Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations

We consider the numerical solution of the stochastic partial differential equation ${\partial u}/{\partial t}={\partial^2u}/{\partial x^2}+\sigma(u)\dot{W}(x,t)$, where $\dot{W}$is space-time white noise, using finite differences. For this equation Gyöngy has obtained an estimate of the rate of convergence for a simple scheme, based on integrals of $\dot{W}$ over a rectangular grid. We investigate the extent to which this order of convergence can be improved, and find that better approximations are possible for the case of additive noise ( $\sigma(u)=1$) if we wish to estimate space averages of the solution rather than pointwise estimates, or if we are permitted to generate other functionals of the noise. But for multiplicative noise ( $\sigma(u)=u$) we show that no such improvements are possible.