4. Fieldaligned derivatives

\( \nabla_\parallel f\) More...

Classes
struct	dg::geo::DS< ProductGeometry, IMatrix, Matrix, container >
	Class for the evaluation of parallel derivatives. More...
struct	dg::geo::Fieldaligned< ProductGeometry, IMatrix, container >
	Create and manage interpolation matrices from fieldline integration. More...

Typedefs
typedef ONE	dg::geo::FullLimiter
	Full Limiter means there is a limiter everywhere. More...
typedef ZERO	dg::geo::NoLimiter
	No Limiter. More...

Enumerations
enum	dg::geo::whichMatrix {   dg::geo::einsPlus = 0 , dg::geo::einsPlusT , dg::geo::einsMinus , dg::geo::einsMinusT ,   dg::geo::zeroPlus , dg::geo::zeroMinus , dg::geo::zeroPlusT , dg::geo::zeroMinusT ,   dg::geo::zeroForw , dg::geo::einsPlus = 0 , dg::geo::einsPlusT , dg::geo::einsMinus ,   dg::geo::einsMinusT , dg::geo::zeroPlus , dg::geo::zeroMinus , dg::geo::zeroPlusT ,   dg::geo::zeroMinusT , dg::geo::zeroBack , dg::geo::einsPlus = 0 , dg::geo::einsPlusT ,   dg::geo::einsMinus , dg::geo::einsMinusT , dg::geo::zeroPlus , dg::geo::zeroMinus ,   dg::geo::zeroPlusT , dg::geo::zeroMinusT , dg::geo::zeroForw }
	Enum for the use in Fieldaligned. More...
enum	dg::geo::whichMatrix {   dg::geo::einsPlus = 0 , dg::geo::einsPlusT , dg::geo::einsMinus , dg::geo::einsMinusT ,   dg::geo::zeroPlus , dg::geo::zeroMinus , dg::geo::zeroPlusT , dg::geo::zeroMinusT ,   dg::geo::zeroForw , dg::geo::einsPlus = 0 , dg::geo::einsPlusT , dg::geo::einsMinus ,   dg::geo::einsMinusT , dg::geo::zeroPlus , dg::geo::zeroMinus , dg::geo::zeroPlusT ,   dg::geo::zeroMinusT , dg::geo::zeroBack , dg::geo::einsPlus = 0 , dg::geo::einsPlusT ,   dg::geo::einsMinus , dg::geo::einsMinusT , dg::geo::zeroPlus , dg::geo::zeroMinus ,   dg::geo::zeroPlusT , dg::geo::zeroMinusT , dg::geo::zeroForw }
	Enum for the use in Fieldaligned. More...
enum	dg::geo::whichMatrix {   dg::geo::einsPlus = 0 , dg::geo::einsPlusT , dg::geo::einsMinus , dg::geo::einsMinusT ,   dg::geo::zeroPlus , dg::geo::zeroMinus , dg::geo::zeroPlusT , dg::geo::zeroMinusT ,   dg::geo::zeroForw , dg::geo::einsPlus = 0 , dg::geo::einsPlusT , dg::geo::einsMinus ,   dg::geo::einsMinusT , dg::geo::zeroPlus , dg::geo::zeroMinus , dg::geo::zeroPlusT ,   dg::geo::zeroMinusT , dg::geo::zeroBack , dg::geo::einsPlus = 0 , dg::geo::einsPlusT ,   dg::geo::einsMinus , dg::geo::einsMinusT , dg::geo::zeroPlus , dg::geo::zeroMinus ,   dg::geo::zeroPlusT , dg::geo::zeroMinusT , dg::geo::zeroForw }
	Enum for the use in Fieldaligned. More...

