equations using the CG Galerkin finite element method in space and implicit/explicit time
stepping (see Luettich and Westerink (2004) for details). As previously mentioned, to achieve
non-oscillatory results the primitive continuity equation is replaced with the GWCE.
We apply the DG method to the sediment continuity equation by multiplying Eq. (1) by
a test function v ∈Vh and integrating over Ωe to obtain:
∂z
∫Ωe ∂t vd Ωe + ∫Ωe ∇ ⋅ qbvd Ωe = 0
(12)
Integrating the second term of this equation by parts gives:
∂z
∫Ωe ∂t vd Ωe - ∫Ωe ∇v ⋅ qb d Ωe + ∫Γe v qb ⋅ nd Γe = 0
(13)
Next we replace the solution z with an approximate solution zh which, using Galerkin's
method, is constructed from a set of basis functions which belong to the same space, Vh, as the
test functions.
Due to the fact that there may be discontinuities along element edges, the
^
formulation, we use a simple upwind flux based on the assumption that the sediment transport is
in the direction of the current:
⎧ qb(in) ⋅ n, U ⋅ n ≥ 0
⎪
qb = ⎨ ex
^
(14)
()
⎪ qb ⋅ n, U ⋅ n < 0
⎩
With the approximate solution and the numerical flux defined, the weak formulation of the
problem now becomes:
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