Radial derivatives and integration

radial_derivatives.f90

Description

Radial derivatives functions

Quick access

Variables

get_dcheb, get_dr, work_1d_real

Routines

bulk_to_ghost(), exch_ghosts(), finalize_der_arrays(), get_bound_vals(), get_dcheb_complex(), get_dcheb_real_1d(), get_ddcheb(), get_dddcheb(), get_ddddr_ghost(), get_dddr(), get_ddr(), get_ddr_ghost(), get_ddr_rloc(), get_dr_complex(), get_dr_real_1d(), get_dr_rloc(), initialize_der_arrays()

Needed modules

  • constants (zero(), one(), three()): module containing constants and parameters used in the code.

  • precision_mod: This module controls the precision used in MagIC

  • mem_alloc: This little module is used to estimate the global memory allocation used in MagIC

  • cosine_transform_odd: This module contains the built-in type I discrete Cosine Transforms. This implementation is based on Numerical Recipes and FFTPACK. This only works for n_r_max-1 = 2**a 3**b 5**c, with a,b,c integers….

  • radial_scheme (type_rscheme()): This is an abstract type that defines the radial scheme used in MagIC

  • logic (l_finite_diff()): Module containing the logicals that control the run

  • parallel_mod: This module contains the blocking information

  • useful (abortrun()): This module contains several useful routines.

Variables

  • radial_der/get_dcheb [public]
  • radial_der/get_dr [public]
  • radial_der/work (*,*) [complex,private/allocatable]
  • radial_der/work_1d_real (*) [real,private/allocatable]

Subroutines and functions

subroutine radial_der/initialize_der_arrays(n_r_max, llm, ulm)

Allocate work arrays to compute derivatives

Parameters

  • n_r_max [integer ,in]

  • llm [integer ,in]

  • ulm [integer ,in]

Called from

magic

subroutine radial_der/finalize_der_arrays()

Deallocate work arrays

Called from

magic

subroutine radial_der/get_dcheb_complex(f, df, n_f_max, n_f_start, n_f_stop, n_r_max, n_cheb_max, d_fac)

Returns Chebyshev coeffitients of first derivative df and second derivative ddf for a function whose cheb-coeff. are given as columns in array f(n_f_max,n_r_max).

Parameters

  • f (n_f_max,n_r_max) [complex ,in]

  • df (n_f_max,n_r_max) [complex ,out]

  • n_f_max [integer ,in,] :: Max no of functions

  • n_f_start [integer ,in] :: No of function to start with

  • n_f_stop [integer ,in] :: No of function to stop with

  • n_r_max [integer ,in,] :: second dimension of f,df,ddf

  • n_cheb_max [integer ,in] :: Number of cheb modes

  • d_fac [real ,in] :: factor for interval mapping

subroutine radial_der/get_dcheb_real_1d(f, df, n_r_max, n_cheb_max, d_fac)
Parameters

  • f (n_r_max) [real ,in]

  • df (n_r_max) [real ,out]

  • n_r_max [integer ,in,optional/default=len(f)] :: Dimension of f,df,ddf

  • n_cheb_max [integer ,in] :: Number of cheb modes

  • d_fac [real ,in] :: factor for interval mapping

subroutine radial_der/get_ddcheb(f, df, ddf, n_f_max, n_f_start, n_f_stop, n_r_max, n_cheb_max, d_fac)

Returns Chebyshev coefficients of first derivative df and second derivative ddf for a function whose cheb-coeff. are given as columns in array f(n_c_tot,n_r_max).

Parameters

  • f (n_f_max,n_r_max) [complex ,in]

  • df (n_f_max,n_r_max) [complex ,out]

  • ddf (n_f_max,n_r_max) [complex ,out]

  • n_f_max [integer ,in,] :: First dimension of f,df,ddf

  • n_f_start [integer ,in] :: No of column to start with

  • n_f_stop [integer ,in] :: No of column to stop with

  • n_r_max [integer ,in,] :: second dimension of f,df,ddf

  • n_cheb_max [integer ,in] :: Number of cheb modes

  • d_fac [real ,in] :: factor for interval mapping

Called from

get_ddr()

subroutine radial_der/get_dddcheb(f, df, ddf, dddf, n_f_max, n_f_start, n_f_stop, n_r_max, n_cheb_max, d_fac)

Returns chebychev coeffitiens of first derivative df and second derivative ddf for a function whose cheb-coeff. are given as columns in array f(n_c_tot,n_r_max).

