An Unfiltered Low-Regularity Integrator for the KdV Equation with Solutions Below $$\mathbf{H^1}$$

IF 2.5 1区数学 Q2 COMPUTER SCIENCE, THEORY & METHODS Foundations of Computational Mathematics Pub Date : 2025-03-04 DOI:10.1007/s10208-025-09702-0

Buyang Li, Yifei Wu

引用次数: 0

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 $\gamma $ (up to a logarithmic factor) in $L^2$ under the regularity condition $u\in C([0,T];H^\gamma )$ for $\gamma \in (0,1]$.

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本文关注环上 KdV 方程对于低于 $H^1$的低规则解的新时间离散的构造和分析。本文建立了新的谐波分析工具，包括指数相位函数的平均近似和 KdV 算子的三线性估计，用于构建和分析低规则性条件下具有更高收敛阶数的时间离散。此外，还引入了新的扰动技术，当能量技术失效时，无需使用滤波器即可建立低规则性条件下时间离散的稳定性估计。在 $\gamma \in C([0,T];H^\gamma )$ 为 $\gamma \in (0,1]$的规则性条件下，所提出的方法被证明在 $L^2$ 中以 $\gamma $阶收敛（达到对数因子）。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Foundations of Computational Mathematics 数学-计算机：理论方法

CiteScore

6.90

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Foundations of Computational Mathematics (FoCM) will publish research and survey papers of the highest quality which further the understanding of the connections between mathematics and computation. The journal aims to promote the exploration of all fundamental issues underlying the creative tension among mathematics, computer science and application areas unencumbered by any external criteria such as the pressure for applications. The journal will thus serve an increasingly important and applicable area of mathematics. The journal hopes to further the understanding of the deep relationships between mathematical theory: analysis, topology, geometry and algebra, and the computational processes as they are evolving in tandem with the modern computer. With its distinguished editorial board selecting papers of the highest quality and interest from the international community, FoCM hopes to influence both mathematics and computation. Relevance to applications will not constitute a requirement for the publication of articles. The journal does not accept code for review however authors who have code/data related to the submission should include a weblink to the repository where the data/code is stored.