Class for the evaluation of parallel derivatives. More...

Public Types
typedef dg::geo::Fieldaligned< ProductGeometry, IMatrix, container >	FA

Public Member Functions
	DS ()
	No memory allocation; all member calls except construct are invalid. More...

template<class Limiter >
	DS (const dg::geo::TokamakMagneticField &vec, const ProductGeometry &grid, dg::bc bcx=dg::NEU, dg::bc bcy=dg::NEU, Limiter limit=FullLimiter(), double eps=1e-5, unsigned mx=10, unsigned my=10, double deltaPhi=-1, std::string interpolation_method="dg", bool benchmark=true)
	Create the magnetic unit vector field and construct. More...

template<class Limiter >
	DS (const dg::geo::CylindricalVectorLvl1 &vec, const ProductGeometry &grid, dg::bc bcx=dg::NEU, dg::bc bcy=dg::NEU, Limiter limit=FullLimiter(), double eps=1e-5, unsigned mx=10, unsigned my=10, double deltaPhi=-1, std::string interpolation_method="dg", bool benchmark=true)
	Use the given vector field to construct. More...

	DS (FA fieldaligned)
	Re-construct from a given `Fieldaligned` object. More...

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

void	set_boundaries (dg::bc bcz, double left, double right)
	Set boundary conditions in the limiter region. More...

void	set_boundaries (dg::bc bcz, const container &left, const container &right)
	Set boundary conditions in the limiter region. More...

void	set_boundaries (dg::bc bcz, const container &global, double scal_left, double scal_right)
	Set boundary conditions in the limiter region. More...

void	forward (double alpha, const container &f, double beta, container &g)
	forward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\) More...

void	forward2 (double alpha, const container &f, double beta, container &g)
	2nd order forward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\) More...

void	backward (double alpha, const container &f, double beta, container &g)
	backward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\) More...

void	backward2 (double alpha, const container &f, double beta, container &g)
	2nd order backward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\) More...

void	centered (double alpha, const container &f, double beta, container &g)
	centered derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\) More...

void	centered_bc_along_field (double alpha, const container &f, double beta, container &g, dg::bc bound, std::array< double, 2 > boundary_value={0, 0})
	Same as `dg::geo::ds_centered` after `dg::geo::ds_assign_bc_along_field_2nd`. More...

void	backward (const container &f, container &g)
	backward derivative \( g = \vec v \cdot \nabla f \) More...

void	forward (const container &f, container &g)
	forward derivative \( g = \vec v \cdot \nabla f \) More...

void	centered (const container &f, container &g)
	centered derivative \( g = \vec v \cdot \nabla f \) More...

void	divForward (double alpha, const container &f, double beta, container &g)
	forward divergence \( g = \alpha \nabla\cdot(\vec v f) + \beta g\) More...

void	divBackward (double alpha, const container &f, double beta, container &g)
	backward divergence \( g = \alpha \nabla\cdot(\vec v f) + \beta g\) More...

void	divCentered (double alpha, const container &f, double beta, container &g)
	centered divergence \( g = \alpha \nabla\cdot(\vec v f) + \beta g\) More...

void	divForward (const container &f, container &g)
	forward divergence \( g = \nabla\cdot(\vec v f)\) More...

void	divBackward (const container &f, container &g)
	backward divergence \( g = \nabla\cdot(\vec v f)\) More...

void	divCentered (const container &f, container &g)
	centered divergence \( g = \nabla\cdot(\vec v f)\) More...

void	ds (dg::direction dir, const container &f, container &g)
	Discretizes \( g = \vec v\cdot \nabla f \). More...

void	ds (dg::direction dir, double alpha, const container &f, double beta, container &g)
	Discretizes \( g = \alpha \vec v\cdot \nabla f + \beta g \). More...

void	div (dg::direction dir, const container &f, container &g)
	Discretizes \( g = \nabla \cdot ( \vec v f) \). More...

void	div (dg::direction dir, double alpha, const container &f, double beta, container &g)
	Discretizes \( g = \alpha \nabla \cdot( \vec v f) + \beta g \). More...

void	symv (const container &f, container &g)
	Discretizes \( g = \nabla\cdot ( \vec v \vec v \cdot \nabla f )\). More...

void	symv (double alpha, const container &f, double beta, container &g)
	Discretizes \( g = \alpha \nabla\cdot ( \vec v \vec v \cdot \nabla f ) + \beta g\) as a symmetric matrix. More...

