Mathematics and Computers in Simulation 76 (2007) 155–160
Simulation of an oil slick movement using a shallow water model
B. Di Martino, M. Peybernes
∗
UMR CNRS 6134, Universit´ e de Corse, 20250 Corte, France
Available online 23 January 2007
Abstract
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.
© 2007 IMAC. Published by Elsevier B.V. All rights reserved.
AMS Classification: 35Q30; 65M60; 76D03
Keywords: Shallow water equations; Free boundary problem; Galerkin and ALE discretization
1. Introduction
This paper presents the modelization and the simulation of an oil slick movement on the sea surface. The objective
is to predict the movement of pollutants generated by marine spills and to have a good representation of the surface
occupied by these pollutants. The oil slick is modelized with a shallow water model written under the following form
∂u
∂t
+ (u ·∇)u + g∇h - νu = f
1
, (1)
∂h
∂t
+ div(uh) = f
2
. (2)
where u is the mean horizontal velocity (integrated on the vertical) of the oil slick, h the thickness of the oil slick, g the
gravity coefficient and ν is the viscosity coefficient. f
1
and f
2
are, respectively, the external forcing (for example, wind
stress, sea surface current stress, sea surface inclination) and a source or a vanishing of oil (for example, by evaporation
or dissolution).
This system is studied in a two-dimensional time dependent domain
t
, with a boundary Ŵ
t
. Physically, this domain
represents the surface occupied by the oil slick at time t. The displacement d of the boundary Ŵ
t
is given by the relation
∂d
∂t
= u on Ŵ
t
.
∗
Corresponding author. Tel.: +33 4 95 45 00 41.
E-mail address: peybernes@univ-corse.fr (M. Peybernes).
0378-4754/$32.00 © 2007 IMAC. Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.matcom.2007.01.022