amun-code/sources/user_problem.F90

!!******************************************************************************
!!
!!  This file is part of the AMUN source code, a program to perform
!!  Newtonian or relativistic magnetohydrodynamical simulations on uniform or
!!  adaptive mesh.
!!
!!  Copyright (C) 2017-2020 Grzegorz Kowal <grzegorz@amuncode.org>
!!
!!  This program is free software: you can redistribute it and/or modify
!!  it under the terms of the GNU General Public License as published by
!!  the Free Software Foundation, either version 3 of the License, or
!!  (at your option) any later version.
!!
!!  This program is distributed in the hope that it will be useful,
!!  but WITHOUT ANY WARRANTY; without even the implied warranty of
!!  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
!!  GNU General Public License for more details.
!!
!!  You should have received a copy of the GNU General Public License
!!  along with this program.  If not, see <http://www.gnu.org/licenses/>.
!!
!!*****************************************************************************
!!
!! module: USER_PROBLEM
!!
!!  This module provides subroutines to setup custom problem.
!!
!!*****************************************************************************
!
module user_problem

#ifdef PROFILE
! include external procedures
!
  use timers, only : set_timer, start_timer, stop_timer
#endif /* PROFILE */

! module variables are not implicit by default
!
  implicit none

#ifdef PROFILE
! timer indices
!
  integer, save :: imi, imp, ims, imu, img, imb
#endif /* PROFILE */

! default problem parameter values
!
  real(kind=8), save :: beta = 1.00d+00
  real(kind=8), save :: zeta = 0.00d+00
  real(kind=8), save :: eta  = 0.00d+00
  real(kind=8), save :: dens = 1.00d+00
  real(kind=8), save :: bamp = 1.00d+00
  real(kind=8), save :: bgui = 0.00d+00

  real(kind=8), save :: pres = 5.00d-01
  real(kind=8), save :: pmag = 5.00d-01
  real(kind=8), save :: ptot = 1.00d+00
  real(kind=8), save :: valf = 1.00d+00
  real(kind=8), save :: lund = 1.00d+00

  real(kind=8), save :: dlta = 1.00d-16
  real(kind=8), save :: blim = 1.00d+00

  integer     , save :: pert = 0
  integer     , save :: nper = 10
  real(kind=8), save :: bper = 0.00d+00
  real(kind=8), save :: vper = 0.00d+00
  real(kind=8), save :: kper = 1.00d+00
  real(kind=8), save :: kvec = 1.00d+00
  real(kind=8), save :: xcut = 1.00d+99
  real(kind=8), save :: ycut = 1.00d+99
  real(kind=8), save :: xdec = 1.00d-01
  real(kind=8), save :: ydec = 1.00d-01

! flag indicating if the gravitational source term is enabled
!
  logical, save :: gravity_enabled_user = .false.

! allocatable arrays for velocity perturbation
!
  real(kind=8), dimension(:), allocatable :: kx, ky, kz, ux, uy, uz, ph

!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
  contains
!
!===============================================================================
!
! subroutine INITIALIZE_USER_PROBLEM:
! ----------------------------------
!
!   Subroutine initializes user problem. It could read problem parameters which
!   are used in all subroutines defining this specific problem.
!
!   Arguments:
!
!     problem - the problem name
!     verbose - a logical flag turning the information printing;
!     status  - an integer flag for error return value;
!
!===============================================================================
!
  subroutine initialize_user_problem(problem, verbose, status)

! include external procedures and variables
!
#if NDIMS == 3
    use constants  , only : pi
#endif /* NDIMS == 3 */
    use constants  , only : pi2
    use coordinates, only : ng => nghosts, ady
    use equations  , only : magnetized, csnd, csnd2
    use helpers    , only : print_section, print_parameter
    use parameters , only : get_parameter
    use random     , only : randuni, randsym

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    character(len=64), intent(in)  :: problem
    logical          , intent(in)  :: verbose
    integer          , intent(out) :: status

! local variables
!
    character(len=64) :: perturbation = "noise"
    integer           :: n
    real(kind=8)      :: thh, fc
#if NDIMS == 3
    real(kind=8)      :: thv, tx, ty, tz, tt
#endif /* NDIMS == 3 */
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! set timer descriptions
!
    call set_timer('user_problem:: initialize'   , imi)
    call set_timer('user_problem:: problem setup', imp)
    call set_timer('user_problem:: shape'        , ims)
    call set_timer('user_problem:: sources'      , imu)
    call set_timer('user_problem:: gravity'      , img)
    call set_timer('user_problem:: boundaries'   , imb)

! start accounting time for module initialization/finalization
!
    call start_timer(imi)
#endif /* PROFILE */

! reset the status flag
!
    status = 0

! this problem does not work with not magnetized set of equations
!
    if (.not. magnetized) then
      if (verbose) then
        write(*,*)
        write(*,"(1x,a)") "ERROR!"
        write(*,"(1x,a)") "The problem " // trim(problem) //                   &
                          " requires magnetized set of equations:" //          &
                          " 'MHD', or 'SR-MHD'."
        write(*,*)
      end if
      status = 1
    end if

