On the mach-uniformity of the Lagrange-projection scheme^∗

Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany.
h.zakerzadeh@igpm.rwth-aachen.de

Received: 1 April 2016
Revised: 22 August 2016
Accepted: 12 September 2016

Abstract

In the present work, we show that the Implicit-Explicit Lagrange-projection scheme applied to the isentropic Euler equations, presented in Coquel et al.’s paper (Math. Comp. 79 (2010) 1493–1533), is asymptotic preserving regarding the Mach number, i.e., it is asymptotically stable in ℓ_∞-norm with unrestrictive CFL condition for all-Mach flows, and asymptotically consistent which means that it gives a consistent discretization to the incompressible Euler equations in the limit, e.g., it preserves the incompressible limit as to satisfy the div-free condition and the analogues of continuous-level asymptotic expansion for the density. This consistency analysis has been done formally as well as rigorously. Moreover, we prove that the scheme is positivity-preserving and entropy-admissible under some Mach-uniform restrictions. The analysis is similar to what has been presented in the original paper, but with the emphasis on the uniformity regarding the Mach number, which is crucial for a scheme to be useful in the low-Mach regime. We then extend the modified (but similar) analysis to the shallow water equations with topography and get similar stability and consistency results.