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 
and 1; 
denotes the average wave strength component; 
is a limiter. The MUSCL type limiter of Van Leer is used, which has moderate dissipative and compressible performance; 
is a dissipative function put forward by Harten. The definitions of all these variables are given in Ref.[10]. The ratio between time and space is
(14)
where 
denotes the distance of the barycenters between element 
and satellite element 1.
Eqs. (12) and (13) concern four satellite elements around the element 
, but the limiter function concerns another four satellite elements, so this scheme concerns eight satellite elements in all.
5. BOUNDARY CONDITIONS
The boundaries of the computational domain have land boundaries (solid boundaries) and water boundaries (open boundaries) for a general shallow water problem. In the case of solid boundaries, no-slip or slip boundary conditions is considered on the basis of whether considering turbulent viscosity or not. Generally speaking, no-slip boundary conditions are given if considering turbulent viscosity, otherwise slip conditions are specified. The open boundary conditions, however, need to have a particular treatment. The local value of Froude number or whether the flow is subcritical or supercritical is the basis of determining the number of boundary conditions. For supercritical flow, three conditions at the inflow boundary and none at the outflow boundary must specified. For subcritical flow, two external conditions are specified at inflow boundary and one is required at the outflow boundary.
6. APPLICATIONS OF DAM-BREAK COMPUTATION
Through the computation of 1D dam-break waves in a horizontal and frictionless channel and the comparison with Stoker's theoretical solution, it is shown that steep and nonoscillatory numerical solutions could be obtained using the hybrid type of TVD scheme 
. Two typical examples of 2D dam-break problems are solved and discussed by solving the shallow water equations using above finite volume TVD scheme.
6.1 Rectangular Dam-Break
Consider a 2D partial dam-break model with a non-symmetrical breach. It is assumed that in the center of a 200m×200m channel, a partial dam breaking takes place instantaneously. The breach is 75m in length, which has distances of 30m from the left bank and 95m from the right. The initial water height is 10m and 5m respectively. No slope and friction are considered. The results displaying the views of the water surface elevation, contour of the surface elevation and velocity field are shown in Figure3 at time t=7.2s after the dam failure. At the instant of breaking of the dam, water is released through the breach, forming a positive wave propagating downstream and a negative wave spreading upstream. These results agree quite well with the results of using finite difference hybrid type of TVD scheme 
and those in Ref. 
.
Fig. 3(a) Water surface elevation for a rectangular dam-break
Fig. 3(b) Contour of surface elevation for a rectangular dam-break
6.2 Circular Dam-Break
Another typical example is based on the hypothetical test case studied by Alcrudo and Garcia-Navarro [7], which involves the breaking of a circular dam. It is an important test example for the analysis of the algorithm performance and solving a complex shallow water problem. The physical model is that two regions of still water are separated by a cylindrical wall of radius 11m. The water depth inside the dam is 10m, whilst outside the dam is 1m. At the instant of dam failure the circular wall is assumed to be removed completely and no slope and friction is considered, then the circular dam-break waves will spread and propagate radially and symmetrically. The results with above method at time t=0.69s are shown in Figures 4 (a), (b) and (c) which denote the water surface elevation, contour of surface elevation and velocity field respectively. It can be clearly seen that the waves spread uniformly and symmetrically. These results agree quite well with those given by Alcrudo and Garcia-Navarro 
, Zhao et al. 
, Alastansiou and Chan 
and they can be tested each other. It demonstrates that the present method is reliable and fine.
Fig. 3(c) Velocity field for a rectangular dam-break
Fig. 4(a) Water surface elevation for a circular dam-break circular dam-break
Fig. 4(b) Contour of surface elevation for a circular dam-break
Fig. 4(c) Velocity field for a circular dam-break
7. SUMMARY AND CONCLUSIONS
TVD scheme is playing an important role in gas dynamics because of its high accuracy, good shock-capturing ability and nonoscillatory numerical performance. But it is constructed based on finite difference method. In this paper a new geometry and topology is defined for the extension of nodes to elements. With the conservative type of the shallow water equations, a hybrid type second order TVD scheme is applied and two-step Runge –Kutta method is adopted in time, then a finite volume TVD scheme for the shallow water equations on arbitrary quadrilateral elements is developed. The numerical results of two types of dam-break problem show that the method is sufficiently robust and can handle discontinuities and complex flow problems efficiently. The results presented in this paper are in excellent agree with those reported recently and even display sharper discontinuities and the maximum values attenuate more slowly. It can be foreseen that this method has much broader application foreground. As for further studies, such as in the cases of a channel having bend, bifurcation and inner islands, will discuss in another paper.
REFERENCES
1. A. Harten, 1983: High Resolution Schemes for Hyperbolic Conservation Laws, Journal of Computational Physics, 49, 357-393.
2. S. Y. Hu, W. Y. Tan, 1990: Numerical Modeling of Bores due to Dam-Break, Journal of Hydrodynamics, Ser. A., 5(2), 90~98 (in Chinese).
3. J. H. Tao, W. D. Zhang, 1993: The Simulation of One and Two Dimensional Dam-Breaking Waves by TVNI Scheme, Journal of Tian Jin University, (1), 7~15 (in Chinese).
4. J. Y. Yang, C. A. Hsu, and S. H. Chang, 1993: Computations of Free Surface Flows, Part 1: 1-D Dam-Break Flow, Journal of Hydraulic Research, 31(1).
5. J. S. Wang, H. G. Ni, S. Jin and J. C. Li, 1998: Simulation of 1D Dam-Break Flood Wave Routing and Reflection by Using TVD Schemes, Journal of Hydraulic Engineering, (5), 7~11 (in Chinese).
6. J. S. Wang, H. G. Ni, and S. Jin, 1998: A High Accurate Numerical Simulation of the Propagation and Diffraction for 2D Dam-Break Bores, Journal of Hydraulic Engineering, (10), 1~6 (in Chinese).
7. F. Alcrudo, P. Garcia-Navarro, 1993: A High Resolution Godunov-Type Scheme in Finite Volumes for the 2D Shallow Water Equation, International Journal for Numerical Method in Fluids, 16, 489-505 1993.
8. D. H. Zhao, H. W. Shen, J. S. Lai, and G. Q. Tabios Ⅲ, 1996: Approximate Riemann Solvers in FVM for 2D Hydraulic Shock Wave Modeling, Journal of Hydraulic Engineering, 692-702.
9. K. Alastansiou, C. T. Chan, 1997: Solution of the 2D Shallow Water Equations Using the Finite Volume Method on Unstructured Triangular Meshes, International Journal for Numerical Method in Fluids, 24, 1225-1245.
10. J. S. Wang, 1998: A Study of Numerical Simulation of Dam-Break Bores by Applying TVD Schemes, Ph.D. Thesis, Dalian University of Technology (in Chinese).
is playing a peculiar role in such studies 
, but it is very little in finite volume discretization.
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. 
(1a)
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’s formula
