Numerical solution of the dirichlet problem for linear parabolic SPDEs based on averaging over characteristics

Abstract

Numerical methods for the Dirichlet problem for linear parabolic stochastic partial differential equations are constructed. The methods are based on the averaging-over-characteristic formula and the weak-sense numerical integration of ordinary stochastic differential equations in bounded domains. Their orders of convergence in the mean-square sense and in the sense of almost sure convergence are obtained. The Monte Carlo technique is used for practical realization of the methods. Results of some numerical experiments are presented. © 2011 Springer-Verlag Berlin Heidelberg.