Hybridizable Discontinuous Galerkin Method for Linear Hyperbolic Integro-Differential Equations

Abstract

This article introduces the hybridizable discontinuous Galerkin (HDG) approach to numerically approximate the solution of a linear hyperbolic integro-differential equation. A priori error estimates for semi-discrete and fully discrete schemes are developed. It is shown that the optimal order of convergence is achieved for both scalar and flux variables. To achieve that, an intermediate projection is introduced for the semi-discrete error analysis, and it also shows that this projection achieves convergence of order $h^{k+3/2}$ for $k ≥ 1.$ Next, superconvergence is achieved for the scalar variable using element-by-element post-processing. For the fully discrete error analysis, the central difference scheme and the mid-point rule approximate the derivative and the integral term, respectively. Hence, the second order of convergence is achieved in the temporal direction. Finally, numerical experiments have been performed to validate the theory developed in this article.