title: Deriving appropriate boundary conditions, and accelerating position-jump simulations, of diffusion using non-local jumping.
abstract: In this paper we explore lattice-based position-jump models of diffusion, and the implications of introducing non-local jumping; particles can jump to a range of nearby boxes rather than only to their nearest neighbours. We begin by deriving conditions for equivalence with traditional local jumping models in the continuum limit. We then generalize a previously postulated implementation of the Robin boundary condition for a non-local process of arbitrary maximum jump length, and present a novel implementation of flux boundary conditions, again generalized for a non-local process of arbitrary maximum jump length. In both these cases we validate our results using stochastic simulation. We then proceed to consider two variations on the basic diffusion model: a hybrid local/non-local scheme suitable for models involving sharp concentration gradients, and the implementation of biased jumping. In all cases we show that non-local jumping can deliver substantial time savings for stochastic simulations.
authors.affiliation: Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.
authors.forename: P R
authors.initials: PR
authors.lastname: Taylor
authors.forename: R E
authors.initials: RE
authors.lastname: Baker
authors.forename: C A
authors.initials: CA
authors.lastname: Yates
mesh_terms: D003198:Computer Simulation; D004058:Diffusion; D008954:Models, Biological; D008956:Models, Chemical; D013269:Stochastic Processes