Localized solutions and filtering mechanisms for the discontinuous Galerkin semi-discretizations of the 1-d wave equation

TL;DR

This paper analyzes the semi-discrete 1-d wave equation using discontinuous Galerkin methods, identifying physical and spurious components, and develops filtering techniques to recover correct wave propagation velocities.

Contribution

It provides a Fourier analysis of DG semi-discretizations, constructs wave packets with small velocities, and introduces filtering mechanisms to improve numerical wave propagation accuracy.

Findings

Identification of physical and spurious wave components.

Construction of wave packets with arbitrarily small velocities.

Development of filtering methods to restore uniform wave velocity.

Abstract

We perform a complete Fourier analysis of the semi-discrete 1-d wave equation obtained through a P1 discontinuous Galerkin (DG) approximation of the continuous wave equation on an uniform grid. The resulting system exhibits the interaction of two types of components: a physical one and a spurious one, related to the possible discontinuities that the numerical solution allows. Each dispersion relation contains critical points where the corresponding group velocity vanishes. Following previous constructions, we rigorously build wave packets with arbitrarily small velocity of propagation concentrated either on the physical or on the spurious component. We also develop filtering mechanisms aimed at recovering the uniform velocity of propagation of the continuous solutions. Finally, some applications to numerical approximation issues of control problems are also presented.