Semi-discretization of stochastic

partial differential equations on

by a finite-difference method

Author:
Hyek Yoo

Journal:
Math. Comp. **69** (2000), 653-666

MSC (1991):
Primary 35R60, 60H15, 65M06, 65M15

DOI:
https://doi.org/10.1090/S0025-5718-99-01150-3

Published electronically:
April 28, 1999

MathSciNet review:
1654030

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The paper concerns finite-difference scheme for the approximation of partial differential equations in , 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 with functions of compact support requires us to introduce a scale of weighted Sobolev spaces. Employing the weighted -theory of SPDE, a sup-norm error estimate is derived and the rate of convergence is given.

**1.**N. Bellomo, Z. Brzezniak, L.M. de Socio,*Nonlinear stochastic evolution problems in applied sciences*, Kluwer, Dordrecht, 1992. MR**94j:60121****2.**N. Bellomo, F. Flandoli,*Stochastic partial differential equations in continuum physics-on the foundations of the stochastic interpolation methods for Itô type equations*, Math. Comput. Simulation, 31 (1989), 3-17. MR**90e:35174****3.**J.F. Bennaton,*Discrete time Galerkin approximations to the nonlinear filtering solution*, J. Math. Anal. and Appl. 110 (1985), 364-383. MR**86k:93147****4.**A. Bensoussan, R. Glowinski, A. R[??]a\c{s}canu,*Approximation of some stochastic differential equations by the splitting up method*, Appl. Math. Optim. 25 (1992), 81-106. MR**92k:60139****5.**A.M. Davie, J.G. Gaines,*Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations*, Preprint**6.**J.G. Gaines,*Numerical experiments with S(P)DE's*, in Stochastic Partial Differential Equations, A.M. Etheridge. ed., London Mathematical Society Lecture Note Series 216, Cambridge Univ. Press., 1995, pp. 55-71. MR**96k:60154****7.**A. Germani, M. Piccioni,*Semi-discretization of stochastic partial differential equations on by a finite- element technique*, Stochastics 23 (1988), 131-148. MR**89f:60063****8.**I. Gyöngy,*Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise I, II*, Potential Analysis 9 (1998), 1-25 and to appear. CMP**99:01****9.**P.E. Kloeden, E. Platen,*Numerical solutions of stochastic differential equations*, Applications of Mathematics series, Vol. 23, Springer-Verlag, Heidelberg, 1992. MR**94b:60069****10.**N.V. Krylov,*On -theory of stochastic partial differential equations in the whole space*, SIAM J. Math. Anal. 27 (1996), 313-340. MR**97b:60107****11.**N.V. Krylov,*An analytic approach to SPDEs*, in Stochastic Partial Differential Equations, Six Perspectives, R. A. Carmona and B. Rozovskii, eds., Mathematical Surveys and Monographs, vol. 64, Amer. Math. Soc., Providence, RI, 1999, pp. 185-242.**12.**N.V. Krylov,*Introduction to the theory of diffusion processes*, Amer. Math. Soc., Providence, RI, 1995. MR**96k:60196****13.**N.V. Krylov,*Lectures on elliptic and parabolic equations in Hölder spaces*, Graduate Studies in Mathematics, Vol. 12, Amer. Math. Soc., 1996. MR**97i:35001****14.**N.V. Krylov, B.L. Rozovskii,*Stochastic partial differential equations and diffusion processes*, Russian Math. Surveys, 37 (1982), no. 6, 81-105. MR**84d:60095****15.**S.V. Lototsky,*Problems in statistics of stochastic differential equations*, Thesis, University of Southern California, 1996.**16.**G. Milstein,*Numerical integration of stochastic differential equations*, Kluwer, Dordrecht, 1995. MR**96e:65003****17.**B.L.Rozovskii,*Stochastic evolution systems*, Kluwer, Dordrecht, 1990. MR**92k:60136****18.**J.C. Strikwerda,*Finite difference schemes and partial differential equations*, Wadsworth & Brook/Cole, Pacific Grove, CA, 1989. MR**90g:65004****19.**H. Yoo,*On -theory of discrete stochastic evolution equations and its application to finite difference approximations for SPDEs*, Preprint.

Retrieve articles in *Mathematics of Computation*
with MSC (1991):
35R60,
60H15,
65M06,
65M15

Retrieve articles in all journals with MSC (1991): 35R60, 60H15, 65M06, 65M15

Additional Information

**Hyek Yoo**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455

Email:
yoo@math.umn.edu

DOI:
https://doi.org/10.1090/S0025-5718-99-01150-3

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