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

Abstract

In this article, we extend a Milstein finite difference scheme introduced in 8 for a certain linear stochastic partial differential equation (SPDE) to semi-implicit and fully implicit time-stepping as introduced by Szpruch 32 for stochastic differential equations (SDEs). We combine standard finite difference Fourier analysis for partial differential equations with the linear stability analysis in 3 for SDEs to analyse the stability and accuracy. The results show that Crank–Nicolson time-stepping for the principal part of the drift with a partially implicit but negatively weighted double Itô 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.

Keywords:

• stochastic partial differential equations
• finite differences
• implicit time-stepping schemes
• Fourier analysis
• local mesh refinement

2010 AMS Subject Classifications:

• 35R60
• 60H15
• 65M06
• 65M12

Acknowledgements

The author thank Lukas Szpruch for helpful discussions on implicit Milstein schemes and Mike Giles for discussions on Fourier analysis in this context.