Asymptotic-Preserving Monte Carlo methods for transport equations in the diffusive limit

TL;DR

This paper introduces an asymptotic-preserving Monte Carlo method for hyperbolic transport equations with stiff terms, enabling stable simulations across different regimes, especially in the diffusive limit where classical methods struggle.

Contribution

The authors develop a novel Monte Carlo approach that remains stable regardless of the scaling parameter and accurately captures the diffusion limit, overcoming limitations of traditional methods.

Findings

Method is stable for all scaling parameters.

It degenerates to a standard approach in the diffusion limit.

Performance comparisons show advantages over classical methods.

Abstract

We develop a new Monte Carlo method that solves hyperbolic transport equations with stiff terms, characterized by a (small) scaling parameter. In particular, we focus on systems which lead to a reduced problem of parabolic type in the limit when the scaling parameter tends to zero. Classical Monte Carlo methods suffer of severe time step limitations in these situations, due to the fact that the characteristic speeds go to infinity in the diffusion limit. This makes the problem a real challenge, since the scaling parameter may differ by several orders of magnitude in the domain. To circumvent these time step limitations, we construct a new, asymptotic-preserving Monte Carlo method that is stable independently of the scaling parameter and degenerates to a standard probabilistic approach for solving the limiting equation in the diffusion limit. The method uses an implicit time discretization to formulate a modified equation in which the characteristic speeds do not grow indefinitely when the scaling factor tends to zero. The resulting modified equation can readily be discretized by a Monte Carlo scheme, in which the particles combine a finite propagation speed with a time-step dependent diffusion term. We show the performance of the method by comparing it with standard (deterministic) approaches in the literature.