An All-Regime Lagrange-Projection Like Scheme for the Gas Dynamics Equations on Unstructured Meshes (Communications in Computational Physics, 2018)
We propose an all regime Lagrange-Projection like numerical scheme for thegas dynamics equations. By all regime, we mean that the numerical scheme is able tocompute accurate approximate solutions with an under-resolved discretization withrespect to the Mach number M, i.e. such that the ratio between the Mach number Mand the mesh size or the time step is small with respect to 1. The key idea is to decoupleacoustic and transport phenomenon and then alter the numerical flux in theacoustic approximation to obtain a uniform truncation error in term of M. This modifiedscheme is conservative and endowed with good stability properties with respectto the positivity of the density and the internal energy. A discrete entropy inequalityunder a condition on the modification is obtained thanks to a reinterpretation of themodified scheme in the Harten Lax and van Leer formalism. A natural extension tomulti-dimensional problems discretized over unstructured mesh is proposed. Thena simple and efficient semi-implicit scheme is also proposed. The resulting schemeis stable under a CFL condition driven by the (slow) material waves and not by the(fast) acoustic waves and so verifies the all regime property. Numerical evidences areproposed and show the ability of the scheme to deal with tests where the flow regimemay vary from low to high Mach values.