In this paper, we have developed numerical calculation of stable solutions for the quasilinear hyperbolic system of Saint-Venant equations, which describes the motion of unsteady river fiows. Carrying out numerical experiments, we took as an example a rectangular channel with a constant coefficient of friction, the slope is not constant. When the slope C is a constant, the steady states are uniform. This particular case was studied in [1] for the case of a linear system. Here, we have used a different approach, namely we applied the difference splitting scheme which is developed in [2] for the case of the slope is not constant. The obtained result, based on the stability theorem [3], allows us to assert the exponential stability of the numerical solution of the system of Saint-Venant equations. We considered how the L2-norm will behave in various cases, for example, when the CFL condition is fulfilled and the boundary conditions are fulfilled, also in the case when the boundary conditions are not fulfilled or the CFL condition is not fulfilled. For all these cases, visual graphs of the L2-norm are given. During the work, many aspects were implied on how to put the initial conditions and boundary conditions. We have achieved the exponential stability of the numerical solution by selecting the best values of the coefficients of the boundary conditions.