Improved error estimates for low-regularity integrators using space-time bounds
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| Format: | Preprint |
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2025
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| _version_ | 1866918444293685248 |
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| author | Ruff, Maximilian |
| author_facet | Ruff, Maximilian |
| contents | We prove optimal convergence rates for certain low-regularity integrators applied to the one-dimensional periodic nonlinear Schrödinger and wave equations under the assumption of $H^1$ solutions. For the Schrödinger equation we analyze the exponential-type scheme proposed by Ostermann and Schratz in 2018, whereas in the wave case we treat the corrected Lie splitting proposed by Li, Schratz, and Zivcovich in 2023. We show that the integrators converge with their full order of one and two, respectively. In this situation only fractional convergence rates were previously known. The crucial ingredients in the proofs are known space-time bounds for the solutions to the corresponding linear problems. More precisely, in the Schrödinger case we use the $L^4$ Strichartz inequality, and for the wave equation a null form estimate. To our knowledge, this is the first time that a null form estimate is exploited in numerical analysis. We apply the estimates for continuous time, thus avoiding potential losses resulting from discrete-time estimates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_22621 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved error estimates for low-regularity integrators using space-time bounds Ruff, Maximilian Numerical Analysis Analysis of PDEs 65M15 (Primary) 35L71, 35Q55, 65M12 (Secondary) We prove optimal convergence rates for certain low-regularity integrators applied to the one-dimensional periodic nonlinear Schrödinger and wave equations under the assumption of $H^1$ solutions. For the Schrödinger equation we analyze the exponential-type scheme proposed by Ostermann and Schratz in 2018, whereas in the wave case we treat the corrected Lie splitting proposed by Li, Schratz, and Zivcovich in 2023. We show that the integrators converge with their full order of one and two, respectively. In this situation only fractional convergence rates were previously known. The crucial ingredients in the proofs are known space-time bounds for the solutions to the corresponding linear problems. More precisely, in the Schrödinger case we use the $L^4$ Strichartz inequality, and for the wave equation a null form estimate. To our knowledge, this is the first time that a null form estimate is exploited in numerical analysis. We apply the estimates for continuous time, thus avoiding potential losses resulting from discrete-time estimates. |
| title | Improved error estimates for low-regularity integrators using space-time bounds |
| topic | Numerical Analysis Analysis of PDEs 65M15 (Primary) 35L71, 35Q55, 65M12 (Secondary) |
| url | https://arxiv.org/abs/2503.22621 |