An Energy Stable Approach for Discretizing Hyperbolic Equations with Nonconforming Discontinuous Galerkin Methods

TL;DR

This paper introduces a skew-symmetric formulation for nonconforming discontinuous Galerkin methods applied to hyperbolic equations, ensuring energy stability even with inexact quadrature, variable materials, and curved elements.

Contribution

It proposes a novel skew-symmetric formulation that guarantees energy stability and conservation in nonconforming DG methods with inexact quadrature and complex geometries.

Findings

The method maintains energy stability with inexact quadrature.

It is conservative and constant preserving with variable materials.

Numerical experiments confirm stability and accuracy.

Abstract

When nonconforming discontinuous Galerkin methods are implemented for hyperbolic equations using quadrature, exponential energy growth can result even when the underlying scheme with exact integration does not support such growth. Using linear elasticity as a model problem, we propose a skew-symmetric formulation that has the same energy stability properties for both exact and inexact quadrature-based integration. These stability properties are maintained even when the material properties are variable and discontinuous, and the elements are non-affine (e.g., curved). Additionally, we show how the nonconforming scheme can be made conservative and constant preserving with variable material properties and curved elements. The analytic stability, conservation, and constant preserving results are confirmed through numerical experiments demonstrating the stability as well as the accuracy of the method.