Finite Elements for the Boussinesq Wave Equations

  • Hans Petter Langtangen
  • Geir Pedersen
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 34)


The propagation and run-up of long surface waves on water (tsunamis, swells etc.) is a problem of great importance in oceanography and marine engineering. The standard simulation models in this field are based on the linear hydrostatic wave equations solved by finite difference methods (Mesinger and Arakawa, 1976; Abott, Petersen and Skovgaard, 1978). In recent years the models have been extended to include also nonlinear and weakly dispersive effects (Ertekin, Webster and Wehausen 1986; Katsis and Akylas 1987; Pedersen 1988a,95; Zelt 1990; Wei, Kirby, Grilli and Subramanya 1995). From a computational point of view the main advantage of such depth integrated long wave models is that the mathematical problem is two-dimensional, which enables simulation in domains of much larger extents than with techniques based on more general wave equations


Surface Elevation Boussinesq Equation Adaptive Grid Krylov Subspace Method Finite Difference Model 
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  1. Abott, M.B., Petersen, H.M, and Skovgaard, O. (1978), On the numerical modelling of short waves in shallow waterJ. Hyd. Res 16(3), 173 – 203.CrossRefGoogle Scholar
  2. Bruaset, A.M. (1995), A Survey of Preconditioned Iterative Methods, volume 328 in Pitman Research Notes in Mathematics Series, Longman House.Google Scholar
  3. Diffpack World Wide Web home page (1995), (Use Mosaic or another WWW browser to load the URL ttp:// The software isavailable from net lib.)Google Scholar
  4. Ertekin,R.C., Webster,W. C., and Wehausen, J.V. (1986), Waves caused by a moving disturbance in a shallow channel of finite widthJ. Fluid Mech 169, 275 – 292.CrossRefGoogle Scholar
  5. Joe, B. (1991), GEOMPACK - a software package for the generation of meshes using geometric algorithms, AdvEng. Software, 13, 325–331. (The software is available from net lib.).Google Scholar
  6. Katsis, C. and Akylas, T.R. (1987), On the excitation of long nonlinear water waves by a moving pressure distribution. Part 2. Three-dimensional effectsJ. Fluid Mech 177, 49 – 65.zbMATHCrossRefGoogle Scholar
  7. Langtangen, H.P. (1989), Conjugate gradient methods and ILU preconditioning of non-symmetric matrix systems with arbitrary sparsity patternsInt. J. Num. Meth. Fluids, vol 9, 213–223.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Mesinger, F., and Arakawa, A. (1976), Numerical methods used in atmospheric modelsGARP, Publ. Ser. WMO 17 64 pp.Google Scholar
  9. Pedersen, G. (1988a), Three-dimensional wave patterns generated by moving disturbances at transcritical speedsJ. Fluid. Mech. vol19639 – 63.zbMATHCrossRefGoogle Scholar
  10. Pedersen, G. (1988b), On the numerical solution of the Boussinesq equationsUniversity of Oslo, Research Report in Mechanics 88–14 Google Scholar
  11. Pedersen G. (1995), Refraction of solitons and wave jumps. This volume.Google Scholar
  12. Peregrine, D.H. (1972), Equations for water waves and the approximation behind them. In: Waves on beaches. Ed. by R.E. Meyer, Academic Press, New York357 – 412.Google Scholar
  13. Wei G., Kirby J.T., Grilli S.T., and Subramanya R. (1995), A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady wavesJ. Fluid Mech 294, 71 – 92.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Wu, T.Y. (1981), Long waves in ocean and coastal watersProc. ASCE, J. Eng. Mech. Div 107,EM3, 501 – 522.Google Scholar
  15. Wu, D. M., and Wu, T. Y. (1982), Three-dimensional nonlinear long waves due to moving surface pressure. In Proc. 14th Symp Naval hydrodyn, 103 – 129.Google Scholar
  16. Zelt, J.A., and Raichlen, F. (1990), A Lagrangian model for wave-induced harbour oscillations J. Fluid Mech 213, 203 – 225.zbMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Hans Petter Langtangen
    • 1
  • Geir Pedersen
    • 1
  1. 1.Mechanics Division, Department of MathematicsUniversity of OsloNorway

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