Stability and Accuracy of Weighted Four-Point Implicit Finite Difference Schemes for Open Channel Flow

Abstract

The unsteady motion of streams, with a free surface, has been described by a system of equations that were proposed by Saint-Venant in 1871. These are universally known as “Saint-Venant’s equations.” Weighted four-point implicit finite difference schemes are largely used for their numerical solution in the case of one-dimensional flow. For these models, stability, dissipation, and dispersion are first investigated by looking at the truncation error and then by using Fourier’s classic linear analysis. Variations of the space and time weighting coefficients, the Courant number, the Froude number, and the frictional resistance term are examined in this paper. In particular, instabilities are analyzed that are due to the progressive accumulation of dispersion errors, which are the consequence of an increasing frictional resistance term. However, in this case the numerical scheme requires a dissipation mechanism. Computational suggestions are given for this mechanism.