Fully discrete low-regularity integrator for the Korteweg–de Vries equation

In this paper we propose a fully discrete low-regularity integrator for the Korteweg-de Vries equation on the torus. This is an explicit scheme and can be computed with a complexity of $\mathcal{O}(N \log N)$ operations by fast Fourier transform, where $N$ is the degrees of freedom in the spatial discretization. We prove that the scheme is first-order convergent in both time and space variables in $H^\gamma$-norm for $H^{\gamma+1}$ initial data under Courant–Friedrichs–Lewy condition $N \geq 1 / \tau$, where $\tau$ denotes the temporal step size. We also carry out numerical experiments that illustrate the convergence behavior.

Keywords

the KdV equation, low regularity, fully discrete, fast Fourier transform

2010 Mathematics Subject Classification

35Q55, 65M12, 65M15

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This research is supported by NSFC key project under the grant number 11831003, NSFC under the grant numbers 1197135, and Fundamental Research Funds for the Central Universities under the grant numbers 2019MS110 and 2019MS112.

Received 7 January 2022

Received revised 12 January 2023

Accepted 23 January 2023

Published 9 October 2023