Research On High-Resolution Finite-Difference Methods For The Shallow Water Equations

The impact of highway traffic for environment is mainly induced by vehicle emissions on atmospheric and pollution traffic accidents for water quality of a river. A model for coupling the shallow water equations with transport equation can be used to simulate the transport of toxic and hazardous materials in water regions and predict water quality caused by traffic pollution accidents on freeway. It can provide the guidance for dealing with traffic pollution accidents. The shallow water equations have important applications in hydraulic, coastal ocean and environmental engineering. A key and difficult problem of numerical methods for such equations is that they can resolve discontinuities. The high-resolution numerical methods are designed for problems containing discontinuities. This thesis concerns with the high-resolution finite-difference methods for the shallow water equations. The application of the present method to dealing with traffic pollution accidents is discussed. The most distinctive parts can be described as following:1. A fourth-order relaxation scheme for the shallow water equations is proposed. The scheme is based on central weighted essentially non-oscillatory(WENO) reconstruction for one space dimension. In the two-dimensional cases, this reconstruction is generalized by dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver R0CK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The presented scheme is tested on a variety of numerical experiments with one-dimensional Euler equations of gas dynamics, the two-dimensional Burgers equation and the shallow water equation in both one and two dimensions. To illustrate the improvement of our method, the results are compared with numerical solutions computed by the third-order relaxation scheme. The numerical experiments demonstrate that the present method has the higher shock resolution and smaller numerical dissipation than the third-order relaxation scheme.2. A new third-order relaxation scheme is proposed. The scheme combines with third-order WENO reconstruction for spatial discretization and third-order implicit-explicit method for time discretization. The new scheme is much simpler and less computationally expensive than the original one. The new scheme is test on...