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ts 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 , the integral form of equation (1a) for the inner region and the boundary can be written as (3) where A represents the area of the region , dl denotes the arc length of the boundary , and n is a unit outward vector normal to the boundary . The vector U is assumed constant over an element. Further discretizing (3), the basic equation of the finite volume method can be obtained (4) where is the length of side k, denotes the outer normal flux vector of side k. satisfies (5) F(U) and G(U) have a rotational invariance property, so they satisfy the relation (6) or (7) where represents the angle between unit vector n and the x axis (along the counter-clockwise from the x axis), and denote transformation and inverse transformation matrices respectively (8) Eq. (4) can be rewritten as (9) Let the right terms of above equation be , then (10) Two-step Runge-Kutta method is used to discretize Eq. (10), then the second-order accuracy in time can be obtained (11) The flux at every side of any element (e.g. at the side 1 of element ) can be given through the following form (12) where is the right eigenvector component (l=1,2,3) by Roe's average state between the element and the satellite element 1. A hybrid type form of is used (13) where represents the characteristic speed component by Roe's average state between element a本论文由英语论文网提供整理，提供论文代写，英语论文代写，代写论文，代写英语论文，代写留学生论文，代写英文论文，留学生论文代写相关核心关键词搜索。