Finite volume schemes for dispersive wave propagation and runup

Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems. We focus mainly on the application of the method to bidirectional nonlinear, dispersive wave propagation in one space dimension. Special emphasis is given to important nonlinear phenomena such as solitary waves interactions, dispersive shock wave formation and the runup of breaking and non-breaking long waves.Comment: 41 pafes, 20 figures. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

Modeling Nonlinear Dispersive Water Waves

An expository review is given on various theories of modeling weakly to strongly nonlinear, dispersive, time-evolving, three-dimensional gravity-capillary waves on a layer of water. It is based on a new model that allows the nonlinear and dispersive effects to operate to the same full extent as in the Euler equations. Its relationships with some existing models are discussed. Various interesting phenomena will be illustrated with applications of these models and with an exposition on the salient features of nonlinear waves in wave-wave interactions and the related processes of transport of mass and energy

Derivation of asymptotic two-dimensional time-dependent equations for ocean wave propagation

A general method for the derivation of asymptotic nonlinear shallow water and deep water models is presented. Starting from a general dimensionless version of the water-wave equations, we reduce the problem to a system of two equations on the surface elevation and the velocity potential at the free surface. These equations involve a Dirichlet-Neumann operator and we show that all the asymptotic models can be recovered by a simple asymptotic expansion of this operator, in function of the shallowness parameter (shallow water limit) or the steepness parameter (deep water limit). Based on this method, a new two-dimensional fully dispersive model for small wave steepness is also derived, which extends to uneven bottom the approach developed by Matsuno \cite{matsuno3} and Choi \cite{choi}. This model is still valid in shallow water but with less precision than what can be achieved with Green-Naghdi model, when fully nonlinear waves are considered. The combination, or the coupling, of the new fully dispersive equations with the fully nonlinear shallow water Green-Naghdi equations represents a relevant model for describing ocean wave propagation from deep to shallow waters

A numerical model for the simulation of a solitary wave in a coastal region

In this paper we propose a numerical model for the simulation of the tsunami wave propagation on coastal region. The model can simulate the wave transformation due to refraction, shoaling, diffraction and breaking phenomena that take place in the surf zone and can simulate the wet front progress on the mainland. The above mentioned model is based on the numerical integration of the Fully Non-linear Boussinesq Equations in the deep water region and of the Non-linear Shallow Water Equations in the surf zone. These equations are expressed in an integral contravariant formulation and are integrated on generalized curvilinear boundary conforming grid that can reproduce the complex morphology of the coast line. The numerical integration of the model equations is implemented by a high order Upwind WENO numerical scheme that involves an exact Riemann Solver. For the simulation of the wet front progress on the dry bed, the exact solution of the Riemann problem for the wet-dry front is used. The capacity of the proposed model to simulate the wet front progress velocity is tested by numerical reproducing the dam-break problem on a dry bed. The capacity of the proposed model to correctly simulate the tsunami wave evolution and propagation on the coastal region is tested by numerical reproducing a benchmark test case about the tsunami wave propagation on a conic island

Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of tsunami waves

Considered here are Boussinesq systems of equations of surface water wave theory over a variable bottom. A simplified such Boussinesq system is derived and solved numerically by the standard Galerkin-finite element method. We study by numerical means the generation of tsunami waves due to bottom deformation and we compare the results with analytical solutions of the linearized Euler equations. Moreover, we study tsunami wave propagation in the case of the Java 2006 event, comparing the results of the Boussinesq model with those produced by the finite difference code MOST, that solves the shallow water wave equations

Wave-structure interaction for long wave models in the presence of a freely moving body on the bottom

In this paper we address a particular fluid-solid interaction problem in which the solid object is lying at the bottom of a layer of fluid and moves under the forces created by waves travelling on the surface of this layer. More precisely, we consider the water waves problem in a fluid of fixed depth with a flat bottom topography and with an object lying on the bottom, allowed to move horizontally under the pressure forces created by the waves. After establishing the physical setting of the problem, namely the dynamics of the fluid and the mechanics of the solid motion, as well as analyzing the nature of the coupling, we examine in detail two particular shallow water regimes: the case of the (nonlinear) Saint-Venant system, and the (weakly nonlinear) Boussinesq system. We prove an existence and uniqueness theorem for the coupled system in both cases. Using the particular structure of the coupling terms we are able to go beyond the standard scale for the existence time of solutions to the Boussinesq system with a moving bottom.Comment: 37 pages, 1 imag

Unsteady undular bores in fully nonlinear shallow-water theory

We consider unsteady undular bores for a pair of coupled equations of Boussinesq-type which contain the familiar fully nonlinear dissipationless shallow-water dynamics and the leading-order fully nonlinear dispersive terms. This system contains one horizontal space dimension and time and can be systematically derived from the full Euler equations for irrotational flows with a free surface using a standard long-wave asymptotic expansion. In this context the system was first derived by Su and Gardner. It coincides with the one-dimensional flat-bottom reduction of the Green-Naghdi system and, additionally, has recently found a number of fluid dynamics applications other than the present context of shallow-water gravity waves. We then use the Whitham modulation theory for a one-phase periodic travelling wave to obtain an asymptotic analytical description of an undular bore in the Su-Gardner system for a full range of "depth" ratios across the bore. The positions of the leading and trailing edges of the undular bore and the amplitude of the leading solitary wave of the bore are found as functions of this "depth ratio". The formation of a partial undular bore with a rapidly-varying finite-amplitude trailing wave front is predicted for ``depth ratios'' across the bore exceeding 1.43. The analytical results from the modulation theory are shown to be in excellent agreement with full numerical solutions for the development of an undular bore in the Su-Gardner system.Comment: Revised version accepted for publication in Phys. Fluids, 51 pages, 9 figure