Kinetic-diffusion asymptotic-preserving Monte Carlo algorithm for Boltzmann-BGK in the diffusive scaling

TL;DR

This paper introduces a hybrid Monte Carlo algorithm for the Boltzmann-BGK model that adapts to both low and high collisional regimes, ensuring accuracy and efficiency across different physical conditions.

Contribution

The authors develop an asymptotic-preserving Monte Carlo method that seamlessly transitions between kinetic and diffusive behaviors based on local collisionality, maintaining statistical properties.

Findings

The method accurately captures mean, variance, and correlation of positional increments.

It reverts to standard velocity-jump in low-collisional regimes.

It simplifies to a random walk in high-collisional regimes.

Abstract

We develop a novel Monte Carlo strategy for the simulation of the Boltzmann-BGK model with both low-collisional and high-collisional regimes present. The presented solution to maintain accuracy in low-collisional regimes and remove exploding simulation costs in high-collisional regimes uses hybridized particles that exhibit both kinetic behaviour and diffusive behaviour depending on the local collisionality. In this work, we develop such a method that maintains the correct mean, variance, and correlation of the positional increments over multiple time steps of fixed step size for all values of the collisionality, under the condition of spatial homogeneity during the time step. In the low-collisional regime, the method reverts to the standard velocity-jump process. In the high-collisional regime, the method collapses to a standard random walk process. We analyze the error of the presented scheme in the low-collisional regime for which we obtain the order of convergence in the time step size. We furthermore provide an analysis in the high-collisional regime that demonstrates the asymptotic-preserving property.