Convergence analysis of a finite difference method for stochastic Cahn--Hilliard equation
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866908955165327360 |
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| author | Hong, Jialin Jin, Diancong Sheng, Derui |
| author_facet | Hong, Jialin Jin, Diancong Sheng, Derui |
| contents | This paper presents the convergence analysis of the spatial finite difference method (FDM) for the stochastic Cahn--Hilliard equation with Lipschitz nonlinearity and multiplicative noise. Based on fine estimates of the discrete Green function, we prove that both the spatial semi-discrete numerical solution and its Malliavin derivative have strong convergence order $1$. Further, by showing the negative moment estimates of the exact solution, we obtain that the density of the spatial semi-discrete numerical solution converges in $L^1(\mathbb R)$ to the exact one. Finally, we apply an exponential Euler method to discretize the spatial semi-discrete numerical solution in time and show that the temporal strong convergence order is nearly $\frac38$, where a difficulty we overcome is to derive the optimal Hölder continuity of the spatial semi-discrete numerical solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_09055 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Convergence analysis of a finite difference method for stochastic Cahn--Hilliard equation Hong, Jialin Jin, Diancong Sheng, Derui Numerical Analysis This paper presents the convergence analysis of the spatial finite difference method (FDM) for the stochastic Cahn--Hilliard equation with Lipschitz nonlinearity and multiplicative noise. Based on fine estimates of the discrete Green function, we prove that both the spatial semi-discrete numerical solution and its Malliavin derivative have strong convergence order $1$. Further, by showing the negative moment estimates of the exact solution, we obtain that the density of the spatial semi-discrete numerical solution converges in $L^1(\mathbb R)$ to the exact one. Finally, we apply an exponential Euler method to discretize the spatial semi-discrete numerical solution in time and show that the temporal strong convergence order is nearly $\frac38$, where a difficulty we overcome is to derive the optimal Hölder continuity of the spatial semi-discrete numerical solution. |
| title | Convergence analysis of a finite difference method for stochastic Cahn--Hilliard equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2202.09055 |