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 ∂ u / ∂ t = ∂ 2 u / ∂ x 2 + σ ( u ) W ˙ ( x , t ) {\partial u}/{\partial t}={\partial ^2u}/{\partial x^2}+\sigma (u)\dot {W}(x,t) , where W ˙ \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 W ˙ \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 ( σ ( u ) = 1 \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 ( σ ( u ) = u \sigma (u)=u ) we show that no such improvements are possible.

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