Are the stationary solution of the system of Saint-Venant equations

1) This presentation is devoted to obtaining an algebraic condition for the exponential stability of the numerical solution of the upwind difference scheme for the mixed problem posed for Saint-Venant equations. Hydrological processes occupy an important place in mathematical modelling and Engineering phenomena. They are used to solve water management problems: assessment of water reserves, distribution and rational use, transformation the functioning of the river system. All these phenomena is related to the unsteady flow of water in the system of rivers and canals, the flow of which is most often described by the hypothetical equations of Saint-Venant.

We consider inhomogeneous case when the characteristic velocities depend on the spatial variable. A priori estimate of the numerical solution of the boundary-value difference problem is obtained. This estimate allows us to state the exponential stability of the numerical solution.

The theorems on exponential stability of the solution of the numerical solution of a boundary-value difference problem are proved. Easily verifiable algebraic conditions of exponential stability are given for solving numerical solution of boundary-value difference problem. Presented numerical examples confirmed theoretical results obtained in the theorems.
2) - are the stationary solution of the system of Saint-Venant equations;

Where - is the water depth, - is the horizontal water velocity are unknown functions of two variables. The slope is the channel bottom slope, g is the constant gravity acceleration and k is a constant friction coefficient.

Conclusion. In conclusion, note that in this presentation we have considered the problem of differential splitting upwind scheme of small members for linear Saint-Venant equations in the general case. For the numerical value of stable solutions of the system of linear San Venant equations, a discrete analogue of the Lyapunov function is constructed. A priori estimate is obtained for the discrete analogue of the Lyapunov function. The obtained a priori estimate allows us to speak about the exponential stability of the numerical solution. A theorem on the exponential stability of the numerical solution of the difference scheme of splitting in lower terms for the system of linear Saint-Venant equations in the general case in the corresponding norms is proved. In this way, this enables us to prove the convergence of a stable numerical solution to a stable solution of the differential problem. Examples of numerical calculations are given, confirming the obtained theoretical results.