Accuracy and efficiency of two numerical methods of solving the potential flow problem for highly nonlinear and dispersive water waves

SummaryThe accuracy and efficiency of two methods of resolving the exact potential flow problem for nonlinear waves are compared using three different one horizontal dimension (1DH) test cases. The two model approaches use high‐order finite difference schemes in the horizontal dimension and differ in the resolution of the vertical dimension. The first model uses high‐order finite difference schemes also in the vertical, while the second model applies a spectral approach. The convergence, accuracy, and efficiency of the two models are demonstrated as a function of the temporal, horizontal, and vertical resolutions for the following: (1) the propagation of regular nonlinear waves in a periodic domain; (2) the motion of nonlinear standing waves in a domain with fully reflective boundaries; and (3) the propagation and shoaling of a train of waves on a slope. The spectral model approach converges more rapidly as a function of the vertical resolution. In addition, with equivalent vertical resolution, the spectral model approach shows enhanced accuracy and efficiency in the parameter range used for practical model applications. Copyright © 2015 John Wiley & Sons, Ltd. Two methods of resolving the exact potential flow problem for nonlinear waves are validated with three 1DH test cases by evaluating the convergence properties and evolution of errors as a function of the temporal, horizontal, and vertical resolution. In comparison to high‐order finite difference schemes, applying a spectral approach to resolve the vertical velocity at the free surface improves the model accuracy and efficiency, and this model is retained for future work.