A HIGH RESOLUTION FINITE VOLUME METHOD FOR SOLVING SHALLOW W(2)
2015-12-08 01:08
导读:(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 manning’
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(1a) where
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(1b) where h is water depth,
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are the discharges per unit width, bottom slopes and friction slopes along x- and y- directions respectively. the friction slopes
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and
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are determined by manning’s formula
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(2) in which n is manning roughness coefficient.
fig. 1. geometric and topological relationship between elements
fig. 2 relationship between elements on land boundaries 3. geometrical and topological relationships of elements the second-order tvd schemes belong to five-point finite difference scheme and the unsolved variables are node-node arrangement. in order to extend them to the finite volume method, it is necessary to define the control volume. the types of traditional control volume have element itself, such as triangle, quadrilateral and other polygons or some kinds of combinations, and polygons made up of the barycenters from the adjacent elements. in this paper we consider that a node corresponds to an element and the middle states between two conjunction nodes correspond to the interface states of public side between two conjunction elements. a new geometrical and topological relationship is presented for convenience to describe and utilize the tvd scheme. an arbitrary quadrilateral element is defined as a main element and the eight elements surrounding this main element are named as satellitic elements. if the number of all the elements and nodes is known, the topological relations between the main elements and the satellite ones can be predetermined (see ref.[10] in detail). then the numerical fluxes of all the sides of the main element can be determined. the relationships between the main and the satellite elements are shown in figure 1. however , the elements on land boundaries have only six satellite ones shown in figure 2. 1. finite volume tvd scheme for the element
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, the integral form of equation (1a) for the inner region
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and the boundary
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can be written as
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(3)where a represents the area of the region