Parameters

  • f (n_f_max,n_r_max) [complex ,in]

  • df (n_f_max,n_r_max) [complex ,out]

  • ddf (n_f_max,n_r_max) [complex ,out]

  • dddf (n_f_max,n_r_max) [complex ,out]

  • n_f_max [integer ,in,] :: First dimension of f,df,ddf

  • n_f_start [integer ,in] :: No of column to start with

  • n_f_stop [integer ,in] :: No of column to stop with

  • n_r_max [integer ,in,] :: second dimension of f,df,ddf

  • n_cheb_max [integer ,in] :: Number of cheb modes

  • d_fac [real ,in] :: factor for interval mapping

Called from

get_dddr()

subroutine radial_der/get_dr_real_1d(f, df, n_r_max, r_scheme)
Parameters

  • f (n_r_max) [real ,in]

  • df (n_r_max) [real ,out] :: first derivative of f

  • n_r_max [integer ,in,optional/default=len(f)] :: number of radial grid points

  • r_scheme [real ]

subroutine radial_der/get_dr_complex(f, df, n_f_max, n_f_start, n_f_stop, n_r_max, r_scheme[, nocopy[, l_dct_in]])

Returns first radial derivative df of the input function f. Array f(n_f_max,*) may contain several functions numbered by the first index. The subroutine calculates the derivaties of the functions f(n_f_start,*) to f(n_f_stop) by transforming to a Chebychev representation using n_r_max radial grid points .

Parameters

  • f (n_f_max,n_r_max) [complex ,inout]

  • df (n_f_max,n_r_max) [complex ,out] :: first derivative of f

  • n_f_max [integer ,in,] :: first dim of f

  • n_f_start [integer ,in] :: first function to be treated

  • n_f_stop [integer ,in] :: last function to be treated

  • n_r_max [integer ,in,] :: number of radial grid points

  • r_scheme [real ]

Options

  • nocopy [logical ,in,]

  • l_dct_in [logical ,in,]

subroutine radial_der/get_ddr(f, df, ddf, n_f_max, n_f_start, n_f_stop, n_r_max, r_scheme[, l_dct_in])

Returns first radial derivative df and second radial derivative ddf of the input function f. Array f(n_f_max,*) may contain several functions numbered by the first index. The subroutine calculates the derivatives of the functions f(n_f_start,*) to f(n_f_stop) by transforming to a Chebychev representation using n_r_max radial grid points.

Parameters

  • f (n_f_max,n_r_max) [complex ,in]

  • df (n_f_max,n_r_max) [complex ,out] :: first derivative of f

  • ddf (n_f_max,n_r_max) [complex ,out] :: second derivative of f

  • n_f_max [integer ,in,] :: first dim of f

  • n_f_start [integer ,in] :: first function to be treated

  • n_f_stop [integer ,in] :: last function to be treated

  • n_r_max [integer ,in,] :: number of radial grid points

  • r_scheme [real ]

Options

l_dct_in [logical ,in,]

Called from

get_mag_rhs_imp(), get_phase_rhs_imp(), get_entropy_rhs_imp(), get_pol_rhs_imp(), assemble_pol(), assemble_single(), get_single_rhs_imp(), get_comp_rhs_imp(), get_tor_rhs_imp()

Call to

get_ddcheb()

subroutine radial_der/get_dddr(f, df, ddf, dddf, n_f_max, n_f_start, n_f_stop, n_r_max, r_scheme[, l_dct_in])

Returns first radial derivative df, the second radial deriv. ddf, and the third radial derivative dddf of the input function f. Array f(n_f_max,*) may contain several functions numbered by the first index. The subroutine calculates the derivaties of the functions f(n_f_start,*) to f(n_f_stop) by transforming to a Chebychev representation using n_r_max radial grid points.

