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Convergence of Numerical Schemes for the Solution of Parabolic Stochastic Partial Differential Equations

A. M. Davie and J. G. Gaines
Mathematics of Computation
Vol. 70, No. 233 (Jan., 2001), pp. 121-134
Stable URL: http://www.jstor.org/stable/2698927
Page Count: 14
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Convergence of Numerical Schemes for the Solution of Parabolic Stochastic Partial Differential Equations
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Abstract

We consider the numerical solution of the stochastic partial differential equation $\partial u/ \partial t = \partial^2 u/ \partial x^2 + \sigma (u) \dot {W}(x, t)$, where Ẇ is space-time white noise, using finite differences. For this equation Gyongy has obtained an estimate of the rate of convergence for a simple scheme, based on integrals of Ẇ 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) 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) we show that no such improvements are possible.

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