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##### A HIGH RESOLUTION FINITE VOLUME METHOD FOR SOLVING SHALLOW WATER EQUATIONS

ТлЮФзїепЃКWang Jia-s ТлЮФЪєадЃКЖЬЮФ essayЕЧГіЪБМфЃК2007-08-06БрМ­ЃКЕуЛїТЪЃК6646

ТлЮФзжЪ§ЃК18530ТлЮФБрКХЃКorg200708061142142607гяжжЃКгЂгяТлЮФ EnglishЕиЧјЃКжаЙњМлИёЃКУтЗбТлЮФ

ABSTRACT: A high-resolution finite volume numerical method for solving the shallow water equations is developed in this paper. In order to extend finite difference TVD scheme to finite volume method, a new geometry and topology of control bodies is defined considering the corresponding the relationships between nodes and elements. This solver is implemented on arbitrary quadrilateral meshes and their satellite elements, and based on a second-order hybrid type TVD scheme in space discretization and a two-step Runge-Kutta method in time discretization. Then it is used to deal with two typical dam-break problems and very satisfactory results are obtained comparing with other numerical solutions. It can be considered as an efficient implement for the computation of shallow water problems, especially concerning those having discontinuities, subcritical and supercritical flows and with complex geometries. KEY WORDS: shallow water equations, finite volume, TVD scheme, dam-break bores 1. INTRODUCTION It is necessary to conduct fluid flow analyses in many areas, such as in environmental and hydraulic Engineering. Numerical method becomes gradually the most important approach. The computation for general shallow water flow problems are successful, but the studies of complex problems, such as having discontinuities, free surface and irregular boundaries are still under development. The analysis of dam-break flows is a very important subject both in science and engineering. For the complex boundaries, the traditional method has usually involved a kind of body-fitted coordinate transformation system, whilst this may make the original equations become more complicated and sometimes the transformation would be difficult. It is naturally desirable to handle arbitrary complex geometries on every control element without having to use coordinate transformations. For the numerical approach, the general methods can be listed as characteristics, implicit and approximate Riemann solver, etc. The TVD finite difference scheme is playing a peculiar role in such studies , but it is very little in finite volume discretization. The traditional TVD schemes have different features in the aspects of constructive form and numerical performance. Some are more dissipative and some are more compressive. Through the numerical studies it is shown that good numerical performance and the complicated flow characteristics, such as the reflection and diffraction of dam-break waves can be demonstrated by using a hybrid type of TVD scheme with a proper limiter. In this paper, such type of scheme is extnded to the 2D shallow water equations. A finite volume method on arbitrary quadrilateral elements is presented to solve shallow water flow problems with complex boundaries and having discontinuities. 2. GOVERNING EQUATIONS The governing equations of shallow water problems can be derived by depth averaging of the Navier-Stokes equations. The conservative form of the shallow water equations is given by (1a) where (1b) where h is water depth, are the discharges per unit width, bottom slopes and friction slopes along x- and y- directions respectively. The friction slopes and are determined by ManningЁЏs formula (2) in which n is Manning roughness coefficient. ЁЁ ЁЁ Fig. 1. Geometric and topological relationship between elemenБОТлЮФгЩгЂгяТлЮФЭјЬсЙЉећРэЃЌЬсЙЉТлЮФДњаДЃЌгЂгяТлЮФДњаДЃЌДњаДТлЮФЃЌДњаДгЂгяТлЮФЃЌДњаДСєбЇЩњТлЮФЃЌДњаДгЂЮФТлЮФЃЌСєбЇЩњТлЮФДњаДЯрЙиКЫаФЙиМќДЪЫбЫїЁЃ ```   Europe ЃЈ24-hoursЃЉ
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