mod_global_parameters.t

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!> This module contains definitions of global parameters and variables and some
!> generic functions/subroutines used in AMRVAC.
!>
module mod_global_parameters
  use mod_physicaldata
  use mod_connectivity
  use mpi
  use mod_constants
  use mod_variables
  use mod_basic_types
 
  implicit none
  public
 
  ! Parameters
  character(len=*), parameter :: undefined = 'undefined'
 
  !> The number of MPI tasks
  integer :: npe
 
  !> The rank of the current MPI task
  integer :: mype
 
  !> The MPI communicator
  integer :: icomm
 
  !> A global MPI error return code
  integer :: ierrmpi
 
  !> MPI file handle for logfile
  integer :: log_fh
  !> MPI type for block including ghost cells and its size
  integer :: type_block, size_block
  !> MPI type for block coarsened by 2, and for its children blocks
  integer :: type_coarse_block, type_sub_block(2^d&)
  !> MPI type for staggered block coarsened by 2, and for its children blocks
  integer :: type_coarse_block_stg(^nd,2^d&), type_sub_block_stg(^ND,2^D&)
  !> MPI type for IO: block excluding ghost cells
  integer :: type_block_io, size_block_io
  !> MPI type for IO of staggered variables
  integer :: type_block_io_stg, size_block_io_stg
  !> MPI type for IO: cell corner (xc) or cell center (xcc) coordinates
  integer :: type_block_xc_io,type_block_xcc_io
  !> MPI type for IO: cell corner (wc) or cell center (wcc) variables
  integer :: type_block_wc_io,type_block_wcc_io
 
 
  ! geometry and domain setups
 
  !> the mesh range (within a block with ghost cells)
  integer :: ixm^ll
 
  !> minimum and maximum domain boundaries for each dimension
  double precision  :: xprob^l
 
  !> Indices for cylindrical coordinates FOR TESTS, negative value when not used:
  integer :: r_ = -1
  integer :: phi_ = -1
  integer :: z_ = -1
 
  !> Number of spatial dimensions for grid variables
  integer, parameter :: ndim=^nd
 
  !> Number of spatial dimensions (components) for vector variables
  integer :: ndir=ndim
 
  !> Cartesian geometry or not
  logical :: slab
 
  !> uniform Cartesian geometry or not (stretched Cartesian)
  logical :: slab_uniform
 
  !> each cell has its own timestep or not
  logical :: local_timestep = .false.
 
  !> number of grid blocks in domain per dimension, in array over levels
  integer, dimension(:), allocatable :: ng^d
  !> extent of grid blocks in domain per dimension, in array over levels
  double precision, dimension(:), allocatable :: dg^d
 
  !> number of cells for each dimension in level-one mesh
  integer :: domain_nx^d
 
  !> number of cells for each dimension in grid block excluding ghostcells
  integer :: block_nx^d
 
  !> Lower index of grid block arrays (always 1)
  integer, parameter :: {ixglo^d = 1|, }
 
  !> Upper index of grid block arrays
  integer :: ixghi^d
 
  !> Lower index of stagger grid block arrays (always 0)
  integer, parameter :: {ixgslo^d = 0|, }
 
  !> Upper index of stagger grid block arrays
  integer :: ixgshi^d
 
  !> Number of ghost cells surrounding a grid
  integer :: nghostcells = 2
 
  integer, parameter :: stretch_none = 0 !< No stretching
  integer, parameter :: stretch_uni  = 1 !< Unidirectional stretching from a side
  integer, parameter :: stretch_symm = 2 !< Symmetric stretching around the center
 
  !> If true, adjust mod_geometry routines to account for grid stretching (but
  !> the flux computation will not)
  logical :: stretch_uncentered
  !> True if a dimension is stretched
  logical :: stretched_dim(ndim)
  !> What kind of stretching is used per dimension
  integer :: stretch_type(ndim)
  !> stretch factor between cells at AMR level 1, per dimension
  double precision ::  qstretch_baselevel(ndim)
  !> (even) number of (symmetrically) stretched
  !> blocks at AMR level 1, per dimension
  integer ::  nstretchedblocks_baselevel(ndim)
  !> (even) number of (symmetrically) stretched blocks per level and dimension
  integer, allocatable ::  nstretchedblocks(:,:)
  !> physical extent of stretched border in symmetric stretching
  double precision :: xstretch^d
  !> Stretching factors and first cell size for each AMR level and dimension
  double precision, allocatable :: qstretch(:,:), dxfirst(:,:),  &
                                   dxfirst_1mq(:,:), dxmid(:,:)
 
