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High Order Accuracy Finite Volume Methods For 2D Shallow Water Equation

Posted on:2007-06-25Degree:MasterType:Thesis
Country:ChinaCandidate:H J ZhuFull Text:PDF
GTID:2120360215470274Subject:Mathematics
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In the present work, the numerical simulation on shallow water equations has attracted more attention, and has become a hot topic in computational mathematics. With the development of the techniques of making unstructured mesh, the research of finite volume method with high order of accuracy becomes more and more important. In this thesis, we mainly discuss the using of finite volume method with high order accuracy in shallow water equations. The following work has been done:On unstructured triangular mesh,basing on the construction of composite finite volume method, by choosing Lax-Friedrich flux function and Roe's Riemann Solver for different reconstruction functions, we give several finite volume schemes with high order of accuracy for 2D shallow water equations. First, based on the monotone scheme of one order accuracy, a limited linear reconstruction for each variable is made on every triangular mesh, thus we obtain a TVD-type finite volume method with second order of accuracy. Secondly, through constructing linear interpolation for variables on every mesh, we bring forward ENO-type finite volume method with second order of accuracy. Thirdly, In order to get schemes with higher order of accuracy, a simplified WENO-type scheme is advanced by giving a weighted quadratic interpolation on each mesh. In addition, we serve space discretization by considering reconstruction, flux function and integral formulation dividually and time discretization by using TVD Runge-Kutta method.We analyze partial dam break problem in the following aspect: treating of boundary condition, construction of dummy mesh, and the possibility of stencil to be not admissible and who to prevent. With the schemes proposed here, we make numerical simulation about partial dam break on two kinds of unstructured grid (regular grid and irregular grid), and compare the results of different schemes. The numerical results show that the proposed schemes are accurate and of high resolution, especially, WENO-type finite volume method which has better ability to capture discontinuities. At last, we make numerical simulation about partial dam break with the varying bottom on irregular grid, which means that the proposed schemes can solve problems with arbitrary geometry.
Keywords/Search Tags:Shallow water equations, Unstructured mesh, TVD-type finite volume method, ENO-type finite volume method, WENO-type finite volume method
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