void	dss (const container &f, container &g)
	Discretizes \( g = (\vec v\cdot \nabla)^2 f \). More...

void	dss (double alpha, const container &f, double beta, container &g)
	Discretizes \( g = \alpha (\vec v\cdot \nabla)^2 f + \beta g \). More...

void	dss_bc_along_field (double alpha, const container &f, double beta, container &g, dg::bc bound, std::array< double, 2 > boundary_value={0, 0})
	Same as `dg::geo::dss_centered` after `dg::geo::ds_assign_bc_along_field_2nd`. More...

void	dssd (double alpha, const container &f, double beta, container &g)
	Same as `dg::geo::dssd_centered`. More...

void	dssd_bc_along_field (double alpha, const container &f, double beta, container &g, dg::bc bound, std::array< double, 2 > boundary_value={0, 0})
	Same as `dg::geo::dssd_centered` after `dg::geo::ds_assign_bc_along_field_2nd`. More...

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

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

const container &	weights () const
	The volume form with dG weights. More...

FA &	fieldaligned ()
	access the underlying Fieldaligned object More...

const FA &	fieldaligned () const

Detailed Description

template<class ProductGeometry, class IMatrix, class Matrix, class container>
struct dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >

Class for the evaluation of parallel derivatives.

This class discretizes the operators \( \nabla_\parallel = \vec{v}\cdot \nabla = v^\zeta\partial_\zeta + v^\eta\partial_\eta + v^\varphi\partial_\varphi \), \(\nabla_\parallel^\dagger = -\nabla\cdot(\vec v .)\) and \(\Delta_\parallel=-\nabla_\parallel^\dagger\cdot\nabla_\parallel\) in arbitrary coordinates

    const dg::CylindricalGrid3d g3d( R_0-a, R_0+a, -a, a, 0, 2.*M_PI, n, Nx, Ny, Nz, dg::NEU, dg::NEU, dg::PER);
    //create magnetic field
    const dg::geo::TokamakMagneticField mag = dg::geo::createCircularField( R_0, I_0);
    auto bhat = dg::geo::createBHat(mag);
    //create Fieldaligned object and construct DS from it
    dg::geo::Fieldaligned<dg::aProductGeometry3d,dg::IDMatrix,dg::DVec>  dsFA(
            bhat, g3d, dg::NEU, dg::NEU, dg::geo::NoLimiter(), 1e-8, mx[0], mx[1],
            -1, method);
    dg::geo::DS<dg::aProductGeometry3d, dg::IDMatrix, dg::DMatrix, dg::DVec>
        ds( dsFA );

Note: The parallel Laplacian cannot be inverted as long as there are closed fieldlines somewhere in the domain (which is virtually always true). There is therefore no norm parameter in the class.; The div and symv member functions are close to but not exactly volume conserving (at least if not the linear interpolation is chosen in Fieldaligned)

Template Parameters

ProductGeometry	must be either `dg::aProductGeometry3d` or `dg::aProductMPIGeometry3d` or any derivative
IMatrix	The type of the interpolation matrix `dg::IHMatrix`, or `dg::IDMatrix`, `dg::MIHMatrix`, or `dg::MIDMatrix`
Matrix	The matrix class of the jump matrix `dg::HMatrix`, or `dg::DMatrix`, `dg::MHMatrix`, or `dg::MDMatrix`
container	The container-class on which the interpolation matrix operates on `dg::HVec`, or `dg::DVec`, `dg::MHVec`, or `dg::MDVec`

See also: The pdf parallel derivative writeup

Member Typedef Documentation

◆ FA

template<class ProductGeometry , class IMatrix , class Matrix , class container >

typedef dg::geo::Fieldaligned<ProductGeometry, IMatrix, container> dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::FA

conveniently abbreviates underlying Fieldaligned type

Constructor & Destructor Documentation

◆ DS() [1/4]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::DS ( )

inline

No memory allocation; all member calls except construct are invalid.

◆ DS() [2/4]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

template<class Limiter >

dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::DS	(	const dg::geo::TokamakMagneticField &	vec,
		const ProductGeometry &	grid,
		dg::bc	bcx = `dg::NEU`,
		dg::bc	bcy = `dg::NEU`,
		Limiter	limit = `FullLimiter()`,
		double	eps = `1e-5`,
		unsigned	mx = `10`,
		unsigned	my = `10`,
		double	deltaPhi = `-1`,
		std::string	interpolation_method = `"dg"`,
		bool	benchmark = `true`
	)

inline

Create the magnetic unit vector field and construct.

