Authors

M.A. Sánchez, C. Ciuca, N.C. Nguyen, J. Peraire, B. Cockburn

Abstract

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 L 2 -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 L 2 -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.

Keywords

Finite element methods; Discontinuous Galerkin methods; Hybrid/mixed methods; Acoustic wave equation; Hamiltonian systems; Symplectic time integrators; Energy conservation

Citation

  • Journal: Journal of Computational Physics
  • Year: 2017
  • Volume: 350
  • Issue:
  • Pages: 951–973
  • Publisher: Elsevier BV
  • DOI: 10.1016/j.jcp.2017.09.010

BibTeX

@article{S_nchez_2017,
  title={{Symplectic Hamiltonian HDG methods for wave propagation phenomena}},
  volume={350},
  ISSN={0021-9991},
  DOI={10.1016/j.jcp.2017.09.010},
  journal={Journal of Computational Physics},
  publisher={Elsevier BV},
  author={Sánchez, M.A. and Ciuca, C. and Nguyen, N.C. and Peraire, J. and Cockburn, B.},
  year={2017},
  pages={951--973}
}

Download the bib file

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