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Calculation of the matter propagation in the river or open channel system

In this paper we consider the following problems: The existence of solution the stability of finite difference scheme and the non-negative property of numerical solution. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 23, 2001, No 1 (39 - 50) CALCULATION OF THE MATTER PROPAGATION IN THE RIVER OR OPEN CHANNEL SYSTEM TRAN GIA LICH Institute of Mathematics, NCST of Vietnam ABSTRACT. In this paper we consider the following problems: The existence of solution the stability of finite difference scheme and the non-negative property of numerical solution'. Introduction The differential equation describing diffusion process of matter S in a river or open channel is as follows: as+ v as ox fJt _.!:_~ (wPas) + bS = ¢ w ox ox (0.1) ' where x is the coordinate along the streamed, t - the time, v - the average velocity, w - the area of cross section, P 2::: 80 > 0 - the general diffusion coefficient, b 2::: 0 - the decay coefficient. . Equation (0.1) is of parabolf~ type. In order to solve this boundary problem, beside the initial condition at the time t = 0: S(x, 0) = S0 (x), one more boundary condition is needed at every boundary S(Li , t) = SL; (t) , (L 1 = x 1 = 0, L2 = xN = L). At the inflow boundary Li the boundary condition will be given by S(Li, t) = fi(t) 2::: 0. At the outflow boundary we will consider that S(Li, t) is resulted from the transport process or ~SI = 0. uX L; In the river or open channel system (see fig. 1) beside the boundary conditions at the boundary nodes A, B, C, it is necessary to give the adjoint conditions at the internal nodes D, E, F . These adjoint conditions are resulted from the law of matter conservati9rt :-a.nd on the supposion that there are no source, no creation, no decay of matter S at the internal nodes (see [5]). · · ata j SidV- + j SivndS-+ j Pi (as) ox dS-'- = i v s s 39 0, i·' I where ID is the set of branches having common internal node D, Qi - the discharge at the node D of the river branch i , Vn - th~ projection of the velocity vector on the external normal vector of the boundary S, a·= { 1 i -1 if Dis right boundary of river branch i, if D is left boundary of .