RECENT DEVELOPMENTS IN THE NUMERICAL-SIMULATION OF SHALLOW-WATER EQUATIONS .1. BOUNDARY-CONDITIONS

Shallow water equations (briefly, SWE) provide a model to describe flu id dynamical processes of various nature, and find therefore widesprea d application in science and engineering. A rigorous mathematical anal ysis is not available, unless for few specific cases under strict assu mptions on the problem's data. In particular, the issue of which kind of boundary conditions are allowed is not completely understood yet. H ere we investigate several sets of boundary conditions of physical int erest that are admissible from the mathematical viewpoint. By that we mean that, when plugged into the integral form of SWE, these boundary conditions allow the proof of a priori estimates for the unknowns of p hysical interest: the velocity field and the elevation on the fluid (o r its pressure). In our investigation we consider the most general cas e in which the physical boundary is partitioned into two sets: one clo sed (this is typically a coast or a shore), the other open (this is a virtual boundary delimiting the domain of investigation). In the latte r we further distinguish among inflow and outflow boundary. Several ki nds of conditions are investigated on each boundary component. The pap er is concluded showing how to achieve a priori estimates correspondin g to three different choices of boundary conditions. The correct treat ment of boundary terms is crucial for both mathematical and numerical analysis of SWE. The characterization of the set of boundary condition s of physical interest that are mathematically admissible is important in view of the numerical simulation of this kind of phenomena. This p aper is the first part of an investigation that the authors have carri ed out in this field. A second one shows how to implement these bounda ry conditions in the framework of discrete methods based on a finite e lement approximation in space, and several kind of time-marching techn iques [11]. In particular, the a priori estimates obtained throughout this paper are extended in order to show stability properties for the approximate solution. Numerical experiments based on test cases corres ponding to the various sets of boundary conditions considered here are presented in [10,12].