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December  2021, 26(12): 6285-6310. doi: 10.3934/dcdsb.2021019

Weak time discretization for slow-fast stochastic reaction-diffusion equations

Chungang Shi 1,, , Wei Wang 1, and  Dafeng Chen 2,

1. 

Department of Mathematics, Nanjing University, Nanjing 210093, China
2. 

School of Information, Nanjing Audit University, 211815, China

* Corresponding author: Chungang Shi

Received  May 2020 Revised  August 2020 Published  December 2021 Early access  January 2021

Fund Project: The first author is supported by the National Nature Science Foundation of China grant 11771207

This paper derives a weak convergence theorem of the time discretization of the slow component for a two-time-scale stochastic evolutionary equations on interval [0, 1]. Here the drift coefficient of the slow component is cubic with linear coupling between slow and fast components.

Keywords: Averaging, invariant measure, Mont–Carlo, asymptotic expansion, numerical scheme, weak convergence.
Mathematics Subject Classification: Primary: 60F10, 60H15; Secondary: 35Q55.
Citation: Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6285-6310. doi: 10.3934/dcdsb.2021019
References:
[1]

J. Bao, G. Yin and C. Yuan, Two-time-scale stochastic partial differential equations driven by $\alpha$-stable noises: averaging principles, Bernoulli., 23 (2017), 645-669.  doi: 10.3150/14-BEJ677.

[2]

R. Bertram and J. E. Rubin, Multi-time scale systems and fast-slow analysis, Math. Biosci., 287 (2017), 105-121.  doi: 10.1016/j.mbs.2016.07.003.

[3]

C.-E. Bréhier, Strong and weak orders in averaging for SPDEs, Stoch. Proc. Appl., 122 (2012), 2553-2593.  doi: 10.1016/j.spa.2012.04.007.

[4]

C.-E. Bréhier, Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise, Potential Analysis., 40 (2014), 1-40.  doi: 10.1007/s11118-013-9338-9.

[5]

S. Cerrai, A Khasminskii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Prob., 19 (2009), 899-948.  doi: 10.1214/08-AAP560.

[6]

S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Prob. Th. Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z.

[7]

Z. Dong, X. Sun, H. Xiao and J. Zhai, Averaging principle for one dimensional stochastic Burgers equation, J. Diff. Equa., 265 (2018), 4749-4797.  doi: 10.1016/j.jde.2018.06.020.

[8]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003, Available from: https://www.researchgate.net/publication/228607920.

[9] J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier Press, London, 2014.  doi: 10.1016/C2013-0-15235-X.
[10]

W. E, Analysis of the heterogeneous multiscale method for ordinary differential equations, Comm. Math. Sci., 1 (2003), 423–436. https://projecteuclid.org/euclid.cms/1250880094.

[11]

W. E and B. Engquist, Multiscale modeling and computations, Notice of AMS, 50 (2003), 1062–1070. http://hdl.handle.net/1903/9877

[12]

W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), 87-132. 

[13]

W. E, D. Liu and E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Comm. Pure Appl. Math., 58 (2005), 1544-1585.  doi: 10.1002/cpa.20088.

[14]

W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), 121-156.  doi: 10.1090/S0894-0347-04-00469-2.

[15]

H. Fu, L. Wan and J. Liu, Weak order in averaging principle for two-time-scale stochastic partial differential equations, preprint, arXiv: 1802.00903.

[16]

H. Fu, L. Wan, Y. Wang, J. Liu and X. Liu, Strong convergence rate in averaging principle for stochastic FitzHug-Nagumo system with two time-scales, J. Math. Anal. Appl., 416 (2014), 609-628.  doi: 10.1016/j.jmaa.2014.02.062.

[17]

E. Harvey, V. Kirk, M. Wechselberger and J. Sneyd, Multiple timescales, mixed mode oscillations and canards in models of intracellular calcium dynamics, J. Nonlinear Sci., 21 (2011), 639-683.  doi: 10.1007/s00332-011-9096-z.

[18]

R. Z. Khasminskiǐ, On the averaging principle of the Itô stochastic differential equations, Kibernetika, 4 (1968), 260-279. 

[19] N. Krylov and N. Bogliubov, Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, 1943. 
[20]

J. A. Sauders and F. Verhulst, Averaging Methods to Nonlinear Dynamical Systems, Springer–Verlag, New York, 1985. doi: 10.1007/978-1-4757-4575-7.

[21]

N. G. Sorokina, On a theorem of N.N. Bogoliubov, Ukrain. Mat. Zh., 4 (1959), 220-222. 

[22]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer–Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[23]

E. Vanden-Eijnden, Numerical techniques for multiscale dynamical systems with stochastic effects, Comm. Math. Sci., 1 (2003), 385-391.  doi: 10.4310/CMS.2003.v1.n2.a11.

