Recent developments in the numerical simulation of shallow water equations I: boundary conditions

Shallow water equations (briefly, SWE) provide a model to describe fluid dynamical processes of various nature, and find therefore widespread application in science and engineering. A rigorous mathematical analysis is not available, unless for few specific cases under strict assumptions on the problem's data. In particular, the issue of which kind of boundary conditions are allowed is not completely understood yet. Here we investigate several sets of boundary conditions of physical interest 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 physical interest: the velocity field and the elevation on the fluid (or its pressure). In our investigation we consider the most general case in which the physical boundary is partitioned into two sets: one closed (this is typically a coast or a shore), the other open (this is a virtual boundary delimiting the domain of investigation). In the latter we further distinguish among inflow and outflow boundary. Several kinds of conditions are investigated on each boundary component. The paper is concluded showing how to achieve a priori estimates corresponding to three different choices of boundary conditions. The correct treatment of boundary terms is crucial for both mathematical and numerical analysis of SWE. The characterization of the set of boundary conditions of physical interest that are mathematically admissible is important in view of the numerical simulation of this kind of phenomena. This paper is the first part of an investigation that the authors have carried out in this field. A second one shows how to implement these boundary conditions in the framework of discrete methods based on a finite element approximation in space, and several kind of time-marching techniques [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 corresponding to the various sets of boundary conditions considered here are presented in [10,12]. © 1994