Postprocessing for stochastic parabolic partial differential equations

SIAM Journal on Numerical Analysis

Volumn 45, Issue 2, 2007, Pages 870-889

  Lord, Gabriel J ^a   Shardlow, Tony ^b  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

^b UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Numerical solution of stochastic PDEs; Postprocessing; Stochastic exponential integrator

Indexed keywords

APPROXIMATION THEORY; CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; GALERKIN METHODS; RANDOM PROCESSES;

EULER-MARUYAMA METHOD; NUMERICAL SOLUTION OF STOCHASTIC PDES; STOCHASTIC EXPONENTIAL INTEGRATOR;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 42649137906     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/050640138     Document Type: Article

References (29)

1. C. CANUTO, M. Y. HUSSAINI, Q. QUATERONI, AND T. A. ZANG, Spectral Methods in Fluid Dynamics, Computational Physics, Springer-Verlag, Berlin, 1988.
2. G. DA PRATO AND A. DEBUSSCHE, Construction of stochastic inertial manifolds using backward integration, Stochastics Stochastics Rep., 59 (1996), pp. 305-324.
3. G. DA PRATO AND J. ZABCZYK, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications 44, Cambridge University Press, Cambridge, UK, 1992.
4. A. M. DAVIE AND J. G. GAINES, Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations, Math. Comp., 70 (2001), pp. 121-134.
5. A. DOELMAN AND E. TITI, Regularity of solutions and the convergence of the Galerkin method in the complex Ginzburg-Landau equation, Numer. Funct. Anal. Optim., 14 (1993), pp. 299-321.
6. C. FOIAS, O. MANLEY, AND R. TEMAM, Modelling of the interaction of small and large eddies in two-dimensional turbulent flows, RAIRO Model. Math. Anal. Numér., 22 (1988), pp. 93-114.
7. B. GARCI'A-ARCHILLA, J. NOVO, AND E. S. TITI, Postprocessing the Galerkin method: A novel approach to approximate inertial manifolds, SIAM J. Numer. Anal., 35 (1998), pp. 941-972.
8. B. GARCÍA-ARCHILLA, J. NOVO, AND E. S. TITI, An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations, Math. Comp., 68 (1999), pp. 893-911.
9. B. GARCÍA-ARCHILLA, J. NOVO, AND E. S. TITI, Postprocessing Fourier spectral methods: The case of smooth solutions, Appl. Numer. Math., 43 (2002), pp. 191-209.
10. W. GRECKSCH AND P. KLOEDEN, Time-discretized Galerkin approximations of parabolic stochastic PDEs, Bull. Austral. Math. Soc., 54 (1996), pp. 79-85.
11. I. GYÖNGY, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise II, Potential Anal., 11 (1999), pp. 1-37.
12. I. GYÖNGY AND D. NUALART, Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space-time white noise, Stochastic Processes Appl., 58 (1995), pp. 57-72.
13. E. HAUSENBLAS, Approximation for semilinear stochastic evolution equations, Potential Anal., 18 (2003), pp. 141-186.
14. E. HAUSENBLAS, Numerical analysis of semilinear stochastic evolution equations in Banach spaces, J. Comput. Appl. Math., 147 (2002), pp. 485-516.
15. Y. HE AND R. M. M. MATTHEU, Stability and convergence for the reform postprocessing Galerkin method, Nonlinear Anal. Real World Appl., 1 (2000), pp. 517-533.
16. D. J. HIGHAM, X. MAO, AND A. M. STUART, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), pp. 1041-1063.
17. M. HOCHBRUCK AND A. OSTERMANN, Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal, 43 (2005), pp. 1069-1090.
18. J. C JIMINEZ, I. SHOJI, AND T. OZAKI, Simulation of stochastic differential equations through the local linearization method. A comparative study, J. Statist. Phys., 94 (1999), pp. 587-602.
19. P. KLOEDEN AND S. SHOTT, Linear-implicit strong schemes for Ito-Galerkin approximations of stochastic PDEs, J. Appl. Math. Stoch. Anal., 14 (2001), pp. 47-53.
20. J. D. LAWSON, Generalized Runge-Kutta processes for stable systems with large Lipschitz con-stants, SIAM J. Numer. Anal., 4 (1967), pp. 372-380.
21. G. J. LORD AND J. ROUGEMONT, A numerical scheme for stochastic PDEs with Gevrey regularity, IMA J. Numer. Anal., 24 (2004), pp. 587-604.
22. L. G. MARGOLIN, E. S. TITI, AND S. WYNNE, The postprocessing Galerkin and nonlinear Galerkin methods - A truncation analysis point of view, SIAM J. Numer. Anal., 41 (2003), pp. 695-714.
23. B. MINCHEV AND W. WRIGHT, A Review of Exponential Integrators for First Order Semi-linear Problems, Technical report, Norwegian University of Science and Technology, Trondheim, Norway, 2005.
24. C. M. MORA, Weak exponential schemes for stochastic differential equations with additive noise, IMA J. Numer. Anal., 25 (2005), pp. 486-506.
25. T. MUELLER-GRONBACH AND K. RITTER, Lower bounds and nonuniform time discretization for approximation of stochastic heat equations, Found. Comput. Math., to appear.
26. T. SHARDLOW, Numerical methods for stochastic parabolic PDEs, Numer. Funct. Anal. Optim., 20 (1999), pp. 121-145.
27. T. SHARDLOW, Numerical simulation of stochastic PDEs for excitable media, J. Comput. Appl. Math., 175 (2005), pp. 429-446.
28. I. SHOJI, Approximation of continuous time stochastic processes by a local linearization method, Math. Comp., 67 (1998), pp. 287-298.
29. Y. YUBIN, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differ-ential equation driven by an additive noise, BIT, 44 (2004), pp. 829-847.