28 research outputs found
Finite volume methods for unidirectional dispersive wave model
We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular, we consider a KdVâBBM-type equation. Explicit and implicitâexplicit RungeâKutta-type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariantsâ conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves, and their various interaction
Finite volume methods for unidirectional dispersive wave models
We extend the framework of the finite volume method to dispersive
unidirectional water wave propagation in one space dimension. In particular we
consider a KdV-BBM type equation. Explicit and IMEX Runge-Kutta type methods
are used for time discretizations. The fully discrete schemes are validated by
direct comparisons to analytic solutions. Invariants conservation properties
are also studied. Main applications include important nonlinear phenomena such
as dispersive shock wave formation, solitary waves and their various
interactions.Comment: 25 pages, 12 figures, 51 references. Other authors papers can be
downloaded at http://www.lama.univ-savoie.fr/~dutykh
The variational 2D Boussinesq model for wave propagation over a shoal
The Variational Boussinesq Model (VBM) for waves (Klopman et al. 2010) is based on the Hamiltonian structure of gravity surface waves. In its approximation, the fluid potential in the kinetic energy is approximated by the sum of its value at the free surface and a linear combination of vertical profiles with horizontal spatially dependent functions as coefficients. The vertical profiles are chosen a priori and determine completely the dispersive property of the model. For coastal applications, the 1D version of the model has been implemented in Finite Element with piecewise linear basis functions and has been compared with experiments from MARIN hydrodynamic laboratory for focusing wave group running above a flat bottom (Lakturov et al. 2011) and for irregular waves running above a sloping bottom (Adytia and Groesen 2011). The 2D version of the model has been derived and implemented using a pseudo-spectral method with a rectangular grid in Klopman et al. 2007, 2010. A limitation of the later implementation is a lack of flexibility when the model deals with a complicated domain such as a harbour. Here, we will present an implementation of the model in 2D Finite Element which consistent with the derivation of the model via the variational formulation. To illustrate the accuracy of wave refraction and diffraction over a complex bathymetry, the experiment of Berkhoff et al. 1982 is used to compare the FE results with measurements
A spectral ansatz for the long-time homogenization of the wave equation
Consider the wave equation with heterogeneous coefficients in the
homogenization regime. At large times, the wave interacts in a nontrivial way
with the heterogeneities, giving rise to effective dispersive effects. The main
achievement of the present work is the introduction of a new ansatz for the
long-time two-scale expansion inspired by spectral analysis. Based on this
spectral ansatz, we extend and refine all previous results in the field,
proving homogenization up to optimal timescales with optimal error estimates,
and covering all the standard sets of assumptions on heterogeneities (both
periodic and stationary random settings).Comment: 60 page
{The New Generalized of exp(-a(b)) Expansion Method And its Application To Some Complex Nonlinear Partial Differential Equations
In this article, the generalized expansion method has been successfully implemented to seek traveling wave solutions of the Eckaus equation and the nonlinear Schr\"{o}dinger equation. The result reveals that the method together with the new ordinary differential equation is a very influential and effective tool for solving nonlinear partial differential equations in mathematical physics and engineering. The obtained solutions have been articulated by the hyperbolic functions, trigonometric functions and rational functions with arbitrary constants
Model Coupling for Environmental Flows, with Applications in Hydrology and Coastal Hydrodynamics.
The aim of this paper is to present an overview of âmodel couplingâ methods and issues in the area of environmental hydrodynamics, particularly coastal hydrodynamics and surface/subsurface hydrology. To this end, we will examine specific coupled phenomena in order to illustrate coupling hypotheses and methods, and to gain new
insights from analyses of modelling results in comparison with experiments. Although this is to some extent a review of recent works, nevertheless, some of the methods and results discussed here were not published before, and some of the analyses are new. Moreover, this study is part of a more general framework concerning various types of environmental interactions, such as: interactions between soil water flow (above the water table) and groundwater flow (below the water table); interactions between surface and subsurface waters in fluvial environments (streams, floodplains); interactions
between coastal flow processes and porous structures (e.g. seaâdriven oscillations and waves through sand beach or a porous dike); feedback effects of flow systems on the geoâenvironmental media. This paper starts with a general review of conceptual coupling approaches, after which we present specific modelling and coupling methods for dealing with hydrological flows with surface water / groundwater interactions, and with coastal flows involving the propagation of
seawater oscillations through a porous beach (vertically and horizontally). The following topics are treated. (1) Coupled
streamâaquifer plane flow in an alluvial river valley (quasiâsteady seasonal flow regime), assuming aquifer/stream continuity,
and using in situ piezometric measurements for comparisons. (2) Water table oscillations induced by sea waves, and propagating through the beach in the crossâshore direction: this phenomenon is studied numerically and experimentally using a wave canal with an inclined beach equipped with capacitive microâpiezometers. (3) Tidally driven vertical oscillations of water flow and capillary pressure in a partially saturated / unsaturated sand beach column, studied numerically and experimentally via a âtide machineâ contraption (described in some detail): the goal is to apprehend the role of
capillary effects, and forcing frequency, on the hydraulic response of a beach column forced by tides from below. At the time of this writing, some of the results from the tide machine are being reinterpreted (ongoing work). We also point out a recent study of vertical flow in the beach, which focuses on the effect of intermittent waves in the swash zone, rather than tidal oscillations
Introduction to the special issue on numerical methods and applications for waves in coastal environments
International audienc
New exact traveling wave solutions for the KleinâGordonâZakharov equations
AbstractBased on the extended hyperbolic functions method, we obtain the multiple exact explicit solutions of the KleinâGordonâZakharov equations. The solutions obtained in this paper include (a) the solitary wave solutions of bell-type for u and n, (b) the solitary wave solutions of kink-type for u and bell-type for n, (c) the solitary wave solutions of a compound of the bell-type and the kink-type for u and n, (d) the singular traveling wave solutions, (e) periodic traveling wave solutions of triangle function types, and solitary wave solutions of rational function types. We not only rederive all known solutions of the KleinâGordonâZakharov equations in a systematic way but also obtain several entirely new and more general solutions. The variety of structures of the exact solutions of the KleinâGordonâZakharov equations is illustrated