The Moment Guided Monte Carlo method for the Boltzmann equation

TL;DR

This paper introduces an enhanced Monte Carlo method for the Boltzmann equation that reduces variance and computational cost by coupling kinetic and macroscopic equations, enabling efficient simulations of non-equilibrium gases.

Contribution

It generalizes the Moment Guided Monte Carlo method to the full Boltzmann operator and introduces a hybrid approach that simplifies the kinetic resolution in high collision regimes.

Findings

Significant variance reduction in particle simulations.

Effective coupling of kinetic and macroscopic models.

Reduced computational cost for non-equilibrium gas simulations.

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.