Stability and convergence analysis of a fully discrete semi-implicit scheme for stochastic Allen-Cahn equations with multiplicative noise
HTML articles powered by AMS MathViewer
- by Can Huang and Jie Shen
- Math. Comp. 92 (2023), 2685-2713
- DOI: https://doi.org/10.1090/mcom/3846
- Published electronically: June 8, 2023
- HTML | PDF | Request permission
Abstract:
We consider a fully discrete scheme for stochastic Allen-Cahn equation in a multi-dimensional setting. Our method uses a polynomial based spectral method in space, so it does not require the elliptic operator $A$ and the covariance operator $Q$ of noise in the equation commute, and thus successfully alleviates a restriction of Fourier spectral method for stochastic partial differential equations pointed out by Jentzen, Kloeden and Winkel [Ann. Appl. Probab. 21 (2011), pp. 908–950]. The discretization in time is a tamed semi-implicit scheme which treats the nonlinear term explicitly while being unconditionally stable. Under regular assumptions which are usually made for SPDEs, we establish strong convergence rates in the one spatial dimension for our fully discrete scheme. We also present numerical experiments which are consistent with our theoretical results.References
- R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
- Adam Andersson and Stig Larsson, Weak convergence for a spatial approximation of the nonlinear stochastic heat equation, Math. Comp. 85 (2016), no. 299, 1335–1358. MR 3454367, DOI 10.1090/mcom/3016
- E. J. Allen, S. J. Novosel, and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics Stochastics Rep. 64 (1998), no. 1-2, 117–142. MR 1637047, DOI 10.1080/17442509808834159
- Dimitra C. Antonopoulou, Georgia Karali, and Annie Millet, Existence and regularity of solution for a stochastic Cahn-Hilliard/Allen-Cahn equation with unbounded noise diffusion, J. Differential Equations 260 (2016), no. 3, 2383–2417. MR 3427670, DOI 10.1016/j.jde.2015.10.004
- S. Becker, B. Gess, A. Jentzen, and P. E. Kloeden, Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations, SAM Research Report, Eidgenössische Technische Hochschule, Zürich, 54, 2017.
- Sebastian Becker and Arnulf Jentzen, Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations, Stochastic Process. Appl. 129 (2019), no. 1, 28–69. MR 3906990, DOI 10.1016/j.spa.2018.02.008
- Christine Bernardi and Yvon Maday, Spectral methods, Handbook of numerical analysis, Vol. V, Handb. Numer. Anal., V, North-Holland, Amsterdam, 1997, pp. 209–485. MR 1470226, DOI 10.1016/S1570-8659(97)80003-8
- Charles-Edouard Bréhier, Jianbo Cui, and Jialin Hong, Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen-Cahn equation, IMA J. Numer. Anal. 39 (2019), no. 4, 2096–2134. MR 4019051, DOI 10.1093/imanum/dry052
- Sandra Cerrai, Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Probab. Theory Related Fields 125 (2003), no. 2, 271–304. MR 1961346, DOI 10.1007/s00440-002-0230-6
- Pao-Liu Chow, Stochastic partial differential equations, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2007. MR 2295103, DOI 10.1201/9781420010305
- Jianbo Cui and Jialin Hong, Strong and weak convergence rates of a spatial approximation for stochastic partial differential equation with one-sided Lipschitz coefficient, SIAM J. Numer. Anal. 57 (2019), no. 4, 1815–1841. MR 3984308, DOI 10.1137/18M1215554
- Jianbo Cui and Jialin Hong, Absolute continuity and numerical approximation of stochastic Cahn-Hilliard equation with unbounded noise diffusion, J. Differential Equations 269 (2020), no. 11, 10143–10180. MR 4123754, DOI 10.1016/j.jde.2020.07.007
- Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136, DOI 10.1017/CBO9780511666223
- Giuseppe Da Prato, Arnulf Jentzen, and Michael Röckner, A mild Itô formula for SPDEs, Trans. Amer. Math. Soc. 372 (2019), no. 6, 3755–3807. MR 4009384, DOI 10.1090/tran/7165
- Arnaud Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89–117. MR 2728973, DOI 10.1090/S0025-5718-2010-02395-6
- J. Dixon and S. McKee, Weakly singular discrete Gronwall inequalities, Z. Angew. Math. Mech. 66 (1986), no. 11, 535–544 (English, with German and Russian summaries). MR 880357, DOI 10.1002/zamm.19860661107
- Xiaobing Feng, Yukun Li, and Yi Zhang, Finite element methods for the stochastic Allen-Cahn equation with gradient-type multiplicative noise, SIAM J. Numer. Anal. 55 (2017), no. 1, 194–216. MR 3600370, DOI 10.1137/15M1022124
- István Gyöngy, Sotirios Sabanis, and David Šiška, Convergence of tamed Euler schemes for a class of stochastic evolution equations, Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016), no. 2, 225–245. MR 3498982, DOI 10.1007/s40072-015-0057-7
- Martin Hutzenthaler, Arnulf Jentzen, and Peter E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab. 22 (2012), no. 4, 1611–1641. MR 2985171, DOI 10.1214/11-AAP803
- Martin Hutzenthaler and Arnulf Jentzen, Convergence of the stochastic Euler scheme for locally Lipschitz coefficients, Found. Comput. Math. 11 (2011), no. 6, 657–706. MR 2859952, DOI 10.1007/s10208-011-9101-9
