3.2
Quadrature Rules
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 2p and 2p+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 , zi ) ⎟
(21)
⎝ i=1
⎠
Ωe
where the wi's are the quadrature weights of the associated quadrature points, which are the
midpoints of each edge. Using this rule, the sediment transport function, qb is easily evaluated at
the quadrature points given the fact that we already have zi, 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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