In this paper, a space–time spectral method for solving the Korteweg–de Vries equation is considered. The discrete schemes of the method are based on the Legendre–Petrov–Galerkin method in spatial direction and the Legendre-tau method in temporal direction with nonperiodic boundary conditions. Stability analysis results and error estimates are obtained in -norm by introducing a cut-off function without Lipschitz condition. The method is also applicable to solve some th-order differential equations. Comparison of our numerical results with those of other spectral methods exhibits the accuracy of our methods for the Korteweg–de Vries equation.

中文翻译：

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求解 Korteweg-de Vries 方程的时空勒让德谱方法的误差估计

本文考虑了一种求解 Korteweg-de Vries 方程的时空谱方法。该方法的离散格式基于空间方向的 Legendre-Petrov-Galerkin 方法和时间方向的 Legendre-tau 方法，具有非周期边界条件。通过引入不带 Lipschitz 条件的截止函数，在范数范围内获得稳定性分析结果和误差估计。该方法也适用于求解一些三阶微分方程。我们的数值结果与其他光谱方法的比较显示了我们的 Korteweg-de Vries 方程方法的准确性。