Non-asymptotic uniform in time error bounds for new and old numerical schemes for SPDEs
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arXiv
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| Natura: | Preprint |
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2026
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| author | Huang, Can Ottobre, Michela Simpson, Gideon |
| author_facet | Huang, Can Ottobre, Michela Simpson, Gideon |
| contents | We study numerical schemes for Stochastic Partial Differential Equations (SPDEs). We introduce a general method of proof of non-asymptotic uniform in time error bounds on numerical integrators for SPDEs, ensuring the schemes capture both the transient and the long term dynamics faithfully. We then consider SPDEs with non-globally Lipshitz nonlinearities, which include for example the stochastic Allen-Cahn equation and some stochastic advection-diffusion equations. For the case of Allen-Cahn type SPDEs we show that the classic semi-implicit Euler time-discretization can exhibit finite time blow up. This motivates analysing other schemes which do not suffer from this blow-up problem. We consider three numerical schemes for SPDEs with non globally Lipshitz nonlinearity: a fully implicit scheme and two tamed schemes. For these schemes we prove non-asymptotic uniform in time error bounds by leveraging our general criterion, and provide numerical comparisons. While the main emphasis in this paper is on the properties of the time-discretization, the schemes we consider are full space-time discretization of the SPDE. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18944 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-asymptotic uniform in time error bounds for new and old numerical schemes for SPDEs Huang, Can Ottobre, Michela Simpson, Gideon Numerical Analysis Probability 65C20, 65C30, 60H10, 65G99, 47D07, 60J60, 60H15 We study numerical schemes for Stochastic Partial Differential Equations (SPDEs). We introduce a general method of proof of non-asymptotic uniform in time error bounds on numerical integrators for SPDEs, ensuring the schemes capture both the transient and the long term dynamics faithfully. We then consider SPDEs with non-globally Lipshitz nonlinearities, which include for example the stochastic Allen-Cahn equation and some stochastic advection-diffusion equations. For the case of Allen-Cahn type SPDEs we show that the classic semi-implicit Euler time-discretization can exhibit finite time blow up. This motivates analysing other schemes which do not suffer from this blow-up problem. We consider three numerical schemes for SPDEs with non globally Lipshitz nonlinearity: a fully implicit scheme and two tamed schemes. For these schemes we prove non-asymptotic uniform in time error bounds by leveraging our general criterion, and provide numerical comparisons. While the main emphasis in this paper is on the properties of the time-discretization, the schemes we consider are full space-time discretization of the SPDE. |
| title | Non-asymptotic uniform in time error bounds for new and old numerical schemes for SPDEs |
| topic | Numerical Analysis Probability 65C20, 65C30, 60H10, 65G99, 47D07, 60J60, 60H15 |
| url | https://arxiv.org/abs/2603.18944 |