Quadrature rules for N-th order consistency of discontinuous Galerkin on rectangles

We want to solve

  ∂tu + Div(f) = s.

Multiply by a test function p and integrate by parts over a mesh cell:

  ∂t(∫ u*p) + ∫∂(n•f*p) = ∫(f•∇ p) + ∫(s*p)

Restrict p to N-th order polynomials. Then we have an exact unclosed system of equations for the coefficients of the Taylor expansion of u out to order N. Replacing f with a numerical flux function F (and specifying a numerical integration rule, which can be viewed as part of specifying F if we project F onto the subspace for which the numerical integration rule is exact) and likewise replacing s with a numerical source term S gives a closed system:

  ∂t(∫ u*p) + ∫∂(n•F*p) = ∫(F•∇ p) + ∫(S*p).

We seek conditions to ensure that the error is order M:=(N+1), i.e. the maximal order of accuracy that the representation space of the solution is capable of.

A k-point integration rule (Gaussian Quadrature) is needed to exactly integrate one-dimensional polynomials of order 2*k-1. That is, to exactly integrate a one-dimensional polynomial of order κ you need an integration rule with ceiling((κ+1)/2) many points. If we use a tensor product integration rule for a rectangular cell with n dimensions then we need M^n points.

Taking F to be an N-th order polynomial (projection), to exactly integrate each term in a cartesian mesh cell in n-dimensional space we need

   term           order of integrand  number of points needed
   ============   ==================  =======================
   ∫∂(n•F*p)       2N                  (N+1)n-1
   ∫(F•∇ p)        2N-1                Nn
   ∫(S*p)          2N                  (N+1)n