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

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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

Abstract
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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.

Keywords: averaging, overlapping finite elements, interelement coupling., Stochastic reaction-diffusion equation.

Mathematics Subject Classification: Primary: 60H15; Secondary: 65M5.

Citation: Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain. Discrete & Continuous Dynamical Systems - A, 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, (1984).
[2]	W. E, X. Li and E. Vanden-Eijnden, Some recent progress in multiscale modeling,, in, 39 (2004), 3.
[3]	P. Imkeller and A. Monahan, editors, "Stochastic Climate Dynamics,", a Special Issue in the journal Stoch. and Dyna., 2 (2002).
[4]	D. Givon, R. Kupferman and A. Stuart, Extracting macroscopic dynamics: Model problems and algorithms,, Nonlinearity, 17 (2004). doi: 10.1088/0951-7715/17/6/R01.
[5]	H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion,, Commun. Math. Phys., 65 (1979), 97. doi: 10.1007/BF01225144.
[6]	G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223.
[7]	A. J. Roberts, A holistic finite difference approach models linear dynamics consistently,, Mathematics of Computation, 72 (2003), 247. doi: 10.1090/S0025-5718-02-01448-5.
[8]	A. J. Roberts, A step towards holistic discretisation of stochastic partial differential equations,, ANZIAM J., 45 ().
[9]	A. J. Roberts, Resolving the multitude of microscale interactions accurately models stochastic partial differential equations,, LMS J. Computation and Math., 9 (2006), 193.
[10]	A. J. Roberts, Subgrid and interelement interactions affect discretisations of stochastically forced diffusion,, ANZIAM J., 48 ().
[11]	A. J. Roberts, Choose interelement coupling to preserve self-adjoint dynamics in multiscale modelling and computation,, Applied Numerical Modelling, 60 (2010), 949.
[12]	Tony MacKenzie and A. J. Roberts, Holistic discretisation ensures fidelity to dynamics in two spatial dimensions,, preprint, (2009).
[13]	J. Simon, Compact sets in the space $L^p(0, T; B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360.
[14]	R. A. Adams and J. J. F. Fournier, "Sobolev Spaces,", Academic Press, (2003).
[15]	S. Engblom, L. Ferm, A. Hellander and P. Lötstedt, Simulation of stochastic reaction diffusion processes on unstructured meshes,, preprint, (2008).
[16]	V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems,", Springer-Verlag, (1997).
[17]	W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations,, J. Math. Phys., 48 (2007).
[18]	W. Wang and J. Duan, Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions,, Comm. Math. Phys., 275 (2007), 163. doi: 10.1007/s00220-007-0301-8.
[19]	W. Wang and A. J. Roberts, Macroscopic reduction for stochastic reaction-diffusion equations,, preprint, (2008).
[20]	W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations,, preprint, (2008).
[21]	W. Wang and A. J. Roberts, Macroscopic discrete modelling of stochastic reaction-diffusion equations,, preprint, (2009).
[22]	H. Watanabe, Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients,, Probab. Th. Rel. Fields, 77 (1988), 359. doi: 10.1007/BF00319294.
[23]	E. Waymire and J. Duan, editors, "Probability and Partial Differential Equations in Modern Applied Mathematics,", IMA, 140 (2005).
[24]	Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations,, SIAM J. Numer. Anal., 43 (2005), 1363. doi: 10.1137/040605278.
[25]	Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations,, SIAM J. Numer. Anal., 40 (2002), 1421. doi: 10.1137/S0036142901387956.

show all references

References:

[1]	H. Attouch, "Variational Convergence for Functions and Operators,", Pitman Publishing Limited, (1984).
[2]	W. E, X. Li and E. Vanden-Eijnden, Some recent progress in multiscale modeling,, in, 39 (2004), 3.
[3]	P. Imkeller and A. Monahan, editors, "Stochastic Climate Dynamics,", a Special Issue in the journal Stoch. and Dyna., 2 (2002).
[4]	D. Givon, R. Kupferman and A. Stuart, Extracting macroscopic dynamics: Model problems and algorithms,, Nonlinearity, 17 (2004). doi: 10.1088/0951-7715/17/6/R01.
[5]	H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion,, Commun. Math. Phys., 65 (1979), 97. doi: 10.1007/BF01225144.
[6]	G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223.
[7]	A. J. Roberts, A holistic finite difference approach models linear dynamics consistently,, Mathematics of Computation, 72 (2003), 247. doi: 10.1090/S0025-5718-02-01448-5.
[8]	A. J. Roberts, A step towards holistic discretisation of stochastic partial differential equations,, ANZIAM J., 45 ().
[9]	A. J. Roberts, Resolving the multitude of microscale interactions accurately models stochastic partial differential equations,, LMS J. Computation and Math., 9 (2006), 193.
[10]	A. J. Roberts, Subgrid and interelement interactions affect discretisations of stochastically forced diffusion,, ANZIAM J., 48 ().
[11]	A. J. Roberts, Choose interelement coupling to preserve self-adjoint dynamics in multiscale modelling and computation,, Applied Numerical Modelling, 60 (2010), 949.
[12]	Tony MacKenzie and A. J. Roberts, Holistic discretisation ensures fidelity to dynamics in two spatial dimensions,, preprint, (2009).
[13]	J. Simon, Compact sets in the space $L^p(0, T; B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360.
[14]	R. A. Adams and J. J. F. Fournier, "Sobolev Spaces,", Academic Press, (2003).
[15]	S. Engblom, L. Ferm, A. Hellander and P. Lötstedt, Simulation of stochastic reaction diffusion processes on unstructured meshes,, preprint, (2008).
[16]	V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems,", Springer-Verlag, (1997).
[17]	W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations,, J. Math. Phys., 48 (2007).
[18]	W. Wang and J. Duan, Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions,, Comm. Math. Phys., 275 (2007), 163. doi: 10.1007/s00220-007-0301-8.
[19]	W. Wang and A. J. Roberts, Macroscopic reduction for stochastic reaction-diffusion equations,, preprint, (2008).
[20]	W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations,, preprint, (2008).
[21]	W. Wang and A. J. Roberts, Macroscopic discrete modelling of stochastic reaction-diffusion equations,, preprint, (2009).
[22]	H. Watanabe, Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients,, Probab. Th. Rel. Fields, 77 (1988), 359. doi: 10.1007/BF00319294.
[23]	E. Waymire and J. Duan, editors, "Probability and Partial Differential Equations in Modern Applied Mathematics,", IMA, 140 (2005).
[24]	Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations,, SIAM J. Numer. Anal., 43 (2005), 1363. doi: 10.1137/040605278.
[25]	Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations,, SIAM J. Numer. Anal., 40 (2002), 1421. doi: 10.1137/S0036142901387956.

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