Hybrid High-Order Methods for the Acoustic Wave Equation in the Time Domain

Abstract

We devise hybrid high-order (HHO) methods for the acoustic wave equation in the time domain. We first consider the second-order formulation in time. Using the Newmark scheme for the temporal discretization, we show that the resulting HHO-Newmark scheme is energy-conservative, and this scheme is also amenable to static condensation at each time step. We then consider the formulation of the acoustic wave equation as a first-order system together with singly-diagonally implicit and explicit Runge-Kutta (SDIRK and ERK) schemes. HHO-SDIRK schemes are amenable to static condensation at each time step. For HHO-ERK schemes, the use of the mixed-order formulation, where the polynomial degree of the cell unknowns is one order higher than that of the face unknowns, is key to benefit from the explicit structure of the scheme. Numerical results on test cases with analytical solutions show that the methods can deliver optimal convergence rates for smooth solutions of order $$\mathcal{O}(h^{k+1})$$ in the $$H^1$$ -norm and of order $$\mathcal{O}(h^{k+2})$$ in the $$L^2$$ -norm. Moreover, test cases on wave propagation in heterogeneous media indicate the benefits of using high-order methods.