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| Convergence of an Embedded Exponential-Type Low-Regularity Integrators for the KdV Equation without Loss of Regularity |
| Yongsheng Li1,,Yifei Wu,Fangyan Yao |
| (School of Mathematical Sciences, South China University of Technology, Guangzhou, Guangdong 510640, China;Center for Applied Mathematics, Tianjin University, Tianjin 300072,China) |
| DOI： |
| Abstract: |
| In this paper, we study the convergence rate of an Embedded exponential-type low-regularity integrator (ELRI) for the Korteweg-de Vries equation. We develop some new harmonic analysis techniques to handle the ``stability" issue. In particular, we use a new stability estimate which allows us to avoid the use of the fractional Leibniz inequality,}
\begin{align*}
\big|\big\langle J^\gamma \partial_x\big( f g\big),J^\gamma f\big\rangle \big|
\lesssim \big\|f\big\|_{H^\gamma}^2 \big\|g\big\|_{H^{\gamma+1}},
\end{align*}
and replace it by suitable inequalities without loss of regularity. Based on these techniques,
we prove that the ELRI scheme proposed in~\cite{wu} provides $\frac12$-order convergence accuracy in $H^\gamma$ for any initial data belonging to $H^\gamma$ with $\gamma >\frac32$, which does not require any additional derivative assumptions. |
| Key words: The KdV equation, numerical solution, convergence analysis, error estimate, low regularity, fast Fourier transform. |