In the first part of this work, we introduce a new relaxation system in order to approximate the solutions to the barotropic Euler equations. We show that the solutions to this two-speed relaxation model can be understood as viscous approximations of the solutions to the barotropic Euler equations under appropriate sub-characteristic conditions. Our relaxation system is a generalization of the well-known Suliciu relaxation system, and it is entropy satisfying. A Godunovtype finite volume scheme based on the exact resolution of the Riemann problem associated with the relaxation system is deduced, as well as its stability properties. In the second part of this work, we show how the new relaxation approach can be successfully applied to the numerical approximation of low Mach number flows. We prove that the underlying scheme satisfies the well-known asymptotic-preserving property in the sense that it is uniformly (first-order) accurate with respect to the Mach number, and at the same time it satisfies a fully discrete entropy inequality. This discrete entropy inequality allows us to prove strong stability properties in the low Mach regime. At last, numerical experiments are given to illustrate the behaviour of our scheme.

Hydrodynamic simulations of high-energy-density plasmas require a detailed description of energy fluxes. For low and intermediate atomic number materials, the leading mechanism is the electron transport, which may be a nonlocal phenomenon requiring a kinetic modeling. In this paper, we present and test the results of a nonlocal model based on the first angular moments of a simplified Fokker-Planck equation. This multidimensional model is closed thanks to an entropic relation (the Boltzman H-theorem). It provides a better description of the electron distribution function, thus enabling studies of small scale kinetic effects within the hydrodynamic framework. Examples of instabilities of electron plasma and ion-acoustic waves, driven by the heat flux, are presented and compared with the classical formula.

Abstract. This work is devoted to the derivation of an asymptotic-preserving scheme for the electronic M1 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 HLL [A. Harten, P.D. Lax and B. Van Leer, SIAM Rev. 25 (1983) 35-61.] scheme usually used for the electronic M1 model. It is shown that the new scheme gives comparable results with respect to the HLL scheme in the classical regime. On the contrary, in the diffusive regime, the asymptotic-preserving scheme coincides with the expected diffusion equation, while the HLL scheme suffers from a severe lack of accuracy because of its unphysical numerical viscosity.Mathematics Subject Classification. 65C20, 65M12.