Template Parameters

Limiter Class that can be evaluated on a 2d grid, returns 1 if there is a limiter and 0 if there isn't. If a field line crosses the limiter in the plane \( \phi=0\) then the limiter boundary conditions apply.

Parameters

vec	The vector field to integrate. Note that you can control how the boundary conditions are represented by changing vec outside the grid domain using e.g. the `periodify` function.
grid	The grid on which to integrate fieldlines.
bcx	This parameter is passed on to `dg::create::interpolation(const thrust::host_vector<real_type>&,const thrust::host_vector<real_type>&,const aRealTopology2d<real_type>&,dg::bc,dg::bc,std::string)` (see there for more details) function and deterimens what happens when the endpoint of the fieldline integration leaves the domain boundaries of `grid`. Note that `bcx` and `grid.bcx()` have to be either both periodic or both not periodic.
bcy	analogous to `bcx`, applies to y direction
limit	Instance of the limiter class (Note that if `grid.bcz()==dg::PER` this parameter is ignored, Default is a limiter everywhere)
eps	Desired accuracy of the fieldline integrator
mx	refinement factor in X of the fine grid relative to grid (Set to 1, if the x-component of `vec` vanishes, else as high as possible, 12 is a good start) If the projection method (parsed from `interpolation_method`) is not "dg" then `mx` must be a multiple of `nx`
my	analogous to `mx`, applies to y direction
deltaPhi	The angular distance that the fieldline-integrator will integrate. Per default this is the distance between planes, which is chosen automatically if you set it <=0, i.e. if deltaPhi <=0 then it will be overwritten to deltaPhi = grid.hz(). Sometimes however, you may want to set it to a different value from `grid.hz()` for example for 2d problems or for a staggered grid.

Note: deltaPhi influences the interpolation matrices and the parallel modulation in the evaluate() member function.; If there is a limiter, the boundary condition on the first/last plane is set by the grid.bcz() variable and can be changed by the set_boundaries function. If there is no limiter, the boundary condition is periodic.

Parameters

interpolation_method parsed according to

interpolation_method	interpolation	projection
"dg"	"dg"	"dg"
"linear"	"linear"	"dg"
"cubic"	"cubic"	"dg"
"nearest"	"nearest"	"dg"
"linear-nearest" (##)	"linear"	"nearest"
"cubic-nearest"	"cubic"	"nearest"
"nearest-nearest"	"nearest"	"nearest"
"dg-linear"	"dg"	"linear"
"linear-linear"	"linear"	"linear"
"cubic-linear"	"cubic"	"linear"
"nearest-linear"	"nearest"	"linear"

(##) Use "linear-nearest" if in doubt. The table yields one parameter passed to create::interpolation (from the given grid to the fine grid) and one parameter to create::projection (from the fine grid to the given grid)

Parameters

benchmark	If true write construction timings to std::cout
eps	Desired accuracy of the fieldline integrator
mx	refinement factor in X of the fine grid relative to grid (Set to 1, if the x-component of `vec` vanishes, else as high as possible, 10 is a good start)
my	analogous to `mx`, applies to y direction
deltaPhi	The angular distance that the fieldline-integrator will integrate. Per default this is the distance between planes, which is chosen automatically if you set it <=0, i.e. if deltaPhi <=0 then it will be overwritten to deltaPhi = grid.hz(). Sometimes however, you may want to set it to a different value from `grid.hz()` for example for 2d problems or for a staggered grid.

Note: deltaPhi influences the interpolation matrices and the parallel modulation in the evaluate() member function.; If there is a limiter, the boundary condition on the first/last plane is set by the grid.bcz() variable and can be changed by the set_boundaries function. If there is no limiter, the boundary condition is periodic.

Parameters

interpolation_method	Several interpolation methods are available: dg uses the native dG interpolation scheme given by the grid, nearest searches for the nearest point and copies its value, linear searches for the two (in 2d four, etc.) closest points and linearly interpolates their values, cubic searches for the four (in 2d 16, etc) closest points and interpolates a cubic polynomial
benchmark	If true write construction timings to std::cout

See also: Fieldaligned

◆ DS() [3/4]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

template<class Limiter >

dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::DS	(	const dg::geo::CylindricalVectorLvl1 &	vec,
		const ProductGeometry &	grid,
		dg::bc	bcx = `dg::NEU`,
		dg::bc	bcy = `dg::NEU`,
		Limiter	limit = `FullLimiter()`,
		double	eps = `1e-5`,
		unsigned	mx = `10`,
		unsigned	my = `10`,
		double	deltaPhi = `-1`,
		std::string	interpolation_method = `"dg"`,
		bool	benchmark = `true`
	)

inline

Use the given vector field to construct.

