Refraction and diffraction of water waves using finite elements with a DNL boundary condition

A discrete non-local (DNL) boundary condition is used to solve diffraction and refraction water waves problems. The finite elements with a DNL boundary condition are used to solve a wide range of unbounded surface wave problems. Two order low strategies are examined in rectangular and/or circular two-dimensional coordinates system. Numerical studies reveal the advantages of using this numerical approach and its limitations in case of variations of wave number in direction to infinity. Two new procedures to improve the accuracy of DNL boundary in case of significant variations of wave number in the domain are therefore developed. These procedures are based on the addition of a term to the numerical scheme, that collects the error away from the open boundary. Such term can be incorporated into the DNL formula as a source term or as an additional layer. This improvement permits the development of a suitable solution method, which is tested against analytical solutions and other methods, for bi-dimensional water waves problems defined on rectangular or circular geometries. Also, implementation details are reported. Satisfactory numerical results confirm the improvement of DNL method in case of dependent range, which allow us to conclude that the DNL method is an achievable method for the solution of unbounded water waves problems governed by the Helmholtz equation.