Improving Numerical Error Bounds Near Sharp Interface Limit for Stochastic Reaction-Diffusion Equations
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909456889020416 |
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| author | Cui, Jianbo Wang, Feng-Yu |
| author_facet | Cui, Jianbo Wang, Feng-Yu |
| contents | In the study of geometric surface evolutions, stochastic reaction-diffusion equation provides a powerful tool for capturing and simulating complex dynamics. A critical challenge in this area is developing numerical approximations that exhibit error bounds with polynomial dependence on $\vv^{-1}$, where the small parameter $\vv>0$ represents the diffuse interface thickness. The existence of such bounds for fully discrete approximations of stochastic reaction-diffusion equations remains unclear in the literature. In this work, we address this challenge by leveraging the asymptotic log-Harnack inequality to overcome the exponential growth of $\vv^{-1}$. Furthermore, we establish the numerical weak error bounds under the truncated Wasserstein distance for the spectral Galerkin method and a fully discrete tamed Euler scheme, with explicit polynomial dependence on $\vv^{-1}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12604 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improving Numerical Error Bounds Near Sharp Interface Limit for Stochastic Reaction-Diffusion Equations Cui, Jianbo Wang, Feng-Yu Numerical Analysis Probability 60H15, 60H35, 60D05, 65M1 In the study of geometric surface evolutions, stochastic reaction-diffusion equation provides a powerful tool for capturing and simulating complex dynamics. A critical challenge in this area is developing numerical approximations that exhibit error bounds with polynomial dependence on $\vv^{-1}$, where the small parameter $\vv>0$ represents the diffuse interface thickness. The existence of such bounds for fully discrete approximations of stochastic reaction-diffusion equations remains unclear in the literature. In this work, we address this challenge by leveraging the asymptotic log-Harnack inequality to overcome the exponential growth of $\vv^{-1}$. Furthermore, we establish the numerical weak error bounds under the truncated Wasserstein distance for the spectral Galerkin method and a fully discrete tamed Euler scheme, with explicit polynomial dependence on $\vv^{-1}$. |
| title | Improving Numerical Error Bounds Near Sharp Interface Limit for Stochastic Reaction-Diffusion Equations |
| topic | Numerical Analysis Probability 60H15, 60H35, 60D05, 65M1 |
| url | https://arxiv.org/abs/2412.12604 |