! proceed if no errors
!
    if (status == 0) then

! get the reconnection equilibrium parameters
!
      call get_parameter("beta", beta)
      if (beta <= 0.0d+00) then
        if (verbose) then
          write(*,*)
          write(*,"(1x,a)") "ERROR!"
          write(*,"(1x,a)") "Parameter 'beta' must be positive (beta > 0.0)!"
          write(*,*)
        end if
        status = 1
      end if
      call get_parameter("zeta", zeta)
      if (zeta < 0.0d+00 .or. zeta > 1.0d+00) then
        if (verbose) then
          write(*,*)
          write(*,"(1x,a)") "ERROR!"
          write(*,"(1x,a)") "Parameter 'zeta' must be between 0.0 and 1.0!"
          write(*,*)
        end if
        status = 1
      end if
      call get_parameter("resistivity", eta)
      if (eta < 0.0d+00) then
        if (verbose) then
          write(*,*)
          write(*,"(1x,a)") "ERROR!"
          write(*,"(1x,a)") "Resistivity cannot be negative!"
          write(*,*)
        end if
        status = 1
      end if
      call get_parameter("dens", dens)
      if (dens <= 0.0d+00) then
        if (verbose) then
          write(*,*)
          write(*,"(1x,a)") "ERROR!"
          write(*,"(1x,a)") "Parameter 'dens' must be positive (dens > 0.0)!"
          write(*,*)
        end if
        status = 1
      end if
      call get_parameter("bamp", bamp)
      call get_parameter("bgui", bgui)

! calculate the maximum magnetic pressure, thermal pressure from the plasma-β
! parameters, and the sound speed in the case of isothermal equations of state
!
      pmag  = 0.5d+00 * (bamp**2 + bgui**2)
      pres  = beta * pmag
      ptot  = pres + pmag
      csnd2 = pres / dens
      csnd  = sqrt(csnd2)
      valf  = sqrt(2.0d+00 * pmag / dens)
      lund  = valf / max(tiny(eta), eta)
      dlta  = lund**(- 1.0d+00 / 3.0d+00)

! get the geometry parameters
!
      call get_parameter("delta", dlta)
      if (dlta < 0.0d+00) then
        if (verbose) then
          write(*,*)
          write(*,"(1x,a)") "ERROR!"
          write(*,"(1x,a)") "Parameter 'delta' must be equal or bigger than zero!"
          write(*,*)
        end if
        status = 1
      end if

      call get_parameter("blimit", blim)

! lower limit for blim
!
      blim = max(blim, ng * ady(1))

    end if ! status

! proceed if no errors
!
    if (status == 0) then

! get the perturbation parameters
!
      call get_parameter("perturbation", perturbation)
      call get_parameter("nper"        , nper)
      call get_parameter("bper"        , bper)
      call get_parameter("vper"        , vper)
      call get_parameter("kper"        , kper)
      call get_parameter("xcut"        , xcut)
      call get_parameter("ycut"        , ycut)
      call get_parameter("xdec"        , xdec)
      call get_parameter("ydec"        , ydec)

! choose the perturbation type
!
      select case(perturbation)
      case('noise', 'random')
        pert = 0
      case('mode', 'one mode', 'one-mode', 'one_mode')
        pert = 1
      case('multi-wave', 'random waves', 'random-waves', 'random_waves')
        pert = 2
      case default
        perturbation = 'mode'
        pert = 1
      end select

! prepare the wave vector of the perturbation
!
      kvec = pi2 * kper

! prepare the wave vectors for multi-wave perturbation
!
      if (pert == 2) then

! allocate phase and wave vector components
!
        allocate(kx(nper), ky(nper), kz(nper), ux(nper), uy(nper), uz(nper),   &
                 ph(nper), stat = status)

        if (status == 0) then

! choose random wave vector directions
!
          fc = 1.0d+00 / sqrt(1.0d+00 * nper)
          do n = 1, nper
            thh   = pi2 * randuni()
#if NDIMS == 3
            thv   = pi  * randsym()
            ux(n) = cos(thh) * cos(thv)
            uy(n) = sin(thh) * cos(thv)
            uz(n) =            sin(thv)
            kx(n) = pi2 * nint(kper * ux(n))
            ky(n) = pi2 * nint(kper * uy(n))
            kz(n) = pi2 * nint(kper * uz(n))
            tt = 0.0d+00
            do while(tt < 1.0d-08)
              thh   = pi2 * randuni()
              thv   = pi  * randsym()
              tx    = cos(thh) * cos(thv)
              ty    = sin(thh) * cos(thv)
              tz    =            sin(thv)
              ux(n) = ty * kz(n) - tz * ky(n)
              uy(n) = tz * kx(n) - tx * kz(n)
              uz(n) = tx * ky(n) - ty * kx(n)
              tt    = sqrt(ux(n)**2 + uy(n)**2 + uz(n)**2)
            end do
            ux(n) = fc * ux(n) / tt
            uy(n) = fc * uy(n) / tt
            uz(n) = fc * uz(n) / tt
#else /* NDIMS == 3 */
            kx(n) = pi2 * nint(kper * cos(thh))
            ky(n) = pi2 * nint(kper * sin(thh))
            kz(n) = 0.0d+00
            ux(n) = fc * sin(thh)
            uy(n) = fc * cos(thh)
            uz(n) = 0.0d+00
#endif /* NDIMS == 3 */
            ph(n) = pi2 * randuni()
          end do