  !> grid hierarchy info (level and grid indices)
  integer, parameter :: nodehi=^nd+1
  integer, parameter :: plevel_=1
  integer, parameter :: pig^d_=plevel_+^d
 
  integer, allocatable :: node(:,:)
  integer, allocatable :: node_sub(:,:)
 
  !> grid location info (corner coordinates and grid spacing)
  integer, parameter :: rnodehi=3*^nd
  integer, parameter :: rpxmin0_=0
  integer, parameter :: rpxmin^d_=rpxmin0_+^d
  integer, parameter :: rpxmax0_=^nd
  integer, parameter :: rpxmax^d_=rpxmax0_+^d
  integer, parameter :: rpdx^d_=2*^nd+^d
 
  !> Corner coordinates
  double precision, allocatable :: rnode(:,:)
  double precision, allocatable :: rnode_sub(:,:)
 
  double precision, allocatable :: dx(:,:)
  double precision :: dxlevel(ndim)
 
  ! IO related quantities
 
  !> Maximum number of saves that can be defined by tsave or itsave
  integer, parameter :: nsavehi=100
 
  !> Number of output methods
  integer, parameter :: nfile = 5
 
  !> index number of the latest existing data file
  integer :: index_latest_data
 
  !> Names of the output methods
  character(len=40), parameter  :: output_names(nfile) = &
       ['log      ', 'normal   ', 'slice    ', 'collapsed', 'analysis ']
 
  !> User parameter file
  character(len=std_len)   :: usr_filename
 
  !> If collapse(DIM) is true, generate output integrated over DIM
  logical :: collapse(ndim)
 
  !> Save output of type N on times tsave(:, N)
  double precision :: tsave(nsavehi,nfile)
 
  double precision :: tsavelast(nfile)
 
  !> Repeatedly save output of type N when dtsave(N) simulation time has passed
  double precision :: dtsave(nfile)
 
  !> Save output of type N on iterations itsave(:, N)
  integer :: itsave(nsavehi,nfile)
 
  integer :: itsavelast(nfile)
 
  !> Repeatedly save output of type N when ditsave(N) time steps have passed
  integer :: ditsave(nfile)
 
  integer :: isavet(nfile)
 
  integer :: isaveit(nfile)
 
  !> Start of read out (not counting specified read outs)
  double precision :: tsavestart(nfile)
 
  !> The level at which to produce line-integrated / collapsed output
  integer :: collapselevel
 
  !> Number of saved files of each type
  integer :: n_saves(1:nfile)
 
  !> whether or not to save an output file
  logical :: save_file(nfile)
 
  !> to monitor timeintegration loop at given wall-clock time intervals
  double precision :: time_between_print
 
  !> accumulated wall-clock time spent on boundary conditions
  double precision :: time_bc
 
  !> IO: snapshot and collapsed views output numbers/labels
  integer :: snapshotnext, collapsenext
 
  !> Constant indicating log output
  integer, parameter :: filelog_      = 1
 
  !> Constant indicating regular output
  integer, parameter :: fileout_      = 2
 
  !> Constant indicating slice output
  integer, parameter :: fileslice_    = 3
 
  !> Constant indicating collapsed output
  integer, parameter :: filecollapse_ = 4
 
  !> Constant indicating analysis output (see @ref analysis.md)
  integer, parameter :: fileanalysis_ = 5
 
  !> Unit for standard input
  integer, parameter :: unitstdin=5
 
  !> Unit for standard output
  integer, parameter :: unitterm=6
 
  !> Unit for error messages
  integer, parameter :: uniterr=6
 
  !> file handle for IO
  integer, parameter :: unitpar=9
  integer, parameter :: unitconvert=10
  integer, parameter :: unitslice=11
  integer, parameter :: unitsnapshot=12
  integer, parameter :: unitcollapse=13
  integer, parameter :: unitanalysis=14
 
  !> Number of auxiliary variables that are only included in output
  integer :: nwauxio
 
  !> IO switches for conversion
  logical          :: nocartesian
  logical, allocatable :: w_write(:)
  logical, allocatable :: writelevel(:)
  double precision :: writespshift(ndim,2)
  integer          :: level_io, level_io_min, level_io_max, type_endian
 
