On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 6, p. 1807-1857
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In this paper, we propose implicit and semi-implicit in time finite volume schemes for the barotropic Euler equations (hence, as a particular case, for the shallow water equations) and for the full Euler equations, based on staggered discretizations. For structured meshes, we use the MAC finite volume scheme, and, for general mixed quadrangular/hexahedral and simplicial meshes, we use the discrete unknowns of the Rannacher-Turek or Crouzeix-Raviart finite elements. We first show that a solution to each of these schemes satisfies a discrete kinetic energy equation. In the barotropic case, a solution also satisfies a discrete elastic potential balance; integrating these equations over the domain readily yields discrete counterparts of the stability estimates which are known for the continuous problem. In the case of the full Euler equations, the scheme relies on the discretization of the internal energy balance equation, which offers two main advantages: first, we avoid the space discretization of the total energy, which involves cell-centered and face-centered variables; second, we obtain an algorithm which boils down to a usual pressure correction scheme in the incompressible limit. Consistency (in a weak sense) with the original total energy conservative equation is obtained thanks to corrective terms in the internal energy balance, designed to compensate numerical dissipation terms appearing in the discrete kinetic energy inequality. It is then shown in the 1D case, that, supposing the convergence of a sequence of solutions, the limit is an entropy weak solution of the continuous problem in the barotropic case, and a weak solution in the full Euler case. Finally, we present numerical results which confirm this theory.

DOI : https://doi.org/10.1051/m2an/2014021
Classification:  35Q31,  65N12,  76M10,  76M12
Keywords: finite volumes, finite elements, staggered, pressure correction, Euler equations, shallow-water equations, compressible flows, analysis
@article{M2AN_2014__48_6_1807_0,
     author = {Herbin, R. and Kheriji, W. and Latch\'e, J.-C.},
     title = {On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {6},
     year = {2014},
     pages = {1807-1857},
     doi = {10.1051/m2an/2014021},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_6_1807_0}
}
Herbin, R.; Kheriji, W.; Latché, J.-C. On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 6, pp. 1807-1857. doi : 10.1051/m2an/2014021. http://www.numdam.org/item/M2AN_2014__48_6_1807_0/

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