Derivation of errors of kinematic-wave and diffusion-wave approximations for space-independent flows

Abstract

Hydrodynamic models of overland flow and channel flow are based on the shallow waterwave theory that is described by the St Venant (SV) equations. These models are derived from either the kinematic-wave (KW) approximation, the diffusion-wave approximation (DW), or the dynamic-wave (DYW) representation of the SV equations. In studies reported to data, different criteria have been established to evaluate the adequacy of the KW and DW approximations, but no explicit relations either in time or in space between these criteria and the errors resulting from these approximations have yet been derived. Furthermore, when doing hydrologic modeling, it is not evident if the KW and the DW approximations are valid for the entire hydrograph or for a portion thereof. In other words, these criteria take on fixed values for a given rainfall-runoff event. This paper attempts to derive, under simplified conditions, error equations for the KW or DW approximations to the DYM equations for space-independent flows, which provide a continuous description of error in the flow discharge hydrograph. The kinematic wave, diffusion wave and dynamic wave solutions are parameterized through a dimensionless parameter γ which reflects the effect of initial depth of flow, channel-bed slope, lateral inflow, and channel roughness. By comparing the kinematic wave and diffusion wave solutions with the dynamic wave solution, equations are derived in terms of γ for the error in the kinematic wave and diffusion wave approximations. The error equations turn out to have the form of the Riccati equation.