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Journal of Engineering Mathematics

, Volume 56, Issue 3, pp 351–370 | Cite as

Nodal DG-FEM solution of high-order Boussinesq-type equations

  • Allan P. Engsig-Karup
  • Jan S. Hesthaven
  • Harry B. Bingham
  • Per A. Madsen
Original Paper

Abstract

A discontinuous Galerkin finite-element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one horizontal dimension is presented. The continuous equations are discretized using nodal polynomial basis functions of arbitrary order in space on each element of an unstructured computational domain. A fourth-order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy and convergence of the model with both h (grid size) and p (order) refinement are confirmed for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar, and reflection of a steep solitary wave from a vertical wall. Test cases for two horizontal dimensions will be considered in future work.

Keywords

Discontinuous Galerkin finite-element method Gravity waves High-order Boussinesq-type equations Unstructured grids 

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Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • Allan P. Engsig-Karup
    • 1
  • Jan S. Hesthaven
    • 2
  • Harry B. Bingham
    • 1
  • Per A. Madsen
    • 1
  1. 1.Department of Mechanical EngineeringTechnical University of DenmarkLyngbyDenmark
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

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