Nonlinear effects in the propagation properties of numerical schemes employed in computational hydraulics and computational hydrology

Abstract

Fourier analysis is a powerful tool for the determination of the propagation properties of numerical schemes employed in computational hydraulics and computational hydrology. Nevertheless, its use is restricted to cases that involve linear and constant coefficient equations, subject to periodic boundary conditions, or to linear and constant coefficient pure initial value problems occurring in infinite spatial domains. Most equations that describe hydraulic and hydrologic phenomena are nonlinear and, therefore, such limitation prevents the rigorous application of Fourier analysis and may even lead to erroneous conclusions when a so-called “frozen coefficients approach” is used. The purpose of this paper is to present a general methodology that removes the above described limitation. In order to accomplish this task, the differential or difference operator of the evolution equation or system of equations, are expanded in a Taylor–Fréchet series around a reference solution, which is assumed to be slowly varying. When only the zeroth and first order terms in the series are kept, the resulting equation is linear in a perturbation function, even though it depends nonlinearly on the reference solution, thereby capturing the dominant nonlinear behavior of the system under analysis. The ensuing step consists of performing a local analysis of the resulting equation or system of equations. Such local analysis is asymptotic in a conveniently defined small parameter. The zeroth order asymptotic approximation of the solution is governed by a linear, pure initial value problem with constant coefficients. Thus, Fourier analysis is applicable to such problem. Examples drawn from computational hydrology, such as the stability and iteration convergence of schemes employed for the solution of Richards equation, that describes the motion of water in partially saturated porous media; and from computational hydraulics, such as the stability of the generalized Preissman scheme, and the consistency of the generalized wave continuity equation formulation of the shallow water equations, are included, showing the advantages of the proposed methodology to explain observations that have defied understanding, when a frozen coefficient approach is employed.