Parameters

  • f (n_f_max,n_r_max) [complex ,in]

  • df (n_f_max,n_r_max) [complex ,out] :: first derivative of f

  • ddf (n_f_max,n_r_max) [complex ,out] :: second derivative of f

  • dddf (n_f_max,n_r_max) [complex ,out] :: third derivative of f

  • n_f_max [integer ,in,] :: first dim of f

  • n_f_start [integer ,in] :: first function to be treated

  • n_f_stop [integer ,in] :: last function to be treated

  • n_r_max [integer ,in,] :: number of radial grid points

  • r_scheme [real ]

Options

l_dct_in [logical ,in,]

Called from

get_pol_rhs_imp(), get_single_rhs_imp()

Call to

get_dddcheb()

subroutine radial_der/get_dr_rloc(f_rloc, df_rloc, lm_max, nrstart, nrstop, n_r_max, r_scheme)

Purpose of this subroutine is to take the first radial derivative of an input complex array distributed over radius. This can only be used with finite differences.

Parameters

  • f_rloc (lm_max,nrstop-nrstart+1) [complex ,in]

  • df_rloc (lm_max,nrstop-nrstart+1) [complex ,out]

  • lm_max [integer ,in,]

  • nrstart [integer ,in]

  • nrstop [integer ,in]

  • n_r_max [integer ,in]

  • r_scheme [real ]

Called from

dtvrms(), dtbrms(), transp_lmloc_to_rloc(), transp_rloc_to_lmloc_io(), finish_exp_mag_rdist(), finish_exp_entropy_rdist(), finish_exp_pol_rdist(), get_pol_rhs_imp_ghost(), finish_exp_smat_rdist(), finish_exp_comp_rdist()

Call to

abortrun(), get_openmp_blocks(), exch_ghosts(), get_bound_vals()

subroutine radial_der/get_ddr_rloc(f_rloc, df_rloc, ddf_rloc, lm_max, nrstart, nrstop, n_r_max, r_scheme)

Purpose of this subroutine is to take the first and second radial derivatives of an input complex array distributed over radius. This can only be used with finite differences.

Parameters
Called from

transp_lmloc_to_rloc()

Call to

abortrun(), get_openmp_blocks(), exch_ghosts(), get_bound_vals()

subroutine radial_der/get_ddr_ghost(f_rloc, df_rloc, ddf_rloc, lm_max, start_lm, stop_lm, nrstart, nrstop, r_scheme)

Purpose of this subroutine is to take the first and second radial derivatives of an input complex array distributed over radius that has the ghost zones properly filled.

Parameters
Called from

get_mag_rhs_imp_ghost(), get_phase_rhs_imp_ghost(), get_entropy_rhs_imp_ghost(), get_comp_rhs_imp_ghost(), get_tor_rhs_imp_ghost()

Call to

abortrun()

subroutine radial_der/get_ddddr_ghost(f_rloc, df_rloc, ddf_rloc, dddf_rloc, ddddf_rloc, lm_max, start_lm, stop_lm, nrstart, nrstop, r_scheme)

Purpose of this subroutine is to take the first and second radial derivatives of an input complex array distributed over radius that has the ghost zones properly filled.