Functions
template<class FieldAligned , class container >
void	dg::geo::assign_bc_along_field_2nd (const FieldAligned &fa, const container &fm, const container &f, const container &fp, container &fmg, container &fpg, dg::bc bound, std::array< double, 2 > boundary_value={0, 0})
	Assign boundary conditions along magnetic field lines interpolating a 2nd order polynomial. More...
template<class FieldAligned , class container >
void	dg::geo::assign_bc_along_field_1st (const FieldAligned &fa, const container &fm, const container &fp, container &fmg, container &fpg, dg::bc bound, std::array< double, 2 > boundary_value={0, 0})
	Assign boundary conditions along magnetic field lines interpolating a 1st order polynomial (a line) More...
template<class FieldAligned , class container >
void	dg::geo::swap_bc_perp (const FieldAligned &fa, const container &fm, const container &fp, container &fmg, container &fpg)
	Swap the perp boundary condition. More...
template<class FieldAligned , class container >
void	dg::geo::ds_forward (const FieldAligned &fa, double alpha, const container &f, const container &fp, double beta, container &g)
	forward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\) More...
template<class FieldAligned , class container >
void	dg::geo::ds_forward2 (const FieldAligned &fa, double alpha, const container &f, const container &fp, const container &fpp, double beta, container &g)
	2nd order forward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\) More...
template<class FieldAligned , class container >
void	dg::geo::ds_backward (const FieldAligned &fa, double alpha, const container &fm, const container &f, double beta, container &g)
	backward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\) More...
template<class FieldAligned , class container >
void	dg::geo::ds_backward2 (const FieldAligned &fa, double alpha, const container &fmm, const container &fm, const container &f, double beta, container &g)
	2nd order backward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\) More...
template<class FieldAligned , class container >
void	dg::geo::ds_centered (const FieldAligned &fa, double alpha, const container &fm, const container &fp, double beta, container &g)
	centered derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\) More...
template<class FieldAligned , class container >
void	dg::geo::dss_centered (const FieldAligned &fa, double alpha, const container &fm, const container &f, const container &fp, double beta, container &g)
	Centered derivative \( g = \alpha (\vec v\cdot \nabla)^2 f + \beta g \). More...
template<class FieldAligned , class container >
void	dg::geo::dssd_centered (const FieldAligned &fa, double alpha, const container &fm, const container &f, const container &fp, double beta, container &g)
	Centered derivative \( g = \alpha \nabla\cdot(\vec v \vec v\cdot \nabla) f + \beta g \). More...
template<class FieldAligned , class container >
void	dg::geo::ds_divBackward (const FieldAligned &fa, double alpha, const container &fm, const container &f, double beta, container &g)
	backward derivative \( g = \alpha \nabla \cdot \vec v f + \beta g\) More...
template<class FieldAligned , class container >
void	dg::geo::ds_divForward (const FieldAligned &fa, double alpha, const container &f, const container &fp, double beta, container &g)
	forward derivative \( g = \alpha \nabla \cdot \vec v f + \beta g\) More...
template<class FieldAligned , class container >
void	dg::geo::ds_divCentered (const FieldAligned &fa, double alpha, const container &fm, const container &fp, double beta, container &g)
	centered derivative \( g = \alpha \nabla \cdot \vec v f + \beta g\) More...
template<class FieldAligned , class container >
void	dg::geo::ds_average (const FieldAligned &fa, double alpha, const container &fm, const container &fp, double beta, container &g)
	Compute average along a fieldline \( g = \alpha \frac{f_{k+1} + f_{k-1}}{2} + \beta g\). More...
template<class FieldAligned , class container >
void	dg::geo::ds_slope (const FieldAligned &fa, double alpha, const container &fm, const container &fp, double beta, container &g)
	Compute simple slope along a fieldline \( g = \alpha v^\varphi\frac{f_{k+1} - f_{k-1}}{2\Delta\varphi} + \beta g\). More...
template<class BinaryOp , class UnaryOp >
thrust::host_vector< double >	dg::geo::fieldaligned_evaluate (const aProductGeometry3d &grid, const CylindricalVectorLvl0 &vec, const BinaryOp &binary, const UnaryOp &unary, unsigned p0, unsigned rounds, double eps=1e-5)
	Evaluate a 2d functor and transform to all planes along the fieldlines More...
template<class BinaryOp , class UnaryOp >
MPI_Vector< thrust::host_vector< double > >	dg::geo::fieldaligned_evaluate (const aProductMPIGeometry3d &grid, const CylindricalVectorLvl0 &vec, const BinaryOp &binary, const UnaryOp &unary, unsigned p0, unsigned rounds, double eps=1e-5)
	Evaluate a 2d functor and transform to all planes along the fieldlines (MPI Version) More...

