A numerical scheme for a kinetic model for mixtures in the diffusive limit using the moment method

Abstract

In this article, we consider a multi‐species kinetic model which leads to the Maxwell–Stefan equations under a standard diffusive scaling (small Knudsen and Mach numbers). We propose a suitable numerical scheme which approximates both the solution of the kinetic model in rarefied regime and the one in the diffusion limit. We prove some a priori estimates (mass conservation and nonnegativity) and well‐posedness of the discrete problem. We also present numerical examples where we observe the asymptotic‐preserving behavior of the scheme.

From the Boltzmann Description for Mixtures to the Maxwell–Stefan Diffusion Equations

“…7 Some related research directions Some other articles consider problems with a strong relationship with the subject of this note. Bondesan, Boudin and Grec [6] derive a numerical method for the Euler equations describing a mixture, stable with respect to the relaxation parameter, whose limit is the Maxwell-Stefan system.…”

On the Maxwell-Stefan diffusion limit for a reactive mixture of polyatomic gases in non-isothermal setting

“…As usual in kinetic theory, the post-collisional velocities can be expressed in terms of pre-collisional velocities and the corresponding expressions for both elastic and reactive collisions are derived from the conservation laws (8), (9) and (10), (11) respectively. In order to give these expressions, we describe first the Borgnakke-Larsen procedure [10].…”

“…Using the conservation of momentum (8) for the two first addends within the brackets and the conservation of energy (9) for the next two addends, we obtain the result stated in equation (59). Therefore, part (a) of Proposition 9 is proven.…”

“…In particular, the first studies have been devoted to the matrix formulation of the gradient-flux relationships (see [23] and the references therein). Subsequently, the wellposedness and the long-time behavior of the solutions of the Maxwell-Stefan system (or some variants) have been studied in [11,14,16,25,28,29], the numerical simulation of the Maxwell-Stefan system has been the subject of [9,14,22,32], and the relationships between Fickian diffusion and the Maxwell-Stefan model have been analyzed in [12,37]. By following the research line initiated by Bardos, Golse and Levermore in the Nineties -whose goal was the derivation of the equations of fluid mechanics starting from the Boltzmann equation [2,3] -several articles have carried out the formal derivation of isothermal multicomponent Maxwell-Stefan type diffusion equations starting from the Boltzmann system for monatomic non-reactive gaseous mixtures [12,13,15,26,27].…”

Projective and Telescopic Projective Integration for Non-Linear Kinetic Mixtures

“…At the numerical level, the most advanced deterministic methods use a Fourier approach to evaluate the full collision operator: a fast spectral algorithm was recently introduced in [59] for the multispecies case. Nevertheless, the method was not asymptotic preserving (AP), namely, stable in the small relaxation parameter limit; the only recent paper on this subject can be found in [7], where the Maxwell-Stephan limit for a multispecies gas is investigated numerically via a moment method (which does not compute the full operator). In this work, we shall introduce a new family of numerical integrators for full kinetic multispecies models that are able to deal with a large range of values of the relaxation parameters uniformly on the numerical parameters.…”