Embedded exponential-type low-regularity integrators for KdV equation under rough data

TL;DR

This paper introduces new explicit exponential-type integrators for the KdV equation that achieve optimal convergence with rough initial data, improving regularity requirements and demonstrating efficiency through theoretical analysis and numerical tests.

Contribution

The paper develops the first-order and second-order embedded exponential-type low-regularity integrators (ELRIs) for KdV, reducing regularity constraints compared to existing methods.

Findings

First-order ELRI achieves optimal convergence in $H^eta$ for initial data in $H^{eta+1}$.

Second-order ELRI attains second-order accuracy in $H^eta$ for initial data in $H^{eta+3}$.

Numerical experiments confirm the theoretical convergence and efficiency of the proposed methods.

Abstract

In this paper, we introduce a novel class of embedded exponential-type low-regularity integrators (ELRIs) for solving the KdV equation and establish their optimal convergence results under rough initial data. The schemes are explicit and efficient to implement. By rigorous error analysis, we first show that the ELRI scheme provides the first order accuracy in $H^\gamma$ for initial data in $H^{\gamma+1}$ for $\gamma>\frac12$. Moreover, by adding two more correction terms to the first order scheme, we show a second order ELRI that provides the second order accuracy in $H^\gamma$ for initial data in $H^{\gamma+3}$ for $\gamma\ge0$. The proposed ELRIs further reduce the regularity requirement of existing methods so far for optimal convergence. The theoretical results are confirmed by numerical experiments, and comparisons with existing methods illustrate the efficiency of the new methods.