Advanced Search
Article Contents
Article Contents

Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain

Abstract Related Papers Cited by
  • Dynamical systems theory provides powerful methods to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. Here we derive and theoretically support a macroscopic, spatially discrete, model for a class of stochastic reaction-diffusion partial differential equations with cubic nonlinearity. Dividing space into overlapping finite elements, a special coupling condition between neighbouring elements preserves the self-adjoint dynamics and controls interelement interactions. When the interelement coupling parameter is small, an averaging method and an asymptotic expansion of the slow modes show that the macroscopic discrete model will be a family of coupled stochastic ordinary differential equations which describe the evolution of the grid values. This modelling shows the importance of subgrid scale interaction between noise and spatial diffusion and provides a new rigorous approach to constructing semi-discrete approximations to stochastic reaction-diffusion partial differential equations.
    Mathematics Subject Classification: Primary: 60H15; Secondary: 65M55.


    \begin{equation} \\ \end{equation}
  • [1]

    H. Attouch, "Variational Convergence for Functions and Operators," Pitman Publishing Limited, London, 1984.


    W. E, X. Li and E. Vanden-Eijnden, Some recent progress in multiscale modeling, in "Multiscale Modelling and Simulation," Lect. Notes Comput. Sci. Eng., 39, 3-21, Springer, Berlin, 2004.


    P. Imkeller and A. Monahan, editors, "Stochastic Climate Dynamics," a Special Issue in the journal Stoch. and Dyna., 2, 2002.


    D. Givon, R. Kupferman and A. Stuart, Extracting macroscopic dynamics: Model problems and algorithms, Nonlinearity, 17 (2004), R55-R127.doi: 10.1088/0951-7715/17/6/R01.


    H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Commun. Math. Phys., 65 (1979), 97-128.doi: 10.1007/BF01225144.


    G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Cambridge University Press, 1992.doi: 10.1017/CBO9780511666223.


    A. J. Roberts, A holistic finite difference approach models linear dynamics consistently, Mathematics of Computation, 72 (2003), 247-262. Available from: http://www.ams.org/mcom/2003-72-241/S0025-5718-02-01448-5.doi: 10.1090/S0025-5718-02-01448-5.


    A. J. RobertsA step towards holistic discretisation of stochastic partial differential equations, ANZIAM J., 45 (2003/04), C1-C15. Available from: http://anziamj.austms.org.au/V45/CTAC2003/Robe.


    A. J. Roberts, Resolving the multitude of microscale interactions accurately models stochastic partial differential equations, LMS J. Computation and Math., 9 (2006), 193-221. Available from: http://www.lms.ac.uk/jcm/9/lms2005-032.


    A. J. RobertsSubgrid and interelement interactions affect discretisations of stochastically forced diffusion, ANZIAM J., 48 (2006/07), C169-C188. Available from: http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/36.


    A. J. Roberts, Choose interelement coupling to preserve self-adjoint dynamics in multiscale modelling and computation, Applied Numerical Modelling, 60 (2010), 949-973. Available from: http://www.sciencedirect.com/science/article/pii/S0168927410001145


    Tony MacKenzie and A. J. Roberts, Holistic discretisation ensures fidelity to dynamics in two spatial dimensions, preprint, 2009.


    J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.doi: 10.1007/BF01762360.


    R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, 2003.


    S. Engblom, L. Ferm, A. Hellander and P. Lötstedt, Simulation of stochastic reaction diffusion processes on unstructured meshes, preprint, 2008. Available from: http://arXiv.org/abs/0804.3288.


    V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Springer-Verlag, Berlin, 1997.


    W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007).


    W. Wang and J. Duan, Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions, Comm. Math. Phys., 275 (2007), 163-186.doi: 10.1007/s00220-007-0301-8.


    W. Wang and A. J. Roberts, Macroscopic reduction for stochastic reaction-diffusion equations, preprint, 2008. Available from: http://arxiv.org/abs/0812.1837.


    W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, preprint, 2008. Available from: http://arxiv.org/abs/0904.1462.


    W. Wang and A. J. Roberts, Macroscopic discrete modelling of stochastic reaction-diffusion equations, preprint, 2009.


    H. Watanabe, Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients, Probab. Th. Rel. Fields, 77 (1988), 359-378.doi: 10.1007/BF00319294.


    E. Waymire and J. Duan, editors, "Probability and Partial Differential Equations in Modern Applied Mathematics," IMA, 140, Springer-Verlag, New York, 2005.


    Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), 1363-1384.doi: 10.1137/040605278.


    Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations, SIAM J. Numer. Anal., 40 (2002), 1421-1445.doi: 10.1137/S0036142901387956.

  • 加载中

Article Metrics

HTML views() PDF downloads(97) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint