The ultra weak variational formulation for the modified mild-slope equation

This paper is devoted to evaluate a new alternative numerical method to capture accurately the diffraction-refraction process of waves in coastal areas. The modified mild-slope equation (MMSE) has been used to predict the water wave transformation when waves approach the shoreline. We perform numerical simulations in order to illustrate the efficiency of the ultra weak variational formulation (UWVF) method in comparison with the finite elements method (FEM). The UWVF method uses plane wave solutions on each element and has been shown to reduce the computational complexity at high wave numbers. We also present an alternative method to seek the angle of attack of the wave front on the domain boundary and show that the UWVF method reproduces effectively the numerical experimental data. We compare the FEM and the UWVF method for the MMSE and show that the UWVF method solves some of the difficulties that arise when the FEM is applied.