Diffusion Wave Equation

In shallow frictional and gravity controlled flow; unsteady, advection, turbulence and Coriolis terms of the momentum equation can be disregarded to arrive at a simplified version. Flow movement is driven by a barotropic pressure gradient balanced by bottom friction. Simplifying the momentum equation results in:

1)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial h}{\partial t} = \nabla \cdot (\beta \nabla z_s) + q\end{array}$

where:

$\begin{array}{l}\displaystyle \beta = \frac{R^{2/3}h}{n \left| \nabla z_s \right| ^{1/2}}\end{array}$

$\begin{array}{l}z_s\end{array}$ is the elevation of the water surface, and $\begin{array}{l}q\end{array}$ denotes a mass source. As the name implies, this is essentially a diffusion equation in which the diffusion coefficient is a function of the slope of the water surface. Since the diffusion coefficient $\begin{array}{l}\beta\end{array}$ includes the water surface slope in the denominator the diffusion coefficient goes to infinity as the water surface approaches flat. The time step limitation of explicit schemes is inversely related to the diffusion coefficient, leading to infinitesimal time steps as the water surface becomes flat. Various approaches such as linearization of $\begin{array}{l}\beta\end{array}$ below some threshold slope are possible solutions, at the expense of a less accurate solution.