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An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations

Published online by Cambridge University Press:  20 August 2015

Jeffrey Haack ,
Shi Jin  and
Jian‐Guo Liu
Jeffrey Haack*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI53706, USA Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA
Shi Jin*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI53706, USA
Jian‐Guo Liu*
Affiliation:
Departments of Physics and Mathematics, Duke University, Durham, NC 27708, USA
*
Corresponding author.Email:haack@math.utexas.edu
Email:jin@math.wisc.edu
Email:jian‐guo.liu@duke.edu
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Abstract

The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes.

Keywords

35Q3565M0865M9976M1276N99Low Mach number limitasymptotic preserving schemesincompressible limitprojection schemeisentropic Euler equation
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 4 , October 2012 , pp. 955 - 980
Copyright
Copyright © Global Science Press Limited 2012

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