Estimation as a post‐processing step for random walk approximations of the Boltzmann‐BGK model

Abstract

Recently, asymptotic‐preserving Monte Carlo simulation methods have been developed to simulate the Boltzmann‐BGK equation with advective–diffusive limiting behaviour over a broad range of regimes. These simulation methods hybridize a particle tracing Monte Carlo scheme for the kinetic equation and a random walk (RW) Monte Carlo simulation for the advection–diffusion limit, combining the precision of the former with the efficiency of the latter. In the RW part of the simulation, details of the travelled path are absent. This complicates the extraction of integral quantities of interest such as mass, momentum, and energy sources. Here, we present a new estimation strategy that couples the RW parts of the particle trajectories with a deterministic simulation of the corresponding fluid model. The contributions of the RW parts to quantities of interest are then computed from a single time step of this fluid model in a post‐processing step. We illustrate this new estimation strategy for a fusion‐relevant test‐case and focus on the bias and variance present when using this estimator.