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

We consider the numerical solution of the stochastic partial differential equation ∂u/∂t = ∂ 2 u/∂x 2 + σ(u)W(x,t), where W is space-time white noise, using finite differences. For this equation Gyongy has obtained an estimate of the rate of convergence for a simple scheme, based on integrals of 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 (σ(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 (σ(u) = u) we show that no such improvements are possible.