Module containing B-spline utilities for solvers. More...

Data Types
interface	rebase

Functions/Subroutines
subroutine, public	initpde (nidbas, ns_, ds_, smin_, nchi_, dchi_, nphip_, dphi_, mmin_, mmax_, iquafor)
	Initialize the module. More...

real function, dimension(0:p)	splines (z)
	Evaluate basis functions on point z in interval [0, 1]. More...

subroutine, public	analytical_rhs (rhs)
	Project an analytical function on B-splines. More...

subroutine	rebase_r1d (v, nsel_which_mat)
	Multiply v by a boundary change of basis matrix. More...

subroutine	rebase_r2d (v, nsel_which_mat)
	Call rebase_r1d for each index of second dimension of v. More...

subroutine	rebase_c1d (v, nsel_which_mat)
	Complex version of rebase_r1d. More...

subroutine	rebase_c2d (v, nsel_which_mat)
	Call rebase_c1d for each index of second dimension of v. More...

subroutine, public	impose_bc (v)
	Impose boundary conditions on a vector before backsolve. More...

real function	func_int_base (zs, zj)
	Flux surface average of theta basis function. More...

Variables
integer, save, public	width_f
	Total width of diagonal filter as defined using `mmin_f` and `mmax_f` in the `def_matrix_filter` subroutine. Note that it may differ from `width_m` defined using the filter array. More...

integer, save, public	nbmatf
	number of matrix per proc More...

integer, save	p
	Spline order. More...

integer, save	nsupp
	support in number of intervals of the spline function More...

integer, dimension(:), allocatable, save	nlocad
	local 1D addresses of contributions of one spline function More...

integer, save	ns
	Number of intervals in radial direction. More...

real, save	ds
	Mesh size in radial direction. More...

real, save	smin
	Lower bound of radial direction. More...

integer, save	nchi
	Number of intervals in poloidal direction. More...

real, save	dchi
	Mesh size in poloidal direction. More...

integer, save	nphip
	Number of local intervals in toroidal direction. More...

real, save	dphi
	Mesh size in toroidal direction. More...

integer, dimension(:,:), allocatable, save	mmin
	minimum m value for Fourier solver More...

integer, dimension(:,:), allocatable, save	mmax
	maximum m value for Fourier solver More...

real, dimension(:), allocatable, save	splines_g
	basis functions at grid points More...

real, dimension(:,:), allocatable, save	splines_q
	basis functions at quadrature points More...

real, dimension(10), save	zqp
	Quadrature points. More...

real, dimension(10), save	zqw
	Quadrature weights. More...

integer, save	nq
	Number of quadrature points. More...

real, dimension(:,:), allocatable, save, public	zwavg
	Weight function surface average. More...

Detailed Description

Module containing B-spline utilities for solvers.

Function/Subroutine Documentation

◆ analytical_rhs()

subroutine, public pdespline::analytical_rhs ( real, dimension(0:ns+p-1, 0:nchi+p-1, 0:nphip+p-1), intent(out) rhs )

Project an analytical function on B-splines.

Date: 02.2018

\( b_{ijk} = \int J_{\theta^\star s\varphi}\Lambda_i(s)\Lambda_j(\theta^\star)\Lambda_k(\varphi)f(s,\theta^\star,\varphi) ds d\theta^\star d\varphi \)

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◆ func_int_base()

real function pdespline::func_int_base	(	real, intent(in)	zs,
		integer, intent(in)	zj
	)

private

Flux surface average of theta basis function.

Parameters

[in]	zs	s position
[in]	zj	Base index

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◆ impose_bc()

subroutine, public pdespline::impose_bc ( complex, dimension(:,:), intent(inout) v )

Impose boundary conditions on a vector before backsolve.

Author: N. Ohana

Date: 12.2017

Parameters

[in,out] v Vector on which boundary conditions are applied, of size (ns+p)*width_f in first dimension

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◆ initpde()

subroutine, public pdespline::initpde	(	integer, intent(in)	nidbas,
		integer, intent(in)	ns_,
		real, intent(in)	ds_,
		real, intent(in)	smin_,
		integer, intent(in)	nchi_,
		real, intent(in)	dchi_,
		integer, intent(in)	nphip_,
		real, intent(in)	dphi_,
		integer, dimension(:,:), intent(in)	mmin_,
		integer, dimension(:,:), intent(in)	mmax_,
		integer, intent(in)	iquafor
	)

Initialize the module.

Parameters

[in]	nidbas	Spline order
[in]	ns_	Number of intervals in radial direction
[in]	ds_	Mesh size in radial direction
[in]	smin_	Lower bound of radial direction
[in]	nchi_	Number of intervals in poloidal direction
[in]	dchi_	Mesh size in poloidal direction
[in]	nphip_	Number of local intervals in toroidal direction
[in]	dphi_	Mesh size in toroidal direction
[in]	mmin_	minimum m value for Fourier solver
[in]	mmax_	maximum m value for Fourier solver
[in]	iquafor	order of Gaussian integration

Note: It is important to use the same quadrature method as in matrix assembly.

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◆ rebase_c1d()

subroutine pdespline::rebase_c1d	(	complex, dimension(:), intent(inout)	v,
		integer, intent(in)	nsel_which_mat
	)

private

Complex version of rebase_r1d.

Author: N. Ohana

Date: 12.2017

Parameters

[in]	nsel_which_mat	Refer to rebase_r1d
[in,out]	v	Vector to be multiplied, of size (ns+p)*width_f

◆ rebase_c2d()

subroutine pdespline::rebase_c2d	(	complex, dimension(:,:), intent(inout)	v,
		integer, intent(in)	nsel_which_mat
	)

private

Call rebase_c1d for each index of second dimension of v.

