Journal of Engineering Mathematics

, Volume 58, Issue 1–4, pp 211–228 | Cite as

On the accuracy of finite-difference solutions for nonlinear water waves

Original Paper

Abstract

This paper considers the relative accuracy and efficiency of low- and high-order finite-difference discretisations of the exact potential-flow problem for nonlinear water waves. The method developed is an extension of that employed by Li and Fleming (Coastal Engng 30: 235–238, 1997) to allow arbitrary-order finite-difference schemes and a variable grid spacing. Time-integration is performed using a fourth-order Runge–Kutta scheme. The linear accuracy, stability and convergence properties of the method are analysed and high-order schemes with a stretched vertical grid are found to be advantageous relative to second-order schemes on an even grid. Comparison with highly accurate periodic solutions shows that these conclusions carry over to nonlinear problems and that the advantages of high-order schemes improve with both increasing nonlinearity and increasing accuracy tolerance. The combination of non-uniform grid spacing in the vertical and fourth-order schemes is suggested as optimal for engineering purposes.

Keywords

Accuracy Convergence Finite-difference methods Nonlinear waves Stability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Newman JN (1977) Marine hydrodynamics. The MIT Press, Cambridge, MassachusettsGoogle Scholar
  2. 2.
    Newman JN (2005) Efficient hydrodynamic analysis of very large floating structures. Marine Struct 18:169–180CrossRefGoogle Scholar
  3. 3.
    Newman JN, Lee C-H (2002) Boundary-element methods in offshore structure analysis. J Offshore Mech Arctic Engng 124:81–89CrossRefGoogle Scholar
  4. 4.
    Sclavounos PD, Borgen H (2004) Seakeeping analysis of a high-speed monohull with a motion-control bow hydrofoil. J Ship Res 48:77–117Google Scholar
  5. 5.
    Bertram V, Yasukawa H, Thiart G (2005) Evaluation of local pressures in ships using potential flow models. In: Oceans 2005 – Europe (IEEE Cat. No. 05EX1042), vol 1. IEEE, pp 113–117Google Scholar
  6. 6.
    Büchmann B, Ferrant P, Skourup J (2000) Run-up on a body in waves and current. fully nonlinear and finite-order calculations. Appl Ocean Res 22:349–360CrossRefGoogle Scholar
  7. 7.
    Grilli ST, Guyenne P, Dias F (2001) A fully non-linear model for three-dimensional overturning waves over an arbitrary bottom. Int J Num Methods Fluids 35:829–867MATHCrossRefGoogle Scholar
  8. 8.
    Dommermuth DG, Yue DKP (1987) A high-order spectral method for the study of nonlinear gravity waves. J Fluid Mech 184:267–288MATHCrossRefADSGoogle Scholar
  9. 9.
    Fructus D, Clamond D, Grue J, Kristiansen Ø (2005) An efficient model for three-dimensional surface wave simulations. part i: Free space problems. J Comput Phys 205:665–685MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Fuhrman DR, Bingham HB, Madsen PA (2005) Nonlinear wave-structure interaction with a high-order Boussinesq model. Coastal Engng 52:655–672CrossRefGoogle Scholar
  11. 11.
    Shi F, Kirby JT, Dalrymple RA, Chen Q (2003) Wave simulations in ponce de leon inlet using Boussinesq model. J Waterway Port Coastal Ocean Engng 129:124–135CrossRefGoogle Scholar
  12. 12.
    Chen H-C, Lee S-K (1999) Rans/laplace calculations of nolinear waves induced by surface-piercing bodies. J Engng Mech 125(11):1231–1242CrossRefGoogle Scholar
  13. 13.
    Wu GX, Ma QW, Eatock-Taylor R (1998) Numerical simulation of sloshing waves in a 3d tank based on a finite element method. Appl Ocean Res 20:337–355CrossRefGoogle Scholar
  14. 14.
    Li B, Fleming CA (1997) A three-dimensional multigrid model for fully nonlinear water waves. Coastal Engng 30:235–258CrossRefGoogle Scholar
  15. 15.
    Kreiss HO, Oliger J (1972) Comparison of accurate methods for the integration of hyperboloic equations. Tellus Series A 24(3):199–215MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fuhrman DR, Bingham HB (2004) Numerical solutions of fully nonlinear and highly dispersive Boussinesq equations. Int J Num Methods Fluids 44(3):231–255MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Fenton JD (1988) The numerical solution of steady water wave problems. Comput Geosci 14(3):357–68CrossRefADSGoogle Scholar
  18. 18.
    Zakharov VE (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. J Appl Mech Tech Phys 9:190–194CrossRefADSGoogle Scholar
  19. 19.
    Iserles A (1996) A first course in the numerical analysis of differential equations. Cambridge University PressGoogle Scholar
  20. 20.
    Duff IS (1989) Multiprocessing a sparse matrix code on the alliant fx/8. J Comput Appl Math 27:229–239CrossRefMATHGoogle Scholar
  21. 21.
    Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7:856–869MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Amestoy PR, Davis TA, Duff IS (1996) An approximate minimum degree ordering algorithm. SIAM J Mat Anal Appl 17:886–905MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Whaley RC, Petitet A, Dongarra JJ (2000) Automated empirical optimization of software and the atlas project, 2000. Available: http://math-atlas.sourceforge.net/Google Scholar
  24. 24.
    Hirsch C (1988) Numerical computation of internal and external flows. Volume 1: Fundamentals of numerical discretizaton. John Wiley & Sons, New YorkGoogle Scholar
  25. 25.
    Trefethen LN (1996) Finite difference and spectral methods for ordinary and partial differential equations. Available: http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/pdetext.html, Unpublished textGoogle Scholar
  26. 26.
    Williams JM (1981) Limiting gravity waves in water of finite depth. Phil Trans R Soy London A 302:139–188MATHCrossRefADSGoogle Scholar
  27. 27.
    Fenton JD (1990) Nonlinear wave theories. In: Le Mehaute B, Hanes DM (eds) The sea. John Wiley & Sons, pp 3–25Google Scholar
  28. 28.
    Jensen TG (1998) Open boundary conditions in stratified ocean models. J Marine Syst 16:297–322CrossRefADSGoogle Scholar
  29. 29.
    Larsen J, Dancy H (1983) Open boundaries in short wave simulations – a new approach. Coastal Engng 7:285–297CrossRefGoogle Scholar
  30. 30.
    Luth HR, Klopman B, Kitou N (1994) Projects 13 g: kinematics of waves breaking partially on an offshore bar; ldv measurements for waves with and without a net onshore current. Technical Report H1573, Delft Hydraulics, Delft, The NetherlandsGoogle Scholar
  31. 31.
    Beji S, Battjes JA (1993) Experimental investigation of wave propagation over a bar. Coastal Engng 19:151–162CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Mechanical EngineeringTechnical University of DenmarkLyngbyDenmark
  2. 2.Marine & Foundation Eng.COWI A/SLyngbyDenmark

Personalised recommendations