Reference documentation for deal.II version GIT relicensing-1373-g41e71ffbc0 2024-08-06 19:50:02+00:00
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QGaussRadau< dim > Class Template Reference

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

Inheritance diagram for QGaussRadau< dim >:
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## Public Types

enum  EndPoint { left , right }

## Private Attributes

const EndPoint end_point

## Detailed Description

template<int dim>

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.

Definition at line 79 of file quadrature_lib.h.

## ◆ EndPoint

template<int dim>

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

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.

template<int dim>
defaultnoexcept

Move constructor.

## ◆ end_point

template<int dim>
private

Definition at line 112 of file quadrature_lib.h.

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