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

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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.

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Authors and Affiliations

  1. Mechanical Engineering, Technical University of Denmark, DK, 2800, Lyngby, Denmark

    Harry B. Bingham

  2. Marine & Foundation Eng., COWI A/S, DK, 2800, Lyngby, Denmark

    Haiwen Zhang

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  1. Harry B. Bingham
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  2. Haiwen Zhang
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Correspondence to Harry B. Bingham.

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Bingham, H.B., Zhang, H. On the accuracy of finite-difference solutions for nonlinear water waves. J Eng Math 58, 211–228 (2007). https://doi.org/10.1007/s10665-006-9108-4

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  • DOI: https://doi.org/10.1007/s10665-006-9108-4

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