A Successive Penalty-Based Asymptotic-Preserving Scheme for Kinetic Equations

Abstract

We propose an asymptotic-preserving (AP) scheme for kinetic equations that is efficient also in the hydrodynamic regimes. This scheme is based on the BGK-penalty method introduced by , but uses the penalization successively to achieve the desired asymptotic property. This method possesses a stronger AP property than the original method of Filbet-Jin, with the additional feature of being also positivity preserving when applied on the Boltzmann equation. It is also general enough to be applicable to several important classes of kinetic equations, including the Boltzmann equation and the Landau equation. Numerical experiments verify these properties.

An asymptotic-preserving Monte Carlo method for the Boltzmann equation

“…As a result, one can obtain a Boltzmann solver uniformly stable with respect to ε, and yet can be implemented explicitly. Later, Yan and Jin [39] proposed a successive penalty AP scheme which was positivitypreserving and has strong AP properties. Their method had been implemented in the finite difference framework in [39].…”

“…Later, Yan and Jin [39] proposed a successive penalty AP scheme which was positivitypreserving and has strong AP properties. Their method had been implemented in the finite difference framework in [39]. Earlier, Pareschi and Caflisch [31] formulated a hybrid Monte Carlo method that performed well in the fluid dynamic regime for the space homogeneous Boltzmann equation.…”

“…The goal of this work is to propose a new asymptotic-preserving direct simulation Monte Carlo method (AP-DSMC) based on the successive penalty AP time-discretization [39]. This line of research is of significance not only in rarefied gas and reentry problems [22,26,27,30,38] governed by the Boltzmann equation, but also in some more complicated transport phenomena, whose master equations are the quantum Boltzmann equation [15] and the Landau equation [25] in plasma physics.…”

“…This line of research is of significance not only in rarefied gas and reentry problems [22,26,27,30,38] governed by the Boltzmann equation, but also in some more complicated transport phenomena, whose master equations are the quantum Boltzmann equation [15] and the Landau equation [25] in plasma physics. For the space homogeneous Boltzmann equation, it was realized in [39] that the successive penalization step can be written as a convex combination of the density distribution of the previous time step, a positive collision operator and the local Maxwellian. This forms the basis for a Monte-Carlo implementation [31].…”

“…This method is based on the successive penalty method [39], which is an improved BGK-penalization method originally proposed by Filbet-Jin [16]. Here we introduce the Monte-Carlo implementation of the method, which, despite of its lower order accuracy, is very efficient in higher dimensions or simulating some complicated chemical processes.…”