Journal of Applied Mathematics and Stochastic Analysis
Volume 14 (2001), Issue 1, Pages 47-53
doi:10.1155/S1048953301000053
Linear-implicit strong schemes for Itô-Galkerin approximations of stochastic PDEs
1Johann Wolfgang Goethe Universität,, Fachbereich Mathematik, Frankfurt D-60054, Germany
2University of Tasmania, Department of Mathematics, Hobart, Tasmania, Australia
Received 1 August 1999; Revised 1 December 1999
Copyright © 2001 P. E. Kloeden and S. Shott. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Linear-implicit versions of strong Taylor numerical schemes for finite
dimensional Itô stochastic differential equations (SDEs) are shown to have
the same order as the original scheme. The combined truncation and
global discretization error of an γ strong linear-implicit Taylor scheme
with time-step Δ applied to the N dimensional Itô-Galerkin SDE for a
class of parabolic stochastic partial differential equation (SPDE) with a
strongly monotone linear operator with eigenvalues λ1≤λ2≤… in its
drift term is then estimated by
K(λN+1−½+Δγ)
where the constant K depends on the initial value, bounds on the other
coefficients in the SPDE and the length of the time interval under consideration.