  !> Which par files are used as input
  character(len=std_len), allocatable :: par_files(:)
 
  !> Base file name for simulation output, which will be followed by a number
  character(len=std_len) :: base_filename
 
  !> If not 'unavailable', resume from snapshot with this base file name
  character(len=std_len) :: restart_from_file
 
  !> Which type of log to write: 'normal', 'special', 'regression_test'
  character(len=std_len) :: typefilelog
 
  !> Resume from the snapshot with this index
  integer :: snapshotini
 
  !> If true, restart a previous run from the latest snapshot
  logical :: resume_previous_run
 
  !> If true and restart_from_file is given, convert snapshots to
  !> other file formats
  logical                :: convert
 
  !> If true, already convert to output format during the run
  logical                :: autoconvert
 
  !> If true, convert from conservative to primitive variables in output
  logical                :: saveprim
 
  !> Which format to use when converting
  !>
  !> Options are: tecplot, tecplotCC, vtu, vtuCC, vtuB, vtuBCC,
  !> tecplotmpi, tecplotCCmpi, vtumpi, vtuCCmpi, vtuBmpi, vtuBCCmpi, pvtumpi, pvtuCCmpi,
  !> pvtuBmpi, pvtuBCCmpi, tecline, teclinempi, onegrid
  character(len=std_len) :: convert_type
 
  character(len=std_len) :: collapse_type
 
  !> Conversion factors the primitive variables
  double precision, allocatable :: w_convert_factor(:)
 
  double precision :: length_convert_factor
 
  !> Conversion factor for time unit
  double precision       :: time_convert_factor
 
  !> Stores the memory and load imbalance, used in printlog
  double precision       :: xload, xmemory
 
  !> Save a snapshot before crash a run met unphysical values
  logical :: crash=.false.
 
  ! Physics factors
 
  !> Physical scaling factor for length
  double precision :: unit_length=1.d0
 
  !> Physical scaling factor for time
  double precision :: unit_time=1.d0
 
  !> Physical scaling factor for density
  double precision :: unit_density=1.d0
 
  !> Physical scaling factor for velocity
  double precision :: unit_velocity=1.d0
 
  !> Physical scaling factor for temperature
  double precision :: unit_temperature=1.d0
 
  !> Physical scaling factor for pressure
  double precision :: unit_pressure=1.d0
 
  !> Physical scaling factor for magnetic field
  double precision :: unit_magneticfield=1.d0
 
  !> Physical scaling factor for number density
  double precision :: unit_numberdensity=1.d0
 
  !> Physical scaling factor for charge
  double precision :: unit_charge=1.d0
 
  !> Physical scaling factor for mass
  double precision :: unit_mass=1.d0
 
  !> Normalised speed of light
  double precision :: c_norm=1.d0
 
  !> Physical scaling factor for Opacity
  double precision :: unit_opacity=1.d0
 
  !> Physical scaling factor for radiation flux
  double precision :: unit_radflux=1.d0
 
  !> error handling
  double precision :: small_temperature,small_pressure,small_density
 
  !> amplitude of background dipolar, quadrupolar, octupolar, user's field
  double precision :: bdip=0.d0
  double precision :: bquad=0.d0
  double precision :: boct=0.d0
  double precision :: busr=0.d0
 
  !> check and optionally fix unphysical small values (density, gas pressure)
  logical :: check_small_values=.true.
  logical :: fix_small_values=.false.
 
  !> split magnetic field as background B0 field
  logical :: b0field=.false.
 
  ! number of equilibrium set variables, besides the mag field
  integer :: number_equi_vars = 0
 
  !> Use SI units (.true.) or use cgs units (.false.)
  logical :: si_unit=.false.
 
  !> Use TRAC (Johnston 2019 ApJL, 873, L22) for MHD or 1D HD
  logical :: phys_trac=.false.
  integer :: phys_trac_type=1
  double precision :: phys_trac_mask
 
  !> Use particles module or not
  logical :: use_particles=.false.
 
  !> Use multigrid (only available in 2D and 3D)
  logical :: use_multigrid = .false.
 
  ! AMR switches
 
  !> The maximum number of grid blocks in a processor
  integer :: max_blocks
 
  !> The maximum number of levels in the grid refinement
  integer, parameter :: nlevelshi = 20
 
  !> Maximal number of AMR levels
  integer :: refine_max_level
 
  !> Weights of variables used to calculate error for mesh refinement
  double precision, allocatable :: w_refine_weight(:)
 
  !> Fix the AMR grid after this time
  double precision :: tfixgrid
 
  !> Whether to apply flux conservation at refinement boundaries
  logical :: fix_conserve_global = .true.
 
