We introduce and analyze the first energy-conservative hybridizable discontinuous Galerkin method for the semidiscretization in space of the acoustic wave equation. We prove optimal convergence and superconvergence estimates for the semidiscrete method. We then introduce a two-step fourth-order-in-time Stormer-Numerov discretization and prove energy conservation and convergence estimates for the fully discrete method. In particular, we show that by using polynomial approximations of degree two, convergence of order four is obtained. Numerical experiments verifying that our theoretical orders of convergence are sharp are presented. We also show experiments comparing the method with dissipative methods of the same order.
Bibliographical noteFunding Information:
Bernardo Cockburn: Partially supported by the National Science Foundation (Grant DMS-1115331). Francisco-Javier Sayas: Partially supported by the National Science Foundation (Grant DMS-1216356).
- Conservation of energy
- Discontinuous Galerkin
- Wave equation