hp-version C1-continuous Petrov–Galerkin method for nonlinear second-order initial value problems with application to wave equations

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.