We devise the first symplectic Hamiltonian hybridizable discontinuous Galerkin (HDG) methods for the acoustic wave equation. We discretize in space by using a Hamiltonian HDG scheme, that is, an HDG method which preserves the Hamiltonian structure of the wave equation, and in time by using symplectic, diagonally implicit and explicit partitioned Runge–Kutta methods. The fundamental feature of the resulting scheme is that the conservation of a discrete energy, which is nothing but a discrete version of the original Hamiltonian, is guaranteed. We present numerical experiments which indicate that the method achieves optimal approximations of order k+1 in the L2-norm when polynomials of degree k≥0 and Runge–Kutta time-marching methods of order k+1 are used. In addition, by means of post-processing techniques and by increasing the order of the Runge–Kutta method to k+2, we obtain superconvergent approximations of order k+2 in the L2-norm for the displacement and the velocity. We also present numerical examples that corroborate that the methods conserve energy and that they compare favorably with dissipative HDG schemes, of similar accuracy properties, for long-time simulations.
Bibliographical noteFunding Information:
N. C. Nguyen and J. Peraire would like to acknowledge the partial support by AFOSR Grant No. FA9550-11-1-0141 and AFOSR Grant No. FA9550-16-1-0214 . B. Cockburn was partially supported by the National Science Foundation (Grant DMS-1522657 ) and by the Minnesota Supercomputing Institute .
- Acoustic wave equation
- Discontinuous Galerkin methods
- Energy conservation
- Finite element methods
- Hamiltonian systems
- Hybrid/mixed methods
- Symplectic time integrators