Development and implementation of a method for calculating stable solutions of the Saint-Venant equation using an upwind implicit difference scheme

The paper investigates the issue of developing a computational algorithm for finding a numerical solution to the shallow water equation system at a one-dimensional level (the Saint-Venant equation) and its implementation on the example of the Big Almaty Canal (in what follows the BAC). The algorithm is based on the use of an implicit upwind difference scheme. The theorem on exponential stability of the difference scheme is proved. The computational algorithm is based on the well-known matrix sweep method. By means of the proposed computational algorithm, calculations of the water level and velocity on the BAC section (10000 m) located in the Almaty region were carried out. The results of the numerical calculation were compared with real data. The efficiency of the proposed algorithm in time is established. Exponential stability of the numerical solution of the Saint-Venant equation is numerically confirmed.