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Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations. (English) Zbl 0956.60064
Summary: 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.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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