Analytical solution of a kinematic wave approximation for channel routing

The kinematic wave approach is often used in hydrological models to describe channel and overland flow. The kinematic wave is suitable for situations where the local and convective acceleration, as well as the pressure term in the dynamic wave model is negligible with respect to the friction and body forces. This is the case when describing runoff processes in the upper parts of catchments, where slopes are generally of the order of 10⁻³. In physical-based hydrological models, the point-scale conservation equations are integrated over model entities, such as grid pixels or control volumes. The integration leads to a set of ordinary differential governing equations, which can be solved numerically by methods such as the Runge–Kutta integrator. Here, we propose an analytical solution of a Taylor-series approximation of the kinematic wave equation, which is presented as non-linear reservoir equation. We show that the analytical solution is numerically robust and third-order accurate. It is compared with the numerical solution and the solution of the complete dynamic wave model. The analytical solution proves to be computationally better performing and more accurate than the numerical solution. The proposed analytical solution can also be generalized to situations of leaking channels.