Mathematics > Numerical Analysis
arXiv:2303.13766 (math)
[Submitted on 24 Mar 2023 (v1), last revised 7 Jul 2023 (this version, v2)]
Title:Higher order time discretization method for a class of semilinear stochastic partial differential equations with multiplicative noise
View a PDF of the paper titled Higher order time discretization method for a class of semilinear stochastic partial differential equations with multiplicative noise, by Yukun Li and 2 other authors
View PDFAbstract:In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly $1$ for the approximate solution. The proof is based on new Hölder continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Then we obtain $H^1$ stability, higher moment $H^1$ stability, $L^2$ stability, and higher moment $L^2$ stability results using numerical and stochastic techniques. The nearly optimal convergence orders in time and space are hence obtained by coupling all previous results. Numerical experiments are presented to implement the proposed numerical scheme and to validate the theoretical results.
| Comments: | 28 pages, 8 figures. arXiv admin note: text overlap witharXiv:1811.05028 |
| Subjects: | Numerical Analysis (math.NA) |
| MSC classes: | 65N30, 65N15, 65N12 |
| Cite as: | arXiv:2303.13766 [math.NA] |
| (orarXiv:2303.13766v2 [math.NA] for this version) | |
| https://doi.org/10.48550/arXiv.2303.13766 arXiv-issued DOI via DataCite |
Submission history
From: Liet Vo [view email][v1] Fri, 24 Mar 2023 02:41:24 UTC (162 KB)
[v2] Fri, 7 Jul 2023 12:00:44 UTC (172 KB)
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View a PDF of the paper titled Higher order time discretization method for a class of semilinear stochastic partial differential equations with multiplicative noise, by Yukun Li and 2 other authors
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