166 research outputs found
Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model
We investigate here the ability of a Green-Naghdi model to reproduce strongly
nonlinear and dispersive wave propagation. We test in particular the behavior
of the new hybrid finite-volume and finite-difference splitting approach
recently developed by the authors and collaborators on the challenging
benchmark of waves propagating over a submerged bar. Such a configuration
requires a model with very good dispersive properties, because of the
high-order harmonics generated by topography-induced nonlinear interactions. We
thus depart from the aforementioned work and choose to use a new Green-Naghdi
system with improved frequency dispersion characteristics. The absence of dry
areas also allows us to improve the treatment of the hyperbolic part of the
equations. This leads to very satisfying results for the demanding benchmarks
under consideration
A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes
In this paper, we introduce a discontinuous Finite Element formulation on
simplicial unstructured meshes for the study of free surface flows based on the
fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a
new class of asymptotically equivalent equations, which have a simplified
analytical structure, we consider a decoupling strategy: we approximate the
solutions of the classical shallow water equations supplemented with a source
term globally accounting for the non-hydrostatic effects and we show that this
source term can be computed through the resolution of scalar elliptic
second-order sub-problems. The assets of the proposed discrete formulation are:
(i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary
order of approximation in space, (iii) the exact preservation of the motionless
steady states, (iv) the preservation of the water height positivity, (v) a
simple way to enhance any numerical code based on the nonlinear shallow water
equations. The resulting numerical model is validated through several
benchmarks involving nonlinear wave transformations and run-up over complex
topographies
A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model
The fully nonlinear and weakly dispersive Green-Naghdi model for shallow
water waves of large amplitude is studied. The original model is first recast
under a new formulation more suitable for numerical resolution. An hybrid
finite volume and finite difference splitting approach is then proposed. The
hyperbolic part of the equations is handled with a high-order finite volume
scheme allowing for breaking waves and dry areas. The dispersive part is
treated with a classical finite difference approach. Extensive numerical
validations are then performed in one horizontal dimension, relying both on
analytical solutions and experimental data. The results show that our approach
gives a good account of all the processes of wave transformation in coastal
areas: shoaling, wave breaking and run-up
On the multi-symplectic structure of the Serre-Green-Naghdi equations
In this short note, we present a multi-symplectic structure of the
Serre-Green-Naghdi (SGN) equations modelling nonlinear long surface waves in
shallow water. This multi-symplectic structure allow the use of efficient
finite difference or pseudo-spectral numerical schemes preserving exactly the
multi-symplectic form at the discrete level.Comment: 10 pages, 1 figure, 30 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
A modified Galerkin/finite element method for the numerical solution of the Serre-Green-Naghdi system
A new modified Galerkin / Finite Element Method is proposed for the numerical
solution of the fully nonlinear shallow water wave equations. The new numerical
method allows the use of low-order Lagrange finite element spaces, despite the
fact that the system contains third order spatial partial derivatives for the
depth averaged velocity of the fluid. After studying the efficacy and the
conservation properties of the new numerical method, we proceed with the
validation of the new numerical model and boundary conditions by comparing the
numerical solutions with laboratory experiments and with available theoretical
asymptotic results
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