[24]

V. M. Volosov, Averaging in systems of ordinary differential equations, Russian Math. Surveys, 17 (1962), 1-126. 

[25]

W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J. Diff. Equa., 253 (2012), 1265-1286.  doi: 10.1016/j.jde.2012.05.011.

[26]

W. Wang and A. J. Roberts, Slow manifold and averaging for slow-fast stochastic system, J. Math. Anal. Appl., 398 (2013), 822-839.  doi: 10.1016/j.jmaa.2012.09.029.

[27]

Y. Xu, J. Q. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Lévy noise, Physica D, 240 (2011), 1395-1401.  doi: 10.1016/j.physd.2011.06.001.

show all references

References:
[1]

J. Bao, G. Yin and C. Yuan, Two-time-scale stochastic partial differential equations driven by $\alpha$-stable noises: averaging principles, Bernoulli., 23 (2017), 645-669.  doi: 10.3150/14-BEJ677.

[2]

R. Bertram and J. E. Rubin, Multi-time scale systems and fast-slow analysis, Math. Biosci., 287 (2017), 105-121.  doi: 10.1016/j.mbs.2016.07.003.

[3]

C.-E. Bréhier, Strong and weak orders in averaging for SPDEs, Stoch. Proc. Appl., 122 (2012), 2553-2593.  doi: 10.1016/j.spa.2012.04.007.

[4]

C.-E. Bréhier, Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise, Potential Analysis., 40 (2014), 1-40.  doi: 10.1007/s11118-013-9338-9.

[5]

S. Cerrai, A Khasminskii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Prob., 19 (2009), 899-948.  doi: 10.1214/08-AAP560.

[6]

S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Prob. Th. Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z.

[7]

Z. Dong, X. Sun, H. Xiao and J. Zhai, Averaging principle for one dimensional stochastic Burgers equation, J. Diff. Equa., 265 (2018), 4749-4797.  doi: 10.1016/j.jde.2018.06.020.

[8]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003, Available from: https://www.researchgate.net/publication/228607920.

[9] J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier Press, London, 2014.  doi: 10.1016/C2013-0-15235-X.
[10]

W. E, Analysis of the heterogeneous multiscale method for ordinary differential equations, Comm. Math. Sci., 1 (2003), 423–436. https://projecteuclid.org/euclid.cms/1250880094.

[11]

W. E and B. Engquist, Multiscale modeling and computations, Notice of AMS, 50 (2003), 1062–1070. http://hdl.handle.net/1903/9877

[12]

W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), 87-132. 

[13]

W. E, D. Liu and E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Comm. Pure Appl. Math., 58 (2005), 1544-1585.  doi: 10.1002/cpa.20088.

[14]

W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), 121-156.  doi: 10.1090/S0894-0347-04-00469-2.

[15]

H. Fu, L. Wan and J. Liu, Weak order in averaging principle for two-time-scale stochastic partial differential equations, preprint, arXiv: 1802.00903.

[16]

H. Fu, L. Wan, Y. Wang, J. Liu and X. Liu, Strong convergence rate in averaging principle for stochastic FitzHug-Nagumo system with two time-scales, J. Math. Anal. Appl., 416 (2014), 609-628.  doi: 10.1016/j.jmaa.2014.02.062.

[17]

E. Harvey, V. Kirk, M. Wechselberger and J. Sneyd, Multiple timescales, mixed mode oscillations and canards in models of intracellular calcium dynamics, J. Nonlinear Sci., 21 (2011), 639-683.  doi: 10.1007/s00332-011-9096-z.

[18]

R. Z. Khasminskiǐ, On the averaging principle of the Itô stochastic differential equations, Kibernetika, 4 (1968), 260-279. 

[19] N. Krylov and N. Bogliubov, Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, 1943. 
[20]

J. A. Sauders and F. Verhulst, Averaging Methods to Nonlinear Dynamical Systems, Springer–Verlag, New York, 1985. doi: 10.1007/978-1-4757-4575-7.

[21]

N. G. Sorokina, On a theorem of N.N. Bogoliubov, Ukrain. Mat. Zh., 4 (1959), 220-222. 

[22]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer–Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[23]

E. Vanden-Eijnden, Numerical techniques for multiscale dynamical systems with stochastic effects, Comm. Math. Sci., 1 (2003), 385-391.  doi: 10.4310/CMS.2003.v1.n2.a11.

[24]

V. M. Volosov, Averaging in systems of ordinary differential equations, Russian Math. Surveys, 17 (1962), 1-126. 

[25]

W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J. Diff. Equa., 253 (2012), 1265-1286.  doi: 10.1016/j.jde.2012.05.011.

[26]

W. Wang and A. J. Roberts, Slow manifold and averaging for slow-fast stochastic system, J. Math. Anal. Appl., 398 (2013), 822-839.  doi: 10.1016/j.jmaa.2012.09.029.

[27]

Y. Xu, J. Q. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Lévy noise, Physica D, 240 (2011), 1395-1401.  doi: 10.1016/j.physd.2011.06.001.

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