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An Unfiltered Low-Regularity Integrator for the KdV Equation with Solutions Below H1.

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Title: An Unfiltered Low-Regularity Integrator for the KdV Equation with Solutions Below H1.
Authors: Li, Buyang1 (AUTHOR) buyang.li@polyu.edu.hk, Wu, Yifei2 (AUTHOR) yerfmath@gmail.com
Source: Foundations of Computational Mathematics. Jun2026, Vol. 26 Issue 3, p1321-1380. 60p.
Subjects: Korteweg-de Vries equation, Time integration scheme, Harmonic analysis (Mathematics), Stability theory, Perturbation theory, Numerical analysis
Abstract: This article is concerned with the construction and analysis of new time discretizations for the KdV equation on a torus for low-regularity solutions below H 1 . New harmonic analysis tools, including averaging approximations to the exponential phase functions and trilinear estimates of the KdV operator, are established for the construction and analysis of time discretizations with higher convergence orders under low-regularity conditions. In addition, new perturbation techniques are introduced to establish stability estimates of time discretizations under low-regularity conditions without using filters when the energy techniques fail. The proposed method is proved to be convergent with order γ (up to a logarithmic factor) in L 2 under the regularity condition u ∈ C ([ 0 , T ] ; H γ) for γ ∈ (0 , 1 ] . [ABSTRACT FROM AUTHOR]
Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="AR" term="%22Li%2C+Buyang%22">Li, Buyang</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> buyang.li@polyu.edu.hk</i><br /><searchLink fieldCode="AR" term="%22Wu%2C+Yifei%22">Wu, Yifei</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> yerfmath@gmail.com</i>
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  Data: <searchLink fieldCode="JN" term="%22Foundations+of+Computational+Mathematics%22">Foundations of Computational Mathematics</searchLink>. Jun2026, Vol. 26 Issue 3, p1321-1380. 60p.
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  Data: <searchLink fieldCode="DE" term="%22Korteweg-de+Vries+equation%22">Korteweg-de Vries equation</searchLink><br /><searchLink fieldCode="DE" term="%22Time+integration+scheme%22">Time integration scheme</searchLink><br /><searchLink fieldCode="DE" term="%22Harmonic+analysis+%28Mathematics%29%22">Harmonic analysis (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Stability+theory%22">Stability theory</searchLink><br /><searchLink fieldCode="DE" term="%22Perturbation+theory%22">Perturbation theory</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink>
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  Data: This article is concerned with the construction and analysis of new time discretizations for the KdV equation on a torus for low-regularity solutions below H 1 . New harmonic analysis tools, including averaging approximations to the exponential phase functions and trilinear estimates of the KdV operator, are established for the construction and analysis of time discretizations with higher convergence orders under low-regularity conditions. In addition, new perturbation techniques are introduced to establish stability estimates of time discretizations under low-regularity conditions without using filters when the energy techniques fail. The proposed method is proved to be convergent with order γ (up to a logarithmic factor) in L 2 under the regularity condition u ∈ C ([ 0 , T ] ; H γ) for γ ∈ (0 , 1 ] . [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Text: English
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      – SubjectFull: Time integration scheme
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              Text: Jun2026
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