Detailed Description

\( \nabla_\parallel f\)

Typedef Documentation

FullLimiter

typedef ONE dg::geo::FullLimiter

Full Limiter means there is a limiter everywhere.

NoLimiter

typedef ZERO dg::geo::NoLimiter

No Limiter.

Enumeration Type Documentation

whichMatrix [1/3]

enum dg::geo::whichMatrix

Enum for the use in Fieldaligned.

Enumerator
einsPlus	plus interpolation in next plane
einsPlusT	transposed plus interpolation in previous plane
einsMinus	minus interpolation in previous plane
einsMinusT	transposed minus interpolation in next plane
zeroPlus	plus interpolation in the current plane
zeroMinus	minus interpolation in the current plane
zeroPlusT	transposed plus interpolation in the current plane
zeroMinusT	transposed minus interpolation in the current plane
zeroForw	from dg to transformed coordinates
einsPlus	plus interpolation in next plane
einsPlusT	transposed plus interpolation in previous plane
einsMinus	minus interpolation in previous plane
einsMinusT	transposed minus interpolation in next plane
zeroPlus	plus interpolation in the current plane
zeroMinus	minus interpolation in the current plane
zeroPlusT	transposed plus interpolation in the current plane
zeroMinusT	transposed minus interpolation in the current plane
zeroBack	from transfomred back to dg coordinates
einsPlus	plus interpolation in next plane
einsPlusT	transposed plus interpolation in previous plane
einsMinus	minus interpolation in previous plane
einsMinusT	transposed minus interpolation in next plane
zeroPlus	plus interpolation in the current plane
zeroMinus	minus interpolation in the current plane
zeroPlusT	transposed plus interpolation in the current plane
zeroMinusT	transposed minus interpolation in the current plane
zeroForw	from dg to transformed coordinates

whichMatrix [2/3]

enum dg::geo::whichMatrix

Enum for the use in Fieldaligned.

Enumerator
einsPlus	plus interpolation in next plane
einsPlusT	transposed plus interpolation in previous plane
einsMinus	minus interpolation in previous plane
einsMinusT	transposed minus interpolation in next plane
zeroPlus	plus interpolation in the current plane
zeroMinus	minus interpolation in the current plane
zeroPlusT	transposed plus interpolation in the current plane
zeroMinusT	transposed minus interpolation in the current plane
zeroForw	from dg to transformed coordinates
einsPlus	plus interpolation in next plane
einsPlusT	transposed plus interpolation in previous plane
einsMinus	minus interpolation in previous plane
einsMinusT	transposed minus interpolation in next plane
zeroPlus	plus interpolation in the current plane
zeroMinus	minus interpolation in the current plane
zeroPlusT	transposed plus interpolation in the current plane
zeroMinusT	transposed minus interpolation in the current plane
zeroBack	from transfomred back to dg coordinates
einsPlus	plus interpolation in next plane
einsPlusT	transposed plus interpolation in previous plane
einsMinus	minus interpolation in previous plane
einsMinusT	transposed minus interpolation in next plane
zeroPlus	plus interpolation in the current plane
zeroMinus	minus interpolation in the current plane
zeroPlusT	transposed plus interpolation in the current plane
zeroMinusT	transposed minus interpolation in the current plane
zeroForw	from dg to transformed coordinates

whichMatrix [3/3]

enum dg::geo::whichMatrix

Enum for the use in Fieldaligned.

Enumerator
einsPlus	plus interpolation in next plane
einsPlusT	transposed plus interpolation in previous plane
einsMinus	minus interpolation in previous plane
einsMinusT	transposed minus interpolation in next plane
zeroPlus	plus interpolation in the current plane
zeroMinus	minus interpolation in the current plane
zeroPlusT	transposed plus interpolation in the current plane
zeroMinusT	transposed minus interpolation in the current plane
zeroForw	from dg to transformed coordinates
einsPlus	plus interpolation in next plane
einsPlusT	transposed plus interpolation in previous plane
einsMinus	minus interpolation in previous plane
einsMinusT	transposed minus interpolation in next plane
zeroPlus	plus interpolation in the current plane
zeroMinus	minus interpolation in the current plane
zeroPlusT	transposed plus interpolation in the current plane
zeroMinusT	transposed minus interpolation in the current plane
zeroBack	from transfomred back to dg coordinates
einsPlus	plus interpolation in next plane
einsPlusT	transposed plus interpolation in previous plane
einsMinus	minus interpolation in previous plane
einsMinusT	transposed minus interpolation in next plane
zeroPlus	plus interpolation in the current plane
zeroMinus	minus interpolation in the current plane
zeroPlusT	transposed plus interpolation in the current plane
zeroMinusT	transposed minus interpolation in the current plane
zeroForw	from dg to transformed coordinates

