dg::mat::TensorElliptic< Geometry, Matrix, Container > Struct Template Reference

Matrix class that represents the arbitrary polarization operator. More...

Public Types
using	container_type = Container
using	geometry_type = Geometry
using	matrix_type = Matrix
using	value_type = get_value_type<Container>

Public Member Functions
	TensorElliptic ()
	empty object ( no memory allocation)
	TensorElliptic (const Geometry &g, direction dir=dg::centered, value_type jfactor=1.)
	Construct TensorElliptic operator.
	TensorElliptic (const Geometry &g, bc bcx, bc bcy, direction dir=dg::centered, value_type jfactor=1.)
	Construct TensorElliptic operator.
template<class ... Params>
void	construct (Params &&...ps)
	Perfect forward parameters to one of the constructors.
const Container &	weights () const
	Return the weights making the operator self-adjoint.
const Container &	precond () const
	Preconditioner to use in conjugate gradient solvers.
template<class ContainerType0 >
void	set_chi (const ContainerType0 &chi)
	Set Chi in the above formula.
template<class ContainerType0 >
void	set_iota (const ContainerType0 &iota)
	Set Iota in the above formula.
void	variation (const Container &phi, const value_type &alpha, const Container &chi, Container &varphi)
	compute the variational of the operator (psi_2 in gf theory):
template<class ContainerType0 , class ContainerType1 >
void	operator() (const ContainerType0 &x, ContainerType1 &y)
template<class ContainerType0 , class ContainerType1 >
void	symv (const ContainerType0 &x, ContainerType1 &y)
template<class ContainerType0 , class ContainerType1 >
void	symv (value_type alpha, const ContainerType0 &x, value_type beta, ContainerType1 &y)
	apply operator

Detailed Description

template<class Geometry, class Matrix, class Container>
struct dg::mat::TensorElliptic< Geometry, Matrix, Container >

Matrix class that represents the arbitrary polarization operator.

Discretization of

\[ (-\nabla \cdot \chi \nabla - \Delta \iota \Delta + \nabla \cdot\nabla \cdot 2\iota \nabla \nabla ) x \]

in two dimensions where \( \chi\) and \(\iota\) are functions

Member Typedef Documentation

container_type

template<class Geometry , class Matrix , class Container >

using dg::mat::TensorElliptic< Geometry, Matrix, Container >::container_type = Container

geometry_type

template<class Geometry , class Matrix , class Container >

using dg::mat::TensorElliptic< Geometry, Matrix, Container >::geometry_type = Geometry

matrix_type

template<class Geometry , class Matrix , class Container >

using dg::mat::TensorElliptic< Geometry, Matrix, Container >::matrix_type = Matrix

value_type

template<class Geometry , class Matrix , class Container >

using dg::mat::TensorElliptic< Geometry, Matrix, Container >::value_type = get_value_type<Container>

Constructor & Destructor Documentation

TensorElliptic() [1/3]

template<class Geometry , class Matrix , class Container >

dg::mat::TensorElliptic< Geometry, Matrix, Container >::TensorElliptic ( )

inline

empty object ( no memory allocation)

TensorElliptic() [2/3]

template<class Geometry , class Matrix , class Container >

dg::mat::TensorElliptic< Geometry, Matrix, Container >::TensorElliptic ( const Geometry & g, direction dir = dg::centered, value_type jfactor = 1. )

inline

Construct TensorElliptic operator.

Parameters

g	The grid to use
dir	Direction of the Laplace operator (Note: only dg::centered tested)
jfactor	The jfactor used in the Laplace operator (probably 1 is always the best factor but one never knows...)

Note

The default value of \(\chi\) and \(\iota\) is one

TensorElliptic() [3/3]

template<class Geometry , class Matrix , class Container >

dg::mat::TensorElliptic< Geometry, Matrix, Container >::TensorElliptic ( const Geometry & g, bc bcx, bc bcy, direction dir = dg::centered, value_type jfactor = 1. )

inline

Construct TensorElliptic operator.

