On the mach-uniformity of the Lagrange-projection scheme

Zakerzadeh, Hamed

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1343-1366.

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 $\ell {}_{\infty }$-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.

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