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 ∈*V*h and integrating over Ωe to obtain:

∂*z*

∫Ωe ∂*t vd *Ωe + ∫Ωe ∇ ⋅ **q**bvd Ωe = 0

(12)

Integrating the second term of this equation by parts gives:

∂*z*

∫Ωe ∂*t vd *Ωe - ∫Ωe ∇*v *⋅ **q**b d Ωe + ∫Γe v **q**b ⋅ **n***d *Γe = 0

(13)

Next we replace the solution *z *with an approximate solution *z*h which, using Galerkin's

method, is constructed from a set of basis functions which belong to the same space, *V*h, 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:

⎧ **q**b(in) ⋅ **n**, ** U **⋅ **n **≥ 0

⎪

^

(14)

()

⎪ **q**b ⋅ **n**, **U **⋅ **n **< 0

⎩

With the approximate solution and the numerical flux defined, the weak formulation of the

problem now becomes:

11

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