Stochastic models of advection-diffusion in layered media

TL;DR

This paper develops stochastic random walk models that approximate continuum advection-diffusion equations in layered media, providing a probabilistic framework that captures inherent randomness and aligns well with traditional models.

Contribution

It introduces a discretization-based method to derive Markov chain models from continuum PDEs, including both local and non-local random walks with explicit transition probabilities.

Findings

Stochastic models closely match continuum PDE solutions in simulations.

Explicit transition probabilities are derived for different discretization schemes.

Constraints ensure non-negativity of transition probabilities.

Abstract

Mathematically modelling diffusive and advective transport of particles in heterogeneous layered media is important to many applications in computational, biological and medical physics. While deterministic continuum models of such transport processes are well established, they fail to account for randomness inherent in many problems and are valid only for a large number of particles. To address this, this paper derives a suite of equivalent stochastic (discrete-time discrete-space random walk) models for several standard continuum (partial differential equation) models of diffusion and advection-diffusion across a fully- or semi-permeable interface. Our approach involves discretising the continuum model in space and time to yield a Markov chain, which governs the transition probabilities between spatial lattice sites during each time step. Discretisation in space is carried out using a standard finite volume method while two options are considered for discretisation in time. A simple forward Euler discretisation yields a stochastic model taking the form of a local (nearest-neighbour) random walk with simple analytical expressions for the transition probabilities while an exact exponential discretisation yields a non-local random walk with transition probabilities defined numerically via a matrix exponential. Constraints on the size of the spatial and/or temporal steps are provided for each option to ensure the transition probabilities are non-negative. MATLAB code comparing the stochastic and continuum models is available on GitHub (https://github.com/elliotcarr/Carr2024c) with simulation results demonstrating good agreement for several example problems.