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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Semi-discretization of stochastic
partial differential equations on $\mathbb{R}^1$
by a finite-difference method


Author: Hyek Yoo
Journal: Math. Comp. 69 (2000), 653-666
MSC (1991): Primary 35R60, 60H15, 65M06, 65M15
Published electronically: April 28, 1999
MathSciNet review: 1654030
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Abstract: The paper concerns finite-difference scheme for the approximation of partial differential equations in $\mathbb{R}^1$, with additional stochastic noise. By replacing the space derivatives in the original stochastic partial differential equation (SPDE, for short) with difference quotients, we obtain a system of stochastic ordinary differential equations. We study the difference between the solution of the original SPDE and the solution to the corresponding equation obtained by discretizing the space variable. The need to approximate the solution in $\mathbb{R}^1$ with functions of compact support requires us to introduce a scale of weighted Sobolev spaces. Employing the weighted $L_p$-theory of SPDE, a sup-norm error estimate is derived and the rate of convergence is given.


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Additional Information

Hyek Yoo
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, MN 55455
Email: yoo@math.umn.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01150-3
PII: S 0025-5718(99)01150-3
Keywords: Stochastic partial differential equations, finite-difference method, weighted spaces of Bessel potentials, embedding theorems, rate of convergence
Received by editor(s): March 3, 1998
Received by editor(s) in revised form: July 10, 1998
Published electronically: April 28, 1999
Article copyright: © Copyright 2000 American Mathematical Society