        end if ! status
      end if
    end if ! status

! proceed if no errors
!
    if (status == 0) then

! print information about the user problem setup
!
      call print_section(verbose, "Parameters (* - set, otherwise calculated)")
      call print_parameter(verbose, '(*) beta (plasma-beta)'   , beta)
      call print_parameter(verbose, '(*) zeta'                 , zeta)
      call print_parameter(verbose, '(*) dens'                 , dens)
      call print_parameter(verbose, '(*) bamp'                 , bamp)
      call print_parameter(verbose, '(*) bgui (guide field)'   , bgui)
      call print_parameter(verbose, '( ) pres (thermal pres.)' , pres)
      call print_parameter(verbose, '( ) pmag (magnetic pres.)', pmag)
      call print_parameter(verbose, '( ) ptot (total pressure)', ptot)
      call print_parameter(verbose, '( ) csnd (sound speed)'   , csnd)
      call print_parameter(verbose, '( ) Valf (Alfven speed)'  , valf)
      if (eta > 0.0d+00) then
        call print_parameter(verbose, '( ) S (Lundquist number)', lund)
      end if
      call print_parameter(verbose, '(*) delta (thickness)', dlta)
      call print_parameter(verbose, '(*) blim'             , blim)
      call print_parameter(verbose, '(*) perturbation', perturbation)
      call print_parameter(verbose, '(*) bper'        , bper)
      call print_parameter(verbose, '(*) vper'        , vper)
      if (pert >= 1) then
        call print_parameter(verbose, '(*) kper'      , kper)
      end if
      if (pert == 2) then
        call print_parameter(verbose, '(*) nper'      , nper)
      end if
      call print_parameter(verbose, '(*) xcut'        , xcut)
      call print_parameter(verbose, '(*) ycut'        , ycut)
      call print_parameter(verbose, '(*) xdec'        , xdec)
      call print_parameter(verbose, '(*) ydec'        , ydec)
    end if ! status

#ifdef PROFILE
! stop accounting time for module initialization/finalization
!
    call stop_timer(imi)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine initialize_user_problem
!
!===============================================================================
!
! subroutine FINALIZE_USER_PROBLEM:
! --------------------------------
!
!   Subroutine releases memory used by the module.
!
!   Arguments:
!
!     status - an integer flag for error return value;
!
!===============================================================================
!
  subroutine finalize_user_problem(status)

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    integer, intent(out) :: status
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! start accounting time for module initialization/finalization
!
    call start_timer(imi)
#endif /* PROFILE */

! reset the status flag
!
    status = 0

! deallocate wave vector components, random directions, and random phase
!
    if (allocated(kx)) deallocate(kx, ky, kz, stat = status)
    if (allocated(ux)) deallocate(ux, uy, uz, stat = status)
    if (allocated(ph)) deallocate(ph, stat = status)

#ifdef PROFILE
! stop accounting time for module initialization/finalization
!
    call stop_timer(imi)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine finalize_user_problem
!
!===============================================================================
!
! subroutine SETUP_PROBLEM_USER:
! -----------------------------
!
!   Subroutine sets the initial conditions for the user specific problem.
!
!   Arguments:
!
!     pdata - pointer to the datablock structure of the currently initialized
!             block;
!
!===============================================================================
!
  subroutine setup_problem_user(pdata)

! include external procedures and variables
!
    use blocks     , only : block_data
    use constants  , only : pi, pi2
    use coordinates, only : nn => bcells
    use coordinates, only : ax, ay, adx, ady, ylen
#if NDIMS == 3
    use coordinates, only : az, adz
#endif /* NDIMS == 3 */
    use equations  , only : prim2cons
    use equations  , only : nv, ns
    use equations  , only : idn, ivx, ivy, ivz, ipr, ibx, iby, ibz, ibp, isl
    use equations  , only : csnd2
    use operators  , only : curl
    use random     , only : randsym

! local variables are not implicit by default
!
    implicit none

! input arguments
!
    type(block_data), pointer, intent(inout) :: pdata

! local variables
!
    integer      :: i, j, k = 1, n
    real(kind=8) :: xpl, xpu, ypl, ypu
    real(kind=8) :: xp, yp
    real(kind=8) :: yrat
    real(kind=8) :: kv, fv
    real(kind=8) :: vx = 0.0d+00, vy = 0.0d+00, vv, va
#if NDIMS == 3
    real(kind=8) :: vz = 0.0d+00
#endif /* NDIMS == 3 */
    real(kind=8) :: pz = 0.0d+00, pp, ba
#if NDIMS == 3
    real(kind=8) :: px = 0.0d+00, py = 0.0d+00
#endif /* NDIMS == 3 */

! local arrays
!
#if NDIMS == 3
    real(kind=8), dimension(nv,nn,nn,nn) :: qpert
    real(kind=8), dimension(3 ,nn,nn,nn) :: pot
#else /* NDIMS == 3 */
    real(kind=8), dimension(nv,nn,nn, 1) :: qpert
    real(kind=8), dimension(3 ,nn,nn, 1) :: pot
#endif /* NDIMS == 3 */
    real(kind=8), dimension(nv,nn)       :: q, u, qprof
    real(kind=8), dimension(nn)          :: xl, xu, xc, fx
    real(kind=8), dimension(nn)          :: yl, yu, yc, fy
#if NDIMS == 3
    real(kind=8), dimension(nn)          :: zc
#endif /* NDIMS == 3 */
    real(kind=8), dimension(3)           :: dh
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! start accounting time for the problem setup
!
    call start_timer(imp)
#endif /* PROFILE */

