Upwind Stabilized Finite Element Modelling of Non-hydrostaticWave Breaking and Run-up
Résumé
In the following report a new methodology is presented to model the propagation, wave breaking and run-up of waves in coastal zones. We represent the different coastal phenomena through the coupling of non-linear shallow water equations with the extended Boussinesq equations of Madsen and Sørensen. Each of the involved equations has a major role in describing a particular physical behaviour of the wave: the latter equations permit to model the propagation, while the non-linear shallow water ones lead waves to locally converge into discontinuities. We start from the third-order stabilized finite element scheme for the Boussinesq equations, developed in a previous scientific work (Ricchiuto and Filippini, J.Comput.Phys. 2014) and develop a non-linear variant, and detach the dispersive from the shallow water terms. A shock-capturing technique based on local non-linear mass lumping that permits in the shallow water regions to degrade locally the scheme to a first-order one across bores (shocks) and dry fronts is proposed. As for the detection of the breaking fronts, the shallow water areas, this involves physics based breaking criteria. We present different definitions of the breaking criterion, including a local implementation of the convective criterion of (Bjørkavåg and H. Kalisch, Phys.Letters A 2011), and the hybrid models of (Kazolea et. al, J.Comput.Phys. 2014), and (Tonelli and Petti, J.Hydr.Res. 2011). The behavior of different breaking criteria is investigated on several cases for which experimental data are available.
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