Stability of the Numerical Solution for the Mixed Problem of the Saint-Venant Equations

Abstract

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.