Template Parameters

Limiter Class that can be evaluated on a 2d grid, returns 1 if there is a limiter and 0 if there isn't. If a field line crosses the limiter in the plane \( \phi=0\) then the limiter boundary conditions apply.

Parameters

vec	The vector field to integrate. Note that you can control how the boundary conditions are represented by changing vec outside the grid domain using e.g. the `periodify` function.
grid	The grid on which to integrate fieldlines.
bcx	This parameter is passed on to `dg::create::interpolation(const thrust::host_vector<real_type>&,const thrust::host_vector<real_type>&,const aRealTopology2d<real_type>&,dg::bc,dg::bc,std::string)` (see there for more details) function and deterimens what happens when the endpoint of the fieldline integration leaves the domain boundaries of `grid`. Note that `bcx` and `grid.bcx()` have to be either both periodic or both not periodic.
bcy	analogous to `bcx`, applies to y direction
limit	Instance of the limiter class (Note that if `grid.bcz()==dg::PER` this parameter is ignored, Default is a limiter everywhere)
eps	Desired accuracy of the fieldline integrator
mx	refinement factor in X of the fine grid relative to grid (Set to 1, if the x-component of `vec` vanishes, else as high as possible, 12 is a good start) If the projection method (parsed from `interpolation_method`) is not "dg" then `mx` must be a multiple of `nx`
my	analogous to `mx`, applies to y direction
deltaPhi	The angular distance that the fieldline-integrator will integrate. Per default this is the distance between planes, which is chosen automatically if you set it <=0, i.e. if deltaPhi <=0 then it will be overwritten to deltaPhi = grid.hz(). Sometimes however, you may want to set it to a different value from `grid.hz()` for example for 2d problems or for a staggered grid.

Note: deltaPhi influences the interpolation matrices and the parallel modulation in the evaluate() member function.; If there is a limiter, the boundary condition on the first/last plane is set by the grid.bcz() variable and can be changed by the set_boundaries function. If there is no limiter, the boundary condition is periodic.

Parameters

interpolation_method parsed according to

interpolation_method	interpolation	projection
"dg"	"dg"	"dg"
"linear"	"linear"	"dg"
"cubic"	"cubic"	"dg"
"nearest"	"nearest"	"dg"
"linear-nearest" (##)	"linear"	"nearest"
"cubic-nearest"	"cubic"	"nearest"
"nearest-nearest"	"nearest"	"nearest"
"dg-linear"	"dg"	"linear"
"linear-linear"	"linear"	"linear"
"cubic-linear"	"cubic"	"linear"
"nearest-linear"	"nearest"	"linear"

(##) Use "linear-nearest" if in doubt. The table yields one parameter passed to create::interpolation (from the given grid to the fine grid) and one parameter to create::projection (from the fine grid to the given grid)

Parameters

benchmark	If true write construction timings to std::cout
eps	Desired accuracy of the fieldline integrator
mx	refinement factor in X of the fine grid relative to grid (Set to 1, if the x-component of `vec` vanishes, else as high as possible, 10 is a good start)
my	analogous to `mx`, applies to y direction
deltaPhi	The angular distance that the fieldline-integrator will integrate. Per default this is the distance between planes, which is chosen automatically if you set it <=0, i.e. if deltaPhi <=0 then it will be overwritten to deltaPhi = grid.hz(). Sometimes however, you may want to set it to a different value from `grid.hz()` for example for 2d problems or for a staggered grid.

Note: deltaPhi influences the interpolation matrices and the parallel modulation in the evaluate() member function.; If there is a limiter, the boundary condition on the first/last plane is set by the grid.bcz() variable and can be changed by the set_boundaries function. If there is no limiter, the boundary condition is periodic.

