Abstract
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.
Highlights
It is known that the behavior of water in rivers, lakes, oceans, as well as in small reservoirs is described by the Saint-Venant equations
The aim of this study is to develop an effective algorithm for finding a numerical solution using an implicit upwind difference scheme for a linear Saint-Venant equations system in the general case
An implicit upwind difference splitting scheme is constructed for the numerical solution of a mixed problem for a linear system of the Saint-Venant equation
Summary
It is known that the behavior of water in rivers, lakes, oceans, as well as in small reservoirs is described by the Saint-Venant equations ( called the shallow water equation in the literature). The application of the Saint-Venant equation to the solution of various practical problems has a rich history. The main difficulties in constructing a numerical approximation of the Saint-Venant equation are associated with obtaining a stable difference solution to the problem. 4/4 ( 112 ) 2021 the stability of the Lyapunov numerical solution (exponential stability) of difference schemes is acute This question is practically not investigated for hyperbolic systems. The task of improving and developing effective algorithms for mathematical modeling of flows in the Saint-Venant approximation is urgent. Great progress has been achieved in studying the general properties and patterns of shallow water modeling in systems with open canals and reservoirs. The task of developing algorithms for mathematical modeling of flows in the Saint-Venant approximation is urgent
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