Development of a Robust Diffusion-Kinematic Flow Algorithm for Regional Hydrologic Models Operating with Large Time Steps

Diffusion flow models used to simulate hydrologic systems can become inaccurate under certain extreme flow conditions. Two such flow conditions include the relatively deep low slope flow condition as in the case of stagnant canals and the relatively shallow steep flow condition as in the case of mountain streams. In the case of relatively deep slow flows and short term simulations, the inertia effects are important, and full shallow water equations are needed to solve this problem. However, in the case of steep slopes, even if inertia terms can be neglected, long term simulations become problematic because of the need of fully implicit diffusion models to use small time steps when using kinematic flow. Many of the long term physically based hydrologic models for surface flow have to deal with meshes or cells when simulating irregular bottom slopes. These models need to trap water in deep ponds and pass water over steep slopes at the same time, creating conditions that are difficult to solve with models using only the diffusion flow approximation. Numerical methods that are stable with large time steps and small cells under both diffusion and kinematic flow conditions are needed in solving such problems. The current paper describes a stable computational approach that can be useful in solving both kinematic and diffusion problems. A fully implicit finite volume formulation of the approximate St. Venant equations is used in the formulation. Newton's method is applied to solve the nonlinear equation resulting from this approach. The paper demonstrates the stability and the boundedness of the approach when using very large time steps of the order of days.