hp-version C^1-continuous Petrov-Galerkin method for nonlinear second-order initial value problems with application to wave equations

03/25/2023

∙

by   Lina Wang, et al.

∙

0

∙

share

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^∞-, 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.