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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.

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
Full Text: DOI
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