dg::ArakawaX< Geometry, Matrix, Container > Struct Template Reference

Arakawa's scheme for Poisson bracket \( \{ f,g\} \). More...

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

Public Member Functions
	ArakawaX ()=default
	ArakawaX (const Geometry &g)
	Create Arakawa on a grid. More...
	ArakawaX (const Geometry &g, bc bcx, bc bcy)
	Create Arakawa on a grid using different boundary conditions. More...
template<class ... Params>
void	construct (Params &&...ps)
	Perfect forward parameters to one of the constructors. More...
template<class ContainerType0 , class ContainerType1 , class ContainerType2 >
void	operator() (const ContainerType0 &lhs, const ContainerType1 &rhs, ContainerType2 &result)
	Compute Poisson bracket. More...
template<class ContainerType0 , class ContainerType1 , class ContainerType2 >
void	operator() (value_type alpha, const ContainerType0 &lhs, const ContainerType1 &rhs, value_type beta, ContainerType2 &result)
template<class ContainerType0 >
void	set_chi (const ContainerType0 &new_chi)
	Change Chi. More...
const Matrix &	dx () const
	Return internally used x - derivative. More...
const Matrix &	dy () const
	Return internally used y - derivative. More...

Detailed Description

template<class Geometry, class Matrix, class Container>
struct dg::ArakawaX< Geometry, Matrix, Container >

Arakawa's scheme for Poisson bracket \( \{ f,g\} \).

Computes

\[ \{f,g\} := \chi/\sqrt{g_{2d}}\left(\partial_x f\partial_y g - \partial_y f\partial_x g\right) \]

where \( g_{2d} = g/g_{zz}\) is the two-dimensional volume element of the plane in 2x1 product space and \( \chi\) is an optional factor. If \( \chi=1\), then the discretization conserves, mass, energy and enstrophy.

const double lx = 2*M_PI;
const double ly = 2*M_PI;
dg::bc bcx = dg::PER;
dg::bc bcy = dg::PER;
 
double left( double x, double y) {
    return sin(x)*cos(y);
}
double right( double x, double y) {
    return sin(y)*cos(x); 
}
double jacobian( double x, double y) {
    return cos(x)*cos(y)*cos(x)*cos(y) - sin(x)*sin(y)*sin(x)*sin(y);
}

    // create a Cartesian grid on the domain [0,lx]x[0,ly]
    const dg::CartesianGrid2d grid( 0, lx, 0, ly, n, Nx, Ny, bcx, bcy);
 
    // evaluate left and right hand side on the grid
    const dg::DVec lhs = dg::construct<dg::DVec>( dg::evaluate( left, grid));
    const dg::DVec rhs = dg::construct<dg::DVec>( dg::evaluate( right, grid));
    dg::DVec jac(lhs);
 
    // create an Arakawa object
    dg::ArakawaX<dg::aGeometry2d, dg::DMatrix, dg::DVec> arakawa( grid);
 
    //apply arakawa scheme
    arakawa( lhs, rhs, jac);

Note

This is the algorithm published in L. Einkemmer, M. Wiesenberger A conservative discontinuous Galerkin scheme for the 2D incompressible Navier-Stokes equations Computer Physics Communications 185, 2865-2873 (2014)

See also

A discussion of this and other advection schemes can also be found here https://mwiesenberger.github.io/advection

Template Parameters

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

Member Typedef Documentation

container_type

template<class Geometry , class Matrix , class Container >

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

geometry_type

template<class Geometry , class Matrix , class Container >

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

matrix_type

template<class Geometry , class Matrix , class Container >

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

value_type

template<class Geometry , class Matrix , class Container >

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

Constructor & Destructor Documentation

ArakawaX() [1/3]

template<class Geometry , class Matrix , class Container >

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

default

ArakawaX() [2/3]

template<class Geometry , class Matrix , class Container >

dg::ArakawaX< Geometry, Matrix, Container >::ArakawaX

(

const Geometry &

g

)

Create Arakawa on a grid.

Parameters

g

The grid

Note

chi defaults to 1

ArakawaX() [3/3]

template<class Geometry , class Matrix , class Container >

dg::ArakawaX< Geometry, Matrix, Container >::ArakawaX	(	const Geometry &	g,
		bc	bcx,
		bc	bcy
	)

Create Arakawa on a grid using different boundary conditions.

Parameters

g	The grid
bcx	The boundary condition in x
bcy	The boundary condition in y

Note

chi defaults to 1

Member Function Documentation

construct()

template<class Geometry , class Matrix , class Container >

template<class ... Params>

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

inline

Perfect forward parameters to one of the constructors.

Template Parameters

Params

deduced by the compiler

Parameters

ps	parameters forwarded to constructors

dx()

template<class Geometry , class Matrix , class Container >

const Matrix & dg::ArakawaX< Geometry, Matrix, Container >::dx ( ) const

inline

Return internally used x - derivative.

The same as a call to dg::create::dx( g, bcx)

Returns

derivative

dy()

template<class Geometry , class Matrix , class Container >

const Matrix & dg::ArakawaX< Geometry, Matrix, Container >::dy ( ) const

inline

Return internally used y - derivative.

The same as a call to dg::create::dy( g, bcy)

Returns

derivative

operator()() [1/2]

template<class Geometry , class Matrix , class Container >

template<class ContainerType0 , class ContainerType1 , class ContainerType2 >

void dg::ArakawaX< Geometry, Matrix, Container >::operator() ( const ContainerType0 &  lhs, const ContainerType1 &  rhs, ContainerType2 &  result  )

inline

Compute Poisson bracket.

Computes

\[ [f,g] := 1/\sqrt{g_{2d}}\left(\partial_x f\partial_y g - \partial_y f\partial_x g\right) \]

where \( g_{2d} = g/g_{zz}\) is the two-dimensional volume element of the plane in 2x1 product space.

Parameters

lhs	left hand side in x-space
rhs	rights hand side in x-space
result	Poisson's bracket in x-space

Note

memops: 30

Template Parameters

ContainerTypes

must be usable with Container in The dg dispatch system

operator()() [2/2]

template<class Geometry , class Matrix , class Container >

template<class ContainerType0 , class ContainerType1 , class ContainerType2 >

void dg::ArakawaX< Geometry, Matrix, Container >::operator()	(	value_type	alpha,
		const ContainerType0 &	lhs,
		const ContainerType1 &	rhs,
		value_type	beta,
		ContainerType2 &	result
	)

set_chi()

template<class Geometry , class Matrix , class Container >

template<class ContainerType0 >

void dg::ArakawaX< Geometry, Matrix, Container >::set_chi ( const ContainerType0 &  new_chi )

inline

Change Chi.

Parameters

new_chi

The new chi

Template Parameters

ContainerTypes

must be usable with Container in The dg dispatch system

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

• arakawa.h