hummingbird::GaussChebyshev Class Reference

Class defines a Gauss-Chebyshev quadrature set, which approximates the integral of the form: More...

#include <gauss_chebyshev.h>

Inheritance diagram for hummingbird::GaussChebyshev:

Collaboration diagram for hummingbird::GaussChebyshev:

Public Member Functions
	GaussChebyshev (const size_t n_points)
	Construct a new Gauss Chebyshev object.
Public Member Functions inherited from hummingbird::QuadratureBase< double >
	QuadratureBase ()=default
double	GetWeight (const unsigned int abscissa_index) const
	Get the weight value corresponding to a given abscissa value.
double	GetAbscissa (const unsigned int index) const
	Get the abscissa corresponding to the index.
double	Integrate (const std::vector< QuadraturePair > &quad_pairs) const
	Integrate the function using the function value and its location on the quadrature grid.
const std::vector< double > &	abscissas () const
	Getter for the abscissas in the quadrature set.
size_t	n_points () const
	Get total number of abscissas.

Private Member Functions
std::vector< double >	ComputeChebyshevAbscissas (const size_t n_points)
	Computes the abscissa values for Gauss Chebyshev quadrature. These are given by the explicit formula:
double	ComputeWeight (const size_t k, const size_t n) override
	Computes the quadrature weight for abscissa k of n total points. In Gauss-Chebyshev quadrature, all weights are equal and are given by:

Additional Inherited Members
Protected Member Functions inherited from hummingbird::QuadratureBase< double >
void	CreateWeightMap ()
	Create the weight map.
Protected Attributes inherited from hummingbird::QuadratureBase< double >
std::vector< double >	abscissas_
	Abscissa values.
std::map< unsigned int, double >	weight_map_
	Map to store the weights of the quadrature set where the keys are the indices corresponding to abscissa values.

Detailed Description

Class defines a Gauss-Chebyshev quadrature set, which approximates the integral of the form:

\[\int^{1}_{-1}\frac{u(x)}{\sqrt{1-x^2}}dx\approx \sum_{k=0}^{N-1}w_k u(\xi_k) \]

Formulae for this class were taken from the textbook "High-Order Methods for Incompressible Fluid Flow" by Deville, Fischer, and Mund. https://doi.org/10.1017/CBO9780511546792 and https://mathworld.wolfram.com/Chebyshev-GaussQuadrature.html

Definition at line 23 of file gauss_chebyshev.h.

Constructor & Destructor Documentation

GaussChebyshev()

hummingbird::GaussChebyshev::GaussChebyshev

(

const size_t

n_points

)

Construct a new Gauss Chebyshev object.

Parameters

n_points

Total number of points desired

Definition at line 5 of file gauss_chebyshev.cc.

Member Function Documentation

ComputeChebyshevAbscissas()

std::vector< double > hummingbird::GaussChebyshev::ComputeChebyshevAbscissas ( const size_t n_points )

private

Computes the abscissa values for Gauss Chebyshev quadrature. These are given by the explicit formula:

\[\xi_k=\cos \left( \frac{(2k+1)\pi}{2N} \right) \]

for N total points.

Parameters

n_points

Total number of points desired

Returns

std::vector<double>

Definition at line 10 of file gauss_chebyshev.cc.

ComputeWeight()

double hummingbird::GaussChebyshev::ComputeWeight ( const size_t k, const size_t n )

overrideprivatevirtual

Computes the quadrature weight for abscissa k of n total points. In Gauss-Chebyshev quadrature, all weights are equal and are given by:

\[w_k=\frac{\pi}{n} \]

Parameters

k	Abscissa number (not used in computation)
n	Total number of points

Returns

Weight for abscissa k

Reimplemented from hummingbird::QuadratureBase< double >.

Definition at line 19 of file gauss_chebyshev.cc.

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

• /home/runner/work/hummingbird/hummingbird/include/quadrature/gauss_chebyshev.h
• /home/runner/work/hummingbird/hummingbird/src/quadrature/gauss_chebyshev.cc