A 1d negative elliptic differential operator \( -\partial_x ( \chi \partial_x ) \). More...

Public Types
using	geometry_type = Geometry

using	matrix_type = Matrix

using	container_type = Container

using	value_type = get_value_type< Container >

Public Member Functions
	Elliptic1d ()=default
	empty object ( no memory allocation) More...

	Elliptic1d (const Geometry &g, direction dir=forward, value_type jfactor=1.)
	Construct from Grid. More...

	Elliptic1d (const Geometry &g, bc bcx, direction dir=forward, value_type jfactor=1.)
	Construct from grid and boundary conditions. More...

template<class ... Params>
void	construct (Params &&...ps)
	Perfect forward parameters to one of the constructors. More...

template<class ContainerType0 >
void	set_chi (const ContainerType0 &sigma)
	Change scalar part Chi. More...

const Container &	weights () const
	Return the weights making the operator self-adjoint. More...

const Container &	precond () const
	Return the default preconditioner to use in conjugate gradient. More...

void	set_jfactor (value_type new_jfactor)
	Set the currently used jfactor ( \( \alpha \)) More...

value_type	get_jfactor () const
	Get the currently used jfactor ( \( \alpha \)) More...

template<class ContainerType0 , class ContainerType1 >
void	operator() (const ContainerType0 &x, ContainerType1 &y)
	Compute elliptic term and store in output. More...

template<class ContainerType0 , class ContainerType1 >
void	symv (const ContainerType0 &x, ContainerType1 &y)
	Compute elliptic term and store in output. More...

template<class ContainerType0 , class ContainerType1 >
void	symv (value_type alpha, const ContainerType0 &x, value_type beta, ContainerType1 &y)
	Compute elliptic term and add to output. More...

Detailed Description

template<class Geometry, class Matrix, class Container>
class dg::Elliptic1d< Geometry, Matrix, Container >

A 1d negative elliptic differential operator \( -\partial_x ( \chi \partial_x ) \).

The term discretized is

\[ -\partial_x ( \chi \partial_x ) \]

where \( \partial_x \) is the one-dimensional derivative and \(\chi\) is a scalar function Per default, \( \chi = 1\) but you can set it to any value you like (in order for the operator to be invertible \(\chi\) should be strictly positive though).

See also: Our theory guide Introduction to dg methods on overleaf holds a detailed derivation

Note that the local discontinuous Galerkin discretization adds so-called jump terms

\[ D^\dagger \chi D + \alpha\chi_{on/off} J \]

where \(\alpha\) is a scale factor ( = jfactor), \( D \) contains the discretizations of the above derivatives, and \( J\) is a self-adjoint matrix. ( \(J\) is added before the volume element is divided). The adjoint of a matrix is defined with respect to the volume element including dG weights. Usually the default \( \alpha=1 \) is a good choice. However, in some cases, e.g. when \( \sigma \) exhibits very large variations \( \alpha=0.1\) or \( \alpha=0.01\) might be better values. In a time dependent problem the value of \(\alpha\) determines the numerical diffusion, i.e. for too low values numerical oscillations may appear. The \( \chi_{on/off} \) in the jump term serves to weight the jump term with \( \chi \). This can be switched either on or off with off being the default. Also note that a forward discretization has more diffusion than a centered discretization.

Template Parameters

Geometry	A type that is or derives from one of the abstract geometry base classes ( `aGeometry2d`, `aGeometry3d`, `aMPIGeometry2d`, ...). `Geometry` determines which `Matrix` and `Container` types can be used:
Matrix	A class for which the `dg::blas2::symv` functions are callable in connection with the `Container` class and to which the return type of `dg::create::dx()` can be converted using `dg::blas2::transfer`. The `Matrix` type can be one of: `dg::HMatrix` with `dg::HVec` and one of the shared memory geometries `dg::DMatrix` with `dg::DVec` and one of the shared memory geometries `dg::MHMatrix` with `dg::MHVec` and one of the MPI geometries `dg::MDMatrix` with `dg::MDVec` and one of the MPI geometries
Container	A data container class for which the `blas1` functionality is overloaded and to which the return type of `blas1::subroutine()` can be converted using `dg::assign`. We assume that `Container` is copyable/assignable and has a swap member function. In connection with `Geometry` this is one of `dg::HVec`, `dg::DVec` when `Geometry` is a shared memory geometry `dg::MHVec` or `dg::MDVec` when `Geometry` is one of the MPI geometries This class has the `SelfMadeMatrixTag` so it can be used in `blas2::symv` functions and thus in a conjugate gradient solver.

Note: The constructors initialize \( \chi=1\) so that a negative laplacian operator results; The inverse of \( \chi\) makes a good general purpose preconditioner

Attention: Pay attention to the negative sign which is necessary to make the matrix positive definite

Member Typedef Documentation

◆ container_type

template<class Geometry , class Matrix , class Container >

using dg::Elliptic1d< Geometry, Matrix, Container >::container_type = Container

◆ geometry_type

template<class Geometry , class Matrix , class Container >

using dg::Elliptic1d< Geometry, Matrix, Container >::geometry_type = Geometry

◆ matrix_type

template<class Geometry , class Matrix , class Container >

using dg::Elliptic1d< Geometry, Matrix, Container >::matrix_type = Matrix

◆ value_type

template<class Geometry , class Matrix , class Container >

using dg::Elliptic1d< Geometry, Matrix, Container >::value_type = get_value_type<Container>

Constructor & Destructor Documentation

◆ Elliptic1d() [1/3]

template<class Geometry , class Matrix , class Container >

dg::Elliptic1d< Geometry, Matrix, Container >::Elliptic1d ( )

default

empty object ( no memory allocation)

◆ Elliptic1d() [2/3]

template<class Geometry , class Matrix , class Container >

dg::Elliptic1d< Geometry, Matrix, Container >::Elliptic1d	(	const Geometry &	g,
		direction	dir = `forward`,
		value_type	jfactor = `1.`
	)

inline

Construct from Grid.

