We introduce and analyze an $hp$-version $C^{1}$-continuous Petrov–Galerkin (CPG) method for nonlinear initial value problems of second-order ordinary differential equations. We derive a-priori error estimates in the $L^{2}$-, $L^{\infty }$-, $H^{1}$- and $H^{2}$-norms that are completely explicit in the local time steps and local approximation degrees. Moreover, we show that the $hp$-version $C^{1}$-CPG method superconverges at the nodal points of the time partition with regard to the time steps and approximation degrees. As an application, we apply the $hp$-version $C^{1}$-CPG method to time discretization of nonlinear wave equations. Several numerical examples are presented to verify the theoretical results.

中文翻译：

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用于非线性二阶初值问题的 hp 版本 C1 连续 Petrov-Galerkin 方法及其在波动方程中的应用


我们介绍并分析了用于二阶常微分方程非线性初值问题的 $hp$ 版本 $C^{1}$-连续 Petrov–Galerkin (CPG) 方法。我们在完全明确的 $L^{2}$-、$L^{\infty }$-、$H^{1}$- 和 $H^{2}$-范数中导出先验误差估计在局部时间步长和局部逼近度中。此外，我们表明 $hp$-version $C^{1}$-CPG 方法在时间步长和逼近度方面在时间分区的节点处超收敛。作为一个应用，我们将$hp$版本的$C^{1}$-CPG方法应用于非线性波动方程的时间离散化。给出了几个数值例子来验证理论结果。