  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.
