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


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.


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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Madsen PA, Schäffer HA (1999) A review of Boussinesq-type equations for gravity waves. Adv Coastal Ocean Engng. 5:1–95CrossRefGoogle Scholar
  2. 2.
    Peregrine DH (1967) Long waves on a beach. J Fluid Mech 27:815–827zbMATHCrossRefADSGoogle Scholar
  3. 3.
    Madsen PA, Sørensen OR (1992) A new form of the Boussinesq equations with improved linear dispersion characteristics Part 2 A slowly varying bathymetry. Coastal Engng 18:183–204CrossRefGoogle Scholar
  4. 4.
    Nwogu O (1993) Alternative form of Boussinesq equations for nearshore wave propagation. J Waterway, Port, Coastal and Ocean Engng, 119(6):618–638CrossRefGoogle Scholar
  5. 5.
    Madsen PA, Agnon Y (2003) Accuracy and convergence of velocity formulations for water waves in the framework of Boussinesq theory. J Fluid Mech 477:285–319zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Madsen PA, Bingham HB, Liu H (2002) A new Boussinesq method for fully nonlinear waves from shallow to deep water. J Fluid Mech 462:1–30zbMATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Jamois E, Fuhrman DR, Bingham HB, Molin B (2006) A numerical study of nonlinear wave run-up on a vertical plate. Coastal Engng (to appear)Google Scholar
  8. 8.
    Fuhrman DR, Bingham HB (2004) Numerical solutions of fully non-linear and highly dispersive Boussinesq equations in two horizontal dimensions. Int J Num Meth Fluids 44:655–672MathSciNetGoogle Scholar
  9. 9.
    Fuhrman DR, Bingham HB, Madsen PA (2005) Nonlinear wave-structure interactions with a high-order Boussinesq model. Coastal Engng 52:655–672CrossRefGoogle Scholar
  10. 10.
    Antunes Do Carmo JS, Seabra Santos FJ, Barthèlemy E (1993) Surface waves propagation in shallow water: a finite element model. Int J Num Meth Fluids 16:447–459CrossRefGoogle Scholar
  11. 11.
    Ambrosi D, Quartapelle L (1998) A Taylor-Galerkin method for simulating nonlinear dispersive water waves. J Comp Phys 146:546–569zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Sherwin SJ, Eskilsson C (2006) Spectral/hp discontinuous Galerkin methods for modelling 2D Boussinesq equations. J Comp Phys 212:566–589zbMATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Langtangen HP, Pedersen G (1998) Computational models for weakly dispersive nonlinear water waves. Comput Methods Appl Engng 160:337–358zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Li YS, Liu S-X, Yu Y-X, Lai G-Z (1999) Numerical modelling of Boussinesq equations by finite element method. Coastal Engng 37:97–122CrossRefGoogle Scholar
  15. 15.
    Beji S, Nadaoka K (1996) A formal derivation and numerical modelling of the improved Boussinesq equations for varying depth. Ocean Engng 23:691–704CrossRefGoogle Scholar
  16. 16.
    Sørensen OR, Schäffer HA, Sørensen LS (2004) Boussinesq-type modelling using an unstructured finite element technique. Coastal Engng 50:181–198CrossRefGoogle Scholar
  17. 17.
    Walkley MA, Berzins M (1999) A finite element method for the one-dimensional extended Boussinesq equations. Int J Num Meth Fluids 29:143–157zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Walkley MA, Berzins M (2002) A finite element method for the two-dimensional extended Boussinesq equations. Int J Num Meth Fluids 39:865–885zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Eskilsson C, Sherwin SJ (2003) An hp/spectral element model for efficient long-time integration of Boussinesq-type equations. Coastal Engng 45:295–320CrossRefGoogle Scholar
  20. 20.
    Eskilsson C, Sherwin SJ (2005) An unstructured spectral/hp element model for enhanced Boussinesq-type equations. Submitted to Coastal EngngGoogle Scholar
  21. 21.
    Eskilsson C, Sherwin SJ (2002) A discontinuous spectral element model for Boussinesq-type equations. J Sci Comp 17:143–152zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Eskilsson C, Sherwin SJ (2005) Discontinuous Galerkin spectral/hp element modelling of dispersive shallow water systems. J Sci Comp 22:269–288CrossRefMathSciNetGoogle Scholar
  23. 23.
    Fuhrman DR, Bingham HB, Madsen PA, Thomsen PG (2004) Linear and non-linear stability analysis for finite difference discretizations of high-order Boussinesq equations. Int J Num Meth Fluids 45:751–773zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Cockburn B, Shu C-W (2001) Runge-Kutta Discontinuous Galerkin methods for convection-dominated Problems. J Sci Comp 16(3):173–261zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Hesthaven JS, Warburton T (2002) High-order nodal methods on unstructured grids. I. Time-domain solution of Maxwell’s equations. J Comp Phys 181(1):186–221zbMATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Sherwin S (1999) Dispersion analysis of the continuous and discontinuous Galerkin formulations. In: Cockburn B, Karniadakis GE, Shu C-W (eds) Lecture notes in computational science and engineering: discontinuous Galerkin methods- theory, computation and applications. Springer, pp 425–432Google Scholar
  27. 27.
    Bassi F, Rebay S (1997) A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J Comp Phys 131:267–279zbMATHCrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Davis TA (2005) UMFPACK version 4.6 user guide. University of Florida pp 1–133Google Scholar
  29. 29.
    Carpenter MH, Kennedy CA (1994) Fourth order 2N-storage Runge-Kutta scheme. Technical report NASA-TM-109112, NASA Langley Research Center, VAGoogle Scholar
  30. 30.
    Fischer P, Mullen JS (2001) Filter-based stabilization of spectral element methods. C R Acad Sci Paris 332:265–270zbMATHMathSciNetADSGoogle Scholar
  31. 31.
    Savitzky A, Golay MJE (1964) Smoothening and differentiation of data by simplified least squares procedures. Anal Chem 36:1627–1639CrossRefADSGoogle Scholar
  32. 32.
    Larsen J, Dancy H (1983) Open boundaries in short wave simulations - a new approach. Coastal Engng 7:285–297CrossRefGoogle Scholar
  33. 33.
    Beji S, Battjes JA (1994) Numerical simulation of nonlinear-wave propagation over a bar. Coastal Engng 23:1–16CrossRefGoogle Scholar
  34. 34.
    Luth HR, Klopman B, Kitou N (1994) Projects 13G: kinematics of waves breaking partially on an offshore bar: LDV measurements for waves with and without a net onshore current. Technical report H1573, Delft HydraulicsGoogle Scholar
  35. 35.
    Gobbi MF, Kirby JT (1999) Wave evolution over submerged sills: tests of a high-order Boussinesq model. Coastal Engng 37:57–96CrossRefGoogle Scholar
  36. 36.
    Tanaka M (1986) The stability of solitary waves. J Phys Fluids 29:650–655zbMATHCrossRefADSGoogle Scholar
  37. 37.
    Cooker MJ, Weidman PD, Bale DS (1997) Reflection of a high-amplitude solitary wave at a vertical wall. J Fluid Mech 342:141–158zbMATHCrossRefADSMathSciNetGoogle Scholar

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

Personalised recommendations