A Flexible 2D Nonlinear Approach for Nonlinear Wave Propagation, Breaking and Run up
Résumé
We present a hybrid solution strategy for the numerical solution of
the two-dimensional (2D) partial differential equations of Green-Nagdhi
(GN), which simulates fully nonlinear, weakly dispersive free surface
waves. We re-write the standard form of the equations by splitting the
original system in its elliptic and hyperbolic parts, through the definition
of a new variable, accounting for the dispersive effects and having the
role of a non-hydrostatic pressure gradient in the shallow water equations.
We consider a two-step solution procedure. In the first step we
compute a source term by inverting the elliptic coercive operator associated
to the dispersive effects; then in a hyperbolic step we evolve the
flow variables by using the non-linear shallow water equations, with all
non-hydrostatic effects accounted by the source computed in the elliptic
phase. The advantages of this procedure are firstly that the GN equations
are used for propagation and shoaling, while locally reverting to the nonlinear
shallow water equations to model energy dissipation in breaking
regions. Secondly and from the numerical point of view, this strategy
allows each step to be solved with an appropriate numerical method on
arbitrary unstructured meshes. We propose a hybrid finite element (FE)
finite volume (FV) scheme, where the elliptic part of the system is discretized
by means of the continuous Galerkin FE method and the hyperbolic
part is discretized using a third-order node-centred finite volume
(FV) technique. The performance of the numerical model obtained is extensively
validated against experimental measurements from a series of
relevant benchmark problems.
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