All speed scheme for the low Mach number limit of the isentropic Euler equations

By Pierre Degond, Min Tang. 2009

192 citations recorded by the source corpus.

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 con- verge to their incompressible counterpart, in which the density becomes a constant. In- creasing 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 treatedimplicitly and an elliptic type equation is derived to obtain the density. In this way, the correct limit can be captured without request- ing the mesh size and time step to be smaller than the Mach number. Compared with previous semi-implicit methods (11,13,29),firstly, nonphysical oscillations can be sup- pressed 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. AMS subject classifications: 65M06, 65Z05, 76N99, 76L05

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/d9324882b5c1011e3139ad16a80391d2ab3b301d

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations (2012, methodology): “We discretize the convective flux terms in (3.2) using a second order central scheme [25], which is a higher-order extension of the Lax-Friedrichs scheme.”
A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources (2009, methodology): “…we only show the results obtained in the kinetic regime (10−2) using a spectral scheme for the discretization of the collision operator [22] (with nv = 32 2 and a truncation of the velocity domain vmax = 7) and second order explicit Runge-Kutta and second order method (2.3) for the time…”