Treatment of Wave Breaking and Total Absorption in a Mild-Slope Equation FEM Model

Abstract

This paper looks at the iterative techniques and the convergence criteria that are necessary to deal with two difficulties in the integration of the full elliptic mild-slope equation, by means of an FEM model. The first difficulty is the nonlinearity due to the inclusion of a wave breaking dissipation factor. This paper gives an overall description of an iterative technique and presents an effective convergence criterion. It shows that this technique, although fully reliable, is time-consuming. Therefore the present paper also sets out to present an accelerated method, which is shown to be up to five times faster than the standard one. The second difficulty arises from the generally adopted boundary reflection condition, which depends on the angle—not known a priori—between the incident wave direction and the normal vector to the reflective boundary. A suitable iterative method and an efficient convergence criterion are presented here, along with validation tests aimed at simulating total absorption. Both nonlinear wave breaking and total absorption are important for coastal-morphology applications. A test on a domain reproducing a sloping beach has therefore been carried out in order to investigate the behavior of both these iterative techniques when used simultaneously.