Accelerated spatial approximations for time discretized stochastic partial differential equations

TL;DR

This paper develops methods to improve the convergence rate of space-time discretization schemes for linear parabolic stochastic PDEs, extending previous results to include schemes that discretize both in space and time.

Contribution

It introduces conditions for accelerating spatial approximation convergence using Richardson's method in combined space-time discretization schemes for SPDEs.

Findings

Higher order accuracy achieved through Richardson's method.

Extended convergence results to schemes discretizing in both space and time.

Applicable to linear parabolic stochastic PDEs on the whole space.

Abstract

The present article investigates the convergence of a class of space-time discretization schemes for the Cauchy problem for linear parabolic stochastic partial differential equations (SPDEs) defined on the whole space. Sufficient conditions are given for accelerating the convergence of the scheme with respect to the spatial approximation to higher order accuracy by an application of Richardson's method. This work extends the results of Gy\"ongy and Krylov [SIAM J. Math. Anal., 42 (2010), pp. 2275--2296] to schemes that discretize in time as well as space.