In this paper, the dual reciprocity boundary element method (DRBEM) based on the perturbation method is presented for calculating run-ups of weakly nonlinear long waves scattered by islands. Under the assumption that the incident waves are harmonic, the time-dependent nonlinear Boussinesq equations are transformed into three time-independent linear equations by using the perturbation method, where, besides nonlinearity εε, the dispersion μ2μ2 is also included in the perturbed expansion. The first-order solution η0η0 is found by using the linear long-wave equations. Then η0η0 is used in two second-order governing equations related to the dispersion and nonlinearity, respectively. Since no any omission and approximation for the seabed slope ∇h∇h and its derivatives is made, there are the third- and fourth-order partial derivatives of η0η0 appeared in the right-hand sides of two governing equations of the second-order. By employing a transformation, those third- and fourth-order partial derivatives are removed therefore large errors in approximating these derivatives are eliminated.
To validate the new model, wave diffractions around a large vertical cylinder for 13 cases are first considered. It is found that the nonlinear contributions to the new model are significant for weakly nonlinear waves with a much better comparison with experimental results obtained than for the linear diffraction theory. It is also found that the dispersive effects play an important role in improving the accuracy of the new model as numerical results obtained from the Boussinesq equations (with dispersion terms) are more accurate than those from the Airy's equations (without dispersion term). Then the combined wave diffraction and refraction by a conical island is also modelled and discussed. Our model is not only accurate as the dispersive effects have been included but also computationally efficient since the domain integrals are merely evaluated by distributing collocation points over that surface.