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Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number. (English) Zbl 1329.76228
Summary: We propose a theoretical framework to clearly explain the inaccuracy of Godunov type schemes applied to the compressible Euler system at low Mach number on a Cartesian mesh. In particular, we clearly explain why this inaccuracy problem concerns the 2D or 3D geometry and does not concern the 1D geometry. The theoretical arguments are based on the Hodge decomposition, on the fact that an appropriate well-prepared subspace is invariant for the linear wave equation and on the notion of first-order modified equation. This theoretical approach allows to propose a simple modification that can be applied to any collocated scheme of Godunov type or not in order to define a large class of collocated schemes accurate at low Mach number on any mesh. It also allows to justify collocated schemes that are accurate at low Mach number as, for example, the Roe-Turkel and the AUSM\(^{+}\)-up schemes, and to find a link with a collocated incompressible scheme stabilized with a Brezzi-Pitkäranta type stabilization. Numerical results justify the theoretical arguments proposed in this paper.

76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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