Function Documentation

assign_bc_along_field_1st()

template<class FieldAligned , class container >

void dg::geo::assign_bc_along_field_1st	(	const FieldAligned &	fa,
		const container &	fm,
		const container &	fp,
		container &	fmg,
		container &	fpg,
		dg::bc	bound,
		std::array< double, 2 >	boundary_value = {0,0}
	)

Assign boundary conditions along magnetic field lines interpolating a 1st order polynomial (a line)

Template Parameters

FieldAligned
container

Parameters

fa
fm	fieldaligned(einsMinus, f, fm)
fp	fieldaligned(einsPlus, f, fp)
fmg	result (can alias fm)
fpg	result (can alias fp)
bound	either dg::NEU or dg::DIR (rest not implemented yet)
boundary_value	first value is for incoming fieldlines, second one for outgoing

Attention

The boundary condition given as a parameter to this function overrules the boundary conditions used in the given dg::geo::Fieldaligned object and the ones implicit in the einsPlus and einsMinus interpolations (Independently of whether the magnetic field was periodified or not). In this way it is possible to apply the same Fieldaligned object to several quantities with different boundary conditions and save quite a bit of memory consumption.

assign_bc_along_field_2nd()

template<class FieldAligned , class container >

void dg::geo::assign_bc_along_field_2nd	(	const FieldAligned &	fa,
		const container &	fm,
		const container &	f,
		const container &	fp,
		container &	fmg,
		container &	fpg,
		dg::bc	bound,
		std::array< double, 2 >	boundary_value = {0,0}
	)

Assign boundary conditions along magnetic field lines interpolating a 2nd order polynomial.

Call this function before one of the freestanding ds functions to replace the default boundary conditions with along field boundary conditions

This function replaces the values of the plus and minus fields that are outside of the domain with ghost values. These values are constructed by fitting a polynomial through the boundary point, the plus or minus point and the center point. For the exact resulting formula consult the parallel derivative writeup. This is achieved using masks that mark the points where fieldlines intersect the domain boundary and replace the interpolated boundary values.

Template Parameters

FieldAligned
container

Parameters

fa	this object will be used to get grid distances and masking regions
fm	fieldaligned(einsMinus, f, fm)
f
fp	fieldaligned(einsPlus, f, fp)
fmg	resulting eMinus field (can alias fm)
fpg	resulting ePlus field (can alias fp)
bound	either dg::NEU or dg::DIR (rest not implemented yet)
boundary_value	first value is for incoming fieldlines, second one for outgoing

Attention

The boundary condition given as a parameter to this function overrules the boundary conditions used in the given dg::geo::Fieldaligned object and the ones implicit in the einsPlus and einsMinus interpolations (Independently of whether the magnetic field was periodified or not). In this way it is possible to apply the same Fieldaligned object to several quantities with different boundary conditions and save quite a bit of memory consumption.

ds_average()

template<class FieldAligned , class container >

void dg::geo::ds_average	(	const FieldAligned &	fa,
		double	alpha,
		const container &	fm,
		const container &	fp,
		double	beta,
		container &	g
	)

Compute average along a fieldline \( g = \alpha \frac{f_{k+1} + f_{k-1}}{2} + \beta g\).

