An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation

M. Stanglmeier, N. C. Nguyen, J. Peraire, B. Cockburn

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26 Scopus citations

Abstract

We present an explicit hybridizable discontinuous Galerkin (HDG) method for numerically solving the acoustic wave equation. The method is fully explicit, high-order accurate in both space and time, and coincides with the classic discontinuous Galerkin (DG) method with upwinding fluxes for a particular choice of its stabilization function. This means that it has the same computational complexity as other explicit DG methods. However, just as its implicit version, it provides optimal convergence of order k+1 for all the approximate variables including the gradient of the solution, and, when the time-stepping method is of order k+2, it displays a superconvergence property which allow us, by means of local postprocessing, to obtain new improved approximations of the scalar field variables at any time levels for which an enhanced accuracy is required. In particular, the new approximations converge with order k+2 in the L2-norm for k≥1. These properties do not hold for all numerical fluxes. Indeed, our results show that, when the HDG numerical flux is replaced by the Lax-Friedrichs flux, the above-mentioned superconvergence properties are lost, although some are recovered when the Lax-Friedrichs flux is used only in the interior of the domain. Finally, we extend the explicit HDG method to treat the wave equation with perfectly matched layers. We provide numerical examples to demonstrate the performance of the proposed method.

Original languageEnglish (US)
Pages (from-to)748-769
Number of pages22
JournalComputer Methods in Applied Mechanics and Engineering
Volume300
DOIs
StatePublished - Mar 1 2016

Bibliographical note

Funding Information:
N.C. Nguyen and J. Peraire would like to acknowledge the partial support by the Air Force Office of Scientific Research under the AFOSR Grant FA9550-12-1-0357 . B. Cockburn was partially supported by the National Science Foundation (Grant DMS-1115331 ) and by the Minnesota Supercomputing Institute.

Keywords

  • Discontinuous Galerkin methods
  • Finite element method
  • Runge-Kutta methods
  • Superconvergence
  • Wave equation

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