Efficient Parallel Solution of Nonlinear Parabolic Partial Differential Equations by a Probabilistic Domain Decomposition

Abstract

Initial- and initial-boundary value problems for nonlinear one-dimensional parabolic partial differential equations are solved numerically by a probabilistic domain decomposition method. This is based on a probabilistic representation of solutions by means of branching stochastic processes. Only few values of the solution inside the space-time domain are generated by a Monte Carlo method, and an interpolation is then made so to approximate suitable interfacial values of the solution inside the domain. In this way, a fully decoupled set of sub-problems is obtained. This method allows for an efficient massively parallel implementation, is scalable and fault tolerant. Numerical examples, including some for the KPP equation and beyond are given to show the performance of the algorithm.