The moment‐guided Monte Carlo method

Abstract

SUMMARYIn this work we propose a new approach for the numerical simulation of kinetic equations through Monte Carlo schemes. We introduce a new technique that permits to reduce the variance of particle methods through a matching with a set of suitable macroscopic moment equations. In order to guarantee that the moment equations provide the correct solutions, they are coupled to the kinetic equation through a nonequilibrium term. The basic idea, on which the method relies, consists in guiding the particle positions and velocities through moment equations so that the concurrent solution of the moment and kinetic models furnishes the same macroscopic quantities.

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

“…Therefore, it does not need a specific velocity and space discretization and can be easily applied to different stochastic and deterministic schemes. Moreover, based on this decomposition, other schemes can be constructed [17].…”

An asymptotic-preserving Monte Carlo method for the Boltzmann equation

“…In order to handle the implicit collision term efficiently, a BGK-penalization method was introduced by Filbet and Jin [16], utilizing the factor that the BGK collision operator 12) can be explicitly inverted. As a result, one can obtain a Boltzmann solver uniformly stable with respect to ε, and yet can be implemented explicitly.…”

“…Then Pareschi and Trazzi [33] extended this class of AP schemes to two dimensions in space, and demonstrated its better performance in efficiency compared with the conventional DSMC method as ε → 0. Later, Degond et al [12] introduced a moment-guided Monte Carlo method to reduce the variance. They solved the kinetic equation and the fluid equations respectively, and matched the moments of both solutions.…”

“…The computational domain is often divided into a great number of cells {m = 1, ..., MNC}, where MNC is the total number of cells, to locate particles and sample the local macroscopic quantities. Given the initial condition ( ρ, υ and T ) in cell m, one can obtain the number of particles N m by 12) here x is the largest integer which is less than or equal to x, and V m is the volume of cell m. Then, the local Maxwellian distribution function M (υ, T ) is applied to generate N m particles and their initial velocities,…”

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

“…We have to ensure that the micro-macro structure f = ρM + g with ρ = f dv is preserved numerically. To do that, we correct the weights ω n+1 k , adapting an idea of [7]. We do not detail this procedure here but refer the reader to [5].…”