! prepare cell sizes
!
    dh(1) = adx(pdata%meta%level)
    dh(2) = ady(pdata%meta%level)
#if NDIMS == 3
    dh(3) = adz(pdata%meta%level)
#endif /* NDIMS == 3 */

! prepare block coordinates
!
    xc(:) = pdata%meta%xmin + ax(pdata%meta%level,:)
    yc(:) = pdata%meta%ymin + ay(pdata%meta%level,:)
#if NDIMS == 3
    zc(:) = pdata%meta%zmin + az(pdata%meta%level,:)
#endif /* NDIMS == 3 */

    xl(:) = pdata%meta%xmin + ax(pdata%meta%level,:) - 5.0d-01 * dh(1)
    xu(:) = xl(:) + dh(1)
    yl(:) = pdata%meta%ymin + ay(pdata%meta%level,:) - 5.0d-01 * dh(2)
    yu(:) = yl(:) + dh(2)

! set the equilibrium profile
!
    do j = 1, nn
      qprof(ibx,j) = bamp * max(-1.0d+00, min(1.0d+00,                         &
                     dlta * (log_cosh(yu(j) / dlta)                            &
                           - log_cosh(yl(j) / dlta)) / dh(2)))
    end do
    qprof(iby,:) = 0.0d+00
    qprof(ibz,:) = zeta * sqrt(bamp**2 - qprof(ibx,:)**2) + bgui
    if (ipr > 0) then
      qprof(idn,:) =  dens
      qprof(ipr,:) =  ptot - 0.5d+00 * sum(qprof(ibx:ibz,:)**2, 1)
    else
      qprof(idn,:) = (ptot - 0.5d+00 * sum(qprof(ibx:ibz,:)**2, 1)) / csnd2
    end if
    qprof(ivx,:) = 0.0d+00
    qprof(ivy,:) = 0.0d+00
    qprof(ivz,:) = 0.0d+00
    qprof(ibp,:) = 0.0d+00
    if (ns > 0) then
      qprof(isl,:) = sign(1.0d+00, yc(:))
    end if

! ratio of the current sheet thickness to the cell size
!
    yrat  = dlta / dh(2)

! prepare decaying factors
!
    fv = 0.5d+00 * pi
    do i = 1, nn
      xp    = fv * min(1.0d+00, max(0.0d+00, abs(xc(i)) - xcut) / xdec)
      fx(i) = cos(xp)**2
    end do ! i = 1, nn
    do j = 1, nn
      yp    = fv * min(1.0d+00, max(0.0d+00, abs(yc(j)) - ycut) / ydec)
      fy(j) = cos(yp)**2
    end do ! i = 1, nn

! reset the perturbation matrix
!
    qpert(:,:,:,:) = 0.0d+00

! the random perturbation
!
    if (pert == 0) then

! of velocity
!
      if (abs(vper) > 0.0d+00) then

! initiate the random velocity components
!
#if NDIMS == 3
        do k = 1, nn
#endif /* NDIMS == 3 */
          do j = 1, nn
            if (fy(j) > 0.0d+00) then
              do i = 1, nn
                if (fx(i) > 0.0d+00) then

! calculate the profile of perturbation amplitude
!
                  va = vper * fx(i) * fy(j)

! get the random direction
!
                  vv = 0.0d+00
                  do while(vv < 1.0d-08)
                    vx = randsym()
                    vy = randsym()
#if NDIMS == 3
                    vz = randsym()
#endif /* NDIMS == 3 */

#if NDIMS == 3
                    vv = sqrt(vx * vx + vy * vy + vz * vz)
#else /* NDIMS == 3 */
                    vv = sqrt(vx * vx + vy * vy)
#endif /* NDIMS == 3 */
                  end do

                  qpert(ivx,i,j,k) = va * (vx / vv)
                  qpert(ivy,i,j,k) = va * (vy / vv)
#if NDIMS == 3
                  qpert(ivz,i,j,k) = va * (vz / vv)
#endif /* NDIMS == 3 */

                end if ! |x| < xcut
              end do ! i = 1, nn
            end if ! |y| < ycut
          end do ! j = 1, nn
#if NDIMS == 3
        end do ! k = 1, nn
#endif /* NDIMS == 3 */

      end if ! vper /= 0.0

! of magnetic field
!
      if (abs(bper) > 0.0d+00) then

! reset the potential
!
        pot(:,:,:,:) = 0.0d+00

! initiate the random magnetic field components
!
#if NDIMS == 3
        do k = 1, nn
#endif /* NDIMS == 3 */
          do j = 1, nn
            if (fy(j) > 0.0d+00) then
              do i = 1, nn
                if (fx(i) > 0.0d+00) then

! calculate the profile of perturbation amplitude
!
                  ba = dh(1) * bper * fx(i) * fy(j)

! get the random direction
!
                  pp = 0.0d+00
                  do while(pp < 1.0d-08)
#if NDIMS == 3
                    px = randsym()
                    py = randsym()
#endif /* NDIMS == 3 */
                    pz = randsym()

#if NDIMS == 3
                    pp = sqrt(px * px + py * py + pz * pz)
#else /* NDIMS == 3 */
                    pp = abs(pz)
#endif /* NDIMS == 3 */
                  end do

