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Article Dans Une Revue IMA Journal of Numerical Analysis Année : 2020

A cell-centered pressure-correction scheme for the compressible Euler equations

Résumé

We propose a robust pressure-correction scheme for the numerical solution of the compressible Euler equations discretized by a colocated finite volume method. The scheme is based on an internal energy formulation, which ensures that the internal energy is positive. More generally, the scheme enjoys fundamental stability properties: without restriction on the time step, both the density and the internal energy are positive, the integral of the total energy over the computational domain is preserved thanks to an estimate on the discrete kinetic energy, and a discrete entropy ineqality is satisfied. These stability properties ensure the existence of a solution to the scheme. The internal energy balance features a corrective source term which is needed for the scheme to compute the correct shock solutions: we are indeed able to prove a Lax-type convergence result, in the sense that, under some compactness assumptions, the limit of a converging sequence of approximate solutions obtained with space and time discretization steps tending to zero is an entropy weak solution of the Euler equations. The obtained theoretical results and the scheme accuracy are verified ny numerical experiments; in particular, the qualitative behaviour of the scheme is assessed on 1D and 2D Riemann problems and compared with other schemes.
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Dates et versions

hal-01861734 , version 1 (24-08-2018)
hal-01861734 , version 2 (31-03-2019)

Identifiants

Citer

Raphaele Herbin, Jean-Claude Latché, Chady Zaza. A cell-centered pressure-correction scheme for the compressible Euler equations. IMA Journal of Numerical Analysis, 2020, 40 (3), pp.1792-1837. ⟨10.1093/IMANUM/DRZ024⟩. ⟨hal-01861734v2⟩
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