A numerical study of nonlinear wave run-up on a vertical plate

A finite difference model based on a recently derived highly-accurate Boussinesq-type formulation is presented. Up to the third-order space derivatives in terms of the velocity variables are retained, and the horizontal velocity variables are re-formulated in terms of a velocity potential. This decreases the total number of unknowns in two horizontal dimensions from seven to five, simplifying the implementation, and leading to increased computational efficiency. Analysis of the embedded properties demonstrates that the resulting model has applications with errors of 2 to 3% for (wavenumber times depth) kh ≤ 10 in terms of dispersion and kh ≤ 4 in terms of internal kinematics. The stability and accuracy of the discrete linearised systems are also analysed for both potential and velocity formulations and the advantages and disadvantages of each are discussed. The velocity potential model is then used to study physically demanding problems involving highly nonlinear wave run-up on a bottom-mounted (surface-piercing) plate. New cases involving oblique incidence are considered. In all cases, comparisons with recent physical experiments demonstrate good quantitative accuracy, even in the most demanding cases, where the local wave steepness can exceed (waveheight divided by wavelength) H / L = 0.20. The velocity potential model is additionally shown to have numerical advantages when dealing with wave–structure interactions, requiring less smoothing around exterior structural corners.