  !> Fix the AMR grid after this many time steps
  integer :: itfixgrid
 
  !> Reconstruct the AMR grid once every ditregrid iteration(s)
  integer :: ditregrid
 
  !> refinement: lohner estimate wavefilter setting
  double precision, allocatable :: amr_wavefilter(:)
 
  integer                       :: refine_criterion
  logical                       :: prolongprimitive=.false.
  logical                       :: coarsenprimitive=.false.
 
  !> Error tolerance for refinement decision
  double precision, allocatable :: refine_threshold(:)
  double precision, allocatable :: derefine_ratio(:)
 
  !> If true, rebuild the AMR grid upon restarting
  logical :: reset_grid
  !> True for using stagger grid
  logical :: stagger_grid=.false.
  !> True for record electric field
  logical :: record_electric_field=.false.
 
  !> Number of cells as buffer zone
  integer :: nbufferx^d
 
  integer :: levmin
  integer :: levmax
  integer :: levmax_sub
 
  ! Miscellaneous
 
  !> problem switch allowing different setups in same usr_mod.t
  integer           :: iprob
 
  !> Kronecker delta tensor
  integer :: kr(3,3)
 
  !> Levi-Civita tensor
  integer :: lvc(3,3,3)
 
  ! Time integration aspects
 
  double precision :: dt
 
  logical :: time_advance
 
  !> The Courant (CFL) number used for the simulation
  double precision :: courantpar
 
  !> How to compute the CFL-limited time step
  integer :: type_courant=1
  !> integer switchers for type courant
  integer, parameter :: type_maxsum=1
  integer, parameter :: type_summax=2
  integer, parameter :: type_minimum=3
 
  !> If dtpar is positive, it sets the timestep dt, otherwise courantpar is used
  !> to limit the time step based on the Courant condition.
  double precision :: dtpar
 
  !> For resistive MHD, the time step is also limited by the diffusion time:
  !> \f$ dt < dtdiffpar \times dx^2/eta \f$
  double precision :: dtdiffpar
 
  !> The global simulation time
  double precision :: global_time
 
  !> Start time for the simulation
  double precision :: time_init
 
  !> End time for the simulation
  double precision :: time_max
 
  !> Ending wall time (in hours) for the simulation
  double precision :: wall_time_max
 
  !> Stop the simulation when the time step becomes smaller than this value
  double precision :: dtmin
 
  !> Force timeloop exit when final dt < dtmin
  logical :: final_dt_exit
 
  !> If true, reset iteration count and global_time to original values, and
  !> start writing snapshots at index 0
  logical :: reset_time
 
  !> If true, reset iteration count to 0
  logical :: reset_it
 
  !> If true, allow final dt reduction for matching time_max on output
  logical :: final_dt_reduction
 
  !> If true, call initonegrid_usr upon restarting
  logical :: firstprocess
 
  !> If true, wall time is up, modify snapshotnext for later overwrite
  logical :: pass_wall_time
 
  !> If true, do H-correction to fix the carbuncle problem at grid-aligned shocks
  logical :: h_correction=.false.
 
  !> Number of time steps taken
  integer :: it
 
  !> Stop the simulation after this many time steps have been taken
  integer :: it_max
 
  !> initial iteration count
  integer :: it_init
 
  !> If > 1, then in the first slowsteps-1 time steps dt is reduced
  !> by a factor \f$ 1 - (1- step/slowsteps)^2 \f$
  integer :: slowsteps
 
  ! Method switches
 
  !> Index of the sub-step in a multi-step time integrator
  integer :: istep
 
  !> How many sub-steps the time integrator takes
  integer :: nstep
 
  !> Which flux scheme of spatial discretization to use (per grid level)
  integer, allocatable :: flux_method(:)
 
  !> The spatial discretization for the predictor step when using a two
  !> step PC method
  integer, allocatable :: typepred1(:)
 
