A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit

Abstract

We propose a new numerical scheme for linear transport equations. It is based on a decomposition of the distribution function into equilibrium and nonequilibrium parts. We also use a projection technique that allows us to reformulate the kinetic equation into a coupled system of an evolution equation for the macroscopic density and a kinetic equation for the nonequilibrium part. By using a suitable time semi-implicit discretization, our scheme is able to accurately approximate the solution in both kinetic and diffusion regimes. It is asymptotic preserving in the following sense: when the mean free path of the particles is small, our scheme is asymptotically equivalent to a standard numerical scheme for the limit diffusion model. A uniform stability property is proved for the simple telegraph model. Various boundary conditions are studied. Our method is validated in onedimensional cases by several numerical tests and comparisons with previous asymptotic preserving schemes.

Multi-revolution composition methods for highly oscillatory differential equations

“…which allows to conclude the proof of the first estimate in (20) by choosing C n =C n e LH 0 in (21). The second estimate is straightforwardly obtained through a change of variables.…”

“…Actually, MRCM are asymptotic preserving, a notion introduced in the context of kinetic equations (see [18], and the recent works [20,14]) and ensuring that a method is uniformly accurate for a large range of values of the parameter ε with a computational cost essentially independent of ε. This is a feature shared by the proposed classes of multi-revolution methods.…”

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

“…This result can be extended to essentially all AP schemes, although the specific proof is problem dependent. We refer to AP schemes for kinetic equations in the fluid dynamic or diffusive regimes [2,7,14,32,[40][41][42]44,45,[47][48][49]. The AP framework has also been extended in [15,16] for the study of the quasi-neutral limit of Euler-Poisson and Vlasov-Poisson systems, and in [19,21,34] for all-speed (Mach number) fluid equations bridging the passage from compressible flows to the incompressible flows.…”

Asymptotic-Preserving Scheme Based on a Finite Volume/Particle-In-Cell Coupling for Boltzmann-BGK-Like Equations in the Diffusion Scaling

“…To that purpose, we extend the Finite Volumes/Particles hybrid scheme developed in [5], based on a micro-macro decomposition technique introduced in [1] or [13]. Whereas a uniform grid was used to approximate both the micro and the macro part of the full distribution function in [13], we use here a particle approximation for the kinetic (micro) part, the fluid (macro) part being always discretized by standard finite volume schemes. There are many advantages in doing so: (i) the so-obtained scheme presents a much less level of noise compared to the standard particle method; (ii) the computational cost of the micro-macro model is reduced in the diffusion limit since a small number of particles is needed for the micro part; (iii) the scheme is asymptotic preserving in the sense that it is consistent with the kinetic equation in the rarefied regime and it degenerates into a uniformly (with respect to the Knudsen number) consistent (and deterministic) approximation of the limiting equation in the diffusion regime.…”

“…To that purpose, we extend the Finite Volumes/Particles hybrid scheme developed in [5], based on a micro-macro decomposition technique introduced in [1] or [13]. Whereas a uniform grid was used to approximate both the micro and the macro part of the full distribution function in [13], we use here a particle approximation for the kinetic (micro) part, the fluid (macro) part being always discretized by standard finite volume schemes.…”