Parameters

g	The grid to use
bcx	boundary condition in x
bcy	boundary contition in y
dir	Direction of the Laplace operator (Note: only dg::centered tested)
jfactor	The jfactor used in the Laplace operator (probably 1 is always the best factor but one never knows...)

Note

The default value of \(\chi\) and \(\iota\) is one

Member Function Documentation

construct()

template<class Geometry , class Matrix , class Container >

template<class ... Params>

void dg::mat::TensorElliptic< Geometry, Matrix, Container >::construct ( Params &&... ps )

inline

Perfect forward parameters to one of the constructors.

Template Parameters

Params

deduced by the compiler

Parameters

ps	parameters forwarded to constructors

operator()()

template<class Geometry , class Matrix , class Container >

template<class ContainerType0 , class ContainerType1 >

void dg::mat::TensorElliptic< Geometry, Matrix, Container >::operator() ( const ContainerType0 & x, ContainerType1 & y )

inline

precond()

template<class Geometry , class Matrix , class Container >

const Container & dg::mat::TensorElliptic< Geometry, Matrix, Container >::precond ( ) const

inline

Preconditioner to use in conjugate gradient solvers.

Returns

1

set_chi()

template<class Geometry , class Matrix , class Container >

template<class ContainerType0 >

void dg::mat::TensorElliptic< Geometry, Matrix, Container >::set_chi ( const ContainerType0 & chi )

inline

Set Chi in the above formula.

Parameters

chi	new container

set_iota()

template<class Geometry , class Matrix , class Container >

template<class ContainerType0 >

void dg::mat::TensorElliptic< Geometry, Matrix, Container >::set_iota ( const ContainerType0 & iota )

inline

Set Iota in the above formula.

Parameters

iota	new container

symv() [1/2]

template<class Geometry , class Matrix , class Container >

template<class ContainerType0 , class ContainerType1 >

void dg::mat::TensorElliptic< Geometry, Matrix, Container >::symv ( const ContainerType0 & x, ContainerType1 & y )

inline

symv() [2/2]

template<class Geometry , class Matrix , class Container >

template<class ContainerType0 , class ContainerType1 >

void dg::mat::TensorElliptic< Geometry, Matrix, Container >::symv ( value_type alpha, const ContainerType0 & x, value_type beta, ContainerType1 & y )

inline

apply operator

Computes

\[ y = W\left[-\nabla \cdot \chi \nabla_\perp - \Delta_\perp \iota \Delta_\perp + 2\nabla \cdot\nabla \cdot \iota \nabla_\perp \nabla_\perp \right] x \]

to make the matrix symmetric

Parameters

alpha
x	lhs (is constant up to changes in ghost cells)
beta
y	rhs contains solution

Note

Note that for Cartesian and Cylindrical coordinate systems (with the straight field line approximation) the following relation holds

\[\nabla \cdot\nabla \cdot (\chi \nabla_\perp^2 f) = \frac{1}{\sqrt{g}} \partial_j \left\{\partial_i \left[\sqrt{g} \chi P^{ni} \left( \partial_n ( P^{jm} \partial_m f)\right) \right]\right\} \]

where P is the projection tensor

variation()

template<class Geometry , class Matrix , class Container >

void dg::mat::TensorElliptic< Geometry, Matrix, Container >::variation ( const Container & phi, const value_type & alpha, const Container & chi, Container & varphi )

inline

compute the variational of the operator (psi_2 in gf theory):

\[ - \frac{\chi}{2} \left\{|\nabla \phi|^2 + \alpha \chi ( | \nabla^2 \phi |^2 - (\Delta \phi)^2 / 2) \right\}\]

Parameters

phi	(e.g. Gamma phi)
alpha	(e.g. tau/2)
chi	(e.g. 1/B^2)
varphi	equals psi_2 in gf theory if phi = gamma_phi

weights()

template<class Geometry , class Matrix , class Container >

const Container & dg::mat::TensorElliptic< Geometry, Matrix, Container >::weights ( ) const

inline

Return the weights making the operator self-adjoint.

Returns

weights

---

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

• tensorelliptic.h