Author: N. Ohana

Date: 12.2017

Parameters

[in]	nsel_which_mat	Refer to rebase_c1d
[in,out]	v	Vector to be multiplied, of size (ns+p)*width_f in first dimension

◆ rebase_r1d()

subroutine pdespline::rebase_r1d	(	real, dimension(:), intent(inout)	v,
		integer, intent(in)	nsel_which_mat
	)

private

Multiply v by a boundary change of basis matrix.

Author: N. Ohana

Date: 12.2017

Parameters

[in]	nsel_which_mat	Which matrix to use at each boundary: `NSEL_T` for \( T = \begin{pmatrix} 1/c_1 & 0 & 0 & \dots & 0 \\\\ c_2' & 1 & 0 & \dots & 0 \\\\ c_3' & 0 & 1 & \ddots & \vdots \\\\ \vdots & \vdots & \ddots & \ddots & 0 \\\\ c_p' & 0 & \dots & 0 & 1 \end{pmatrix} \) `NSEL_T_TRANSP` for \( T^{t} = \begin{pmatrix} 1/c_1 & c_2' & c_3' & \dots & c_p' \\\\ 0 & 1 & 0 & \dots & 0 \\\\ 0 & 0 & 1 & \ddots & \vdots \\\\ \vdots & \vdots & \ddots & \ddots & 0 \\\\ 0 & 0 & \dots & 0 & 1 \end{pmatrix} \) `NSEL_T_INV` for \( T^{-1} = \begin{pmatrix} c_1 & 0 & 0 & \dots & 0 \\\\ c_2 & 1 & 0 & \dots & 0 \\\\ c_3 & 0 & 1 & \ddots & \vdots \\\\ \vdots & \vdots & \ddots & \ddots & 0 \\\\ c_p & 0 & \dots & 0 & 1 \end{pmatrix} \) where \( c_i = \Lambda_i^{(p)}(0) \) and \( c_i' = -c_i/c_1 \)
[in,out]	v	Vector to be multiplied, of size (ns+p)*nchi

◆ rebase_r2d()

subroutine pdespline::rebase_r2d	(	real, dimension(:,:), intent(inout)	v,
		integer, intent(in)	nsel_which_mat
	)

private

Call rebase_r1d for each index of second dimension of v.

Author: N. Ohana

Date: 12.2017

Parameters

[in]	nsel_which_mat	Refer to rebase_r1d
[in,out]	v	Vector to be multiplied, of size (ns+p)*nchi in first dimension

◆ splines()

real function, dimension(0:p) pdespline::splines ( real, intent(in) z )

private

Evaluate basis functions on point z in interval [0, 1].

Parameters

[in] z Point in [0, 1] where the functions are evaluated

Returns: The values of the contributing functions

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Variable Documentation

◆ dchi

real, save pdespline::dchi

private

Mesh size in poloidal direction.

◆ dphi

real, save pdespline::dphi

private

Mesh size in toroidal direction.

◆ ds

real, save pdespline::ds

private

Mesh size in radial direction.

◆ mmax

integer, dimension(:,:), allocatable, save pdespline::mmax

private

maximum m value for Fourier solver

◆ mmin

integer, dimension(:,:), allocatable, save pdespline::mmin

private

minimum m value for Fourier solver

◆ nbmatf

integer, save, public pdespline::nbmatf

number of matrix per proc

◆ nchi

integer, save pdespline::nchi

private

Number of intervals in poloidal direction.

◆ nlocad

integer, dimension(:), allocatable, save pdespline::nlocad

private

local 1D addresses of contributions of one spline function

◆ nphip

integer, save pdespline::nphip

private

Number of local intervals in toroidal direction.

◆ nq

integer, save pdespline::nq

private

Number of quadrature points.

◆ ns

integer, save pdespline::ns

private

Number of intervals in radial direction.

◆ nsupp

integer, save pdespline::nsupp

private

support in number of intervals of the spline function

◆ p

integer, save pdespline::p

private

Spline order.

◆ smin

real, save pdespline::smin

private

Lower bound of radial direction.

◆ splines_g

real, dimension(:), allocatable, save pdespline::splines_g

private

basis functions at grid points

◆ splines_q

real, dimension(:,:), allocatable, save pdespline::splines_q

private

basis functions at quadrature points

◆ width_f

integer, save, public pdespline::width_f

Total width of diagonal filter as defined using mmin_f and mmax_f in the def_matrix_filter subroutine. Note that it may differ from width_m defined using the filter array.

◆ zqp

real, dimension(10), save pdespline::zqp

private

Quadrature points.

◆ zqw

real, dimension(10), save pdespline::zqw

private

Quadrature weights.

◆ zwavg

real, dimension(:,:), allocatable, save, public pdespline::zwavg

Weight function surface average.

Data Types

Functions/Subroutines

Variables

Detailed Description

Function/Subroutine Documentation

◆ analytical_rhs()

◆ func_int_base()

◆ impose_bc()

◆ initpde()

◆ rebase_c1d()

◆ rebase_c2d()

◆ rebase_r1d()

◆ rebase_r2d()

◆ splines()

Variable Documentation

◆ dchi

◆ dphi

◆ ds

◆ mmax

◆ mmin

◆ nbmatf

◆ nchi

◆ nlocad

◆ nphip

◆ nq

◆ ns

◆ nsupp

◆ p

◆ smin

◆ splines_g

◆ splines_q

◆ width_f

◆ zqp

◆ zqw

◆ zwavg