∫Ωe ∂t vd Ωe - ∫Ωe ∇v ⋅ qb d Ωe + ∫Γe v qb d Γe = 0
Note that the method is locally or elementally conservative in the following sense: setting v = 1
on Ωe and zero elsewhere we have:
∫Ωe ∂t d Ωe + ∫Γe qb d Γe = 0
That is, the time rate of change of zh over Ωe is balanced by the net flux of sediment into Ωe.
We proceed by describing the details of the implementation of the scheme including the
choice of basis functions, the quadrature rules employed to compute the integrals, the time
discretization, the application of a slope limiter to eliminate local undershoots or overshoots in
the solution in the presence of steep gradients, and the continuous projection procedure used to
project the discontinuous approximation zh into the space of continuous, piecewise linear
functions which are fed back into ADCIRC as updated bathymetry.