Uniform-in-time weak error estimates of explicit full-discretization schemes for SPDEs with non-globally Lipschitz coefficients
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915387583496192 |
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| author | Jiang, Yingsong Wang, Xiaojie |
| author_facet | Jiang, Yingsong Wang, Xiaojie |
| contents | This article is devoted to long-time weak approximations of stochastic partial differential equations (SPDEs) evolving in a bounded domain $\mathcal{D} \subset \mathbb{R}^d$, $d \leq 3$, with non-globally Lipschitz and possibly non-contractive coefficients. Both the space-time white noise ($d=1$) and the trace-class noise in multiple dimensions $d=2,3$ are examined for the considered SPDEs. Based on a spectral Galerkin spatial semi-discretization, we propose a class of novel full-discretization schemes of exponential type, which are explicit, easily implementable and preserve the ergodicity of the original dissipative SPDEs with possibly non-contractive coefficients. The uniform-in-time weak approximation errors are carefully analyzed in a low regularity and non-contractive setting, with uniform-in-time weak convergence rates obtained. A key ingredient is to establish the uniform-in-time moment bounds (in $L^{4q-2}$-norm, $q \geq 1$) for the proposed fully discrete schemes in a super-linear setting. This is highly non-trivial for the explicit full-discretization schemes and new arguments are elaborated by fully exploiting a contractive property of the semi-group in $L^{4q-2}$, the dissipativity of the nonlinearity and the particular benefit of the taming strategy. Numerical experiments are finally reported to verify the theoretical findings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_21364 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform-in-time weak error estimates of explicit full-discretization schemes for SPDEs with non-globally Lipschitz coefficients Jiang, Yingsong Wang, Xiaojie Numerical Analysis 60H35, 60H15, 65C30 This article is devoted to long-time weak approximations of stochastic partial differential equations (SPDEs) evolving in a bounded domain $\mathcal{D} \subset \mathbb{R}^d$, $d \leq 3$, with non-globally Lipschitz and possibly non-contractive coefficients. Both the space-time white noise ($d=1$) and the trace-class noise in multiple dimensions $d=2,3$ are examined for the considered SPDEs. Based on a spectral Galerkin spatial semi-discretization, we propose a class of novel full-discretization schemes of exponential type, which are explicit, easily implementable and preserve the ergodicity of the original dissipative SPDEs with possibly non-contractive coefficients. The uniform-in-time weak approximation errors are carefully analyzed in a low regularity and non-contractive setting, with uniform-in-time weak convergence rates obtained. A key ingredient is to establish the uniform-in-time moment bounds (in $L^{4q-2}$-norm, $q \geq 1$) for the proposed fully discrete schemes in a super-linear setting. This is highly non-trivial for the explicit full-discretization schemes and new arguments are elaborated by fully exploiting a contractive property of the semi-group in $L^{4q-2}$, the dissipativity of the nonlinearity and the particular benefit of the taming strategy. Numerical experiments are finally reported to verify the theoretical findings. |
| title | Uniform-in-time weak error estimates of explicit full-discretization schemes for SPDEs with non-globally Lipschitz coefficients |
| topic | Numerical Analysis 60H35, 60H15, 65C30 |
| url | https://arxiv.org/abs/2504.21364 |