Catherine Giacomoni

Mathematical and Numerical Analysis of a Tsunami Problem

In this paper, we present a tsunami model based on the displacement of the lithosphere and the mathematical and numerical analysis of this model. More precisely, we give an existence and uniqueness result for a problem which models the flow and formation of waves at the time of a submarine earthquake in the vicinity of the coast. We propose a model which describes the behavior of the fluid using a bi-dimensional shallow-water model by means of a depth-mean velocity formulation. The ocean is coupled to the Earths crust whose movement is assumed to be controlled on a large scale by plate equations. Finally, we give some numerical results showing the formation of a tsunami.

ANALYSIS OF SOME SHALLOW WATER PROBLEMS WITH RIGID-LID HYPOTHESIS

We present in this paper the analysis and the comparison of two two-dimensional geophysical flow problems using a rigid-lid approximation (i.e. we do not take into account the variation of surface elevation ζ). A first rigid-lid shallow water model (noted SWRL) is obtained by neglecting the variation of the surface in a weak formulation of a usual viscous shallow water model in depth-mean velocity formulation (noted SWFS for shallow water with free surface). We establish some existence results for this model and we propose a numerical resolution method. The second model we consider is an adaptation of the lake equations proposed by Levermore, Oliver and Titi,4 in which we take into account the viscous effects, in order to compare the two approaches. For the numerical resolution, we apply the curl operator on these equations and we propose a numerical algorithm to solve this problem, that we note SWLV (shallow water for the lake with viscosity). We present finally some comparative results in an idealized configuration between the SWRL, SWLV and SWFS models (in the case where the rigid-lid approximation seems to be reasonable).

A Numerical Approach to Marine Circulation Using Free Surface or Rigid-lid Modeling: Application in the Bay of Calvi

When establishing a model of fluid flow in marine modeling, a key issue is the choice between a rigid-lid approach or a free surface level model. This is not a trivial issue as it plays an important role, not only in the choice of the numerical techniques, but also in the qualitative and quantitative aspects of the numerical results. Most software use either free surface or rigid-lid hypotheses, but comparing their results is difficult, since the numerical tools used are in general, extremely different. In this work, some numerical investigations comparing rigid-lid and free surface models are presented. A numerical method using Galerkins method, but with a new basis, is applied to solve the rigid-lid equations in a realistic domain with varying bottom. A numerical method, similar to the one already used for free surface equations (the same truncating method and precision level) is applied, where the main differences between the simulation results depend only on the model employed. In addition, a comparative simulation between rigid-lid and free surface models to study marine circulation in the bay of Calvi (Corsica) is presented, and numerical results in the non-stratified case only (fluid with constant density) are described, as no further difficulties appear in the stratified case.

Convergence of a characteristic-Galerkin scheme for a shallow water problem

In this paper, we present a numerical method based on a mixed characteristic-Galerkin (or Lagrangian-Galerkin) scheme to solve a shallow water problem with dirichlet boundary conditions. In a first part, we prove an L^2-bound on the water elevation. This bound is obtained by Lions in the case of a linearized momentum equation ^[^1^] and we extend it to the nonlinear case for which the existence is shown for small data ^[^2^]. This bound allows us to construct solutions as limits of the solutions of a regularized problem and to prove the convergence of the discrete problem towards the continuous. We give a numerical criteria connecting the Lagragian discretization and the number of Galerkin eigenvectors to solve the discrete equations with a fixed-point procedure. We present a few numerical results in the case of a fixed domain showing the coherence of the scheme which seems to be adaptable to a domain depending on time.

Simulation of the spread of a viscous fluid using a bidimensional shallow water model

In this paper we propose a numerical method to solve the Cauchy problem based on the viscous shallow water equations in an horizontally moving domain. More precisely, we are interested in a flooding and drying model, used to modelize the overflow of a river or the intrusion of a tsunami on ground. We use a nonconservative form of the two-dimensional shallow water equations, in eight velocity formulation and we build a numerical approximation, based on the Arbitrary lagrangian eulerian formulation, in order to compute the solution in the moving domain.