hexed::Basis Class Reference

Represents a basis of Lagrange polynomials. More...

#include <Basis.hpp>

Inheritance diagram for hexed::Basis:

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

Public Attributes
const int	row_size
	row size

Protected Member Functions
virtual double	min_eig_convection () const =0
	nondimensional minimum real part of eigenvals of 1D convection operator
virtual double	quadratic_safety () const =0

Detailed Description

Represents a basis of Lagrange polynomials.

Interpolates between a set of quadrature points, a.k.a. nodes. The ith basis vector is the polynomial with a value of 1 at the ith node and 0 at all other nodes.

Constructor & Destructor Documentation

Basis()

hexed::Basis::Basis

(

int

row_size_arg

)

Parameters

row_size_arg

specifies member row_size.

Member Function Documentation

boundary()

virtual Mat< dyn, dyn > hexed::Basis::boundary ( ) const

pure virtual

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

Implemented in hexed::Equidistant, hexed::Gauss_legendre, and hexed::Gauss_lobatto.

diff_mat()

virtual Mat< dyn, dyn > hexed::Basis::diff_mat ( ) const

pure virtual

Differentiation matrix.

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

Implemented in hexed::Equidistant, hexed::Gauss_legendre, and hexed::Gauss_lobatto.

filter()

virtual Mat< dyn, dyn > hexed::Basis::filter ( ) const

pure virtual

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

Implemented in hexed::Equidistant, hexed::Gauss_legendre, and hexed::Gauss_lobatto.

interpolate()

Mat< dyn, dyn > hexed::Basis::interpolate

(

const Mat<> &

sample

)

const

Interpolate to arbitrary points.

Current implementation is not very performance-optimized. interpolate(sample)(i, j) is the jth basis polynomial evaluated at sample(i)

min_eig_convection()

virtual double hexed::Basis::min_eig_convection ( ) const

protectedpure virtual

nondimensional minimum real part of eigenvals of 1D convection operator

Implemented in hexed::Equidistant, hexed::Gauss_legendre, and hexed::Gauss_lobatto.

min_eig_diffusion()

virtual double hexed::Basis::min_eig_diffusion ( ) const

pure virtual

nondimensional minimum eigenvalue of 1D diffusion operator

Implemented in hexed::Equidistant, hexed::Gauss_legendre, and hexed::Gauss_lobatto.

node()

virtual double hexed::Basis::node ( int i ) const

pure virtual

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

Implemented in hexed::Equidistant, hexed::Gauss_legendre, and hexed::Gauss_lobatto.

node_weights()

virtual Mat hexed::Basis::node_weights ( ) const

pure virtual

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

Implemented in hexed::Equidistant, hexed::Gauss_legendre, and hexed::Gauss_lobatto.

orthogonal()

virtual Mat hexed::Basis::orthogonal ( int degree ) const

pure virtual

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.

Implemented in hexed::Equidistant, hexed::Gauss_legendre, and hexed::Gauss_lobatto.

prolong()

Mat< dyn, dyn > hexed::Basis::prolong ( int i_half ) const

virtual

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 in hexed::Gauss_legendre, and hexed::Gauss_lobatto.

quadratic_safety()

virtual double hexed::Basis::quadratic_safety ( ) const

protectedpure virtual

Implemented in hexed::Equidistant.

restrict()

virtual Mat< dyn, dyn > hexed::Basis::restrict ( int i_half ) const

pure virtual

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.

Implemented in hexed::Equidistant, hexed::Gauss_legendre, and hexed::Gauss_lobatto.

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

• /home/micaiah/Desktop/hexed/include/Basis.hpp
• /home/micaiah/Desktop/hexed/libhexed/Basis.cpp