A kinetic approach of the bi-temperature Euler model

Stéphane Brull; Bruno Dubroca; Corentin Prigent

doi:10.3934/krm.2020002

Article Contents

2020, Volume 13, Issue 1: 33-61. Doi: 10.3934/krm.2020002

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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: October 31, 2018

Revised: August 31, 2019

Published: January 2020

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Abstract

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).

Keywords:

Mathematics Subject Classification: 82B40, 82D10, 34K28, 76N99.

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Figure 1. 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

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Figure 2. 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

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Figure 3. 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

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Figure 4. Electric field obtained by the kinetic scheme and by Ohm's law for the Sod tube test case with identical initial temperatures

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Figure 5. 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

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Figure 6. 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

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Figure 7. 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

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Figure 8. Electric field obtained by the kinetic scheme and by Ohm's law for the Sod tube test case with different initial temperatures

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Figure 9. 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

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Figure 10. 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

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Figure 11. 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

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Figure 12. Electric field obtained by the kinetic scheme and by Ohm's law for the double rarefaction test case

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Figure 13. 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

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Figure 14. 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

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Figure 15. 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

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Figure 16. Electric field obtained by the kinetic scheme and by Ohm's law for the double shock test case

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Figure 17. 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

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Figure 18. 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

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Figure 19. 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

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Figure 20. 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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