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.
real(kind=r8) function, dimension(0:p)	splines (z)
	Evaluate basis functions on point z in interval [0, 1].
subroutine, public	analytical_rhs (rhs)
	Project an analytical function on B-splines.
subroutine	rebase_r1d (v, nsel_which_mat)
	Multiply v by a boundary change of basis matrix.
subroutine	rebase_r2d (v, nsel_which_mat)
	Call rebase_r1d for each index of second dimension of v.
subroutine	rebase_c1d (v, nsel_which_mat)
	Complex version of rebase_r1d.
subroutine	rebase_c2d (v, nsel_which_mat)
	Call rebase_c1d for each index of second dimension of v.
subroutine, public	impose_bc (v)
	Impose boundary conditions on a vector before backsolve.
real(kind=r8) function	func_int_base (zs, zj)
	Flux surface average of theta basis function.

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.
integer, save, public	nbmatf
	number of matrix per proc
integer, save	p
	Spline order.
integer, save	nsupp
	support in number of intervals of the spline function
integer, dimension(:), allocatable, save	nlocad
	local 1D addresses of contributions of one spline function
integer, save	ns
	Number of intervals in radial direction.
real(kind=r8), save	ds
	Mesh size in radial direction.
real(kind=r8), save	smin
	Lower bound of radial direction.
integer, save	nchi
	Number of intervals in poloidal direction.
real(kind=r8), save	dchi
	Mesh size in poloidal direction.
integer, save	nphip
	Number of local intervals in toroidal direction.
real(kind=r8), save	dphi
	Mesh size in toroidal direction.
integer, dimension(:,:), allocatable, save	mmin
	minimum m value for Fourier solver
integer, dimension(:,:), allocatable, save	mmax
	maximum m value for Fourier solver
real(kind=r8), dimension(:), allocatable, save	splines_g
	basis functions at grid points
real(kind=r8), dimension(:,:), allocatable, save	splines_q
	basis functions at quadrature points
real(kind=r8), dimension(10), save	zqp
	Quadrature points.
real(kind=r8), dimension(10), save	zqw
	Quadrature weights.
integer, save	nq
	Number of quadrature points.
real(kind=r8), dimension(:,:), allocatable, save, public	zwavg
	Weight function surface average.

Detailed Description

Module containing B-spline utilities for solvers.

Function/Subroutine Documentation

◆ analytical_rhs()

subroutine, public pdespline::analytical_rhs ( real(kind=r8), 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(kind=r8) function pdespline::func_int_base	(	real(kind=r8), 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(kind=r8), 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(kind=r8), intent(in)	ds_,
		real(kind=r8), intent(in)	smin_,
		integer, intent(in)	nchi_,
		real(kind=r8), intent(in)	dchi_,
		integer, intent(in)	nphip_,
		real(kind=r8), 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(kind=r8), 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(kind=r8), 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(kind=r8), 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(kind=r8), 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(kind=r8) function, dimension(0:p) pdespline::splines ( real(kind=r8), 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(kind=r8), save pdespline::dchi

private

Mesh size in poloidal direction.

◆ dphi

real(kind=r8), save pdespline::dphi

private

Mesh size in toroidal direction.

◆ ds

real(kind=r8), 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(kind=r8), save pdespline::smin

private

Lower bound of radial direction.

◆ splines_g

real(kind=r8), dimension(:), allocatable, save pdespline::splines_g

private

basis functions at grid points

◆ splines_q

real(kind=r8), 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(kind=r8), dimension(10), save pdespline::zqp

private

Quadrature points.

◆ zqw

real(kind=r8), dimension(10), save pdespline::zqw

private

Quadrature weights.

◆ zwavg

real(kind=r8), 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