Note

The idea is to use this function on staggered grid implementations. There it can be used to convert a function from one grid to the staggered grid and back

Parameters

fa	this object will be used to get grid distances
alpha	Scalar
fm	fieldaligned(einsMinus, f, fm)
fp	fieldaligned(einsPlus, f, fp)
beta	Scalar
g	contains result on output (may alias input vectors)

Note

This function computes the same as the corresponding member function of dg::geo::DS but you have to compute the einsPlus and einsMinus interpolations from dg::geo::Fieldaligned yourself. The reasoning for this function is that you can re-use the latter interpolations if you compute for example both first and second derivative of a function or even if you compute it for difference boundary conditions using dg::geo::assign_bc_along_field_2nd, dg::geo::assign_bc_along_field_1st or dg::geo::swap_bc_perp

ds_backward()

template<class FieldAligned , class container >

void dg::geo::ds_backward	(	const FieldAligned &	fa,
		double	alpha,
		const container &	fm,
		const container &	f,
		double	beta,
		container &	g
	)

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

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

Parameters

fa	this object will be used to get grid distances
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

Parameters

fm	fieldaligned(einsMinus, f, fm)

Note

This function computes the same as the corresponding member function of dg::geo::DS but you have to compute the einsPlus and einsMinus interpolations from dg::geo::Fieldaligned yourself. The reasoning for this function is that you can re-use the latter interpolations if you compute for example both first and second derivative of a function or even if you compute it for difference boundary conditions using dg::geo::assign_bc_along_field_2nd, dg::geo::assign_bc_along_field_1st or dg::geo::swap_bc_perp

ds_backward2()

template<class FieldAligned , class container >

void dg::geo::ds_backward2	(	const FieldAligned &	fa,
		double	alpha,
		const container &	fmm,
		const container &	fm,
		const container &	f,
		double	beta,
		container &	g
	)

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

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

Parameters

fa	this object will be used to get grid distances
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

Parameters

fm	fieldaligned(einsMinus, f, fm)
fmm	twice apply fieldaligned(einsMinus, f, fm)

Note

This function computes the same as the corresponding member function of dg::geo::DS but you have to compute the einsPlus and einsMinus interpolations from dg::geo::Fieldaligned yourself. The reasoning for this function is that you can re-use the latter interpolations if you compute for example both first and second derivative of a function or even if you compute it for difference boundary conditions using dg::geo::assign_bc_along_field_2nd, dg::geo::assign_bc_along_field_1st or dg::geo::swap_bc_perp

ds_centered()

template<class FieldAligned , class container >

void dg::geo::ds_centered	(	const FieldAligned &	fa,
		double	alpha,
		const container &	fm,
		const container &	fp,
		double	beta,
		container &	g
	)

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

The formula used is \( g_i = \alpha \frac{v^\varphi}{2\Delta\varphi}(f_{i+1} - f_{i-1}) + \beta g_i \)

Parameters

fa	this object will be used to get grid distances
alpha	Scalar
fm	fieldaligned(einsMinus, f, fm)
fp	fieldaligned(einsPlus, f, fp)
beta	Scalar
g	contains result on output (may alias input vectors)

Note

This function computes the same as the corresponding member function of dg::geo::DS but you have to compute the einsPlus and einsMinus interpolations from dg::geo::Fieldaligned yourself. The reasoning for this function is that you can re-use the latter interpolations if you compute for example both first and second derivative of a function or even if you compute it for difference boundary conditions using dg::geo::assign_bc_along_field_2nd, dg::geo::assign_bc_along_field_1st or dg::geo::swap_bc_perp

ds_divBackward()

template<class FieldAligned , class container >

void dg::geo::ds_divBackward	(	const FieldAligned &	fa,
		double	alpha,
		const container &	fm,
		const container &	f,
		double	beta,
		container &	g
	)

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

backward derivative \( g_i = \alpha \frac{1}{\Delta\varphi\sqrt{G_i}}(\sqrt{G_{i}}v^\varphi_{i}f_{i} - \sqrt{G_{i-1}}v^\varphi_{i-1}f_{i-1}) + \beta g_i\)

Parameters

fa	this object will be used to get grid distances
alpha	Scalar
fm	fieldaligned(einsMinus, f, fm)
f	The vector to derive
beta	Scalar
g	contains result on output (may alias input vectors)

