dg::geo::DS< ProductGeometry, IMatrix, container > Struct Template Reference

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.
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.
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.
	DS (FA fieldaligned)
	Re-construct from a given Fieldaligned object.
template<class ... Params>
void	construct (Params &&...ps)
	Perfect forward parameters to one of the constructors.
void	set_boundaries (dg::bc bcz, double left, double right)
	Set boundary conditions in the limiter region.
void	set_boundaries (dg::bc bcz, const container &left, const container &right)
	Set boundary conditions in the limiter region.
void	set_boundaries (dg::bc bcz, const container &global, double scal_left, double scal_right)
	Set boundary conditions in the limiter region.
void	forward (double alpha, const container &f, double beta, container &g)
	forward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)
void	forward2 (double alpha, const container &f, double beta, container &g)
	2nd order forward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)
void	backward (double alpha, const container &f, double beta, container &g)
	backward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)
void	backward2 (double alpha, const container &f, double beta, container &g)
	2nd order backward derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)
void	centered (double alpha, const container &f, double beta, container &g)
	centered derivative \( g = \alpha \vec v \cdot \nabla f + \beta g\)
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.
void	backward (const container &f, container &g)
	backward derivative \( g = \vec v \cdot \nabla f \)
void	forward (const container &f, container &g)
	forward derivative \( g = \vec v \cdot \nabla f \)
void	centered (const container &f, container &g)
	centered derivative \( g = \vec v \cdot \nabla f \)
void	divForward (double alpha, const container &f, double beta, container &g)
	forward divergence \( g = \alpha \nabla\cdot(\vec v f) + \beta g\)
void	divBackward (double alpha, const container &f, double beta, container &g)
	backward divergence \( g = \alpha \nabla\cdot(\vec v f) + \beta g\)
void	divCentered (double alpha, const container &f, double beta, container &g)
	centered divergence \( g = \alpha \nabla\cdot(\vec v f) + \beta g\)
void	divForward (const container &f, container &g)
	forward divergence \( g = \nabla\cdot(\vec v f)\)
void	divBackward (const container &f, container &g)
	backward divergence \( g = \nabla\cdot(\vec v f)\)
void	divCentered (const container &f, container &g)
	centered divergence \( g = \nabla\cdot(\vec v f)\)
void	ds (dg::direction dir, const container &f, container &g)
	Discretizes \( g = \vec v\cdot \nabla f \).
void	ds (dg::direction dir, double alpha, const container &f, double beta, container &g)
	Discretizes \( g = \alpha \vec v\cdot \nabla f + \beta g \).
void	div (dg::direction dir, const container &f, container &g)
	Discretizes \( g = \nabla \cdot ( \vec v f) \).
void	div (dg::direction dir, double alpha, const container &f, double beta, container &g)
	Discretizes \( g = \alpha \nabla \cdot( \vec v f) + \beta g \).
void	symv (const container &f, container &g)
	Discretizes \( g = \nabla\cdot ( \vec v \vec v \cdot \nabla f )\).
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.
void	dss (const container &f, container &g)
	Discretizes \( g = (\vec v\cdot \nabla)^2 f \).
void	dss (double alpha, const container &f, double beta, container &g)
	Discretizes \( g = \alpha (\vec v\cdot \nabla)^2 f + \beta g \).
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.
void	dssd (double alpha, const container &f, double beta, container &g)
	Same as dg::geo::dssd_centered.
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.
const container &	weights () const
	The volume form with dG weights.
FA &	fieldaligned ()
	access the underlying Fieldaligned object
const FA &	fieldaligned () const

Detailed Description

template<class ProductGeometry, class IMatrix, class container>
struct dg::geo::DS< ProductGeometry, IMatrix, 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::x::CylindricalGrid3d g3d( R_0-a, R_0+a, -a, a, 0, 2.*M_PI, n, Nx, Ny, Nz, dg::NEU, dg::NEU, dg::PER
#ifdef WITH_MPI
    , comm
#endif
    );
    //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::x::aProductGeometry3d,dg::x::IDMatrix,dg::x::DVec>  dsFA(
            bhat, g3d, dg::NEU, dg::NEU, dg::geo::NoLimiter(), 1e-8, mx[0], mx[1],
            -1, method);
    dg::geo::DS<dg::x::aProductGeometry3d, dg::x::IDMatrix, dg::x::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).

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
container	The container-class on which the interpolation matrix operates on dg::HVec, or dg::DVec, dg::MHVec, or dg::MDVec

Note

The dg::TensorTraits are defined for this class and thus DS is usable as a Matrix class (calling symv )

See also

The pdf parallel derivative writeup

Member Typedef Documentation

FA

template<class ProductGeometry , class IMatrix , class container >

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

conveniently abbreviates underlying Fieldaligned type

Constructor & Destructor Documentation

DS() [1/4]

template<class ProductGeometry , class IMatrix , class container >

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

inline

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

DS() [2/4]

template<class ProductGeometry , class IMatrix , class container >

template<class Limiter >

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

Parameters

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

See also

Fieldaligned

DS() [3/4]

template<class ProductGeometry , class IMatrix , class container >

template<class Limiter >

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

Parameters

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

See also

Fieldaligned

DS() [4/4]

template<class ProductGeometry , class IMatrix , class container >

dg::geo::DS< ProductGeometry, IMatrix, 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 container >

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

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

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

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

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

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

template<class ... Params>

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

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

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

Note

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 container >

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

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

Note

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 container >

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

Note

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 container >

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

Note

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 container >

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

Note

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 container >

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

Note

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 container >

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

Note

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 container >

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

Note

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 container >

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

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

ds() [2/2]

template<class ProductGeometry , class IMatrix , class container >

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

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() [1/2]

template<class ProductGeometry , class IMatrix , class container >

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

inline

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

Same as forward followed by backward

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 container >

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

inline

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

Same as forward followed by backward

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 container >

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

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

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

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

inline

access the underlying Fieldaligned object

Returns

acces to Fieldaligned object

fieldaligned() [2/2]

template<class ProductGeometry , class IMatrix , class container >

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

inline

forward() [1/2]

template<class ProductGeometry , class IMatrix , class container >

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

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

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

set_boundaries() [1/3]

template<class ProductGeometry , class IMatrix , class container >

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

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

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

symv() [1/2]

template<class ProductGeometry , class IMatrix , class container >

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

inline

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

using the dg::geo::dssd_centered method.

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

Note

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)

The dg::TensorTraits are defined for this class and thus DS is usable as a Matrix class (calling symv )

symv() [2/2]

template<class ProductGeometry , class IMatrix , class container >

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

using the dg::geo::dssd_centered method.

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

Note

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)

The dg::TensorTraits are defined for this class and thus DS is usable as a Matrix class (calling symv )

weights()

template<class ProductGeometry , class IMatrix , class container >

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

inline

The volume form with dG weights.

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

• ds.h