Stable Discontinuous Galerkin Spectral Element Methods for Higher Order Wave Equations

Abstract

In this thesis, we focus on discontinuous Galerkin spectral element methods (DGSEMs), which have recently gained considerable interest due to their high order accuracy and scalability. In particular, we discuss the development of a DGSEM spatial discretization framework that satisfies the summation-by-parts property. This property is the discrete analogue of the continuous integration-by-parts property, and it plays a crucial role in achieving numerical stability. Based on this approach, we develop provably stable DGSEMs for three higher order wave equations, namely the shifted wave equation, the Camassa-Holm equation, and the Serre equations. For each of these equations, we start by analyzing well-posed boundary conditions in a bounded domain using the energy method. The boundary conditions yield energy stable initial boundary value problems, facilitating the design of robust and arbitrarily accurate numerical methods. We propose provably stable DGSEMs to obtain the numerical solutions of the initial boundary value problems. The stability of our numerical approximations is rigorously proved by deriving discrete energy estimates mimicking the continuous energy estimates. Numerical experiments are presented to verify the theoretical results.