#if NDIMS == 3
                  pot(1,i,j,k) = ba * (px / pp)
                  pot(2,i,j,k) = ba * (py / pp)
#endif /* NDIMS == 3 */
                  pot(3,i,j,k) = ba * (pz / pp)

                end if ! |x| < xcut
              end do ! i = 1, nn
            end if ! |y| < ycut
          end do ! j = 1, nn
#if NDIMS == 3
        end do ! k = 1, nn
#endif /* NDIMS == 3 */

! calculate magnetic field perturbation components from vector potential
!
        call curl(dh(1:3), pot(:,:,:,:), qpert(ibx:ibz,:,:,:))

      end if ! bper /= 0.0

    end if ! pert == 0

! one-mode perturbation
!
    if (pert == 1) then

! of velocity
!
      if (abs(vper) > 0.0d+00) then

! reset the potential
!
        pot(:,:,:,:) = 0.0d+00

! calculate the perturbation factor
!
        fv = vper * ylen / (pi2 * pi2 * pi * dh(1) * dh(2) * kper)

! calculate the perturbation profile
!
#if NDIMS == 3
        do k = 1, nn
#endif /* NDIMS == 3 */
          do j = 1, nn
            if (fy(j) > 0.0d+00) then
              ypl = pi * yl(j) / ylen
              ypu = pi * yu(j) / ylen
              do i = 1, nn
                if (fx(i) > 0.0d+00) then
                  xpl = pi2 * kper * xl(i)
                  xpu = pi2 * kper * xu(i)

                  pot(3,i,j,k) = fv * fx(i) * fy(j) * (cos(xpl) - cos(xpu))    &
                                                    * (cos(ypl) - cos(ypu))

                end if ! |x| < xcut
              end do ! i = 1, nn
            end if ! |y| < ycut
          end do ! j = 1, nn
#if NDIMS == 3
        end do ! k = 1, nn
#endif /* NDIMS == 3 */

! calculate magnetic field components from vector potential
!
        call curl(dh(1:3), pot(:,:,:,:), qpert(ivx:ivz,:,:,:))

      end if ! vper /= 0.0

! of magnetic field
!
      if (abs(bper) > 0.0d+00) then

! reset the potential
!
        pot(:,:,:,:) = 0.0d+00

! calculate the perturbation factor
!
        fv = bper * ylen / (pi2 * pi2 * pi * dh(1) * dh(2) * kper)

! calculate the perturbation profile
!
#if NDIMS == 3
        do k = 1, nn
#endif /* NDIMS == 3 */
          do j = 1, nn
            if (fy(j) > 0.0d+00) then
              ypl = pi * yl(j) / ylen
              ypu = pi * yu(j) / ylen
              do i = 1, nn
                if (fx(i) > 0.0d+00) then
                  xpl = pi2 * kper * xl(i)
                  xpu = pi2 * kper * xu(i)

                  pot(3,i,j,k) = fv * fx(i) * fy(j) * (sin(xpl) - sin(xpu))    &
                                                    * (sin(ypl) - sin(ypu))

                end if ! |x| < xcut
              end do ! i = 1, nn
            end if ! |y| < ycut
          end do ! j = 1, nn
#if NDIMS == 3
        end do ! k = 1, nn
#endif /* NDIMS == 3 */

! calculate magnetic field components from vector potential
!
        call curl(dh(1:3), pot(:,:,:,:), qpert(ibx:ibz,:,:,:))

      end if ! bper /= 0.0

    end if ! pert == 1

! prepare the random perturbation of velocity
!
    if (pert == 2) then

      if (abs(vper) > 0.0d+00) then

! iterate over the block position and initiate the velocity perturbation
!
#if NDIMS == 3
        do k = 1, nn
#endif /* NDIMS == 3 */
          do j = 1, nn
            if (fy(j) > 0.0d+00) then
              do i = 1, nn
                if (fx(i) > 0.0d+00) then

! calculate the velocity amplitude profile
!
                  fv = vper * fx(i) * fy(j)

! add perturbation components
!
                  do n = 1, nper
#if NDIMS == 3
                    kv = kx(n) * xc(i) + ky(n) * yc(j) + kz(n) * zc(k) + ph(n)
#else /* NDIMS == 3 */
                    kv = kx(n) * xc(i) + ky(n) * yc(j)                 + ph(n)
#endif /* NDIMS == 3 */
                    va = fv * sin(kv)

                    qpert(ivx,i,j,k) = qpert(ivx,i,j,k) + va * ux(n)
                    qpert(ivy,i,j,k) = qpert(ivy,i,j,k) + va * uy(n)
#if NDIMS == 3
                    qpert(ivz,i,j,k) = qpert(ivz,i,j,k) + va * uz(n)
#endif /* NDIMS == 3 */
                  end do

                end if ! fx > 0.0
              end do ! i = 1, nn
            end if ! fy > 0.0
          end do ! j = 1, nn
#if NDIMS == 3
        end do ! k = 1, nn
#endif /* NDIMS == 3 */

      end if ! vper /= 0.0

    end if ! pert == 2

! iterate over all positions in the XZ plane
!
#if NDIMS == 3
    do k = 1, nn
#endif /* NDIMS == 3 */
      do i = 1, nn

! set the variable profiles with perturbations
!
        q(:,:) = qprof(:,:) + qpert(:,i,:,k)

! convert the primitive variables to the conservative ones
!
        call prim2cons(q(:,:), u(:,:), .true.)