Parameters

g	The Grid, boundary conditions are taken from here
dir	Direction of the right first derivative in x (i.e. `dg::forward`, `dg::backward` or `dg::centered`),
jfactor	( \( = \alpha \) ) scale jump terms (1 is a good value but in some cases 0.1 or 0.01 might be better)

Note: chi is one by default

◆ Elliptic1d() [3/3]

template<class Geometry , class Matrix , class Container >

dg::Elliptic1d< Geometry, Matrix, Container >::Elliptic1d	(	const Geometry &	g,
		bc	bcx,
		direction	dir = `forward`,
		value_type	jfactor = `1.`
	)

inline

Construct from grid and boundary conditions.

Parameters

g	The Grid
bcx	boundary condition in x
dir	Direction of the right first derivative in x (i.e. `dg::forward`, `dg::backward` or `dg::centered`),
jfactor	( \( = \alpha \) ) scale jump terms (1 is a good value but in some cases 0.1 or 0.01 might be better)

Note: chi is one by default

Member Function Documentation

◆ construct()

template<class Geometry , class Matrix , class Container >

template<class ... Params>

void dg::Elliptic1d< 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

◆ get_jfactor()

template<class Geometry , class Matrix , class Container >

value_type dg::Elliptic1d< Geometry, Matrix, Container >::get_jfactor ( ) const

inline

Get the currently used jfactor ( \( \alpha \))

Returns: The current scale factor for jump terms

◆ operator()()

template<class Geometry , class Matrix , class Container >

template<class ContainerType0 , class ContainerType1 >

void dg::Elliptic1d< Geometry, Matrix, Container >::operator()	(	const ContainerType0 &	x,
		ContainerType1 &	y
	)

inline

Compute elliptic term and store in output.

i.e. y=M*x

Parameters

x	left-hand-side
y	result

Template Parameters

ContainerTypes must be usable with Container in The dg dispatch system

◆ precond()

template<class Geometry , class Matrix , class Container >

const Container & dg::Elliptic1d< Geometry, Matrix, Container >::precond ( ) const

inline

Return the default preconditioner to use in conjugate gradient.

Currently returns 1 divided by the scalar part \( \sigma\). This is especially good when \( \sigma\) exhibits large amplitudes or variations

Returns: the inverse of \(\sigma\).

◆ set_chi()

template<class Geometry , class Matrix , class Container >

template<class ContainerType0 >

void dg::Elliptic1d< Geometry, Matrix, Container >::set_chi ( const ContainerType0 & sigma )

inline

Change scalar part Chi.

Parameters

sigma The new scalar part \(\chi\)

Attention: If some or all elements of sigma are zero the preconditioner is invalidated and the operator can no longer be inverted (due to divide by zero) until set_chi is called with a positive sigma again. The symv function can always be called, however, if sigma is zero you likely also want to set the jfactor to 0 because the jumps in phi may not vanish and then pollute the result.

Template Parameters

ContainerType0 must be usable with Container in The dg dispatch system

◆ set_jfactor()

template<class Geometry , class Matrix , class Container >

void dg::Elliptic1d< Geometry, Matrix, Container >::set_jfactor ( value_type new_jfactor )

inline

Set the currently used jfactor ( \( \alpha \))

Parameters

new_jfactor The new scale factor for jump terms

◆ symv() [1/2]

template<class Geometry , class Matrix , class Container >

template<class ContainerType0 , class ContainerType1 >

void dg::Elliptic1d< Geometry, Matrix, Container >::symv	(	const ContainerType0 &	x,
		ContainerType1 &	y
	)

inline

Compute elliptic term and store in output.

i.e. y=M*x

Parameters

x	left-hand-side
y	result

Template Parameters

ContainerTypes must be usable with Container in The dg dispatch system

◆ symv() [2/2]

template<class Geometry , class Matrix , class Container >

template<class ContainerType0 , class ContainerType1 >

void dg::Elliptic1d< Geometry, Matrix, Container >::symv	(	value_type	alpha,
		const ContainerType0 &	x,
		value_type	beta,
		ContainerType1 &	y
	)

inline

Compute elliptic term and add to output.

i.e. y=alpha*M*x+beta*y

Parameters

alpha	a scalar
x	left-hand-side
beta	a scalar
y	result

Template Parameters

ContainerTypes must be usable with Container in The dg dispatch system

◆ weights()

template<class Geometry , class Matrix , class Container >

const Container & dg::Elliptic1d< Geometry, Matrix, Container >::weights ( ) const

inline

Return the weights making the operator self-adjoint.

Returns: weights

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

elliptic.h

Public Types

Public Member Functions

Detailed Description

Member Typedef Documentation

◆ container_type

◆ geometry_type

◆ matrix_type

◆ value_type

Constructor & Destructor Documentation

◆ Elliptic1d() [1/3]

◆ Elliptic1d() [2/3]

◆ Elliptic1d() [3/3]

Member Function Documentation

◆ construct()

◆ get_jfactor()

◆ operator()()

◆ precond()

◆ set_chi()

◆ set_jfactor()

◆ symv() [1/2]

◆ symv() [2/2]

◆ weights()