American Institute of Mathematical Sciences

February  2020, 13(1): 33-61. doi: 10.3934/krm.2020002

A kinetic approach of the bi-temperature Euler model

 1 Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France 2 Univ. Bordeaux, Laboratoire des Composites ThermoStructuraux (LCTS), UMR 5801: CNRS-Herakles(Safran)-CEA-UBx, 3, Allée de La Boétie, 33600 Pessac, France

Received  November 2018 Revised  September 2019 Published  December 2019

We are interested in the numerical approximation of the bi-temperature Euler equations, which is a non conservative hyperbolic system introduced in [4]. We consider a conservative underlying kinetic model, the Vlasov-BGK-Poisson system. We perform a scaling on this system in order to obtain its hydrodynamic limit. We present a deterministic numerical method to approximate this kinetic system. The method is shown to be Asymptotic-Preserving in the hydrodynamic limit, which means that any stability condition of the method is independant of any parameter $\varepsilon$, with $\varepsilon \rightarrow 0$. We prove that the method is, under appropriate choices, consistant with the solution for bi-temperature Euler. Finally, our method is compared to methods for the fluid model (HLL, Suliciu).

Citation: Stéphane Brull, Bruno Dubroca, Corentin Prigent. A kinetic approach of the bi-temperature Euler model. Kinetic & Related Models, 2020, 13 (1) : 33-61. doi: 10.3934/krm.2020002
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References:
Density and velocity solutions of shock tube test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
Electronic and ionic temperatures of a shock tube test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
Different jump relations through shocks of electronic and ionic temperatures for a shock tube test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
Electric field obtained by the kinetic scheme and by Ohm's law for the Sod tube test case with identical initial temperatures
Density and velocity solutions of shock tube test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
Electronic and ionic temperatures of a shock tube test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
Different jump relations of electronic and ionic temperatures for a shock tube test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
Electric field obtained by the kinetic scheme and by Ohm's law for the Sod tube test case with different initial temperatures
Density and velocity solution of a rarefaction wave test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
Temperature solutions of a rarefaction wave test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
Zoom on temperature solutions of a rarefaction wave test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
Electric field obtained by the kinetic scheme and by Ohm's law for the double rarefaction test case
Density and veloctiy solutions of a shock wave test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
Electronic and ionic temperature of a shock wave test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
Electronic and ionic temperature of a shock wave test case with a mass ratio of 10 with 120000 space points, 40 velocity points and a domain length of 8
Electric field obtained by the kinetic scheme and by Ohm's law for the double shock test case
Density and velocity solutions of shock tube test case with a mass ratio of 1000 with 100000 space points, 40 velocity points and a domain length of 8
Electronic and ionic temperatures of a shock tube test case with a mass ratio of 1000 with 100000 space points, 40 velocity points and a domain length of 8
Different jump relations of electronic and ionic temperatures for a shock tube test case with a mass ratio of 1000 with 100000 space points, 40 velocity points and a domain length of 8
Electric field obtained by the kinetic scheme and by Ohm's law for a Sod tube test with different initial temperatures
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