- Arnulf Jentzen, Peter Kloeden, and Georg Winkel, Efficient simulation of nonlinear parabolic SPDEs with additive noise, Ann. Appl. Probab. 21 (2011), no. 3, 908–950. MR 2830608, DOI 10.1214/10-AAP711
- Arnulf Jentzen and Michael Röckner, A Milstein scheme for SPDEs, Found. Comput. Math. 15 (2015), no. 2, 313–362. MR 3320928, DOI 10.1007/s10208-015-9247-y
- Arnulf Jentzen and Primož Pušnik, Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities, IMA J. Numer. Anal. 40 (2020), no. 2, 1005–1050. MR 4092277, DOI 10.1093/imanum/drz009
- Mihály Kovács, Stig Larsson, and Fredrik Lindgren, On the backward Euler approximation of the stochastic Allen-Cahn equation, J. Appl. Probab. 52 (2015), no. 2, 323–338. MR 3372078, DOI 10.1239/jap/1437658601
- Mihály Kovács, Stig Larsson, and Fredrik Lindgren, On the discretisation in time of the stochastic Allen-Cahn equation, Math. Nachr. 291 (2018), no. 5-6, 966–995. MR 3795566, DOI 10.1002/mana.201600283
- Raphael Kruse, Strong and weak approximation of semilinear stochastic evolution equations, Lecture Notes in Mathematics, vol. 2093, Springer, Cham, 2014. MR 3154916, DOI 10.1007/978-3-319-02231-4
- Stig Larsson and Ali Mesforush, Finite-element approximation of the linearized Cahn-Hilliard-Cook equation, IMA J. Numer. Anal. 31 (2011), no. 4, 1315–1333. MR 2846757, DOI 10.1093/imanum/drq042
- Zhihui Liu and Zhonghua Qiao, Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise, Stoch. Partial Differ. Equ. Anal. Comput. 9 (2021), no. 3, 559–602. MR 4297233, DOI 10.1007/s40072-020-00179-2
- Wei Liu and Michael Röckner, Stochastic partial differential equations: an introduction, Universitext, Springer, Cham, 2015. MR 3410409, DOI 10.1007/978-3-319-22354-4
- Ananta K. Majee and Andreas Prohl, Optimal strong rates of convergence for a space-time discretization of the stochastic Allen-Cahn equation with multiplicative noise, Comput. Methods Appl. Math. 18 (2018), no. 2, 297–311. MR 3776047, DOI 10.1515/cmam-2017-0023
- Gabriel J. Lord, Catherine E. Powell, and Tony Shardlow, An introduction to computational stochastic PDEs, Cambridge Texts in Applied Mathematics, Cambridge University Press, New York, 2014. MR 3308418, DOI 10.1017/CBO9781139017329
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- Giuseppe Da Prato and Arnaud Debussche, Stochastic Cahn-Hilliard equation, Nonlinear Anal. 26 (1996), no. 2, 241–263. MR 1359472, DOI 10.1016/0362-546X(94)00277-O
- Ruisheng Qi and Xiaojie Wang, Optimal error estimates of Galerkin finite element methods for stochastic Allen-Cahn equation with additive noise, J. Sci. Comput. 80 (2019), no. 2, 1171–1194. MR 3977202, DOI 10.1007/s10915-019-00973-8
- Ruisheng Qi and Xiaojie Wang, Error estimates of semidiscrete and fully discrete finite element methods for the Cahn-Hilliard-Cook equation, SIAM J. Numer. Anal. 58 (2020), no. 3, 1613–1653. MR 4102717, DOI 10.1137/19M1259183
- Jie Shen, Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput. 15 (1994), no. 6, 1489–1505. MR 1298626, DOI 10.1137/0915089
- J. Shen, T. Tang, and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin, Heidelberg, 2011.
- Vidar Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR 1479170, DOI 10.1007/978-3-662-03359-3
- M. V. Tretyakov and Z. Zhang, A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM J. Numer. Anal. 51 (2013), no. 6, 3135–3162. MR 3129758, DOI 10.1137/120902318
- Xiaojie Wang, Strong convergence rates of the linear implicit Euler method for the finite element discretization of SPDEs with additive noise, IMA J. Numer. Anal. 37 (2017), no. 2, 965–984. MR 3649432, DOI 10.1093/imanum/drw016
- Xiaojie Wang, An efficient explicit full-discrete scheme for strong approximation of stochastic Allen-Cahn equation, Stochastic Process. Appl. 130 (2020), no. 10, 6271–6299. MR 4140034, DOI 10.1016/j.spa.2020.05.011
- Yubin Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal. 43 (2005), no. 4, 1363–1384. MR 2182132, DOI 10.1137/040605278
Bibliographic Information
- Can Huang
- Affiliation: School of Mathematical Sciences, Xiamen university, People’s Republic of China; and Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing, Xiamen University, People’s Republic of China
- Jie Shen
- Affiliation: Eastern Institute for Advanced Study, Eastern Institute of Technology, Ningbo, Zhejiang 315200, P. R. China; and Department of Mathematics, Purdue University, West Lafayette, IN, US
- MR Author ID: 257933
- ORCID: 0000-0002-4885-5732
- Received by editor(s): January 6, 2022
- Received by editor(s) in revised form: December 3, 2022
- Published electronically: June 8, 2023
- Additional Notes: This work was partially supported by NSFC grants 11971407, 12271457 and Science Foundation of the Fujian Province grant 2022J01033.
The second author is the corresponding author. - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 2685-2713
- MSC (2020): Primary 65N35, 65E05, 65N12, 41A10, 41A25, 41A30, 41A58
- DOI: https://doi.org/10.1090/mcom/3846