An hp -version of the C 0 -continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems

We present an hp -version of the C 0 -continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems. We derive a priori error bound in the H 1 -norm that is fully explicit with respect to the local time steps and the local approximation orders. Moreover, we prove that the hp -version of the C 0 -continuous Petrov-Galerkin time stepping method based on geometrically refined time steps and on linearly increasing approximation orders yields exponential rates of convergence for solutions with start-up singularities. Numerical examples confirm the theoretical results.