copyright
Copyright © Global Science Press Limited 2011

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