60 research outputs found
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
On the relevance of the dam break problem in the context of nonlinear shallow water equations
The classical dam break problem has become the de facto standard in
validating the Nonlinear Shallow Water Equations (NSWE) solvers. Moreover, the
NSWE are widely used for flooding simulations. While applied mathematics
community is essentially focused on developing new numerical schemes, we tried
to examine the validity of the mathematical model under consideration. The main
purpose of this study is to check the pertinence of the NSWE for flooding
processes. From the mathematical point of view, the answer is not obvious since
all derivation procedures assumes the total water depth positivity. We
performed a comparison between the two-fluid Navier-Stokes simulations and the
NSWE solved analytically and numerically. Several conclusions are drawn out and
perspectives for future research are outlined.Comment: 20 pages, 15 figures. Accepted to Discrete and Continuous Dynamical
Systems. Other author's papers can be downloaded at
http://www.lama.univ-savoie.fr/~dutyk
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
Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics
For surface gravity waves propagating in shallow water, we propose a variant
of the fully nonlinear Serre-Green-Naghdi equations involving a free parameter
that can be chosen to improve the dispersion properties. The novelty here
consists in the fact that the new model conserves the energy, contrary to other
modified Serre's equations found in the literature. Numerical comparisons with
the Euler equations show that the new model is substantially more accurate than
the classical Serre equations, specially for long time simulations and for
large amplitudes.Comment: 24 pages, 4 figures, 41 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Boussinesq Systems of Bona-Smith Type on Plane Domains: Theory and Numerical Analysis
We consider a class of Boussinesq systems of Bona-Smith type in two space
dimensions approximating surface wave flows modelled by the three-dimensional
Euler equations. We show that various initial-boundary-value problems for these
systems, posed on a bounded plane domain are well posed locally in time. In the
case of reflective boundary conditions, the systems are discretized by a
modified Galerkin method which is proved to converge in at an optimal
rate. Numerical experiments are presented with the aim of simulating
two-dimensional surface waves in complex plane domains with a variety of
initial and boundary conditions, and comparing numerical solutions of
Bona-Smith systems with analogous solutions of the BBM-BBM system
Extended water wave systems of Boussinesq equations on a finite interval: Theory and numerical analysis
Considered here is a class of Boussinesq systems of Nwogu type. Such systems
describe propagation of nonlinear and dispersive water waves of significant
interest such as solitary and tsunami waves. The initial-boundary value problem
on a finite interval for this family of systems is studied both theoretically
and numerically. First, the linearization of a certain generalized Nwogu system
is solved analytically via the unified transform of Fokas. The corresponding
analysis reveals two types of admissible boundary conditions, thereby
suggesting appropriate boundary conditions for the nonlinear Nwogu system on a
finite interval. Then, well-posedness is established, both in the weak and in
the classical sense, for a regularized Nwogu system in the context of an
initial-boundary value problem that describes the dynamics of water waves in a
basin with wall-boundary conditions. In addition, a new modified Galerkin
method is suggested for the numerical discretization of this regularized system
in time, and its convergence is proved along with optimal error estimates.
Finally, numerical experiments illustrating the effect of the boundary
conditions on the reflection of solitary waves by a vertical wall are also
provided
Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling
In the present article we describe a few simple and efficient finite volume
type schemes on moving grids in one spatial dimension combined with appropriate
predictor-corrector method to achieve higher resolution. The underlying finite
volume scheme is conservative and it is accurate up to the second order in
space. The main novelty consists in the motion of the grid. This new dynamic
aspect can be used to resolve better the areas with large solution gradients or
any other special features. No interpolation procedure is employed, thus
unnecessary solution smearing is avoided, and therefore, our method enjoys
excellent conservation properties. The resulting grid is completely
redistributed according the choice of the so-called monitor function. Several
more or less universal choices of the monitor function are provided. Finally,
the performance of the proposed algorithm is illustrated on several examples
stemming from the simple linear advection to the simulation of complex shallow
water waves. The exact well-balanced property is proven. We believe that the
techniques described in our paper can be beneficially used to model tsunami
wave propagation and run-up.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to
Geosciences. Other author's papers can be downloaded at
http://www.denys-dutykh.com
Dispersive waves generated by an underwater landslide
In this work we study the generation of water waves by an underwater sliding
mass. The wave dynamics are assumed to fell into the shallow water regime.
However, the characteristic wavelength of the free surface motion is generally
smaller than in geophysically generated tsunamis. Thus, dispersive effects need
to be taken into account. In the present study the fluid layer is modeled by
the Peregrine system modified appropriately and written in conservative
variables. The landslide is assumed to be a quasi-deformable body of mass whose
trajectory is completely determined by its barycenter motion. A differential
equation modeling the landslide motion along a curvilinear bottom is obtained
by projecting all the forces acting on the submerged body onto a local moving
coordinate system. One of the main novelties of our approach consists in taking
into account curvature effects of the sea bed.Comment: 12 pages; 5 figures. Other author papers can be downloaded at
http://www.lama.univ-savoie.fr/~dutykh
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