Parameters
Called from

get_pol_rhs_imp_ghost()

Call to

abortrun()

subroutine radial_der/exch_ghosts(f, lm_max, nrstart, nrstop, nghosts)
Parameters

  • f (lm_max,nrstop+nghosts-(nrstart-nghosts)+1) [complex ,inout]

  • lm_max [integer ,in,]

  • nrstart [integer ,in]

  • nrstop [integer ,in]

  • nghosts [integer ,in]

Called from

get_dr_rloc(), get_ddr_rloc(), getstartfields(), start_from_another_scheme(), assemble_mag_rloc(), assemble_phase_rloc(), assemble_entropy_rloc(), assemble_pol_rloc(), assemble_comp_rloc(), assemble_tor_rloc()

subroutine radial_der/get_bound_vals(fbot, ftop, lm_max, nrstart, nrstop, n_r_max, nbounds)
Parameters

  • fbot (lm_max,n_r_max-(n_r_max-nbounds+1)+1) [complex ,inout]

  • ftop (lm_max,nbounds) [complex ,inout]

  • lm_max [integer ,in,]

  • nrstart [integer ,in]

  • nrstop [integer ,in]

  • n_r_max [integer ,in]

  • nbounds [integer ,in,]

Called from

get_dr_rloc(), get_ddr_rloc()

subroutine radial_der/bulk_to_ghost(x, x_g, ng, nrstart, nrstop, lm_max, start_lm, stop_lm)

This subroutine is used to copy an array that is defined from nRstart to nRstop to an array that is defined from nRstart-1 to nRstop+1

Parameters

  • x (lm_max,nrstop-nrstart+1) [complex ,in]

  • x_g (lm_max,nrstop+ng-(nrstart-ng)+1) [complex ,out]

  • ng [integer ,in] :: Number of ghost zones

  • nrstart [integer ,in]

  • nrstop [integer ,in]

  • lm_max [integer ,in,]

  • start_lm [integer ,in]

  • stop_lm [integer ,in]

Called from

getstartfields(), start_from_another_scheme(), assemble_mag_rloc(), assemble_phase_rloc(), assemble_entropy_rloc(), assemble_pol_rloc(), assemble_comp_rloc(), assemble_tor_rloc()

radial_derivatives_even.f90

Quick access

Routines

get_dcheb_even(), get_ddcheb_even(), get_ddr_even(), get_ddrns_even(), get_drns_even()

Needed modules

  • constants (zero()): module containing constants and parameters used in the code.

  • precision_mod: This module controls the precision used in MagIC

  • cosine_transform_odd: This module contains the built-in type I discrete Cosine Transforms. This implementation is based on Numerical Recipes and FFTPACK. This only works for n_r_max-1 = 2**a 3**b 5**c, with a,b,c integers….

  • cosine_transform_even

Variables

Subroutines and functions

subroutine radial_der_even/get_ddr_even(f, df, ddf, n_f_max, n_f_start, n_f_stop, n_r_max, n_cheb_max, dr_fac, work1, work2, chebt_odd, chebt_even)

Returns first rarial derivative df and second radial derivative ddf of the input function f. Array f(n_f_max,*) may contain several functions numbered by the first index. The subroutine calculates the derivaties of the functions f(n_f_start,*) to f(n_f_stop) by transforming to a Chebychev representation using n_r_max radial grid points. The cheb transforms have to be initialized by calling init_costf1 and init_costf2.

Parameters

  • f (n_f_max,n_r_max) [complex ,in]

  • df (n_f_max,n_r_max) [complex ,out] :: first derivative of f

  • ddf (n_f_max,n_r_max) [complex ,out] :: second derivative of f

  • n_f_max [integer ,in,] :: first dim of f

  • n_f_start [integer ,in] :: first function to be treated

  • n_f_stop [integer ,in] :: last function to be treated

  • n_r_max [integer ,in,] :: number of radial grid points

  • n_cheb_max [integer ,in] :: number of cheb modes

  • dr_fac [real ,in] :: mapping factor

  • work1 (n_f_max,n_r_max) [complex ,out] :: work array needed for costf

  • work2 (n_f_max,n_r_max) [complex ,out] :: work array needed for costf

  • chebt_odd [costf_odd_t ,in]

  • chebt_even [costf_even_t ,in]

Called from

get_mag_ic_rhs_imp()

Call to

get_ddcheb_even()

subroutine radial_der_even/get_drns_even(f, df, n_f_max, n_f_start, n_f_stop, n_r_max, n_cheb_max, dr_fac, work1, chebt_odd, chebt_even)

Returns first rarial derivative df and second radial derivative ddf of the input function f. Array f(n_f_max,*) may contain several functions numbered by the first index. The subroutine calculates the derivaties of the functions f(n_f_start,*) to f(n_f_stop) by transforming to a Chebychev representation using n_r_max radial grid points. The cheb transforms have to be initialized by calling init_costf1 and init_costf2.

