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.

翻译：我们引入并分析了一种用于求解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。我们推导了在$L^2$、$L^\infty$、$H^1$和$H^2$范数下的先验误差估计，这些估计完全显式地依赖于局部时间步长和局部逼近次数。此外，我们证明，在时间剖分的节点上，$hp$版本$C^1$-CPG方法关于时间步长和逼近次数具有超收敛性。作为应用，我们将$hp$版本$C^1$-CPG方法应用于非线性波动方程的时间离散化。文中给出了若干数值算例以验证理论结果。