Galerkin-collocation approximation in time for the wave equation and its post-processing

TL;DR

This paper introduces Galerkin-collocation schemes for the wave equation, combining accuracy and efficiency, with proven optimal error estimates and a post-processing method for improved solutions, validated by numerical experiments.

Contribution

It presents a novel Galerkin-collocation discretization approach for the wave equation, linking Galerkin and collocation methods, with error analysis and a post-processing technique.

Findings

Optimal order error estimates established

Discrete solutions are twice differentiable in time

Numerical experiments confirm theoretical results

Abstract

We introduce and analyze a class of Galerkin-collocation discretization schemes in time for the wave equation. Its conceptual basis is the establishment of a direct connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex linear algebraic systems. Continuously differentiable in time discrete solutions are obtained by the application of a special quadrature rule involving derivatives. Optimal order error estimates are proved for fully discrete approximations based on the Galerkin-collocation approach. Further, the concept of Galerkin-collocation approximation is extended to twice continuously differentiable in time discrete solutions. A direct connection between the two families by a computationally cheap post-processing is presented. The error estimates are illustrated by numerical experiments.