! copy the primitive variables to the current block
!
        pdata%q(:,i,:,k) = q(:,:)

! copy the conserved variables to the current block
!
        pdata%u(:,i,:,k) = u(:,:)

      end do ! i = 1, nn
#if NDIMS == 3
    end do ! k = 1, nn
#endif /* NDIMS == 3 */

#ifdef PROFILE
! stop accounting time for the problems setup
!
    call stop_timer(imp)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine setup_problem_user
!
!===============================================================================
!
! subroutine LOG_COSH:
! -------------------
!
!   Function calculates the logarithm of the hyperbolic cosine, which is
!   the result of the integration of tanh(x). Direct calculation using
!   Fortran intrinsic subroutines fails for large values of x, therefore
!   the logarithm of cosh is approximated as |x| + log(1/2) for
!   |x| > threshold.
!
!   Arguments:
!
!     x - function argument;
!
!===============================================================================
!
  function log_cosh(x) result(y)

! local variables are not implicit by default
!
    implicit none

! function arguments
!
    real(kind=8), intent(in) :: x
    real(kind=8)             :: y

! local parameters
!
    real(kind=8), parameter :: th = acosh(huge(x)), lh = log(0.5d+00)
!
!-------------------------------------------------------------------------------
!
    if (abs(x) < th) then
      y = log(cosh(x))
    else
      y = abs(x) + lh
    end if

!-------------------------------------------------------------------------------
!
  end function log_cosh
!
!===============================================================================
!
! subroutine UPDATE_SHAPES_USER:
! -----------------------------
!
!   Subroutine defines the regions updated by user.
!
!   Arguments:
!
!     pdata - pointer to the data block structure of the currently initialized
!             block;
!     time  - time at the moment of update;
!     dt    - time step since the last update;
!
!===============================================================================
!
  subroutine update_shapes_user(pdata, time, dt)

! include external procedures and variables
!
    use blocks, only : block_data

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    type(block_data), pointer, intent(inout) :: pdata
    real(kind=8)             , intent(in)    :: time, dt
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! start accounting time for the shape update
!
    call start_timer(ims)
#endif /* PROFILE */

#ifdef PROFILE
! stop accounting time for the shape update
!
    call stop_timer(ims)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine update_shapes_user
!
!===============================================================================
!
! subroutine UPDATE_SOURCES_USER:
! ------------------------------
!
!   Subroutine adds the user defined source terms.
!
!   Arguments:
!
!     pdata - the pointer to a data block;
!     t, dt - the time and time increment;
!     du    - the array of variable increment;
!
!===============================================================================
!
  subroutine update_sources_user(pdata, t, dt, du)

! include external variables
!
    use blocks, only : block_data

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    type(block_data), pointer       , intent(inout) :: pdata
    real(kind=8)                    , intent(in)    :: t, dt
    real(kind=8), dimension(:,:,:,:), intent(inout) :: du
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! start accounting time for source terms
!
    call start_timer(imu)
#endif /* PROFILE */

#ifdef PROFILE
! stop accounting time for source terms
!
    call stop_timer(imu)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine update_sources_user
!
!===============================================================================
!
! subroutine GRAVITATIONAL_ACCELERATION_USER:
! ------------------------------------------
!
!   Subroutine returns the user defined gravitational acceleration.
!
!   Arguments:
!
!     t, dt   - time and the time increment;
!     x, y, z - rectangular coordinates;
!     acc     - vector of the gravitational acceleration;
!
!===============================================================================
!
  subroutine gravitational_acceleration_user(t, dt, x, y, z, acc)

! include external procedures and variables
!
    use parameters, only : get_parameter

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    real(kind=8)              , intent(in)  :: t, dt
    real(kind=8)              , intent(in)  :: x, y, z
    real(kind=8), dimension(3), intent(out) :: acc
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! start accounting time for the gravitational acceleration calculation
!
    call start_timer(img)
#endif /* PROFILE */

! reset gravitational acceleration
!
    acc(:) = 0.0d+00

#ifdef PROFILE
! stop accounting time for the gravitational acceleration calculation
!
    call stop_timer(img)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine gravitational_acceleration_user
!
!===============================================================================
!
! subroutine BOUNDARY_USER_X:
! --------------------------
!
!   Subroutine updates ghost zones within the specific region along
!   the X direction.
!
!   Arguments:
!
!     ic      - the block side along the X direction for the ghost zone update;
!     jl, ju  - the cell index limits for the Y direction;
!     kl, ku  - the cell index limits for the Z direction;
!     t, dt   - time and time increment;
!     x, y, z - the block coordinates;
!     qn      - the array of variables to update;
!
!===============================================================================
!
  subroutine boundary_user_x(ic, jl, ju, kl, ku, t, dt, x, y, z, qn)

! import external procedures and variables
!
    use coordinates    , only : nn => bcells, nb, ne, nbl, neu
    use equations      , only : ivx, ibx, iby, ibp
#if NDIMS == 3
    use equations      , only : ibz
#endif /* NDIMS == 3 */

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    integer                         , intent(in)    :: ic
    integer                         , intent(in)    :: jl, ju
    integer                         , intent(in)    :: kl, ku
    real(kind=8)                    , intent(in)    :: t, dt
    real(kind=8), dimension(:)      , intent(in)    :: x
    real(kind=8), dimension(:)      , intent(in)    :: y
    real(kind=8), dimension(:)      , intent(in)    :: z
    real(kind=8), dimension(:,:,:,:), intent(inout) :: qn

