Density convergence of a fully discrete finite difference method for stochastic Cahn–Hilliard equation
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- by Jialin Hong, Diancong Jin and Derui Sheng;
- Math. Comp. 93 (2024), 2215-2264
- DOI: https://doi.org/10.1090/mcom/3928
- Published electronically: November 28, 2023
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Abstract:
This paper focuses on investigating the density convergence of a fully discrete finite difference method when applied to numerically solve the stochastic Cahn–Hilliard equation driven by multiplicative space-time white noises. The main difficulty lies in the control of the drift coefficient that is neither globally Lipschitz nor one-sided Lipschitz. To handle this difficulty, we propose a novel localization argument and derive the strong convergence rate of the numerical solution to estimate the total variation distance between the exact and numerical solutions. This along with the existence of the density of the numerical solution finally yields the convergence of density in $L^1(\mathbb {R})$ of the numerical solution. Our results partially answer positively to the open problem posed by J. Cui and J. Hong [J. Differential Equations 269 (2020), pp. 10143–10180] on computing the density of the exact solution numerically.References
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Bibliographic Information
- Jialin Hong
- Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
- MR Author ID: 258322
- Email: hjl@lsec.cc.ac.cn
- Diancong Jin
- Affiliation: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China; and Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
- MR Author ID: 1410227
- Email: jindc@hust.edu.cn
- Derui Sheng
- Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China; and Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
- MR Author ID: 1491841
- Email: sdr@lsec.cc.ac.cn
- Received by editor(s): January 9, 2023
- Received by editor(s) in revised form: August 16, 2023
- Published electronically: November 28, 2023
- Additional Notes: This work was supported by the National Key R&D Program of China under Grant No. 2020YFA0713701, National Natural Science Foundation of China (Nos. 11971470, 12031020, 12201228, 12171047), and the Fundamental Research Funds for the Central Universities 3004011142.
The third author is the corresponding author. - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 2215-2264
- MSC (2020): Primary 60H35; Secondary 65C30, 60H15, 60H07
- DOI: https://doi.org/10.1090/mcom/3928
- MathSciNet review: 4759374