# Numerical study of Galerkin-collocation approximation in time for the wave equation

## TL;DR

This paper introduces a higher order Galerkin-collocation method for time discretization of wave equations, combining accuracy with computational efficiency, and validates its performance through numerical experiments.

## Contribution

It establishes a novel connection between Galerkin and collocation methods for wave equations, enabling high-order accuracy with reduced computational costs.

## Key findings

- Achieves high-order temporal accuracy in wave simulations.
- Reduces computational complexity compared to traditional methods.
- Demonstrates improved efficiency and accuracy through numerical tests.

## Abstract

The elucidation of many physical problems in science and engineering is subject to the accurate numerical modelling of complex wave propagation phenomena. Over the last decades, high-order numerical approximation for partial differential equations has become a well-established tool. Here we propose and study numerically the implicit approximation in time of wave equations by a Galerkin--collocation approach that relies on a higher order space-time finite element approach. The 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. For the fully discrete solution, higher order regularity in time is further ensured which can be advantageous in the discretization of multi-physics systems. The accuracy and efficiency of the variational collocation approach is carefully studied by numerical experiments.