This paper presents a brief description of three finite element numerical models based on different versions of the time dependent nonlinear mild-slope equation, presented by Nadaoka et al. (1994), namely the one-dimensional, unidirectional and the two-dimensional versions of that equation. The Finite Element Method (FEM) is used for spatial discretisation while for the time integration a predictor-corrector finite difference scheme is used. The problem is formulated for the free surface local acceleration or the vertical velocity according to the version used. The resulting linear system of equations is solved by a direct method. The free-surface elevations are computed either by a double or a single integration in time, depending on the order of the time partial derivative. Implemented boundary conditions include prescribed freesurface displacement, wave generation by normal-velocity specification, radiation conditions and generation-radiation condition (applied to open boundaries).
The models are suitable for the propagation of surface waves over gently varying depths, taking into account the combined effects of nonlinear refraction and diffraction of waves, not only for shallow waters but also for deep and intermediate waters.
The models are validated with simple test cases of wave propagation over different depths – varying between deep to shallow waters – and their results are presented and discussed. Further, some numerical techniques for implementing the outgoing boundary conditions are investigated and discussed.
