American Institute of Mathematical Sciences

March  2011, 31(1): 253-273. doi: 10.3934/dcds.2011.31.253

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

 1 School of Mathematical Sciences, University of Adelaide, South Australia 5005, Australia, Australia

Received  March 2010 Revised  January 2011 Published  June 2011

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.
Citation: Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 253-273. doi: 10.3934/dcds.2011.31.253
References:
 [1] H. Attouch, "Variational Convergence for Functions and Operators," Pitman Publishing Limited, London, 1984. Google Scholar [2] 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.  Google Scholar [3] P. Imkeller and A. Monahan, editors, "Stochastic Climate Dynamics," a Special Issue in the journal Stoch. and Dyna., 2, 2002. Google Scholar [4] 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.  Google Scholar [5] H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Commun. Math. Phys., 65 (1979), 97-128. doi: 10.1007/BF01225144.  Google Scholar [6] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar [7] 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.  Google Scholar [8] A. J. Roberts, A step towards holistic discretisation of stochastic partial differential equations,, ANZIAM J., 45 ().   Google Scholar [9] 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.  Google Scholar [10] A. J. Roberts, Subgrid and interelement interactions affect discretisations of stochastically forced diffusion,, ANZIAM J., 48 ().   Google Scholar [11] 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 Google Scholar [12] Tony MacKenzie and A. J. Roberts, Holistic discretisation ensures fidelity to dynamics in two spatial dimensions, preprint, 2009. Google Scholar [13] J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar [14] R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, 2003. Google Scholar [15] 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. Google Scholar [16] V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Springer-Verlag, Berlin, 1997. Google Scholar [17] W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007).  Google Scholar [18] 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.  Google Scholar [19] W. Wang and A. J. Roberts, Macroscopic reduction for stochastic reaction-diffusion equations, preprint, 2008. Available from: http://arxiv.org/abs/0812.1837. Google Scholar [20] 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. Google Scholar [21] W. Wang and A. J. Roberts, Macroscopic discrete modelling of stochastic reaction-diffusion equations, preprint, 2009. Google Scholar [22] 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.  Google Scholar [23] E. Waymire and J. Duan, editors, "Probability and Partial Differential Equations in Modern Applied Mathematics," IMA, 140, Springer-Verlag, New York, 2005. Google Scholar [24] Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), 1363-1384. doi: 10.1137/040605278.  Google Scholar [25] 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.  Google Scholar

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References:
 [1] H. Attouch, "Variational Convergence for Functions and Operators," Pitman Publishing Limited, London, 1984. Google Scholar [2] 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.  Google Scholar [3] P. Imkeller and A. Monahan, editors, "Stochastic Climate Dynamics," a Special Issue in the journal Stoch. and Dyna., 2, 2002. Google Scholar [4] 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.  Google Scholar [5] H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Commun. Math. Phys., 65 (1979), 97-128. doi: 10.1007/BF01225144.  Google Scholar [6] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar [7] 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.  Google Scholar [8] A. J. Roberts, A step towards holistic discretisation of stochastic partial differential equations,, ANZIAM J., 45 ().   Google Scholar [9] 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.  Google Scholar [10] A. J. Roberts, Subgrid and interelement interactions affect discretisations of stochastically forced diffusion,, ANZIAM J., 48 ().   Google Scholar [11] 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 Google Scholar [12] Tony MacKenzie and A. J. Roberts, Holistic discretisation ensures fidelity to dynamics in two spatial dimensions, preprint, 2009. Google Scholar [13] J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar [14] R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, 2003. Google Scholar [15] 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. Google Scholar [16] V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Springer-Verlag, Berlin, 1997. Google Scholar [17] W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007).  Google Scholar [18] 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.  Google Scholar [19] W. Wang and A. J. Roberts, Macroscopic reduction for stochastic reaction-diffusion equations, preprint, 2008. Available from: http://arxiv.org/abs/0812.1837. Google Scholar [20] 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. Google Scholar [21] W. Wang and A. J. Roberts, Macroscopic discrete modelling of stochastic reaction-diffusion equations, preprint, 2009. Google Scholar [22] 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.  Google Scholar [23] E. Waymire and J. Duan, editors, "Probability and Partial Differential Equations in Modern Applied Mathematics," IMA, 140, Springer-Verlag, New York, 2005. Google Scholar [24] Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), 1363-1384. doi: 10.1137/040605278.  Google Scholar [25] 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.  Google Scholar
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