The moment guided Monte Carlo method for the Boltzmann equation

Giacomo Dimarco

doi:10.3934/krm.2013.6.291

Article Contents

2013, Volume 6, Issue 2: 291-315. Doi: 10.3934/krm.2013.6.291

This issue Previous Article Nonlinear stability of a Vlasov equation for magnetic plasmas Next Article Spectral decompositions and $\mathbb{L}^2$-operator norms of toy hypocoercive semi-groups

The moment guided Monte Carlo method for the Boltzmann equation

Giacomo Dimarco^1,

1.
Institut de Mathématiques de Toulouse, UMR 5219, Université Paul Sabatier, 118, route de Narbonne 31062 TOULOUSE Cedex

Received: July 2012

Revised: October 2012

Published: June 2013

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Abstract

In this work we propose a generalization of the Moment Guided Monte Carlo method developed in [11]. This approach permits to reduce the variance of the 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 non equilibrium term. Here, at the contrary to the previous work in which we considered the simplified BGK operator, we deal with the full Boltzmann operator. Moreover, we introduce an hybrid setting which permits to entirely remove the resolution of the kinetic equation in the limit of infinite number of collisions and to consider only the solution of the compressible Euler equation. This modification additionally reduce the statistical error with respect to our previous work and permits to perform simulations of non equilibrium gases using only a few number of particles. We show at the end of the paper several numerical tests which prove the efficiency and the low level of numerical noise of the method.

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Mathematics Subject Classification: Primary: 76P05, 65C20, 65C05.

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