9.4 Gauss quadratures and summation by parts When computing a discrete expansion in terms of
one question is how to efficiently numerically approximate the coefficients given in
Eq. (9.33). This involves computing weighted integrals of the form
If approximating the weighted integral (9.62) by a quadrature rule,
where the points are given but having the freedom to choose the coefficients , by a counting
argument one would expect to be able to choose the latter in such a way that Eq. (9.63) is
exact for all polynomials of degree at most . That is indeed the case, and the answer is
obtained by approximating by its polynomial interpolant (8.1) and integrating the latter,
where the are the Lagrange polynomials (8.3) and the coefficients
are independent of the integrand . If the weight function is nontrivial, they might not be known in
closed form, but since they are independent of the function being integrated they need to be computed only
once for each set of nodal points .
Suppose now that, in addition to having the freedom to choose the coefficients , we can choose
the nodal points . Then we have points and , i.e., degrees of
freedom. Therefore, we expect that we can make the quadrature exact for all polynomials of degree at
most . This is indeed true and is referred to as Gauss quadratures. Furthermore, the
optimal choice of remains the same as in Eq. (9.65), and only the nodal points need to be
The following remarks are in order:
- The roots of are referred to as Gauss points or nodes.
- Suppose that . Then the Gauss points, i.e., the roots of
the Chebyshev polynomial [see Eq. (9.68)], are exactly the points that minimize
the infinity norm of the nodal polynomial in the interpolation problem, as discussed in
One can see that the Gauss points actually lie inside the interval , and do not contain the
endpoints or . Now suppose that for some reason we want the nodes to include the end points of
One reason for including the end points of the interval in the set of nodes is when applying boundary
conditions in the collocation approach, as discussed in Section 10. Then we are left with two less
degrees of freedom compared to Gauss quadratures and therefore expect to be able to make
the quadrature exact for polynomials of order up to . This leads
Note that the coefficients and in the previous equations are obtained by solving the simple
One can similarly enforce that only one of the end points coincides with a quadrature one, leading to
Gauss–Radau quadratures. The proofs of Theorems 20 and 21 can be found in most numerical analysis
books, in particular .
For Chebyshev polynomials there are closed form expressions for the nodes and weights in Eqs. (9.63)
Summation by parts.
For any two polynomials of degree , in the Legendre case SBP follows for
Gauss, Gauss–Lobatto or Gauss–Radau quadratures, in analogy with the FD case described in
Since both products and are polynomials of degree , their
quadratures are exact (in fact, the equality holds for each term separately):
where we have introduced the discrete counterpart of the weighted scalar product (9.17),
with the nodes and discrete weights those of the corresponding quadrature.
On the other hand, in the Legendre case,
Property (9.75) will be used in Section 9.7 when discussing stability through the energy method, much as
in the FD case.