Accelerated simulation of Boltzmann-BGK equations near the diffusive limit with asymptotic-preserving multilevel Monte Carlo

TL;DR

This paper introduces an asymptotic-preserving multilevel Monte Carlo scheme that accelerates the simulation of Boltzmann-BGK equations near the diffusive limit by leveraging the diffusive approximation to reduce computational costs.

Contribution

The authors develop a novel multilevel Monte Carlo particle scheme that uses the diffusive limit for efficient simulation of kinetic equations at high collision rates.

Findings

Significant reduction in simulation cost in high-collision regimes.

Effective correlation of particle trajectories across multiple discretization levels.

Validation through analysis and numerical experiments showing improved efficiency.

Abstract

Kinetic equations model the position-velocity distribution of particles subject to transport and collision effects. Under a diffusive scaling, these combined effects converge to a diffusion equation for the position density in the limit of an infinite collision rate. Despite this well-defined limit, numerical simulation is expensive when the collision rate is high but finite, as small time steps are then required. In this work, we present an asymptotic-preserving multilevel Monte Carlo particle scheme that makes use of this diffusive limit to accelerate computations. In this scheme, we first sample the diffusive limiting model to compute a biased initial estimate of a Quantity of Interest, using large time steps. We then perform a limited number of finer simulations with transport and collision dynamics to correct the bias. The efficiency of the multilevel method depends on being able to perform correlated simulations of particles on a hierarchy of discretization levels. We present a method for correlating particle trajectories and present both an analysis and numerical experiments. We demonstrate that our approach significantly reduces the cost of particle simulations in high-collisional regimes, compared with prior work, indicating significant potential for adopting these schemes in various areas of active research.