Numerical stability of implicit four-point scheme applied to inverse linear flow routing

Abstract

In this paper an analysis of stability and accuracy for numerical solution of the inverse problem for the Saint Venant equations is presented. For inverse flow routing a four-point implicit difference scheme with two weighting parameters θ and ψ is used. This is the same scheme which with ψ = 1 2 and θ ⩾ 1 2 is used as the Preissmann scheme for the conventional solution of the Saint Venant equations. The numerical experiments presented by Szymkiewicz ( J. Hydrol., 147: 105–120, 1993) showed that the numerical stability of an inverse flow routing requires different values of parameters ψ and θ than for a conventional solution. This problem can be explained by Fourier analysis carried out for a linear hyperbolic system. From this analysis it is found that the four-point implicit scheme applied to inverse flow routing is unconditionally stable if ψ ⩾ 1 2 and θ ⩽ 1 2 . These conditions of stability are the inverse of the conventional solution.