Numerical Approximation of Some Linear Stochastic Partial Differential Equations Driven by Special Additive Noises

This paper is concerned with the numerical approximation of some linear stochastic partial differential equations with additive noises. A special representation of the noise is considered, and it is compared with general representations of noises in the infinite dimensional setting. Convergence analysis and error estimates are presented for the numerical solution based on the standard finite difference and finite element methods. The effects of the noises on the accuracy of the approximations are illustrated. Results of the numerical experiments are provided.

[1]  E. Platen.  An introduction to numerical methods for stochastic differential equations , 1999, Acta Numerica.

[2]  Fox,et al.  Second-order algorithm for the numerical integration of colored-noise problems. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[3]  Tianyu Zhang,et al.  Numerical approximation of stochastic differential equations , 2000 .

[4]  Jessica G. Gaines,et al.  Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations , 2001, Math. Comput..

[5]  Denis Talay,et al.  Simulation and numerical analysis of stochastic differential systems : a review , 1990 .

[6]  J. B. Walsh,et al.  An introduction to stochastic partial differential equations , 1986 .

[7]  D. Dawson.  Stochastic evolution equations , 1972 .

[8]  Rainer Buckdahn,et al.  Monotonicity Methods for White Noise Driven Quasi-Linear SPDEs , 1990 .

[9]  Vanden Eijnden E,et al.  Models for stochastic climate prediction. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[10]  J. M. Sancho,et al.  Non-equilibrium Ginzburg-Landau model driven by colored noise , 1998 .

[11]  J. Bennaton,et al.  Discrete time Galerkin approximations to the nonlinear filtering solution , 1985 .

[12]  T. Shardlow.  Numerical methods for stochastic parabolic PDEs , 1999 .

[13]  Erika Hausenblas,et al.  Error Analysis for Approximation of Stochastic Differential Equations Driven by Poisson Random Measures , 2002, SIAM J. Numer. Anal..

[14]  Claes Johnson.  Numerical solution of partial differential equations by the finite element method , 1988 .

[15]  Joseph B. Keller,et al.  Stochastic equations and wave propagation in random media , 1964 .

[16]  I. Gyöngy.  Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II , 1999 .

[17]  E. Platen,et al.  Balanced Implicit Methods for Stiff Stochastic Systems , 1998 .

[18]  Alison Etheridge.  Stochastic partial differential equations , 1995 .

[19]  Philip Protter,et al.  The Euler scheme for Lévy driven stochastic differential equations , 1997 .

[20]  Esteban Moro,et al.  Stochastic vortex dynamics in two-dimensional easy-plane ferromagnets: Multiplicative versus additive noise , 1999 .

[21]  Sujit K. Ghosh,et al.  Essential Wavelets for Statistical Applications and Data Analysis , 2001, Technometrics.

[22]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[23]  Evgenii A. Novikov,et al.  Functionals and the random-force method in turbulence theory , 1965 .

[24]  Zhimin Zhang,et al.  Finite element and difference approximation of some linear stochastic partial differential equations , 1998 .

[25]  J. Gaines.  Stochastic Partial Differential Equations: Numerical experiments with S(P)DE's , 1995 .

[26]  Hyek Yoo,et al.  Semi-discretization of stochastic partial differential equations on R1 by a finite-difference method , 2000, Math. Comput..

[27]  Michel Deville,et al.  Numerical Simulation of Randomly Forced Turbulent Flows , 1998 .

[28]  Desmond J. Higham,et al.  Mean-Square and Asymptotic Stability of the Stochastic Theta Method , 2000, SIAM J. Numer. Anal..

[29]  Gilbert Strang,et al.  Wavelets and Dilation Equations: A Brief Introduction , 1989, SIAM Rev..

[30]  Peter D. Drummond,et al.  Robust Algorithms for Solving Stochastic Partial Differential Equations , 1997 .

[31]  J García-Ojalvo,et al.  Noise-induced phase separation: mean-field results. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[32]  Yoshihiro Saito,et al.  Stability Analysis of Numerical Schemes for Stochastic Differential Equations , 1996 .