HYPERBOLIC EQUATIONS ANALYSIS FOR HYDRAULIC RISK ASSESS-MENT

Abstract

The shallow water equations describe a thin layer of liquid of constant density in hydrostatic equilibrium, limited from below by the bottom topography and from above by the free surface. They have a rich variety of properties, since they have infinitely many conservation laws. The propagation of a tsunami can be accurately described by the equations of shallow water until the wave approaches the shore. Shallow water equations are often used to model both oceanographic and atmospheric fluid flows. Numerical modeling of two-dimensional shallow currents with complex geometry, including unsteady flows and moving boundaries, has become a challenge for researchers in recent years. There is a wide range of physical situations of interest to the environment, such as flow in open channels and rivers, tsunami and flood simulations, which can be mathematically represented by non-linear systems of equations. Models of such systems make it possible to predict areas that will ultimately suffer from pollution, coastal erosion and melting of polar glaciers. Complex modeling of such phenomena using physical descriptions, such as the Navier-Stokes equations, can often be problematic due to the scale of the modeling areas, as well as the selection of free surfaces. The purpose of this work is to derive and analyze computational hyperbolic partial differential equations, called shallow water equations, which describe non-dispersive waves, often used to simulate physical phenomena in the field of hydrodynamics, introduced into mathematical models for subsequent implementation of the capabilities of computer modeling and forecasting. With the help of these studies, the problem of mathematical modeling of the Kelvin and Rossby waves is solved not only in lakes and rivers, but also in oceans. It is also possible to consider special cases of this phenomenon for pools, and also using the derived equations, it is possible to simulate tides. The article deduces completely nonlinear shallow water equations (and their linearized analogue). These equations are also amenable to vertical sampling.