Note

This function computes the same as the corresponding member function of dg::geo::DS but you have to compute the einsPlus and einsMinus interpolations from dg::geo::Fieldaligned yourself. The reasoning for this function is that you can re-use the latter interpolations if you compute for example both first and second derivative of a function or even if you compute it for difference boundary conditions using dg::geo::assign_bc_along_field_2nd, dg::geo::assign_bc_along_field_1st or dg::geo::swap_bc_perp

ds_divCentered()

template<class FieldAligned , class container >

void dg::geo::ds_divCentered	(	const FieldAligned &	fa,
		double	alpha,
		const container &	fm,
		const container &	fp,
		double	beta,
		container &	g
	)

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

centered derivative \( g_i = \alpha \frac{1}{2\Delta\varphi\sqrt{G_i}}(\sqrt{G_{i+1}}v^\varphi_{i+1}f_{i+1} - \sqrt{G_{i-1}}v^\varphi_{i-1}f_{i-1}) + \beta g_i\)

Parameters

fa	this object will be used to get grid distances
alpha	Scalar
fm	fieldaligned(einsMinus, f, fm)
fp	fieldaligned(einsPlus, f, fp)
beta	Scalar
g	contains result on output (may alias input vectors)

Note

This function computes the same as the corresponding member function of dg::geo::DS but you have to compute the einsPlus and einsMinus interpolations from dg::geo::Fieldaligned yourself. The reasoning for this function is that you can re-use the latter interpolations if you compute for example both first and second derivative of a function or even if you compute it for difference boundary conditions using dg::geo::assign_bc_along_field_2nd, dg::geo::assign_bc_along_field_1st or dg::geo::swap_bc_perp

ds_divForward()

template<class FieldAligned , class container >

void dg::geo::ds_divForward	(	const FieldAligned &	fa,
		double	alpha,
		const container &	f,
		const container &	fp,
		double	beta,
		container &	g
	)

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

forward derivative \( g_i = \alpha \frac{1}{\Delta\varphi\sqrt{G_i}}(\sqrt{G_{i+1}}v^\varphi_{i+1}f_{i+1} - \sqrt{G_{i}}v^\varphi_{i}f_{i}) + \beta g_i\)

Parameters

fa	this object will be used to get grid distances
alpha	Scalar
f	The vector to derive
fp	fieldaligned(einsPlus, f, fp)
beta	Scalar
g	contains result on output (may alias input vectors)

Note

This function computes the same as the corresponding member function of dg::geo::DS but you have to compute the einsPlus and einsMinus interpolations from dg::geo::Fieldaligned yourself. The reasoning for this function is that you can re-use the latter interpolations if you compute for example both first and second derivative of a function or even if you compute it for difference boundary conditions using dg::geo::assign_bc_along_field_2nd, dg::geo::assign_bc_along_field_1st or dg::geo::swap_bc_perp

ds_forward()

template<class FieldAligned , class container >

void dg::geo::ds_forward	(	const FieldAligned &	fa,
		double	alpha,
		const container &	f,
		const container &	fp,
		double	beta,
		container &	g
	)

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

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

Parameters

fa	this object will be used to get grid distances
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

Parameters

fp	fieldaligned(einsPlus, f, fp)

Note

This function computes the same as the corresponding member function of dg::geo::DS but you have to compute the einsPlus and einsMinus interpolations from dg::geo::Fieldaligned yourself. The reasoning for this function is that you can re-use the latter interpolations if you compute for example both first and second derivative of a function or even if you compute it for difference boundary conditions using dg::geo::assign_bc_along_field_2nd, dg::geo::assign_bc_along_field_1st or dg::geo::swap_bc_perp

ds_forward2()

template<class FieldAligned , class container >

void dg::geo::ds_forward2	(	const FieldAligned &	fa,
		double	alpha,
		const container &	f,
		const container &	fp,
		const container &	fpp,
		double	beta,
		container &	g
	)

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

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

Parameters

fa	this object will be used to get grid distances
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

Parameters

fp	fieldaligned(einsPlus, f, fp)
fpp	twice apply fieldaligned(einsPlus, f, fp)

Note

This function computes the same as the corresponding member function of dg::geo::DS but you have to compute the einsPlus and einsMinus interpolations from dg::geo::Fieldaligned yourself. The reasoning for this function is that you can re-use the latter interpolations if you compute for example both first and second derivative of a function or even if you compute it for difference boundary conditions using dg::geo::assign_bc_along_field_2nd, dg::geo::assign_bc_along_field_1st or dg::geo::swap_bc_perp

ds_slope()

template<class FieldAligned , class container >

void dg::geo::ds_slope	(	const FieldAligned &	fa,
		double	alpha,
		const container &	fm,
		const container &	fp,
		double	beta,
		container &	g
	)

Compute simple slope along a fieldline \( g = \alpha v^\varphi\frac{f_{k+1} - f_{k-1}}{2\Delta\varphi} + \beta g\).

