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.
On the surface plasmonic waves excited by a dipole above anisotropic and spatially dispersive two-dimensional surfaces of infinite extent in planarly layered media
Domain decomposition for quasi-periodic scattering by layered media via robust boundary-integral equations at all frequencies
MIT（麻省理工学院）；Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）；Louisiana State University（路易斯安那州立大学）；New Jersey Inst. of Technology（新泽西理工学院）；Duke University（杜克大学）
Pseudo-spectral frequency-domain method with background field decomposition and Green's function preconditioner for electromagnetic scattering problem in EUV lithography
