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

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.