Quadratures

QuadratureStyle{Nq}

Quadrature style supertype. See sub-types:

• GLL
• GL
• Uniform

GLL{Nq}()

Gauss-Legendre-Lobatto quadrature using Nq quadrature points.

GL{Nq}()

Gauss-Legendre quadrature using Nq quadrature points.

Uniform{Nq}()

Uniformly-spaced quadrature.

degrees_of_freedom(QuadratureStyle) -> Int

Returns the degreesoffreedom of the QuadratureStyle concrete type

polynomial_degree(QuadratureStyle) -> Int

Returns the polynomial degree of the QuadratureStyle concrete type

points, weights = quadrature_points(::Type{FT}, quadrature_style)

The points and weights of the quadrature rule in floating point type FT.

barycentric_weights(x::SVector{Nq}) where {Nq}

The barycentric weights associated with the array of point locations x:

\[w_j = \frac{1}{\prod_{k \ne j} (x_i - x_j)}\]

See [14], equation 3.2.

interpolation_matrix(x::SVector, r::SVector{Nq})

The matrix which interpolates the Lagrange polynomial of degree Nq-1 through the points r, to points x. The matrix coefficients are computed using the Barycentric formula of [14], section 4:

\[I_{ij} = \begin{cases} 1 & \text{if } x_i = r_j, \\ 0 & \text{if } x_i = r_k \text{ for } k \ne j, \\ \frac{\displaystyle \frac{w_j}{x_i - r_j}}{\displaystyle \sum_k \frac{w_k}{x_i - r_k}} & \text{otherwise,} \end{cases}\]

where $w_j$ are the barycentric weights, see barycentric_weights.

differentiation_matrix(r::SVector{Nq, T}) where {Nq, T}

The spectral differentiation matrix for the Lagrange polynomial of degree Nq-1 interpolating at points r.

The matrix coefficients are computed using the [14], section 9.3:

\[D_{ij} = \begin{cases} \displaystyle \frac{w_j}{w_i (x_i - x_j)} &\text{ if } i \ne j \\ -\sum_{k \ne j} D_{kj} &\text{ if } i = j \end{cases}\]

where $w_j$ are the barycentric weights, see barycentric_weights.

differentiation_matrix(FT, quadstyle::QuadratureStyle)

The spectral differentiation matrix at the quadrature points of quadstyle, using floating point types FT.

V = orthonormal_poly(points, quad)

V_{ij} contains the j-1th Legendre polynomial evaluated at points[i]. i.e. it is the mapping from the modal to the nodal representation.