Stability Limits for Preissmann's Scheme

Abstract

The St. Venant equations form the basis of many one‐dimensional river models. The Preissmann box scheme is often used to generate approximate solutions, chiefly because the modeler has considerable flexibility in the choice of space and time steps. This paper contains a comprehensive Fourier stability analysis of the Preissmann scheme applied to the St. Venant equations, including the general form of the friction term. Previous analyses have either neglected or simplified the form of the flow resistance. No such approximations are made in this paper. It shows analytically that the linearized, numerical equations are stable when weighted towards the new time level and when the magnitude of the Vedernikov number is less than one. This second condition is the limit of hydrodynamic stability at which roll waves form and the numerical scheme mirrors this physical stability limit. Difficulties that occur in practical applications of this computational method must have their origin in factors omitted from the Fourier analysis, such as the treatment of boundary conditions and the nonlinear terms in the flow equations.