This model was originally proposed by Exner (1925). Assuming a smooth initial condition,
z ( x, 0) = z0 , the classical solution is given implicitly by:
(ζ - z )
z ( x, t ) = z0 ( x - czt ) ,
2
cz = Aq f
(11)
where cz is the propagation speed of the bed. As is well known, non-linear hyperbolic equations
such as Eq. (10), depending on the initial conditions, will develop steep gradients (and eventually
discontinuities or shocks) which provide a rigorous test for a numerical method. A similar model
was examined by Johnson and Zyserman (2002) in the context of testing a finite difference
scheme.
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