3.4
Slope Limiter
In order to prevent spurious oscillations at sharp fronts, a slope limiter is applied at each
step of the Runge-Kutta method described above. We apply a simple slope limiter in which the
degrees of freedom zi for a given element e are compared to the average of the approximate
e′
e
solution over e, zavg and the average of the neighboring element e' of the given edge, zavg . If zi
e′
e
does not fall in between the values zavg and zavg for the given edge i then the degrees of freedom
e
for element e are set equal to zavg . In this way, the average of the element is maintained while
setting its slope equal to zero, and sediment mass is still conserved over the element. It should be
noted this slope limiter is very easy to implement, but it can cause some numerical smoothing of
the solution. More sophisticated limiters that are less dissipative are currently being investigated.
We remark that for sufficiently smooth bathymetries, in practice it is often unnecessary to
apply the limiter. However, as the bed evolves, steep gradients may develop in the bed, and it has
been observed that without the use of a limiter oscillations develop in the neighborhood of the
steep gradient. Typically, however, these oscillations seem to remain localized and do not
degrade the solution globally. The role of the slope limiter then, at least for the problems
examined, is that of a mechanism to eliminate local oscillations rather than for stabilizing the
scheme.
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