Parameters

interpolation_method	Several interpolation methods are available: dg uses the native dG interpolation scheme given by the grid, nearest searches for the nearest point and copies its value, linear searches for the two (in 2d four, etc.) closest points and linearly interpolates their values, cubic searches for the four (in 2d 16, etc) closest points and interpolates a cubic polynomial
benchmark	If true write construction timings to std::cout

See also: Fieldaligned

◆ DS() [4/4]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::DS ( FA fieldaligned )

Re-construct from a given Fieldaligned object.

Parameters

fieldaligned this object will be used in all further member calls

Member Function Documentation

◆ backward() [1/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::backward	(	const container &	f,
		container &	g
	)

inline

backward derivative \( g = \vec v \cdot \nabla f \)

backward derivative \( g_i = \frac{v^\varphi}{\Delta\varphi}(f_{i} - f_{i-1}) \)

Parameters

f	The vector to derive
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor

◆ backward() [2/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::backward	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g
	)

inline

backward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)

backward derivative \( g_i = \alpha \frac{1}{h_z^-}(f_{i} - f_{i-1}) + \beta g_i \)

Parameters

alpha	Scalar
f	The vector to derive
beta	Scalar
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor

◆ backward2()

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::backward2	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g
	)

inline

2nd order backward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)

backward derivative \( g_i = \alpha \frac{1}{2h_z^-}(3f_{i} - 4f_{i-1} + f_{i-2}) + \beta g_i \)

Parameters

alpha	Scalar
f	The vector to derive
beta	Scalar
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor

◆ centered() [1/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::centered	(	const container &	f,
		container &	g
	)

inline

centered derivative \( g = \vec v \cdot \nabla f \)

centered derivative \( g_i = \frac{v^\varphi}{2\Delta\varphi}(f_{i+1} - f_{i-1})\)

Parameters

f	The vector to derive
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor

◆ centered() [2/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::centered	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g
	)

inline

centered derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)

The centered derivative is constructed by fitting a polynomial through the plus point the minus point and the center point and evaluating its derivative at the center point. For the exact resulting formula consult the parallel derivative writeup

Parameters

alpha	Scalar
f	The vector to derive
beta	Scalar
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor

◆ centered_bc_along_field()

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::centered_bc_along_field	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g,
		dg::bc	bound,
		std::array< double, 2 >	boundary_value = `{0,0}`
	)

inline

Same as dg::geo::ds_centered after dg::geo::ds_assign_bc_along_field_2nd.

◆ construct()

template<class ProductGeometry , class IMatrix , class Matrix , class container >

template<class ... Params>

void dg::geo::DS< ProductGeometry, IMatrix, 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

◆ div() [1/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::div	(	dg::direction	dir,
		const container &	f,
		container &	g
	)

inline

Discretizes \( g = \nabla \cdot ( \vec v f) \).

Parameters

dir	redirects to either `divForward()`, `divBackward()` or `divCentered()`
f	The vector to derive
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor; The div and symv member functions are close to but not exactly volume conserving (at least if not the linear interpolation is chosen in Fieldaligned)

◆ div() [2/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::div	(	dg::direction	dir,
		double	alpha,
		const container &	f,
		double	beta,
		container &	g
	)

Discretizes \( g = \alpha \nabla \cdot( \vec v f) + \beta g \).

Parameters

dir	redirects to either `divForward()`, `divBackward()` or `divCentered()`
alpha	Scalar
f	The vector to derive
beta	Scalar
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor; The div and symv member functions are close to but not exactly volume conserving (at least if not the linear interpolation is chosen in Fieldaligned)

◆ divBackward() [1/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::divBackward	(	const container &	f,
		container &	g
	)

inline

backward divergence \( g = \nabla\cdot(\vec v f)\)

Parameters

f	The vector to derive
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor; The div and symv member functions are close to but not exactly volume conserving (at least if not the linear interpolation is chosen in Fieldaligned)

◆ divBackward() [2/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::divBackward	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g
	)

inline

backward divergence \( g = \alpha \nabla\cdot(\vec v f) + \beta g\)

Parameters

alpha	Scalar
f	The vector to derive
beta	Scalar
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor; The div and symv member functions are close to but not exactly volume conserving (at least if not the linear interpolation is chosen in Fieldaligned)

◆ divCentered() [1/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::divCentered	(	const container &	f,
		container &	g
	)

inline

centered divergence \( g = \nabla\cdot(\vec v f)\)

Parameters

f	The vector to derive
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor; The div and symv member functions are close to but not exactly volume conserving (at least if not the linear interpolation is chosen in Fieldaligned)