Note

The idea is to use this function on staggered grid implementations. There it can be used to compute the gradient from one grid onto the staggered grid and vice versa

Parameters

fa	this object will be used to get grid distances
alpha	Scalar
fm	fieldaligned(einsMinus, f, fm)
fp	fieldaligned(einsPlus, f, fp)
beta	Scalar
g	contains result on output (may alias input vectors)

Note

This function computes the same as the corresponding member function of dg::geo::DS but you have to compute the einsPlus and einsMinus interpolations from dg::geo::Fieldaligned yourself. The reasoning for this function is that you can re-use the latter interpolations if you compute for example both first and second derivative of a function or even if you compute it for difference boundary conditions using dg::geo::assign_bc_along_field_2nd, dg::geo::assign_bc_along_field_1st or dg::geo::swap_bc_perp

See also

Does exactly the same as ds_centered

dss_centered()

template<class FieldAligned , class container >

void dg::geo::dss_centered	(	const FieldAligned &	fa,
		double	alpha,
		const container &	fm,
		const container &	f,
		const container &	fp,
		double	beta,
		container &	g
	)

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

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 the boundary conditions are implemented by mirroring points perpendicular to the boundary, which has some drawbacks as to the numerical stability and toroidal resolution.

Parameters

fa	this object will be used to get grid distances
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

Parameters

fm	fieldaligned(einsMinus, f, fm)
fp	fieldaligned(einsPlus, f, fp)

Note

This function computes the same as the corresponding member function of dg::geo::DS but you have to compute the einsPlus and einsMinus interpolations from dg::geo::Fieldaligned yourself. The reasoning for this function is that you can re-use the latter interpolations if you compute for example both first and second derivative of a function or even if you compute it for difference boundary conditions using dg::geo::assign_bc_along_field_2nd, dg::geo::assign_bc_along_field_1st or dg::geo::swap_bc_perp

dssd_centered()

template<class FieldAligned , class container >

void dg::geo::dssd_centered	(	const FieldAligned &	fa,
		double	alpha,
		const container &	fm,
		const container &	f,
		const container &	fp,
		double	beta,
		container &	g
	)

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

The formula used is

\[ \Delta_\parallel f = \nabla\cdot (\vec v \nabla_\parallel f )\]

Parameters

fa	this object will be used to get grid distances
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

Parameters

fm	fieldaligned(einsMinus, f, fm)
fp	fieldaligned(einsPlus, f, fp)

Note

This function computes the same as the corresponding member function of dg::geo::DS but you have to compute the einsPlus and einsMinus interpolations from dg::geo::Fieldaligned yourself. The reasoning for this function is that you can re-use the latter interpolations if you compute for example both first and second derivative of a function or even if you compute it for difference boundary conditions using dg::geo::assign_bc_along_field_2nd, dg::geo::assign_bc_along_field_1st or dg::geo::swap_bc_perp

fieldaligned_evaluate() [1/2]

template<class BinaryOp , class UnaryOp >

thrust::host_vector< double > dg::geo::fieldaligned_evaluate	(	const aProductGeometry3d &	grid,
		const CylindricalVectorLvl0 &	vec,
		const BinaryOp &	binary,
		const UnaryOp &	unary,
		unsigned	p0,
		unsigned	rounds,
		double	eps = 1e-5
	)

Evaluate a 2d functor and transform to all planes along the fieldlines

The algorithm does the equivalent of the following:

• Evaluate the given BinaryOp on a 2d plane
• integrate fieldlines \( r N_z \Delta\varphi\) in both directions where \( N_z\) is the number of planes in the global 3d grid and \( r\) is the number of rounds. Then evaluate BinaryOp at the resulting points.
• Scale the transformations with \( u ( \pm (iN_z + j)\Delta\varphi) \), where u is the given UnarayOp, i in [0..r] is the round index and j in [0..Nz] is the plane index and \(\Delta\varphi\) is the angular distance given in the constructor (can be different from the actual grid distance hz!).
• Sum all transformations with the same plane index j , where the minus transformations get the inverted index \( N_z - j\).
• Shift the index by \( p_0\)

