Arbitrary High-Order Explicit Hybridizable Discontinuous Galerkin Methods for the Acoustic Wave Equation

Abstract

We propose a new formulation of explicit time integration for the hybridizable discontinuous Galerkin (HDG) method in the context of the acoustic wave equation based on the arbitrary derivative approach. The method is of arbitrary high order in space and time without restrictions such as the Butcher barrier for Runge–Kutta methods. To maintain the superconvergence property characteristic for HDG spatial discretizations, a special reconstruction step is developed, which is complemented by an adjoint consistency analysis. For a given time step size, this new method is twice as fast compared to a low-storage Runge–Kutta scheme of order four with five stages at polynomial degrees between two and four. Several numerical examples are performed to demonstrate the convergence properties, reveal dispersion and dissipation errors, and show solution behavior in the presence of material discontinuities. Also, we present the combination of local time stepping with h-adaptivity on three-dimensional meshes with curved elements.