Numerical Scheme for Solution of an Approximation of Saint-Venant Equation

In the work [4] the Saint-Venant's one-dimensional differential model and kinematic approximation describing the free movement of water in an open channel is under consideration. This model is widely used for the calculation of the flow of rivers in the various fields of engineering. Quality of engineering projects depends on the accuracy and speed of the mathematical calculations. This paper presents a promising method for the numerical solution of one approximation of the Saint-Venant's model.


Introduction
In [4] we derived kinematic approximation of the Saint-Venant's model described by the equation: (1) also one particular solution was founded:

Ilya Starodumov
Using these results we construct a numerical scheme to solve the kinematic equation (1).

Physical model
We represent the linear portion of the river in the form of a chain of open water tanks with a free surface; throughout experiencing the same atmospheric pressure (see Fig. 1).Each reservoir i has a finite length .Each tank has two inlet ports in the left side -through which water is pumped into the tank, and one outlet opening in the wall of the rightthrough which water is pumped from the tank.The outlet of any reservoir i , except the last, through the storage tank is connected to one of the tank inputs 1 i  (in Fig. 6 the one upper), a pump mounted between the tanks can pump the water from i -th reservoir to the storage and from the storage to the 1 i  -th reservoir in any given mode.
Modes of operation of pumps -the pump, pumping water through the upper inlet of the first tank; the pump, pumping water through the tank drives; pumps, pumping water into the tanks through their lower inputs, as well as the pump to pump water from the last tankwill consider on the assumption that the conditions for every tank i : 1. Water Movement in a reservoir is described by the equation (9) and formulas (7), (8); 2. At moment of time j t tank i fills to the level ij h .It's enough to, before the moment j t at point i x (at the left wall) the following condition are reasonable for a long time , where the constant intensity of inflow (consumption) water in the tank i until the , where the constant of the intensity of the inflow into the tank i from moment

After finding the function  
, If we know function   , h x t , we can find


its average amount within the limits for every tank i at the moment 1 j t  , corresponding to steady state, similar to described above.We came to the situation for 1 j t  , a similar for the situation at the moment j t … By such repeating the paragraphs.3-5, we can determine the hydraulic condition of the river at any arbitrarily large time.

Numerical scheme
D -is rectangular area of the independent variables x and t , consider that the counting grid is uniform on x , and on t .Each cell of the grid is identifying indices of her left lower node: iindex on x and jindex on t ; the number of intervals on x is assumed to be I, number of intervals on t is equally Ilya Starodumov J ; position of the node in the grid in D will be specified of its spatial coordinate , and the same is for time coordinate jJ  , consider to call the bottom edge is the edge with the index i , the upper edgethe edge with the index 10 i  , left edgethe edge with the index j , right edgethe edge with the index 10 j  .The length of the horizontal edges of the cell is equal to  and does not depend on i , length of the vertical edges is equal to Consider that source   , q x t is centered on the vertical edges of the cells, so that, , where  

Results
Using described above method we have solved several test tasks from [4] and compare the results with analytical solutions.Results on the Fig. 3 and Fig. 4 shows the function   , h x t distribution for each of 10 cell in comparing with pictures given in [4].


Fig. 1 Chain of tanks moment jj tt  .In ii xx  all this time is provided by the free flow condition, which corresponds to the outflow of water in the tank with the intensity

1 iQ
time amount of the flow near the right wall of the tank i .With this constant flow of water through the upper input, we will look for a solution for the tank  -, we obtain the total magnitude of the influx water in the tank 1 i  on the interval from 0 j t  to jj tt  .Repeat steps 3, 4 for the tank 1 i  .Similarly on the time interval from 0 j t  to jj tt  the task can be solved for all tanks.

,
and does not depend on j .Consider that the original data   are positive and constant functions inside each cell   , ij, piecewise constant function in the area D .Values w , Fig. 2 Cases of changes in depth within the cell of the computational grid

Fig. 3 .Fig. 4 .
Fig. 3.  hx depending on the sign of q for constants w , n , 0 S , q