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Numerical approximation of nonlinear Klein-Gordon (KG) equation with quadratic and cubic nonlinearity is performed using the element-free improved moving least squares Ritz (IMLS-Ritz) method. A regular arrangement of nodes is employed in this study for the numerical integration to compute the system equation. A functional formulation for the KG equation is established and discretized by the Ritz minimization procedure. Newmark’s integration scheme combined with an iterative technique is applied to the resulting nonlinear system equations. The effectiveness and efficiency of the IMLS-Ritz method for the KG equation have been testified through convergence analyses and comparison study between the present results and the exact solutions.

In this paper, a new method for predicting wave overtopping discharges of Accropode armored breakwaters using the non-hydrostatic wave model Simulating WAves till SHore (SWASH) is presented. The apparent friction coefficient concept is proposed to allow the bottom shear stress term calculated in the momentum equation to reasonably represent the effect of comprehensive energy dissipation caused by the roughness and seepage during the wave overtopping process. A large number of wave overtopping cases are simulated with a calibrated SWASH model to determine the values of equivalent roughness coefficients so that the apparent friction coefficients can be estimated to achieve the conditions with good agreement between numerical overtopping discharges and those from the EurOtop neural network model. The relative crest freeboard and the wave steepness are found to be the two main factors affecting the equivalent roughness coefficient. A derived empirical formula for the estimation of an equivalent roughness coefficient is presented. The simulated overtopping discharges by the SWASH model using the values of the equivalent roughness coefficient estimated from the empirical formula are compared with the physical model test results. It is found that the mean error rate from the present model predictions is 0.24, which is slightly better than the mean error rate of 0.26 from the EurOtop neural network model.