  !> flux schemes
  integer, parameter :: fs_hll=1
  integer, parameter :: fs_hllc=2
  integer, parameter :: fs_hlld=3
  integer, parameter :: fs_hllcd=4
  integer, parameter :: fs_tvdlf=5
  integer, parameter :: fs_tvdmu=6
  integer, parameter :: fs_tvd=7
  integer, parameter :: fs_hancock=8
  integer, parameter :: fs_cd=9
  integer, parameter :: fs_cd4=10
  integer, parameter :: fs_fd=11
  integer, parameter :: fs_source=12
  integer, parameter :: fs_nul=13
 
  !> time stepper type
  integer :: t_stepper=0
  integer, parameter :: onestep=1
  integer, parameter :: twostep=2
  integer, parameter :: threestep=3
  integer, parameter :: fourstep=4
  integer, parameter :: fivestep=5
 
  !> time integrator method
  integer :: t_integrator=0
  integer, parameter :: forward_euler=1
  integer, parameter :: predictor_corrector=2
  integer, parameter :: ssprk3=3
  integer, parameter :: ssprk4=4
  integer, parameter :: ssprk5=5
 
  integer, parameter :: imex_euler=6
  integer, parameter :: imex_sp=7
  integer, parameter :: rk2_alf=8
  integer, parameter :: ssprk2=9
  integer, parameter :: imex_midpoint=10
  integer, parameter :: imex_trapezoidal=11
  integer, parameter :: imex_222=12
 
  integer, parameter :: rk3_bt=13
  integer, parameter :: imex_ars3=14
  integer, parameter :: imex_232=15
  integer, parameter :: imex_cb3a=16
 
  integer, parameter :: rk4=17
 
  !> Type of slope limiter used for reconstructing variables on cell edges
  integer, allocatable :: type_limiter(:)
 
  !> Type of slope limiter used for computing gradients or divergences, when
  !> typegrad or typediv are set to 'limited'
  integer, allocatable :: type_gradient_limiter(:)
 
  !> background magnetic field location indicator
  integer :: b0i=0
 
  !> Limiter used for prolongation to refined grids and ghost cells
  integer :: prolong_limiter=0
 
  !> Which type of entropy fix to use with Riemann-type solvers
  character(len=std_len), allocatable :: typeentropy(:)
 
  !> Which type of TVD method to use
  character(len=std_len) :: typetvd
 
  !> bound (left/min and right.max) speed of Riemann fan
  integer :: boundspeed
 
  character(len=std_len) :: typeaverage
  character(len=std_len) :: typedimsplit
  character(len=std_len) :: geometry_name='default'
  character(len=std_len) :: typepoly
 
  integer                       :: nxdiffusehllc
  double precision, allocatable :: entropycoef(:)
  double precision              :: tvdlfeps
  logical, allocatable          :: loglimit(:), logflag(:)
  logical                       :: flathllc,flatcd,flatsh
  !> Use split or unsplit way to add user's source terms, default: unsplit
  logical                       :: source_split_usr
  !> if any normal source term is added in split fasion
  logical                       :: any_source_split=.false.
  logical                       :: dimsplit
 
  !> RK2(alfa) method parameters from Butcher tableau
  double precision              :: rk_a21,rk_b1,rk_b2
  !> IMEX-222(lambda) one-parameter family of schemes
  double precision              :: imex222_lambda
  !> SSPRK choice of methods (both threestep and fourstep, Shu-Osher 2N* implementation)
  !> also fivestep SSPRK54
  integer                       :: ssprk_order
  double precision              :: rk_beta11,rk_beta22,rk_beta33,rk_beta44,rk_c2,rk_c3,rk_c4
  double precision              :: rk_alfa21,rk_alfa22,rk_alfa31,rk_alfa33,rk_alfa41,rk_alfa44
  double precision              :: rk_beta54,rk_beta55,rk_alfa53,rk_alfa54,rk_alfa55,rk_c5
  !> RK3 Butcher table
  integer                       :: rk3_switch
  double precision              :: rk3_a21,rk3_a31,rk3_a32,rk3_b1,rk3_b2,rk3_b3,rk3_c2,rk3_c3
  !> IMEX_ARS3 parameter ars_gamma
  double precision              :: ars_gamma
  !> IMEX_232 choice and parameters
  integer                       :: imex_switch
  double precision              :: imex_a21,imex_a31,imex_a32,imex_b1,imex_b2,imex_ha21,imex_ha22
  double precision              :: imex_b3,imex_c2,imex_c3
  !> IMEX_CB3a extra parameters
  double precision              :: imex_a22, imex_a33, imex_ha32
  !> whether IMEX in use or not
  logical                       :: use_imex_scheme
 
  character(len=std_len) :: typediv,typegrad
 
  !> global fastest wave speed needed in fd scheme and glm method
  double precision :: cmax_global
 
  !> global fastest flow speed needed in glm method
  double precision :: vmax_global
 
  !> global largest a2 for schmid scheme
  double precision :: a2max_global(ndim)
 
  !> need global maximal wave speed
  logical :: need_global_cmax=.false.
 
