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In this paper, we build and analyze the stability and consistency of decoupled schemes, involving only explicit steps, for the isentropic Euler equations and for the full Euler equations. These schemes are based on staggered space discretizations, with an upwinding performed with respect to the material velocity only. The pressure gradient is defined as the transpose of the natural velocity divergence, and is thus centered. The velocity convection term is built in such a way that the solutions satisfy a discrete kinetic energy balance, with a remainder term at the left-hand side which is shown to be non-negative under a CFL condition. In the case of the full Euler equations, we solve the internal energy balance, to avoid the space discretization of the total energy, whose expression involves cell-centered and face-centered variables. However, since the residual terms in the kinetic energy balance (probably) do not tend to zero with the time and space steps when computing shock solutions, we compensate them by corrective terms in the internal energy equation, to make the scheme consistent with the conservative form of the continuous problem. We then show, in one space dimension, that, if the scheme converges, the limit is indeed an entropy weak solution of the system. In any case, the discretization preserves by construction the convex of admissible states (positivity of the density and, for Euler equations, of the internal energy), under a CFL condition. Finally, we present numerical results which confort this theory.
Herbin, Raphaèle 1 ; Latché, Jean-Claude 1 ; Nguyen, Trung Tan 1
@article{M2AN_2018__52_3_893_0,
author = {Herbin, Rapha\`ele and Latch\'e, Jean-Claude and Nguyen, Trung Tan},
title = {Consistent segregated staggered schemes with explicit steps for the isentropic and full {Euler} equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {893--944},
publisher = {EDP-Sciences},
volume = {52},
number = {3},
year = {2018},
doi = {10.1051/m2an/2017055},
mrnumber = {3865553},
zbl = {1405.35149},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017055/}
}
TY - JOUR AU - Herbin, Raphaèle AU - Latché, Jean-Claude AU - Nguyen, Trung Tan TI - Consistent segregated staggered schemes with explicit steps for the isentropic and full Euler equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 893 EP - 944 VL - 52 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017055/ DO - 10.1051/m2an/2017055 LA - en ID - M2AN_2018__52_3_893_0 ER -
%0 Journal Article %A Herbin, Raphaèle %A Latché, Jean-Claude %A Nguyen, Trung Tan %T Consistent segregated staggered schemes with explicit steps for the isentropic and full Euler equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 893-944 %V 52 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017055/ %R 10.1051/m2an/2017055 %G en %F M2AN_2018__52_3_893_0
Herbin, Raphaèle; Latché, Jean-Claude; Nguyen, Trung Tan. Consistent segregated staggered schemes with explicit steps for the isentropic and full Euler equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 893-944. doi: 10.1051/m2an/2017055
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