The DG method outlined above has been applied to a number of problems. In this
section, we show the results for three idealized test cases.
Morphological evolution of a symmetric mound
In this test case, we apply the DG method outlined above to the Exner model introduced
in section 2.
The Exner model is examined in order to verify the numerical method
independently of the hydrodynamic model. It also affords us the opportunity to compare our
numerical results to exact solutions so that we may check the order of convergence of the method.
We solve a problem originally posed by Exner (1925). The problem examines the
evolution of an initially symmetric mound subjected to steady, uni-directional flow with a rigid-
lid assumption for the flow. The initial condition is given as:
⎛ 2π x ⎞
z ( x, y, 0) = z0 ( x, y ) = A0 + A1 cos ⎜
⎝ λ ⎠
where the parameters A0 , A1 , and λ are as defined in Fig. 5 which shows a cross section of the
mound along the x-axis. We take A0 = A1 = 1, λ = 20 in Equation (25) and ζ = 3, Aqf = 1 in Eq.
(10). The flow
to be in the x direction only,
and we use periodic boundary conditions,
z ( x = -λ 2 , y ) = z ( x = +λ 2 , y ) and z ( x, y = -λ 2) = z ( x, y = +λ 2) The exact
solution is given by Eq. (11).