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-infinity-, 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.