In this paper, we establish the optimal convergence for a second-order exponential-type integrator from Hofmanová & Schratz (2017, An exponential-type integrator for the KdV equation. Numer. Math., 136, 1117–1137) for solving the Korteweg–de Vries equation with rough initial data. The scheme is explicit and efficient to implement. By rigorous error analysis, we show that the scheme provides second-order accuracy in $H^\gamma $ for initial data in $H^{\gamma +4}$ for any $\gamma \geq 0$, where the regularity requirement is lower than for classical methods. The result is confirmed by numerical experiments, and comparisons are made with the Strang splitting scheme.

中文翻译：

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KdV 方程的二阶低正则积分器的最优收敛

在本文中，我们为来自 Hofmanová & Schratz（2017，KdV 方程的指数型积分器。Numer. Math., 136, 1117–1137）的二阶指数型积分器建立了最优收敛，用于求解 Korteweg –de Vries 方程与粗略的初始数据。该方案是明确和有效的实施。通过严格的误差分析，我们表明该方案为任何 $\gamma \geq 0$ 的 $H^{\gamma +4}$ 中的初始数据提供了 $H^\gamma $ 中的二阶精度，其中规律性要求低于经典方法。结果通过数值实验得到证实，并与Strang分裂方案进行了比较。