Commun. Comput. Phys., 12 (2012), pp. 955-980. |
An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations Jeffrey Haack ^{1*}, Shi Jin ^{2}, Jian-Guo Liu ^{3} 1 Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA; and Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA.2 Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA. 3 Departments of Physics and Mathematics, Duke University, Durham, NC 27708, USA. Received 25 September 2010; Accepted (in revised version) 13 October 2011 Available online 28 March 2012 doi:10.4208/cicp.250910.131011a 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. AMS subject classifications: 35Q35, 65M08, 65M99, 76M12, 76N99Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Low Mach number limit, asymptotic preserving schemes, incompressible limit, projection scheme, isentropic Euler equation. *Corresponding author. Email: haack@math.utexas.edu (J. Haack), jin@math.wisc.edu (S. Jin), jian-guo.liu@duke.edu (J.-G. Liu) |