A new method valid for highly dispersive and highly nonlinear water waves is presented. It combines a time-stepping of the exact surface boundary conditions with an approximate series expansion solution to the Laplace equation in the interior domain. The starting point is an exact solution to the Laplace equation given in terms of infinite series expansions from an arbitrary z-level. We replace the infinite series operators by finite series (Boussinesq-type) approximations involving up to fifth-derivative operators. The finite series are manipulated to incorporate Padé approximants providing the highest possible accuracy for a given number of terms. As a result, linear and nonlinear wave characteristics become very accurate up to wavenumbers as high as kh = 40, while the vertical variation of the velocity field becomes applicable for kh up to 12. These results represent a major improvement over existing Boussinesq-type formulations in the literature. A numerical model is developed in a single horizontal dimension and it is used to study phenomena such as solitary waves and their impact on vertical walls, modulational instability in deep water involving recurrence or frequency downshift, and shoaling of regular waves up to breaking in shallow water.

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Boussinesq formulations valid for highly dispersive and highly nonlinear water waves are derived with the objective of improving the accuracy of the vertical variation of the velocity eld as well as the linear and nonlinear properties. First, an exact solution to the Laplace equation is given in terms of in nite-series expansions from an arbitrary z-level which is a constant fraction of the still-water depth. This de nes the fully dispersive and fully nonlinear water-wave problem in terms of ve variables: the free-surface elevation and the horizontal and vertical velocities evaluated at the free surface and at the arbitrary z-level. Next, the in nite series operators are replaced by nite-series (Boussinesq-type) approximations. Three di¬erent approximations are introduced, each involving up to fth-derivative operators, and these formulations are analysed with respect to the linear-velocity pro le, linear dispersion and linear shoaling. Nonlinear characteristics are investigated by a perturbation analysis to third order for regular waves and to second order for bichromatic waves. Finally, a numerical spectral solution is made for highly nonlinear steady waves in deep and shallow water. It can be concluded that the best of the new formulations (method III) allows an accurate description of dispersive nonlinear waves for kh (wavenumber times water depth) as high as 40, while accurate velocity pro les are restricted to kh < 10. These results represent a major improvement over existing Boussinesq formulations from the literature.

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 [1] 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 highorder 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 are suggested as optimal for engineering purposes.

Forcing by steep regular water waves on a vertical circular cylinder at finite depth was investigated numerically by solving the two-phase incompressible Navier-Stokes equations. Consistently with potential flow theory, boundary layer effects were neglected at the sea bed and at the cylinder surface, but the strong nonlinear motion of the free surface was included. The numerical model was verified and validated by grid convergence and by comparison to relevant experimental measurements. First-order convergence towards an analytical solution was demonstrated and an excellent agreement with the experimental data was found. Time-domain computations of the normalized inline force history on the cylinder were analysed as a function of dimensionless wave height, water depth and wavelength. Here the dependence on depth was weak, while an increase in wavelength or wave height both lead to the formation of secondary load cycles. Special attention was paid to this secondary load cycle and the flow features that cause it. By visual observation and a simplified analytical model it was shown that the secondary load cycle was caused by the strong nonlinear motion of the free surface which drives a return flow at the back of the cylinder following the passage of the wave crest. The numerical computations were further analysed in the frequency domain. For a representative example, the secondary load cycle was found to be associated with frequencies above the fifthand sixth-harmonic force component. For the third-harmonic force, a good agreement with the perturbation theories of Faltinsen, Newman & Vinje (J. was found. It was shown that the third-harmonic forces were estimated well by a Morison force formulation in deep water but start to deviate at decreasing depth.