A conservative and energy stable discontinuous spectral element method for the shifted wave equation in second order form

TL;DR

This paper introduces a new discontinuous spectral element method for the shifted wave equation that is provably energy stable and conservative, combining advantages of several established numerical techniques.

Contribution

The paper presents a novel energy-stable, conservative spectral element method for the shifted wave equation, with proven stability, conservation, and optimal error estimates in multiple dimensions.

Findings

Method is energy stable and conservative.

Numerical experiments confirm optimal convergence.

Applicable in multiple dimensions and regimes.

Abstract

In this paper, we develop a provably energy stable and conservative discontinuous spectral element method for the shifted wave equation in second order form. The proposed method combines the advantages and central ideas of very successful numerical techniques, the summation-by-parts finite difference method, the spectral method and the discontinuous Galerkin method. We prove energy-stability, discrete conservation principle, and derive error estimates in the energy norm for the (1+1)-dimensions shifted wave equation in second order form. The energy-stability results, discrete conservation principle, and the error estimates generalise to multiple dimensions using tensor products of quadrilateral and hexahedral elements. Numerical experiments, in (1+1)-dimensions and (2+1)-dimensions, verify the theoretical results and demonstrate optimal convergence of $L^2$ numerical errors at subsonic, sonic and supersonic regimes.