Both of the integrals appearing in Eq. (15) are evaluated using suitable numerical

quadrature rules. We note that by using numerical quadrature and the simple upwind numerical

flux defined previously, we can easily implement a number of different sediment transport

formulas into the scheme without making any changes to the base algorithm itself (provided that

the formula meets the requirements as specified in Section 2). Cockburn and Shu (1998) note

that for a DG spatial discretization of degree *p*, quadrature rules that are exact for polynomials of

degree 2*p *and 2*p*+1 must be used for the area and boundary integrals, respectively. Thus for the

linear elements used here (*p *= 1) we use a three point quadrature rule for the triangle so the area

integral of Equation (15) is approximated by (noting that ∇*v *is constant over the element):

)

(

⎛ 3

⎞

∇*v *⋅ ∫ qb d Ωe ≈ ∇*v *⋅ ⎜ ∑ wiqb ( Ui ,ζ i , *z*i ) ⎟

(21)

⎝ i=1

⎠

Ωe

where the *w*i's are the quadrature weights of the associated quadrature points, which are the

midpoints of each edge. Using this rule, the sediment transport function, **q**b is easily evaluated at

the quadrature points given the fact that we already have *z*i, which are the degrees of freedom, and

we need only to compute **U **and *ζ *at the mid-point of each edge. We note that these values are

easily obtained by averaging the two vertices for the given edge (owing to the fact that **U **and *ζ*

are approximated using linear functions over the element as well). The boundary integrals, which

must integrate a third degree polynomial exactly, are evaluated using the two-point Legendre-

Gauss quadrature rule.

15

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