! local variables
!
    integer      :: im2, im1, i    , ip1, ip2
    integer      :: jm2, jm1, j    , jp1, jp2
    integer      :: k = 1
#if NDIMS == 3
    integer      :: km2, km1, kp1, kp2
#endif /* NDIMS == 3 */
    real(kind=8) :: dx, dy, dxy
#if NDIMS == 3
    real(kind=8) :: dz, dxz
#endif /* NDIMS == 3 */
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! start accounting time for the boundary update
!
    call start_timer(imb)
#endif /* PROFILE */

! process case with magnetic field, otherwise revert to standard outflow
!
    if (ibx > 0) then

! get the cell sizes and their ratios
!
      dx  = x(2) - x(1)
      dy  = y(2) - y(1)
#if NDIMS == 3
      dz  = z(2) - z(1)
#endif /* NDIMS == 3 */
      dxy = dx / dy
#if NDIMS == 3
      dxz = dx / dz
#endif /* NDIMS == 3 */

! process left and right side boundary separatelly
!
      if (ic == 1) then

! iterate over left-side ghost layers
!
        do i = nbl, 1, -1

! calculate neighbor cell indices
!
          ip1 = min(nn, i + 1)
          ip2 = min(nn, i + 2)

! iterate over boundary layer
!
#if NDIMS == 3
          do k = kl, ku
            km2 = max( 1, k - 2)
            km1 = max( 1, k - 1)
            kp1 = min(nn, k + 1)
            kp2 = min(nn, k + 2)
#endif /* NDIMS == 3 */
            do j = jl, ju
              jm2 = max( 1, j - 2)
              jm1 = max( 1, j - 1)
              jp1 = min(nn, j + 1)
              jp2 = min(nn, j + 2)

! make the normal derivative zero
!
              qn(:,i,j,k) = qn(:,nb,j,k)

! prevent the inflow
!
              qn(ivx,i,j,k) = min(0.0d+00, qn(ivx,nb,j,k))

! update the normal component of magnetic field from divergence-free condition
!
              qn(ibx,i,j,k) = qn(ibx,ip2,j,k)                                  &
                            + (qn(iby,ip1,jp1,k) - qn(iby,ip1,jm1,k)) * dxy
#if NDIMS == 3
              qn(ibx,i,j,k) = qn(ibx,i  ,j,k)                                  &
                            + (qn(ibz,ip1,j,kp1) - qn(ibz,ip1,j,km1)) * dxz
#endif /* NDIMS == 3 */
              qn(ibp,i,j,k) = 0.0d+00
            end do ! j = jl, ju
#if NDIMS == 3
          end do ! k = kl, ku
#endif /* NDIMS == 3 */
        end do ! i = nbl, 1, -1

      else ! ic == 1

! iterate over right-side ghost layers
!
        do i = neu, nn

! calculate neighbor cell indices
!
          im1 = max( 1, i - 1)
          im2 = max( 1, i - 2)

! iterate over boundary layer
!
#if NDIMS == 3
          do k = kl, ku
            km1 = max( 1, k - 1)
            kp1 = min(nn, k + 1)
            km2 = max( 1, k - 2)
            kp2 = min(nn, k + 2)
#endif /* NDIMS == 3 */
            do j = jl, ju
              jm1 = max( 1, j - 1)
              jp1 = min(nn, j + 1)
              jm2 = max( 1, j - 2)
              jp2 = min(nn, j + 2)

! make the normal derivative zero
!
              qn(:,i,j,k) = qn(:,ne,j,k)

! prevent the inflow
!
              qn(ivx,i,j,k) = max(0.0d+00, qn(ivx,ne,j,k))

! update the normal component of magnetic field from divergence-free condition
!
              qn(ibx,i,j,k) = qn(ibx,im2,j,k)                              &
                            + (qn(iby,im1,jm1,k) - qn(iby,im1,jp1,k)) * dxy
#if NDIMS == 3
              qn(ibx,i,j,k) = qn(ibx,i  ,j,k)                              &
                            + (qn(ibz,im1,j,km1) - qn(ibz,im1,j,kp1)) * dxz
#endif /* NDIMS == 3 */
              qn(ibp,i,j,k) = 0.0d+00
            end do ! j = jl, ju
#if NDIMS == 3
          end do ! k = kl, ku
#endif /* NDIMS == 3 */
        end do ! i = neu, nn
      end if ! ic == 1
    else ! ibx > 0
      if (ic == 1) then
        do i = nbl, 1, -1
#if NDIMS == 3
          qn( : ,i,jl:ju,kl:ku) =     qn( : ,nb,jl:ju,kl:ku)
          qn(ivx,i,jl:ju,kl:ku) = min(qn(ivx,nb,jl:ju,kl:ku), 0.0d+00)
#else /* NDIMS == 3 */
          qn( : ,i,jl:ju,  :  ) =     qn( : ,nb,jl:ju,  :  )
          qn(ivx,i,jl:ju,  :  ) = min(qn(ivx,nb,jl:ju,  :  ), 0.0d+00)
#endif /* NDIMS == 3 */
        end do ! i = nbl, 1, -1
      else
        do i = neu, nn
#if NDIMS == 3
          qn( : ,i,jl:ju,kl:ku) =     qn( : ,ne,jl:ju,kl:ku)
          qn(ivx,i,jl:ju,kl:ku) = max(qn(ivx,ne,jl:ju,kl:ku), 0.0d+00)
#else /* NDIMS == 3 */
          qn( : ,i,jl:ju,  :  ) =     qn( : ,ne,jl:ju,  :  )
          qn(ivx,i,jl:ju,  :  ) = max(qn(ivx,ne,jl:ju,  :  ), 0.0d+00)
#endif /* NDIMS == 3 */
        end do ! i = neu, nn
      end if
    end if ! ibx > 0