◆ divCentered() [2/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::divCentered	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g
	)

inline

centered divergence \( g = \alpha \nabla\cdot(\vec v f) + \beta g\)

Parameters

alpha	Scalar
f	The vector to derive
beta	Scalar
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor; The div and symv member functions are close to but not exactly volume conserving (at least if not the linear interpolation is chosen in Fieldaligned)

◆ divForward() [1/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::divForward	(	const container &	f,
		container &	g
	)

inline

forward divergence \( g = \nabla\cdot(\vec v f)\)

Parameters

f	The vector to derive
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor; The div and symv member functions are close to but not exactly volume conserving (at least if not the linear interpolation is chosen in Fieldaligned)

◆ divForward() [2/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::divForward	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g
	)

inline

forward divergence \( g = \alpha \nabla\cdot(\vec v f) + \beta g\)

Parameters

alpha	Scalar
f	The vector to derive
beta	Scalar
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor; The div and symv member functions are close to but not exactly volume conserving (at least if not the linear interpolation is chosen in Fieldaligned)

◆ ds() [1/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::ds	(	dg::direction	dir,
		const container &	f,
		container &	g
	)

inline

Discretizes \( g = \vec v\cdot \nabla f \).

Parameters

dir	redirects to either `forward()`, `backward()` or `centered()`
f	The vector to derive
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor

◆ ds() [2/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::ds	(	dg::direction	dir,
		double	alpha,
		const container &	f,
		double	beta,
		container &	g
	)

Discretizes \( g = \alpha \vec v\cdot \nabla f + \beta g \).

Parameters

dir	redirects to either `forward()`, `backward()` or `centered()`
alpha	Scalar
f	The vector to derive
beta	Scalar
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor

◆ dss() [1/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::dss	(	const container &	f,
		container &	g
	)

inline

Discretizes \( g = (\vec v\cdot \nabla)^2 f \).

The formula used is

\[ \nabla_\parallel^2 f = 2\left(\frac{f^+}{h_z^+ h_z^0} - \frac{f^0}{h_z^- h_z^+} + \frac{f^-}{h_z^-h_z^0}\right) \]

which is the second derivative of a 2nd order polynomial fitted through the plus, minus and centre points

Parameters

f	The vector to derive
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor

◆ dss() [2/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::dss	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g
	)

inline

Discretizes \( g = \alpha (\vec v\cdot \nabla)^2 f + \beta g \).

The formula used is

\[ \nabla_\parallel^2 f = 2\left(\frac{f^+}{h_z^+ h_z^0} - \frac{f^0}{h_z^- h_z^+} + \frac{f^-}{h_z^-h_z^0}\right) \]

which is the second derivative of a 2nd order polynomial fitted through the plus, minus and centre points

Parameters

alpha	Scalar
f	The vector to derive
beta	Scalar
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor

◆ dss_bc_along_field()

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::dss_bc_along_field	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g,
		dg::bc	bound,
		std::array< double, 2 >	boundary_value = `{0,0}`
	)

inline

Same as dg::geo::dss_centered after dg::geo::ds_assign_bc_along_field_2nd.

◆ dssd()

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::dssd	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g
	)

inline

Same as dg::geo::dssd_centered.

◆ dssd_bc_along_field()

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::dssd_bc_along_field	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g,
		dg::bc	bound,
		std::array< double, 2 >	boundary_value = `{0,0}`
	)

inline

Same as dg::geo::dssd_centered after dg::geo::ds_assign_bc_along_field_2nd.

◆ fieldaligned() [1/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

FA & dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::fieldaligned ( )

inline

access the underlying Fieldaligned object

Returns: acces to Fieldaligned object

◆ fieldaligned() [2/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

const FA & dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::fieldaligned ( ) const

inline

◆ forward() [1/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::forward	(	const container &	f,
		container &	g
	)

inline

forward derivative \( g = \vec v \cdot \nabla f \)

forward derivative \( g_i = \frac{v^\varphi}{\Delta\varphi}(f_{i+1} - f_{i})\)

Parameters

f	The vector to derive
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor

◆ forward() [2/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::forward	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g
	)

inline

forward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)

forward derivative \( g_i = \alpha \frac{1}{h_z^+}(f_{i+1} - f_{i}) + \beta g_i\)

Parameters

alpha	Scalar
f	The vector to derive
beta	Scalar
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor

◆ forward2()

