On a class of numerical schemes for the numerical simulation of compressible flows

Raphaèle Herbin (Université d'Aix-Marseille)

Frank Adams Room 1, Alan Turing Building,

We focus on the discretization of the compressible Euler and Navier Stokes equations at all Mach numbers. The numerical schemes that we use are based on a discretization of the internal energy equation; they also feature an upwinding of the density and the internal energy based on the fluid velocity, which ensures the positivity of the density and of the internal energy. In the case of the Euler equations, an additional term must be present in the discrete internal energy equation so as to ensure consistency of the scheme in the Lax-Wendroff sense, which yields in particular that the correct shock speeds are computed. This additional term is obtained through a discrete kinetic inequality, which is deduced from the discrete mass and momentum balance equations equations. These schemes have been developed using either collocated meshes or staggered meshes; each mesh type has advantages and disadvantages. In the case of staggered grids, the discretization of the nonlinear convection term is a bit more tricky, since a discrete mass balance must hold on the dual cells so as to perform a compatible discretization of mass and momentum in order to a discrete kinetic energy inequality. We are able to show existence of a solution to the schemes, stability estimates, and the above mentioned weak consistency. Numerical tests were performed and demonstrate the efficiency of the proposed approach.

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