Hybridized discontinuous Galerkin methods for wave propagation

TL;DR

This paper reviews hybridizable and embedded discontinuous Galerkin methods tailored for wave propagation in various physical domains, highlighting their design, stability, efficiency, and performance through numerical results.

Contribution

It introduces hybridized DG methods with a focus on their formulation, stability, and efficiency for wave propagation problems across multiple fields.

Findings

Effective numerical flux choices enhance stability and accuracy.

Hybridization allows for more efficient computational implementations.

Numerical results demonstrate the methods' performance across different wave problems.

Abstract

We present the recent development of hybridizable and embedded discontinuous Galerkin (DG) methods for wave propagation problems in fluids, solids, and electromagnetism. In each of these areas, we describe the methods, discuss their main features, display numerical results to illustrate their performance, and conclude with bibliography notes. The main ingredients in devising these DG methods are (i) a local Galerkin projection of the underlying partial differential equations at the element level onto spaces of polynomials of degree k to parametrize the numerical solution in terms of the numerical trace; (ii) a judicious choice of the numerical flux to provide stability and consistency; and (iii) a global jump condition that enforces the continuity of the numerical flux to obtain a global system in terms of the numerical trace. These DG methods are termed hybridized DG methods, because they are amenable to hybridization (static condensation) and hence to more efficient implementations. They share many common advantages of DG methods and possess some unique features that make them well-suited to wave propagation problems.