Rigid Lid
u
H =ζ - z
ζ
z
Bed Datum
Fig. 1: Definition sketch for Exner's model
2.1
A simplified model
For purposes of verifying our numerical scheme, we use a simplified mathematical model
that essentially uncouples the sediment continuity equation from the hydrodynamics. This allows
us to verify the underlying numerics of the model in a simplified setting by comparing it to
analytical solutions. Assume that the flow is unidirectional (say in the x directional only) and
quasi-steady with a rigid lid. With these assumptions, the flow velocity is given by:
qf
qf
u=
=
(9)
ζ -z
H
where qf is a constant flow discharge and ζ is the elevation of the rigid lid measured from the bed
datum (see Fig. 1). Furthermore, assume that the sediment transport is given by Eq. (3) with A =
constant and n = 1. With these assumptions the sediment continuity equation can be written as:
∂z ∂ ⎛
qf ⎞
+ ⎜A
⎟=0
(10)
∂t ∂x ⎝ ζ - z ⎠
8
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