Random walk methods for scalar transport problems subject to Dirichlet, Neumann and mixed boundary conditions

Abstract

Three stochastic-based methods are proposed for solving unsteady scalar transport problems in bounded, single-phase domains. The first (Method I), a local solution appropriate to problems having Dirichlet conditions, adapts a well-known local stochastic solution of a backward Fokker–Planck equation to scalar transport. Method II, a local solution applicable to Dirichlet, Neumann and/or mixed initial boundary value problems (IBVPs), and representing a time-dependent extension of a recently reported heuristic steady solution, provides a straightforward addition to the limited collection of techniques available for Neumann and mixed problems. This approach is shown to be equivalent to a long-standing, rigorous low-order solution and, in addition, allows development of a probabilistic-based analytical solution to Neumann problems, stated in terms of an exit probability. Method III, a global solution, likewise suitable for Neumann and mixed IBVPs, follows by combined application of domain boundary Taylor expansions and Method I. This approach is shown to be computationally equivalent to a global version of Method II.