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A well-balanced scheme for the Euler equation with a gravitational potential. (English) Zbl 1304.76027
Fuhrmann, Jürgen (ed.) et al., Finite volumes for complex applications VII – methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, Germany, June 15–20, 2014. Vol. I. Cham: Springer (ISBN 978-3-319-05683-8/hbk; 978-3-319-05684-5/ebook; 978-3-319-06402-4/set). Springer Proceedings in Mathematics & Statistics 77, 217-226 (2014).
Summary: The aim of this work is to derive a well-balanced numerical scheme to approximate the solutions of the Euler equations with a gravitational potential. This system admits an infinity of steady state solutions which are not all known in an explicit way. Among all these solutions, the hydrostatic atmosphere has a special physical interest. We develop an approximate Riemann solver using the formalism of Harten, Lax and van Leer, which takes into account the source term. The resulting numerical scheme is proven to be robust, to preserve exactly the hydrostatic atmosphere and to preserve an approximation of all the other steady state solutions.
For the entire collection see [Zbl 1291.65004].

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
76N99 Compressible fluids and gas dynamics
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