SPECTRWM: Spectral Random Walk Method for the Numerical Solution of Stochastic Partial Differential Equations

摘要

The numerical solution of stochastic partial differential equations (SPDE)presents challenges not encountered in the simulation of PDEs or SDEs. Indeed,the roughness of the noise in conjunction with nonlinearities in the drifttypically make these equations particularly stiff. In practice, this means thatit is tricky to construct, operate, and validate numerical methods for SPDEs.This is especially true if one is interested in path-dependent expected values,long-time simulations, or in the simulation of SPDEs whose solutions haveconstraints on their domains. To address these numerical issues, this paperintroduces a Markov jump process approximation for SPDEs, which we refer to asthe spectral random walk method (SPECTRWM). The accuracy and ergodicity ofSPECTRWM are verified in the context of a heat and overdamped Langevin SPDE,respectively. We also apply the method to Burgers and KPZ SPDEs.