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

Abstract:

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.