The problem of numerical determination of Lyapunov-stable (exponential stability) solutions of the Saint-Venant equations system has remained open until now. The authors of this paper previously proposed an implicit upwind difference splitting scheme, but its practical applicability was not indicated there. In this paper, the problem is solved successfully, namely, an algorithm for calculating Lyapunov-stable solutions of the Saint-Venant equations system is developed and implemented using an upwind implicit difference splitting scheme on the example of the Big Almaty Canal (hereinafter BAC). As a result of the proposed algorithm application, it was established that: 1) we were able to perform a computational calculation of the numerical determination problem of the water level and velocity on a part of the BAC (10,000 meters) located in the Almaty region; 2) the numerical values of the water level height and horizontal velocity are consistent with the actual measurements of the parameters of the water flow in the BAC; 3) the proposed computational algorithm is stable; 4) the numerical stationary solution of the system of Saint-Venant equations on the example of the BAC is Lyapunov-stable (exponentially stable); 5) the obtained results (according to the BAC) show the efficiency of the developed algorithm based on an implicit upwind difference scheme according to the calculated time. Since we managed to increase the values of the difference grid time step up to 0.8 for calculating the numerical solution according to the proposed implicit scheme.

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This article is devoted to the construction and study of the exponential stability of an explicit upwind difference scheme for a mixed problem for the linear system of the Saint Venant equation. For the numerical solution of the mixed problem for the linear system of the Saint Venant equation, an explicit upwind difference scheme is constructed. For a numerical solution, a discrete Lyapunov function is constructed and an a priori estimate for it is obtained. On the basis of the discrete Lyapunov function, the exponential stability of the numerical solution of the initial-boundary-value difference problem of the mixed problem for the linear system of the Saint Venantequation is proved. A theorem on the exponential stability of the numerical solution of the initial-boundary-value difference problem is proved. The behavior of the discrete Lyapunov function is numerically investigated depending on the algebraic condition of exponential stability of the numerical solution of the mixed problem. The results of the theorem on the exponential stability of the numerical solution are confirmed by a specific example of an open channel flow problem.