An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations (Communications in Computational Physics, 2012)
The computation of compressible flows becomes more challenging when theMach number has different orders of magnitude. When the Mach number is of orderone, modern shock capturing methods are able to capture shocks and other complexstructures with high numerical resolutions. However, if the Mach number is small, theacoustic waves lead to stiffness in time and excessively large numerical viscosity, thusdemanding 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-Stokesequations that is uniformly stable and accurate for all Mach numbers. Our idea is tosplit 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 alinear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projectionsystem and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes aprojection method-like incompressible solver. We present numerical results in one andtwo dimensions in both compressible and incompressible regimes.