-
Previous Article
Preservation of homoclinic orbits under discretization of delay differential equations
- DCDS Home
- This Issue
-
Next Article
Attractors for the three-dimensional incompressible Navier-Stokes equations with damping
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 |
References:
[1] |
H. Attouch, "Variational Convergence for Functions and Operators," Pitman Publishing Limited, London, 1984. |
[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. |
[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), R55-R127.
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), 97-128.
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), 247-262. Available from: http://www.ams.org/mcom/2003-72-241/S0025-5718-02-01448-5.
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 (2003/04), C1-C15. Available from: http://anziamj.austms.org.au/V45/CTAC2003/Robe. |
[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. |
[10] |
A. J. Roberts, Subgrid 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. |
[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 |
[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), 65-96.
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. Available from: http://arXiv.org/abs/0804.3288. |
[16] |
V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Springer-Verlag, Berlin, 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), 163-186.
doi: 10.1007/s00220-007-0301-8. |
[19] |
W. Wang and A. J. Roberts, Macroscopic reduction for stochastic reaction-diffusion equations, preprint, 2008. Available from: http://arxiv.org/abs/0812.1837. |
[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. |
[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), 359-378.
doi: 10.1007/BF00319294. |
[23] |
E. Waymire and J. Duan, editors, "Probability and Partial Differential Equations in Modern Applied Mathematics," IMA, 140, Springer-Verlag, New York, 2005. |
[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. |
[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. |
show all references
References:
[1] |
H. Attouch, "Variational Convergence for Functions and Operators," Pitman Publishing Limited, London, 1984. |
[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. |
[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), R55-R127.
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), 97-128.
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), 247-262. Available from: http://www.ams.org/mcom/2003-72-241/S0025-5718-02-01448-5.
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 (2003/04), C1-C15. Available from: http://anziamj.austms.org.au/V45/CTAC2003/Robe. |
[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. |
[10] |
A. J. Roberts, Subgrid 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. |
[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 |
[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), 65-96.
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. Available from: http://arXiv.org/abs/0804.3288. |
[16] |
V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Springer-Verlag, Berlin, 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), 163-186.
doi: 10.1007/s00220-007-0301-8. |
[19] |
W. Wang and A. J. Roberts, Macroscopic reduction for stochastic reaction-diffusion equations, preprint, 2008. Available from: http://arxiv.org/abs/0812.1837. |
[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. |
[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), 359-378.
doi: 10.1007/BF00319294. |
[23] |
E. Waymire and J. Duan, editors, "Probability and Partial Differential Equations in Modern Applied Mathematics," IMA, 140, Springer-Verlag, New York, 2005. |
[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. |
[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. |
[1] |
Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246 |
[2] |
Hongyong Cui, Yangrong Li. Asymptotic $ H^2$ regularity of a stochastic reaction-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021290 |
[3] |
M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079 |
[4] |
Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations and Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43 |
[5] |
Wilhelm Stannat, Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations and Control Theory, 2021, 10 (4) : 701-722. doi: 10.3934/eect.2020087 |
[6] |
Anouar El Harrak, Hatim Tayeq, Amal Bergam. A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2183-2197. doi: 10.3934/dcdss.2021062 |
[7] |
Lu Yang, Meihua Yang. Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2627-2650. doi: 10.3934/dcdsb.2017102 |
[8] |
Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382 |
[9] |
Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085 |
[10] |
Razvan Gabriel Iagar, Ariel Sánchez. Eternal solutions for a reaction-diffusion equation with weighted reaction. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1465-1491. doi: 10.3934/dcds.2021160 |
[11] |
Yuncheng You. Random attractors and robustness for stochastic reversible reaction-diffusion systems. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 301-333. doi: 10.3934/dcds.2014.34.301 |
[12] |
Yangyang Shi, Hongjun Gao. Homogenization for stochastic reaction-diffusion equations with singular perturbation term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2401-2426. doi: 10.3934/dcdsb.2021137 |
[13] |
Markus Gahn. Multi-scale modeling of processes in porous media - coupling reaction-diffusion processes in the solid and the fluid phase and on the separating interfaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6511-6531. doi: 10.3934/dcdsb.2019151 |
[14] |
Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029 |
[15] |
Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033 |
[16] |
Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407 |
[17] |
Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875 |
[18] |
Elena Beretta, Cecilia Cavaterra. Identifying a space dependent coefficient in a reaction-diffusion equation. Inverse Problems and Imaging, 2011, 5 (2) : 285-296. doi: 10.3934/ipi.2011.5.285 |
[19] |
Michio Urano, Kimie Nakashima, Yoshio Yamada. Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity. Conference Publications, 2005, 2005 (Special) : 868-877. doi: 10.3934/proc.2005.2005.868 |
[20] |
Tarik Mohammed Touaoula, Mohammed Nor Frioui, Nikolay Bessonov, Vitaly Volpert. Dynamics of solutions of a reaction-diffusion equation with delayed inhibition. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2425-2442. doi: 10.3934/dcdss.2020193 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]