An all-regime, well-balanced, positive and entropy satisfying one-step finite volume scheme for the Euler's equations of gas dynamics with gravity
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In this paper, we propose a flux splitting finite volume method for the approximation of the Euler equations with source terms derived from a potential. The flux splitting strategy that we adopt here relies on a separate treatment of the terms related to pressure effects from the terms related to transport. We show that this approach can be recast into a relaxation approximation that shares similarities with the Lagrange-Projection method, so that the present flux splitting method can be viewed as an "unsplit Lagrange-Projection" algorithm. We perform a truncation analysis error in the low Mach regime that suggests a flux modification in order to preserve the accuracy of the method in this regime. We show that the resulting method is well-balanced in the sense that it preserves hydrostatic equilibrium profiles. Under a CFL condition, the numerical scheme is also positivity preserving for the mass and the internal energy and it also verifies a discrete entropy inequality. One-dimensional and two-dimensional numerical experiments show the ability of the method to deal with a wide range of Mach regime, shocks, rarefactions and to preserves hydrostatic equilibrium states.
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