On the nonlinear behavior of Boussinesq type models: amplitude-velocity vs amplitude-flux forms

Andrea Gilberto Filippini 1 Stevan Bellec 1 Mathieu Colin 1 Mario Ricchiuto 1
1 BACCHUS - Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems
Inria Bordeaux - Sud-Ouest, UB - Université de Bordeaux, CNRS - Centre National de la Recherche Scientifique : UMR5800
Abstract : In this paper we consider the modeling of nonlinear wave transformation by means of weakly nonlinear Boussinesq models. We show that for a given linear dispersion relation and linear shoaling parameter one can derive, within the same asymptotic truncation, two system of PDEs differing only in the form of the linear dispersive operators. In particular, these can either be formulated in terms of derivatives of the velocity, or in terms of derivatives of the flux. In the first case we speak of amplitude-velocity form of the model, in the second case of amplitude flux form. We show examples of these couples for several linear relations, including a new amplitude-flux variant of the model of Nwogu. We then show, both analytically and by numerical nonlinear shoaling tests, that while for small amplitude waves it is important to have accurate linear dispersion and shoaling characteristics, when approaching breaking conditions it is only the amplitude-velocity or amplitude-flux form of the equations which determines the behavior of the model, and in particular the shape and the height of the waves. This knowledge has tremendous importance when considering the use of these models in conjunction with wave breaking detection and dissipation mechanisms.
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Andrea Gilberto Filippini, Stevan Bellec, Mathieu Colin, Mario Ricchiuto. On the nonlinear behavior of Boussinesq type models: amplitude-velocity vs amplitude-flux forms. [Research Report] RR-8573, INRIA. 2014. 〈hal-01053036〉

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