Development of an Accurate Time Integration Technique for the Assessment of Q-Based versus h-Based Formulations of the Diffusion Wave Equation for Flow Routing

This research provides a new numerical assessment of the diffusion wave model regarding the effect of selecting different state variables (water depth versus discharge) on the accuracy of the results. It also demonstrates the detailed implementation of a rapid and efficient method (differential quadrature) for this model. The diffusion wave model is still an active area of research in hydraulic and hydrologic engineering because of its wide range of applicability, low computational cost, and high efficiency. The diffusion wave equation can be cast in five different forms, depending upon the state variable selected, whereas Q-based (discharge) and h-based (water depth) can be considered as the main alternative formulations. A rigorous mathematical assessment of these different forms is lacking in existing literature. Numerical experiments were carried out to assess the numerical accuracy of the h-based and Q-based formulations. The fully dynamic wave results were used as benchmarks. Several parameters such as bed resistance, upstream hydrograph shape, and channel bed slope were changed to compare the accuracy of different formulations of the diffusion wave model. The results show that two formulations can give accurate results for steep slopes. However, h-based results were subjected to more errors compared with Q-based methods in very mild slopes. Conversion of the downstream boundary condition from kinematic to diffusion reduces the discrepancy dramatically. DQM can be considered as a powerful alternative numerical scheme to explore such avenues in diffusive flood routing on the basis of its efficiency and accuracy.