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

Authors:
Mohammed Lemou;Luc Mieussens
Affiliations:
mohammed.lemou@univ-rennes1.fr;Luc.Mieussens@math.univ-toulouse.fr
Venue:
SIAM Journal on Scientific Computing
Year:
2008

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Analysis of an Asymptotic Preserving Scheme for Linear Kinetic Equations in the Diffusion Limit

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A Micro-Macro Decomposition-Based Asymptotic-Preserving Scheme for the Multispecies Boltzmann Equation

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Microscopically implicit-macroscopically explicit schemes for the BGK equation

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Asymptotic-preserving Projective Integration Schemes for Kinetic Equations in the Diffusion Limit

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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 one-dimensional cases by several numerical tests and comparisons with previous asymptotic preserving schemes.