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

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