Parameters

  • f (n_f_max,n_r_max) [complex ,inout]

  • df (n_f_max,n_r_max) [complex ,out] :: first derivative of f

  • n_f_max [integer ,in,] :: first dim of f

  • n_f_start [integer ,in] :: first function to be treated

  • n_f_stop [integer ,in] :: last function to be treated

  • n_r_max [integer ,in,] :: number of radial grid points

  • n_cheb_max [integer ,in] :: number of cheb modes

  • dr_fac [real ,in] :: mapping factor

  • work1 (n_f_max,n_r_max) [complex ,out] :: work array needed for costf

  • chebt_odd [costf_odd_t ,in]

  • chebt_even [costf_even_t ,in]

Called from

fields_average(), write_dtb_frame()

Call to

get_dcheb_even()

subroutine radial_der_even/get_ddrns_even(f, df, ddf, n_f_max, n_f_start, n_f_stop, n_r_max, n_cheb_max, dr_fac, work1, chebt_odd, chebt_even)

Returns first rarial derivative df and second radial derivative ddf of the input function f. Array f(n_f_max,*) may contain several functions numbered by the first index. The subroutine calculates the derivaties of the functions f(n_f_start,*) to f(n_f_stop) by transforming to a Chebychev representation using n_r_max radial grid points. The cheb transforms have to be initialized by calling init_costf1 and init_costf2.

Parameters

  • f (n_f_max,n_r_max) [complex ,inout]

  • df (n_f_max,n_r_max) [complex ,out] :: first derivative of f

  • ddf (n_f_max,n_r_max) [complex ,out] :: second derivative of f

  • n_f_max [integer ,in,] :: first dim of f

  • n_f_start [integer ,in] :: first function to be treated

  • n_f_stop [integer ,in] :: last function to be treated

  • n_r_max [integer ,in,] :: number of radial grid points

  • n_cheb_max [integer ,in] :: number of cheb modes

  • dr_fac [real ,in] :: mapping factor

  • work1 (n_f_max,n_r_max) [complex ,out] :: work array needed for costf

  • chebt_odd [costf_odd_t ,in]

  • chebt_even [costf_even_t ,in]

Called from

fields_average()

Call to

get_ddcheb_even()

subroutine radial_der_even/get_dcheb_even(f, df, n_f_max, n_f_start, n_f_stop, n_r_max, n_cheb_max, d_fac)

Returns Chebyshev coefficients of first derivative df and second derivative ddf for a function whose cheb-coeff. are given as columns in array f(n_f_max,n_r_max).

Parameters

  • f (n_f_max,n_r_max) [complex ,in]

  • df (n_f_max,n_r_max) [complex ,out]

  • n_f_max [integer ,in,] :: First dimension of f,df

  • n_f_start [integer ,in] :: No of function to start with

  • n_f_stop [integer ,in] :: No of function to stop with

  • n_r_max [integer ,in,] :: second dimension of f,df

  • n_cheb_max [integer ,in] :: Number of cheb modes

  • d_fac [real ,in] :: factor for interval mapping

Called from

get_drns_even()

subroutine radial_der_even/get_ddcheb_even(f, df, ddf, n_f_max, n_f_start, n_f_stop, n_r_max, n_cheb_max, d_fac)

Returns Chebyshev coefficients of first derivative df and second derivative ddf for a function whose cheb-coeff. are given as columns in array f(n_f_max,n_r_max).

