Asymptotic expansion for steady state overland flow

The full Saint Venant equations of overland flow on a plane are often replaced by simpler models. The errors in the kinematic and the diffusion models are estimated by comparing their predictions with the exact numerical solution of the Saint Venant equations under steady state conditions. It is shown that the two approximate models can have significant errors even for critical flow and fairly large kinematic wave numbers. When the kinematic approximation is inaccurate, the improvement of the diffusion approximation seems modest. For the same level of mathematical complexity as the diffusion approximation, a far more accurate approximation is proposed when the kinematic wave number is large. It is then possible to split the solution of the Saint Venant equation in two regions, one near the downstream end of the plane and the other covering most of the plane.