Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative

TL;DR

This paper extends implicit time-stepping schemes for linear parabolic SPDEs with multiplicative noise, demonstrating unconditional stability and mean-square convergence, enabling adaptive spatial mesh refinement.

Contribution

It introduces a semi- and fully implicit Milstein finite difference scheme for SPDEs, analyzing its stability and accuracy using combined PDE and SDE techniques.

Findings

Crank-Nicolson timestepping achieves unconditional stability.

The scheme converges with the expected mean-square order.

Local mesh refinement benefits are demonstrated experimentally.

Abstract

In this article, we extend a Milstein finite difference scheme introduced in [Giles & Reisinger(2011)] for a certain linear stochastic partial differential equation (SPDE), to semi- and fully implicit timestepping as introduced by [Szpruch(2010)] for SDEs. We combine standard finite difference Fourier analysis for PDEs with the linear stability analysis in [Buckwar & Sickenberger(2011)] for SDEs, to analyse the stability and accuracy. The results show that Crank-Nicolson timestepping for the principal part of the drift with a partially implicit but negatively weighted double It\^o integral gives unconditional stability over all parameter values, and converges with the expected order in the mean-square sense. This opens up the possibility of local mesh refinement in the spatial domain, and we show experimentally that this can be beneficial in the presence of reduced regularity at boundaries.