All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations

Abstract

An all speed scheme for the Isentropic Euler equation is presented in this paper. When the Mach number tends to zero, the compressible Euler equation converges to its 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 [11,13,27], nonphysical oscillations can be suppressed. We develop this semi-implicit time discretization in the framework of a first order local Lax-Friedrich (LLF) scheme and numerical tests are displayed to demonstrate its performances.

Modelling of phase transitions in granular flows

“…Thanks to (11), (13) and (15) we are then able to pass to the limit in the weak formulations of the equations. Except the convergence of the viscous term λ ε (ρ ε )∂ x u ε , the procedure is standard and follow the steps of [7] or [22] that we reproduce here for the convenience of the reader.…”

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

“…We refer to AP schemes for kinetic equations in the fluid dynamic or diffusive regimes [2,7,14,32,[40][41][42]44,45,[47][48][49]. The AP framework has also been extended in [15,16] for the study of the quasi-neutral limit of Euler-Poisson and Vlasov-Poisson systems, and in [19,21,34] for all-speed (Mach number) fluid equations bridging the passage from compressible flows to the incompressible flows. One should note that under-resolved computation may not yield accurate or even physically correct approximations in areas with sharp transitions, such as shock and boundary layers.…”

A Relaxation Scheme for the Simulation of Low Mach Number Flows

“…Nevertheless, the upwind discretization is needed in presence of Mach numbers of order one, in order to introduce enough numerical viscosity. Since the adopted spatial discretization is a convex combination of these two, it is able to approximate flows in different regimes and thus the scheme can be seen as an "all-speed" scheme, as the ones proposed in [2,3]. Moreover, thanks to the absolute stability of implicit schemes, the CFL is not limited by the acoustic constraint, which becomes extremely demanding in low Mach regimes.…”