$hp$-version $C^1$-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 $L^2$-, $L^{\mathrm{\infty }}$-, $H^1$- 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.