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A staggered scheme for the Euler equations. (English) Zbl 1366.76060

Cancès, Clément (ed.) et al., Finite volumes for complex applications VIII – hyperbolic, elliptic and parabolic problems. FVCA 8, Lille, France, June 12–16, 2017. Cham: Springer (ISBN 978-3-319-57393-9/hbk; 978-3-319-57394-6/ebook; 978-3-319-58818-6/set). Springer Proceedings in Mathematics & Statistics 200, 91-99 (2017).
Summary: We extend to the full Euler system the scheme introduced in [F. Berthelin et al., Math. Comput. 84, No. 295, 2221–2262 (2015; Zbl 1329.76198)] for solving the barotropic Euler equations. This finite volume scheme is defined on staggered grids with numerical fluxes derived in the spirit of kinetic schemes. The difficulty consists in finding a suitable treatment of the energy equation while density and internal energy on the one hand, and velocity on the other hand, are naturally defined on dual locations. The proposed scheme uses the density, the velocity and the internal energy as computational variables and stability conditions are identified in order to preserve the positivity of the discrete density and internal energy. Moreover, we define averaged energies which satisfy local conservation equations. Finally, we provide numerical simulations of Riemann problems to illustrate the behaviour of the scheme.
For the entire collection see [Zbl 1371.65001].

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35Q31 Euler equations

Citations:

Zbl 1329.76198
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References:

[1] Berthelin, F., Goudon, T., Minjeaud, S.: Kinetic schemes on staggered grids for barotropic Euler models: entropy-stability analysis. Math. Comput. 84, 2221–2262 (2015) · Zbl 1329.76198 · doi:10.1090/S0025-5718-2015-02957-3
[2] Herbin, R., Kheriji, W., Latche, J.C.: Staggered schemes for all speed flows. ESAIM Proc. 35, 122–150 (2012) · Zbl 1357.76038 · doi:10.1051/proc/201235008
[3] Herbin, R., Latche, J.C., Nguyen, T.: Consistent Explicit Staggered Schemes for Compressible Flows. Part II: The Euler Equation. hal-00821069 (2013)
[4] Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edn. Springer, Berlin (2009) · Zbl 1227.76006 · doi:10.1007/b79761
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