QGaussRadau< dim > Class Template Reference

#include <deal.II/base/quadrature_lib.h>

Inheritance diagram for QGaussRadau< dim >:

Public Types
enum	EndPoint { left , right }

Public Member Functions
	QGaussRadau (const unsigned int n, const EndPoint end_point=QGaussRadau::EndPoint::left)
	QGaussRadau (QGaussRadau< dim > &&) noexcept=default

Private Attributes
const EndPoint	end_point

Detailed Description

template<int dim>
class QGaussRadau< dim >

The Gauss-Radau family of quadrature rules for numerical integration.

This modification of the Gauss quadrature uses one of the two interval end points as well. Being exact for polynomials of degree \(2n-2\), this formula is suboptimal by one degree.

This formula is often used in the context of discontinuous Galerkin discretizations of ODEs and the temporal part of PDEs.

The quadrature points are the left interval end point plus the \(n-1\) roots of the polynomial

\[ \frac{P_{n-1}(x)+P_n(x)}{1+x} \]

where \(P_{n-1}\) and \(P_n\) are Legendre polynomials. The quadrature weights are

\[ w_0=\frac{2}{n^2}\quad\text{and} \quad w_i=\frac{1-x_i}{n^2(P_{n-1}(x_i))^2}\text{ for }i>0 \]

For the right Gauss-Radau formula the quadrature points are \(\tilde{x}_i=1-x_{n-i-1}\) and the weights are \(\tilde{w}_i=w_{n-i-1}\), with \((x_i,w_i)\) as quadrature points and weights of the left Gauss-Radau formula.

See also

https://mathworld.wolfram.com/RadauQuadrature.html

Definition at line 79 of file quadrature_lib.h.

Member Enumeration Documentation

EndPoint

template<int dim>

enum QGaussRadau::EndPoint

EndPoint is used to specify which of the two endpoints of the unit interval is used also as quadrature point.

Enumerator
left	Left end point.
right	Right end point.

Definition at line 86 of file quadrature_lib.h.

Constructor & Destructor Documentation

QGaussRadau() [1/2]

template<int dim>

QGaussRadau< dim >::QGaussRadau	(	const unsigned int	n,
		const EndPoint	end_point = QGaussRadau< dim >::EndPoint::left
	)

Generate a formula with n quadrature points (in each space direction). ep defines whether the left/lower/front endpoint(s) (default) or the right/upper/back endpoint(s) are part of the quadrature points.

Definition at line 1055 of file quadrature_lib.cc.

QGaussRadau() [2/2]

template<int dim>

QGaussRadau< dim >::QGaussRadau ( QGaussRadau< dim > &&  )

defaultnoexcept

Move constructor.

Member Data Documentation

end_point

template<int dim>

const EndPoint QGaussRadau< dim >::end_point

private

Definition at line 112 of file quadrature_lib.h.

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The documentation for this class was generated from the following files:

• include/deal.II/base/quadrature_lib.h
• source/base/quadrature_lib.cc