Attention

This version of the algorithm is more exact but much slower than the evaluate member contained in dg::geo::Fieldaligned

Template Parameters

BinaryOp	Binary Functor
UnaryOp	Unary Functor

Parameters

grid	The grid on which to integrate fieldlines.
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.
binary	Functor to evaluate in x-y
unary	Functor to evaluate in z

Note

unary is evaluated such that p0 corresponds to z=0, p0+1 corresponds to z=hz, p0-1 to z=-hz, ...

Parameters

p0	The index of the plane to start
rounds	The number of rounds r to follow a fieldline; can be zero, then the fieldlines are only followed within the current box ( no periodicity)
eps	The accuracy of the fieldline integrator

Returns

3d vector

fieldaligned_evaluate() [2/2]

template<class BinaryOp , class UnaryOp >

MPI_Vector< thrust::host_vector< double > > dg::geo::fieldaligned_evaluate	(	const aProductMPIGeometry3d &	grid,
		const CylindricalVectorLvl0 &	vec,
		const BinaryOp &	binary,
		const UnaryOp &	unary,
		unsigned	p0,
		unsigned	rounds,
		double	eps = 1e-5
	)

Evaluate a 2d functor and transform to all planes along the fieldlines (MPI Version)

The algorithm does the equivalent of the following:

• Evaluate the given BinaryOp on a 2d plane
• integrate fieldlines \( r N_z \Delta\varphi\) in both directions where \( N_z\) is the number of planes in the global 3d grid and \( r\) is the number of rounds. Then evaluate BinaryOp at the resulting points.
• Scale the transformations with \( u ( \pm (iN_z + j)\Delta\varphi) \), where u is the given UnarayOp, i in [0..r] is the round index and j in [0..Nz] is the plane index and \(\Delta\varphi\) is the angular distance given in the constructor (can be different from the actual grid distance hz!).
• Sum all transformations with the same plane index j , where the minus transformations get the inverted index \( N_z - j\).
• Shift the index by \( p_0\)

Attention

This version of the algorithm is more exact but much slower than the evaluate member contained in dg::geo::Fieldaligned

Template Parameters

BinaryOp	Binary Functor
UnaryOp	Unary Functor

Parameters

grid	The grid on which to integrate fieldlines.
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.
binary	Functor to evaluate in x-y
unary	Functor to evaluate in z

Note

unary is evaluated such that p0 corresponds to z=0, p0+1 corresponds to z=hz, p0-1 to z=-hz, ...

Parameters

p0	The index of the plane to start
rounds	The number of rounds r to follow a fieldline; can be zero, then the fieldlines are only followed within the current box ( no periodicity)
eps	The accuracy of the fieldline integrator

Returns

3d vector

swap_bc_perp()

template<class FieldAligned , class container >

void dg::geo::swap_bc_perp	(	const FieldAligned &	fa,
		const container &	fm,
		const container &	fp,
		container &	fmg,
		container &	fpg
	)

Swap the perp boundary condition.

This function multiplies (-1) to every value that lies outside the box. This effectively swaps the boundary conditions in the Fourier boundary mode, i.e. if NEU was used in Fieldaligned, then now they are DIR and vice versa.

Template Parameters

FieldAligned
container

Parameters

fa	this object will be used to get masking regions
fm	fieldaligned(einsMinus, f, fm)
fp	fieldaligned(einsPlus, f, fp)
fmg	resulting eMinus field (can alias fm)
fpg	resulting ePlus field (can alias fp)

Note

The motivation for this function is to avoid reconstructing a fieldaligned object since its constructor is quite expensive in runtime and memory. The idea is to construct Fieldaligned with NEU boundary conditions and then use this function to swap them to DIR whenever needed.

The swapping of bc is not exactly (but almost) the same as re-constructing the fieldaligned object in case the support operator is used (mx, my > 1), this is because the swap is performed only on the coarse grid. In all tests the error is negligible so far.