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

Published electronically:
April 28, 1999

MathSciNet review:
1654030

Full-text PDF Free Access

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. Brzeźniak, and L. M. De Socio,*Nonlinear stochastic evolution problems in applied sciences*, Mathematics and its Applications, vol. 82, Kluwer Academic Publishers Group, Dordrecht, 1992. MR**1233385****2.**N. Bellomo and F. Flandoli,*Stochastic partial differential equations in continuum physics—on the foundations of the stochastic interpolation method for Ito’s type equations*, Math. Comput. Simulation**31**(1989), no. 1-2, 3–17. MR**983314**, 10.1016/0378-4754(89)90049-9**3.**Jocelyn Freitas Bennaton,*Discrete time Galerkin approximations to the nonlinear filtering solution*, J. Math. Anal. Appl.**110**(1985), no. 2, 364–383. MR**805259**, 10.1016/0022-247X(85)90299-9**4.**A. Bensoussan, R. Glowinski, and A. Răşcanu,*Approximation of some stochastic differential equations by the splitting up method*, Appl. Math. Optim.**25**(1992), no. 1, 81–106. MR**1133253**, 10.1007/BF01184157**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*, Stochastic partial differential equations (Edinburgh, 1994) London Math. Soc. Lecture Note Ser., vol. 216, Cambridge Univ. Press, Cambridge, 1995, pp. 55–71. MR**1352735**, 10.1017/CBO9780511526213.005**7.**A. Germani and M. Piccioni,*Semidiscretization of stochastic partial differential equations on 𝑅^{𝑑} by a finite-element technique*, Stochastics**23**(1988), no. 2, 131–148. MR**928351**, 10.1080/17442508808833486**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.**Peter E. Kloeden and Eckhard Platen,*Numerical solution of stochastic differential equations*, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR**1214374****10.**N. V. Krylov,*On 𝐿_{𝑝}-theory of stochastic partial differential equations in the whole space*, SIAM J. Math. Anal.**27**(1996), no. 2, 313–340. MR**1377477**, 10.1137/S0036141094263317**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*, Translations of Mathematical Monographs, vol. 142, American Mathematical Society, Providence, RI, 1995. Translated from the Russian manuscript by Valim Khidekel and Gennady Pasechnik. MR**1311478****13.**N. V. Krylov,*Lectures on elliptic and parabolic equations in Hölder spaces*, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996. MR**1406091****14.**N. V. Krylov and B. L. Rozovskiĭ,*Stochastic partial differential equations and diffusion processes*, Uspekhi Mat. Nauk**37**(1982), no. 6(228), 75–95 (Russian). MR**683274****15.**S.V. Lototsky,*Problems in statistics of stochastic differential equations*, Thesis, University of Southern California, 1996.**16.**G. N. Milstein,*Numerical integration of stochastic differential equations*, Mathematics and its Applications, vol. 313, Kluwer Academic Publishers Group, Dordrecht, 1995. Translated and revised from the 1988 Russian original. MR**1335454****17.**B. L. Rozovskiĭ,*Stochastic evolution systems*, Mathematics and its Applications (Soviet Series), vol. 35, Kluwer Academic Publishers Group, Dordrecht, 1990. Linear theory and applications to nonlinear filtering; Translated from the Russian by A. Yarkho. MR**1135324****18.**John C. Strikwerda,*Finite difference schemes and partial differential equations*, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1989. MR**1005330****19.**H. Yoo,*On -theory of discrete stochastic evolution equations and its application to finite difference approximations for SPDEs*, Preprint.

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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

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