Hitting time distributions for efficient simulations of drift‐diffusion processes

Abstract

Numerical solutions to partial differential equations in anisotropic, heterogeneous media obtained by their probabilistic representations are useful for a number of purposes, including our own interests in biomedical simulations. These solutions are obtained by a walk with both random and deterministic components hitting a boundary. Hitting time distributions are required to efficiently simulate such processes. The distributions for hitting timeTand placeXTon a surface for a particle undergoing both diffusion and drift are presented here, generalizing the previous work. Importantly, the distributions directly obtained are, on the whole, not useful for numerical work, being expressed as very poorly and nonuniformly convergent series. Resummed series with rapid convergence for probabilities over a useful range of the hitting time to the boundary are here obtained in different ways. A numerical inverse is constructed to implement the random walk. We have used these hitting time distributions, combined with efficient inverse function construction, to obtain speed‐up of simulations of fluid and particle transport by permitting the steps allowed for a given accuracy to be as large as possible.