Implicit MAC scheme for compressible Navier-Stokes equations: Low Mach asymptotic error estimates

Thierry Gallouët (1) , Raphaele Herbin (1) , David Maltese (2) , Antonin Novotny (2)

1 I2M - Institut de Mathématiques de Marseille
2 IMATH - Institut de Mathématiques de Toulon - EA 2134

We investigate error between any discrete solution of the implicit Marker and Cell (MAC) numerical scheme for compressible Navier-Stokes equations in low Mach number regime and an exact strong solution of the incompressible Navier-Stokes equations. The main tool is the relative energy method suggested on the continuous level in [7], whose discrete numerical version has been developed in [19]. We get unconditional error estimate in terms of explicitly determined positive powers of the space-time disretization parameters and Mach number in the case of well prepared initial data, and the boundedness of the error if the initial data are ill prepared. The multiplicative constant in the error estimate depends on the suitable norm of the strong solution but is independent on the numerical solution itself (and of course, on the disretization parameters and the Mach number). This is the first proof ever that the MAC scheme is unconditionally and uniformly asymptotically stable at the low Mach number regime.

Mots clés: Navier-Stokes system; finite difference numerical method; finite volume numerical; method; Marker and Cell scheme; error estimate; AMS classification 35Q30; 65N12; 65N30; 76N10; 76N15; 76M10; 76M12

Domaines: Analyse numérique [math.NA]

Thierry Gallouët, Raphaele Herbin, David Maltese, Antonin Novotny. Implicit MAC scheme for compressible Navier-Stokes equations: Low Mach asymptotic error estimates. 2017. ⟨hal-01462822⟩