  !> global value for schmid scheme
  logical :: need_global_a2max=.false.
 
  ! Boundary region parameters
 
  !> True for dimensions with periodic boundaries
  logical :: periodb(ndim)
 
  !> Indicates whether there is a pole at a boundary
  logical :: poleb(2,ndim)
 
  !> True for dimensions with aperiodic boundaries
  logical :: aperiodb(ndim)
 
  !> True for save physical boundary cells in dat files
  logical :: save_physical_boundary
 
  !> True if a block has any physical boundary
  logical, allocatable :: phyboundblock(:)
 
  !> Array indicating the type of boundary condition per variable and per
  !> physical boundary
  integer, allocatable :: typeboundary(:, :)
  !> boundary condition types
  integer, parameter :: bc_special=1
  integer, parameter :: bc_cont=2
  integer, parameter :: bc_symm=3
  integer, parameter :: bc_asymm=4
  integer, parameter :: bc_periodic=5
  integer, parameter :: bc_aperiodic=6
  integer, parameter :: bc_noinflow=7
  integer, parameter :: bc_data=8
  integer, parameter :: bc_character=9
 
  !> whether copy values instead of interpolation in ghost cells of finer blocks
  logical :: ghost_copy=.false.
 
  !> if there is an internal boundary
  logical :: internalboundary
 
  !> Base file name for synthetic EUV emission output
  character(len=std_len) :: filename_euv
  !> wavelength for output
  integer :: wavelength
  !> times for enhancing spatial resolution for EUV image/spectra
  double precision :: instrument_resolution_factor
  !> use arcsec as length unit of images/spectra
  logical :: activate_unit_arcsec
  !> Base file name for synthetic SXR emission output
  character(len=std_len) :: filename_sxr
  ! minimum and maximum energy of SXR (keV)
  integer :: emin_sxr,emax_sxr
  !> Base file name for synthetic white light
  character(len=std_len) :: filename_whitelight
  !> white light observation instrument
  character(len=std_len) :: whitelight_instrument
  !> the white light emission below it (unit=Rsun) is not visible 
  double precision :: r_occultor
  !> direction of the line of sight (LOS)
  double precision :: los_theta,los_phi
  !> rotation of image
  double precision :: image_rotate
  !> where the is the origin (X=0,Y=0) of image
  double precision :: x_origin(1:3)
  !> big image
  logical :: big_image
  !> Base file name for synthetic EUV spectrum output
  character(len=std_len) :: filename_spectrum
  !> wave length for spectrum
  integer :: spectrum_wl
  !> spectral window
  double precision :: spectrum_window_min,spectrum_window_max
  !> location of the slit
  double precision :: location_slit
  !> direction of the slit (for dat resolution only)
  integer :: direction_slit
  !> for spherical coordinate, region below it (unit=Rsun) is treated as not transparent
  double precision :: r_opt_thick
  !> resolution of the images
  logical :: dat_resolution
 
  !> Block pointer for using one block and its previous state
  type(state), pointer :: block
 
  !$OMP THREADPRIVATE(block,dxlevel,b0i)
 
contains
 
  !> Cross product of two vectors
  pure subroutine cross_product(ixI^L,ixO^L,a,b,axb)
    integer, intent(in) :: ixi^l, ixo^l
    double precision, intent(in) :: a(ixi^s,3), b(ixi^s,3)
    double precision, intent(out) :: axb(ixi^s,3)
 
    axb(ixo^s,1)=a(ixo^s,2)*b(ixo^s,3)-a(ixo^s,3)*b(ixo^s,2)
    axb(ixo^s,2)=a(ixo^s,3)*b(ixo^s,1)-a(ixo^s,1)*b(ixo^s,3)
    axb(ixo^s,3)=a(ixo^s,1)*b(ixo^s,2)-a(ixo^s,2)*b(ixo^s,1)
  end subroutine cross_product
 
end module mod_global_parameters