hexed::Gauss_legendre Class Reference

Basis based on Gauss-Legendre quadrature. More...

#include <Gauss_legendre.hpp>

Inheritance diagram for hexed::Gauss_legendre:

Public Member Functions
	Gauss_legendre (int row_size_arg)
	see hexed::Basis::Basis.
double	node (int i) const override
	the ith interpolation node (i.e. quadrature point). Should be in interval [0, 1].
Eigen::VectorXd	node_weights () const override
	node_weights()(i) is the quadrature weight associated with node(i) Integrates in domain [0, 1].
Eigen::MatrixXd	diff_mat () const override
	Differentiation matrix.
Eigen::MatrixXd	boundary () const override
	boundary()(i, j) is the jth basis polynomial evaluated at i for i in {0, 1}.
Eigen::VectorXd	orthogonal (int degree) const override
	orthogonal(degree)(i) is the degreeth-degree Legendre polynomial evaluated at node(i).
Eigen::MatrixXd	filter () const override
	Matrix that applies a low-pass filter to the Legendre modes of a polynomial.
Eigen::MatrixXd	prolong (int i_half) const override
	Matrix to prolong into polynomial space of one higher refinement level.
Eigen::MatrixXd	restrict (int i_half) const override
	Restrict from refined space above to polynomial space spanned by this basis.
double	min_eig_diffusion () const override
	nondimensional minimum eigenvalue of 1D diffusion operator
Public Member Functions inherited from hexed::Basis
	Basis (int row_size_arg)
Mat	nodes () const
	Get vector of nodes.
Mat< dyn, dyn >	interpolate (const Mat<> &sample) const
	Interpolate to arbitrary points.
double	max_cfl () const
	maximum stable CFL number for 1D convection
double	step_ratio () const
	ratio of time step for convection stage 2 to stage 1

Protected Member Functions
double	min_eig_convection () const override
	nondimensional minimum real part of eigenvals of 1D convection operator
double	quadratic_safety () const override

Additional Inherited Members
Public Attributes inherited from hexed::Basis
const int	row_size
	row size

Detailed Description

Basis based on Gauss-Legendre quadrature.

The nodes and node_weights of this basis are those of the Gauss-Legendre quadrature, which is exact for polynomials of degree 2*row_size - 1.

Member Function Documentation

boundary()

Eigen::MatrixXd hexed::Gauss_legendre::boundary ( ) const

overridevirtual

boundary()(i, j) is the jth basis polynomial evaluated at i for i in {0, 1}.

Implements hexed::Basis.

diff_mat()

Eigen::MatrixXd hexed::Gauss_legendre::diff_mat ( ) const

overridevirtual

Differentiation matrix.

diff_mat()(i, j) is the derivative of interpolating polynomial j evaluated at node(i).

Implements hexed::Basis.

filter()

Eigen::MatrixXd hexed::Gauss_legendre::filter ( ) const

overridevirtual

Matrix that applies a low-pass filter to the Legendre modes of a polynomial.

Implements hexed::Basis.

min_eig_convection()

double hexed::Gauss_legendre::min_eig_convection ( ) const

overrideprotectedvirtual

nondimensional minimum real part of eigenvals of 1D convection operator

Implements hexed::Basis.

min_eig_diffusion()

double hexed::Gauss_legendre::min_eig_diffusion ( ) const

overridevirtual

nondimensional minimum eigenvalue of 1D diffusion operator

Implements hexed::Basis.

node()

double hexed::Gauss_legendre::node ( int i ) const

overridevirtual

the ith interpolation node (i.e. quadrature point). Should be in interval [0, 1].

Implements hexed::Basis.

node_weights()

Eigen::VectorXd hexed::Gauss_legendre::node_weights ( ) const

overridevirtual

node_weights()(i) is the quadrature weight associated with node(i) Integrates in domain [0, 1].

Implements hexed::Basis.

orthogonal()

Eigen::VectorXd hexed::Gauss_legendre::orthogonal ( int degree ) const

overridevirtual

orthogonal(degree)(i) is the degreeth-degree Legendre polynomial evaluated at node(i).

Polynomial is scaled so that its norm is 1 with respect to the quadrature rule of the basis.

Implements hexed::Basis.

prolong()

Eigen::MatrixXd hexed::Gauss_legendre::prolong ( int i_half ) const

overridevirtual

Matrix to prolong into polynomial space of one higher refinement level.

prolong(i_half)(i, j) is the jth basis polynomial evaluated at the ith node in the refined space. If i_half is 0, refined space is interval [0, 0.5]. If i_half is 1, then [0.5, 1]. This is a basic, naive implementation. Derived classes may override for performance and/or precision.

Reimplemented from hexed::Basis.

quadratic_safety()

double hexed::Gauss_legendre::quadratic_safety ( ) const

inlineoverrideprotectedvirtual

Implements hexed::Basis.

restrict()

Eigen::MatrixXd hexed::Gauss_legendre::restrict ( int i_half ) const

overridevirtual

Restrict from refined space above to polynomial space spanned by this basis.

restrict(i_half)(i, j) is the value at the ith standard node of the orthogonal projection of the jth polynomial in the refined space into the space of this basis.

Implements hexed::Basis.

---

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

• /home/micaiah/Desktop/hexed/include/Gauss_legendre.hpp