deal.II version GIT relicensing-2206-gaa53ff9447 2024-12-02 09:10:00+00:00
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#include <deal.II/base/quadrature_lib.h>
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 |
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.
Definition at line 79 of file quadrature_lib.h.
enum QGaussRadau::EndPoint |
EndPoint is used to specify which of the two endpoints of the unit interval is used also as quadrature point.
Enumerator | |
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left | Left end point. |
right | Right end point. |
Definition at line 86 of file quadrature_lib.h.
QGaussRadau< dim >::QGaussRadau | ( | const unsigned int | n, |
const EndPoint | end_point = QGaussRadau< dim >::EndPoint::left |
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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.
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defaultnoexcept |
Move constructor.
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private |
Definition at line 112 of file quadrature_lib.h.