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.

This paper is devoted to the study of the low Mach number limit for the isentropic Euler system with axisymmetric initial data without swirl. In the first part of the paper we analyze the problem corresponding to the subcritical regularities, that is ${H}^{s} $ with $s\gt \frac{5}{2} $. Taking advantage of the Strichartz estimates and using the special structure of the vorticity we show that the lifespan ${T}_{\varepsilon } $ of the solutions is bounded below by $\log \log \log \frac{1}{\varepsilon } $, where $\varepsilon $ denotes the Mach number. Moreover, we prove that the incompressible parts converge to the solution of the incompressible Euler system when the parameter $\varepsilon $ goes to zero. In the second part of the paper we address the same problem but for the Besov critical regularity ${ B}_{2, 1}^{\frac{5}{2} } $. This case turns out to be more subtle because of at least two features. The first one is related to the Beale–Kato–Majda criterion which is not known to be valid for rough regularities. The second one concerns the critical aspect of the Strichartz estimate ${ L}_{T}^{1} {L}^{\infty } $ for the acoustic parts $(\nabla {\Delta }^{- 1} \mathrm{div} \hspace{0.167em} {v}_{\varepsilon } , {c}_{\varepsilon } )$: it scales in the space variables like the space of the initial data.

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.

The mathematical theory of the passage from compressible to incompressible fluid flow is reviewed.