Numerical modeling of water flow in an open channel using an explicit difference scheme

The present work is devoted to the numerical calculation of Lyapunov stable solutions to the linear Saint-Venant equations. Calculations are carried out with the help of an explicit upwind difference splitting scheme in terms of lower terms using the example of a channel with a rectangular cross section. In the paper, we carry out a numerical calculation of the discrete Lyapunov function. We study the influence of the Courant-Friedirichs-Levy condition on the discrete Lyapunov function. It is calculated the dependence of the discrete Lyapunov function on the values of the coefficients of the boundary conditions. We consider numerical calculation of the solutions of linear Saint-Venant equations, stable in the sense of Lyapunov, with boundary conditions and initial data with the help of an upwind explicit difference splitting scheme in terms of lower-order terms using the example of an open channel with a rectangular cross section. Moreover we have investigated the proposed explicit upwind difference splitting scheme and obtained some theoretical results. Here we have given only a part of theorems and the necessary definitions and the numerical experiment has been carried out. As a numerical experiment, a channel with a rectangular cross section is considered. The channel width is W=80 meters and domain 1000 meters long with a period of 1 second. The stability conditions of Theorem are verified numerically. In the case, when the conditions of Theorem are satisfied, the graph of the L² -norm of the numerical solution of the initial-boundary-value problem is shown, which exponentially tends to zero, that confirms the reliability Theorem. If at least one of the stability conditions {(1) the Courant-Friedirichs-Levy condition, (2) the condition on the parameters of the boundary conditions} is not satisfied, then the L²-norm of the numerical solution of the initial-boundary-value difference problem tends to infinity, that means the instability of the scheme.