Parameters

  • f (n_f_max,n_r_max) [complex ,in]

  • df (n_f_max,n_r_max) [complex ,out]

  • ddf (n_f_max,n_r_max) [complex ,out]

  • n_f_max [integer ,in,] :: First dimension of f,df,ddf

  • n_f_start [integer ,in] :: No of function to start with

  • n_f_stop [integer ,in] :: No of function to stop with

  • n_r_max [integer ,in,] :: second dimension of f,df,ddf

  • n_cheb_max [integer ,in] :: Number of cheb modes

  • d_fac [real ,in] :: factor for interval mapping

Called from

get_ddr_even(), get_ddrns_even()

integration.f90

Description

Radial integration functions

Quick access

Routines

cylmean_itc(), cylmean_otc(), rint_r(), rintic(), simps()

Needed modules

  • precision_mod: This module controls the precision used in MagIC

  • constants (half(), one(), two(), pi()): module containing constants and parameters used in the code.

  • radial_scheme (type_rscheme()): This is an abstract type that defines the radial scheme used in MagIC

  • cosine_transform_odd: This module contains the built-in type I discrete Cosine Transforms. This implementation is based on Numerical Recipes and FFTPACK. This only works for n_r_max-1 = 2**a 3**b 5**c, with a,b,c integers….

Variables

Subroutines and functions

function integration/rintic(f, nrmax, drfac, chebt)

This function performs the radial integral over a function f that is given on the appropriate nRmax radial Chebychev grid points.

Parameters

  • f (nrmax) [real ,inout] :: This is zero order contribution

  • nrmax [integer ,in,optional/default=len(f)] :: Only even chebs for IC

  • drfac [real ,in]

  • chebt [costf_odd_t ,in]

Return

rintic [real ]

Called from

get_e_mag(), get_power(), spectrum_vec_ic()

function integration/rint_r(f, r, r_scheme)

Same as function rInt but for a radial dependent mapping function dr_fac2.

Parameters

  • f (*) [real ,in] :: Input function

  • r (*) [real ,in] :: Radius

  • r_scheme [real ] :: Radial scheme (FD or Cheb)

Return

rint [real ]

Called from

get_force(), dtbrms(), get_poltorrms(), getdlm(), get_e_kin(), get_u_square(), get_e_mag(), outhemi(), outhelicity(), outheat(), outphase(), get_onset(), outperppar(), get_angular_moment(), get_angular_moment_rloc(), output(), get_power(), precalc(), spectrum_scal(), spectrum_vec(), get_amplitude(), updatewp()

Call to

simps()

function integration/simps(f, r)

Simpson’s method to integrate a function

Parameters

  • f (*) [real ,in] :: Input function

  • r (*) [real ,in] :: Radius

Return

rint [real ]

Called from

rint_r(), initialize_geos(), outgeos(), initialize_outto_mod(), outto()

subroutine integration/cylmean_otc(a, v, n_s_max, n_s_otc, r, s, theta[, zdensin])

This routine computes a z-averaging using Simpsons’s rule outside T.C.

Parameters

  • a (*,*) [real ,in]

  • v (n_s_max) [real ,out]

  • n_s_max [integer ,in,optional/default=len(s)]

  • n_s_otc [integer ,in]

  • r (*) [real ,in] :: Spherical radius

  • s (n_s_max) [real ,in] :: Cylindrical radius

  • theta (*) [real ,in] :: Colatitude

Options

zdensin [real ,in,]

Called from

outgeos(), outomega(), cylmean()

subroutine integration/cylmean_itc(a, vn, vs, n_s_max, n_s_otc, r, s, theta[, zdensin])

This routine computes a z-averaging using Simpsons’s rule inside T.C.

Parameters

  • a (*,*) [real ,in]

  • vn (n_s_max) [real ,out]

  • vs (n_s_max) [real ,out]

  • n_s_max [integer ,in,optional/default=len(s)]

  • n_s_otc [integer ,in]

  • r (*) [real ,in] :: Spherical radius

  • s (n_s_max) [real ,in] :: Cylindrical radius

  • theta (*) [real ,in] :: Colatitude

Options

zdensin [real ,in,]

Called from

outomega(), cylmean()