A Conservative numerical approach for dispersive waves in 1-D channels

  • Jouy, Bastien (EDF R&D / LNHE and Saint-Venant Laboratory for Hydraulics)
  • Le, Minh (Saint-Venant Laboratory for Hydraulics)
  • Violeau, Damien (EDF R&D / LNHE and Saint-Venant Laboratory for Hydraulics)
  • Ricchiuto, Mario (Inria Bordeaux - Sud-Ouest, Team CARDAMOM)

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The use of Boussinesq-type models has increased in recent years due to the physical representations they consider: non-hydrostatic phenomena are taken into account via an assumption of weakly non-linear and weakly dispersive free-surface waves. Among these models, the Peregrine (1968) equations model the propagation of such waves in prismatic channels of arbitrary cross-section. More recently, many extensions have been made including that of Winckler and Liu (2015), who present a model that allows significant longitudinal variation of the channel cross-section, modeling more accurately the channel geometries that can be found in engineering applications. We propose a reformulation of the Winckler–Liu system based on the set of conserved variables: wet surface and total volume flux across the section. We recast the equations in a form greatly simplifying the numerical implementation by writing independent coupled problems for the non-hydrostatic effects, and for the free surface wave dynamics. The first are modeled by an elliptic problem providing a source term which is added to the right-hand side of a section averaged version of the Saint-Venant equations. The latter allows to compute the time evolution of the free surface waves. The coupled model is solved numerically using a finite element method for the elliptic equation, and a finite volume scheme for the shallow water system. We show the existence of an analytical solitary wave solution for a prismatic channel with a rectangular section, which we compare to that of Serre (1953), and which we can use to verify our implementation. Numerical results modeling weakly dispersive wave trains such as Favre (1935) waves are also presented and compared to those in the literature. The impact of the well-balanced treatment of the dispersive source is investigated.