B. Di Martino

ON A BI-LAYER SHALLOW WATER MODEL WITH RIGID-LID HYPOTHESIS

In this paper, we present a new model for a bi-layer shallow water problem using the rigid-lid hypothesis. This model follows from the usual bi-layer model and can drastically decrease the computational time of simulation. But some mathematical and numerical difficulties appear. Particularly, we observe in the equations some terms in the form of 1/hi (where hi is the thickness of the layer) and we are not able to prove that hi > 0. To circumvent this difficulty, we replace in these terms hi by Hi > β > 0, where Hi is a characteristic thickness of the layer. This hypothesis is realistic if the fluctuations of hi are small, which is generally the case. Then, we prove existence and regularity results for this approximated problem which shows the convergence of the numerical scheme. Next, we present some comparative results in an idealized configuration between this model and the classical bi-layer shallow water model.

THE NONLINEAR GALERKIN METHOD APPLIED TO SHALLOW WATER EQUATIONS

In this work, we present some numerical approximations for a shallow water problem with a depth-mean velocity formulation and we give, where possible, an error bound. To prove the existence of solutions, we build a sequence of approximated solutions with the Galerkin method for the momentum equation and solve the continuity equation with the method of the characteristics. This leads to an expensive natural numerical scheme. Then, in order to reduce the CPU time, we present other numerical approximations based on the linear or nonlinear Galerkin method.

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).

Resolution by Galerkin method with a special basis of a geophysical flow in open sea: a Calvi's bay simulation

Abstract We present in this work the numerical resolution of a geophysical flow by the Galerkin method. On the one hand, we expose briefly the three-dimensional model proposed by Nihoul and deduce a shallow water model. On the other hand, we give some theoretical results in order to justify the numerical approach presented. Lastly, we describe the different steps of the resolution as well as the numerical tools used. We apply this method to simulate the Calvis bay flow and give some comparison results between depth-average velocity obtained by a shallow-water and a three-dimensional model.

SIMULATION OF A SPILLED OIL SLICK WITH A SHALLOW WATER MODEL WITH FREE BOUNDARY

In this paper we present a new approach to describe the behaviour of a pollutant slick at the sea surface. To this end, we consider that the pollutant and the water are immiscible and we propose a two-layer model where the lower layer corresponds to the water and the upper layer represents the pollutant. Since the dimension of the pollutant slick is generally much smaller than the domain occupied by the sea, we propose to compute the motion of the pollutant with a shallow water model with free boundary only in the domain occupied by the pollutant. To discretize in time the problem with free boundary, we use an ALE formulation coupled with the characteristic method. Then, to solve the space discretized problem, we approximate the pollutant velocity by using a Galerkin method with a special basis which verifies the boundary conditions and simplifies significantly the resolution. Finally we test this work in a real situation: the dam of Calacuccia (Corsica).

Simulation of an oil slick movement using a shallow water model

The behaviour of an oil slick on the sea surface is studied using a shallow water model with free boundary. This model is solved by coupling the Arbitrary Lagrangian Eulerian method (ALE) with the characteristic method and by using a Galerkin spatial discretization. A special basis, which permits to obtain easily the solution by solving a scalar eigenvalue problem, is then proposed. Finally, a numerical experiment is presented in a realistic configuration.

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.

The Generalized Nonlinear Galerkin Method Applied to Hydrodynamics of the Open Sea

In this paper we present new numerical algorithms based on a generalized nonlinear Galerkin method in order to solve coastal and oceanic circulation problems. The equations system is based on the primitive equations of the ocean under Boussinesq and hydrostatic approximations. These equations are transformed using, at the same time, the classical σ transformation and an original homogenization of the boundary conditions. We use a well adapted special basis to apply the usual Galerkin method and the nonlinear Galerkin method. This basis is built on a modelization of the energetic transfers through the different scales of flow. Two approaches are proposed to solve the continuity equation: the (nonlinear) Galerkin method and the method of the characteristics. We present the advantages and drawbacks of both methods.

A numerical method for a blood-artery interaction problem

We present a numerical method for the resolution of a bidimensional blood flow problem and more generally for a fluid flow surrounded by a time dependent domain. Our approach is based on an ALE formulation which is solved using a Galerkin method with an eigenvectors basis set on the initial fixed domain.