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::forward2	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g
	)

inline

2nd order forward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)

forward derivative \( g_i = \alpha \frac{1}{2h_z^+}(-f_{i+2} + 4f_{i+1} - 3f_{i}) + \beta g_i\)

Parameters

alpha	Scalar
f	The vector to derive
beta	Scalar
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor

◆ get_jfactor()

template<class ProductGeometry , class IMatrix , class Matrix , class container >

double dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::get_jfactor ( ) const

inline

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

Returns: The current scale factor for jump terms

◆ set_boundaries() [1/3]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::set_boundaries	(	dg::bc	bcz,
		const container &	global,
		double	scal_left,
		double	scal_right
	)

inline

Set boundary conditions in the limiter region.

if Dirichlet boundaries are used the left value is the left function value, if Neumann boundaries are used the left value is the left derivative value

Parameters

bcz	boundary condition
global	3D vector containing boundary values
scal_left	left scaling factor
scal_right	right scaling factor

◆ set_boundaries() [2/3]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::set_boundaries	(	dg::bc	bcz,
		const container &	left,
		const container &	right
	)

inline

Set boundary conditions in the limiter region.

if Dirichlet boundaries are used the left value is the left function value, if Neumann boundaries are used the left value is the left derivative value

Parameters

bcz	boundary condition
left	spatially variable left boundary value (2d size)
right	spatially variable right boundary value (2d size)

◆ set_boundaries() [3/3]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::set_boundaries	(	dg::bc	bcz,
		double	left,
		double	right
	)

inline

Set boundary conditions in the limiter region.

if Dirichlet boundaries are used the left value is the left function value, if Neumann boundaries are used the left value is the left derivative value

Parameters

bcz	boundary condition
left	constant left boundary value
right	constant right boundary value

◆ set_jfactor()

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::set_jfactor ( double new_jfactor )

inline

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

Parameters

new_jfactor The new scale factor for jump terms

◆ symv() [1/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::symv	(	const container &	f,
		container &	g
	)

inline

Discretizes \( g = \nabla\cdot ( \vec v \vec v \cdot \nabla f )\).

Parameters

f	The vector to derive
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor; The div and symv member functions are close to but not exactly volume conserving (at least if not the linear interpolation is chosen in Fieldaligned)

◆ symv() [2/2]

template<class ProductGeometry , class IMatrix , class Matrix , class container >

void dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::symv	(	double	alpha,
		const container &	f,
		double	beta,
		container &	g
	)

Discretizes \( g = \alpha \nabla\cdot ( \vec v \vec v \cdot \nabla f ) + \beta g\) as a symmetric matrix.

Parameters

alpha	Scalar
f	The vector to derive
beta	Scalar
g	contains result on output (may alias input vectors)

Note: the vector sizes need to equal the grid size in the constructor; The div and symv member functions are close to but not exactly volume conserving (at least if not the linear interpolation is chosen in Fieldaligned)

◆ weights()

template<class ProductGeometry , class IMatrix , class Matrix , class container >

const container & dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >::weights ( ) const

inline

The volume form with dG weights.

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

ds.h

Public Types

Public Member Functions

Detailed Description

Member Typedef Documentation

◆ FA

Constructor & Destructor Documentation

◆ DS() [1/4]

◆ DS() [2/4]

◆ DS() [3/4]

◆ DS() [4/4]

Member Function Documentation

◆ backward() [1/2]

◆ backward() [2/2]

◆ backward2()

◆ centered() [1/2]

◆ centered() [2/2]

◆ centered_bc_along_field()

◆ construct()

◆ div() [1/2]

◆ div() [2/2]

◆ divBackward() [1/2]

◆ divBackward() [2/2]

◆ divCentered() [1/2]

◆ divCentered() [2/2]

◆ divForward() [1/2]

◆ divForward() [2/2]

◆ ds() [1/2]

◆ ds() [2/2]

◆ dss() [1/2]

◆ dss() [2/2]

◆ dss_bc_along_field()

◆ dssd()

◆ dssd_bc_along_field()

◆ fieldaligned() [1/2]

◆ fieldaligned() [2/2]

◆ forward() [1/2]

◆ forward() [2/2]

◆ forward2()

◆ get_jfactor()

◆ set_boundaries() [1/3]

◆ set_boundaries() [2/3]

◆ set_boundaries() [3/3]

◆ set_jfactor()

◆ symv() [1/2]

◆ symv() [2/2]

◆ weights()