An all-Mach cell-centered multi-dimensional nite volume numerical scheme for the Euler equations
Résumé
In the context of the numerical approximation of Euler equations, great efforts have been devoted to developing schemes that can accurately reproduce solutions in low Mach number ows. Solutions of classic Finite Volume (FV) schemes are usually plagued by an excessive diusion as the numerical scheme is not consistent with the limit equations for the Mach number that tends to zero. Instead, a numerical scheme that satises such a property is called Asymptotic-Preserving (AP). In this paper, we propose an AP FV scheme for the multi-dimensional Euler equations. In classic FV methods, the numerical approximation of the face ux is obtained by means of a two-state 1D approximate Riemann Solver (RS) in the normal direction to the face. Here, we rely on a node-based ux approximation that exploits a particular RS involving a nodal quantity which depends on all the cells around a given node. Such an idea has been exploited by Barsukow et al. (2023) for the linear acoustic equations. Their method is vorticity-preserving, but its extension to the Euler equations proved to be far from trivial. For such a reason, a change of perspective is needed in the denition of the RS.
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