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


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.


Accuracy Convergence Finite-difference methods Nonlinear waves Stability 


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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

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