Asymptotic-preserving well-balanced scheme for the electronic $M_{1}$ model in the diffusive limit: Particular cases

Guisset, Sébastien; Brull, Stéphane; D’Humières, Emmanuel; Dubroca, Bruno

doi:10.1051/m2an/2016079

Geodesic

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Asymptotic-preserving well-balanced scheme for the electronic

M_{1}

model in the diffusive limit: Particular cases

Guisset, Sébastien ^1,² ; Brull, Stéphane ¹ ; D’Humières, Emmanuel ² ; Dubroca, Bruno ²

¹ Université Bordeaux, IMB, UMR 5251, 33405 Talence, France.
² Université Bordeaux, CELIA, UMR 5107, 33400 Talence, France.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1805-1826

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Résumé

This work is devoted to the derivation of an asymptotic-preserving scheme for the electronic $M_{1}$ model in the diffusive regime. The case without electric field and the homogeneous case are studied. The derivation of the scheme is based on an approximate Riemann solver where the intermediate states are chosen consistent with the integral form of the approximate Riemann solver. This choice can be modified to enable the derivation of a numerical scheme which also satisfies the admissible conditions and is well-suited for capturing steady states. Moreover, it enjoys asymptotic-preserving properties and handles the diffusive limit recovering the correct diffusion equation. Numerical tests cases are presented, in each case, the asymptotic-preserving scheme is compared to the classical $H L L$ [A. Harten, P.D. Lax and B. Van Leer, SIAM Rev. 25 (1983) 35–61.] scheme usually used for the electronic $M_{1}$ model. It is shown that the new scheme gives comparable results with respect to the $H L L$ scheme in the classical regime. On the contrary, in the diffusive regime, the asymptotic-preserving scheme coincides with the expected diffusion equation, while the $H L L$ scheme suffers from a severe lack of accuracy because of its unphysical numerical viscosity.

MR Zbl

DOI : 10.1051/m2an/2016079

Classification : 65C20, 65M12
Keywords: Electronic M₁moment model, approximate Riemann solvers, Godunov type schemes, asymptotic preserving schemes, diffusive limit, plasma physics

Affiliations des auteurs :

Guisset, Sébastien ^{1, 2} ; Brull, Stéphane ¹ ; D’Humières, Emmanuel ² ; Dubroca, Bruno ²

¹ Université Bordeaux, IMB, UMR 5251, 33405 Talence, France.
² Université Bordeaux, CELIA, UMR 5107, 33400 Talence, France.

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     title = {Asymptotic-preserving well-balanced scheme for the electronic $M_{1}$ model in the diffusive limit: {Particular} cases},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1805--1826},
     publisher = {EDP-Sciences},
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Guisset, Sébastien; Brull, Stéphane; D’Humières, Emmanuel; Dubroca, Bruno. Asymptotic-preserving well-balanced scheme for the electronic $M_{1}$ model in the diffusive limit: Particular cases. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1805-1826. doi: 10.1051/m2an/2016079

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