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


Authors: A. M. Davie and J. G. Gaines
Journal: Math. Comp. 70 (2001), 121-134
MSC (2000): Primary 60H15, 60H35, 65M06
DOI: https://doi.org/10.1090/S0025-5718-00-01224-2
Published electronically: February 23, 2000
MathSciNet review: 1803132
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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.


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Additional Information

A. M. Davie
Affiliation: Department of Mathematics and Statistics, University of Edinburgh
Email: sandy@ed.ac.uk

J. G. Gaines
Affiliation: Department of Mathematics and Statistics, University of Edinburgh
Email: jessica@ed.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-00-01224-2
Received by editor(s): January 6, 1999
Published electronically: February 23, 2000
Article copyright: © Copyright 2000 American Mathematical Society