#ifdef PROFILE
! stop accounting time for the boundary update
!
    call stop_timer(imb)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine boundary_user_x
!
!===============================================================================
!
! subroutine BOUNDARY_USER_Y:
! --------------------------
!
!   Subroutine updates ghost zones within the specific region along
!   the Y direction.
!
!   Arguments:
!
!     jc      - the block side along the Y direction for the ghost zone update;
!     il, iu  - the cell index limits for the X direction;
!     kl, ku  - the cell index limits for the Z direction;
!     t, dt   - time and time increment;
!     x, y, z - the block coordinates;
!     qn      - the array of variables to update;
!
!===============================================================================
!
  subroutine boundary_user_y(jc, il, iu, kl, ku, t, dt, x, y, z, qn)

! import external procedures and variables
!
    use coordinates    , only : nn => bcells, nb, ne, nbl, neu
    use equations      , only : nv
    use equations      , only : idn, ivy, ipr, ibx, iby, ibz, ibp

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    integer                         , intent(in)    :: jc
    integer                         , intent(in)    :: il, iu
    integer                         , intent(in)    :: kl, ku
    real(kind=8)                    , intent(in)    :: t, dt
    real(kind=8), dimension(:)      , intent(in)    :: x
    real(kind=8), dimension(:)      , intent(in)    :: y
    real(kind=8), dimension(:)      , intent(in)    :: z
    real(kind=8), dimension(:,:,:,:), intent(inout) :: qn

! local variables
!
    integer      :: i, im1, ip1
    integer      :: j, jm1, jp1, jm2, jp2
    integer      :: k = 1
#if NDIMS == 3
    integer      :: km1, kp1
#endif /* NDIMS == 3 */
    real(kind=8) :: dx, dy, dyx
#if NDIMS == 3
    real(kind=8) :: dz, dyz
#endif /* NDIMS == 3 */
    real(kind=8) :: fl, fr
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! start accounting time for the boundary update
!
    call start_timer(imb)
#endif /* PROFILE */

! process case with magnetic field, otherwise revert to standard outflow
!
    if (ibx > 0) then

! get the cell sizes and their ratios
!
      dx  = x(2) - x(1)
      dy  = y(2) - y(1)
#if NDIMS == 3
      dz  = z(2) - z(1)
#endif /* NDIMS == 3 */
      dyx = dy / dx
#if NDIMS == 3
      dyz = dy / dz
#endif /* NDIMS == 3 */

! process left and right side boundary separatelly
!
      if (jc == 1) then

! iterate over left-side ghost layers
!
        do j = nbl, 1, -1

! calculate neighbor cell indices
!
          jp1 = min(nn, j + 1)
          jp2 = min(nn, j + 2)

! calculate variable decay coefficients
!
          fr  = (dy * (nb - j - 5.0d-01)) / blim
          fl  = 1.0d+00 - fr

! iterate over boundary layer
!
#if NDIMS == 3
          do k = kl, ku
            km1 = max( 1, k - 1)
            kp1 = min(nn, k + 1)
#endif /* NDIMS == 3 */
            do i = il, iu
              im1 = max( 1, i - 1)
              ip1 = min(nn, i + 1)

! make normal derivatives zero
!
              qn(1:nv,i,j,k) = qn(1:nv,i,nb,k)

! decay density and pressure to their limits
!
              qn(idn,i,j,k) = fl * qn(idn,i,nb,k) + fr * dens
              if (ipr > 0) qn(ipr,i,j,k) = fl * qn(ipr,i,nb,k) + fr * pres

! decay magnetic field to its limit
!
              qn(ibx,i,j,k) = fl * qn(ibx,i,nb,k) - fr * bamp
              qn(ibz,i,j,k) = fl * qn(ibz,i,nb,k) + fr * bgui

! update By from div(B)=0
!
              qn(iby,i,j,k) = qn(iby,i,jp2,k)                              &
                           + (qn(ibx,ip1,jp1,k) - qn(ibx,im1,jp1,k)) * dyx
#if NDIMS == 3
              qn(iby,i,j,k) = qn(iby,i,j  ,k)                              &
                           + (qn(ibz,i,jp1,kp1) - qn(ibz,i,jp1,km1)) * dyz
#endif /* NDIMS == 3 */
              qn(ibp,i,j,k) = 0.0d+00
            end do ! i = il, iu
#if NDIMS == 3
          end do ! k = kl, ku
#endif /* NDIMS == 3 */
        end do ! j = nbl, 1, -1
      else ! jc = 1

! iterate over right-side ghost layers
!
        do j = neu, nn

! calculate neighbor cell indices
!
          jm1 = max( 1, j - 1)
          jm2 = max( 1, j - 2)

! calculate variable decay coefficients
!
          fr  = (dy * (j - ne - 5.0d-01)) / blim
          fl  = 1.0d+00 - fr

! iterate over boundary layer
!
#if NDIMS == 3
          do k = kl, ku
            km1 = max( 1, k - 1)
            kp1 = min(nn, k + 1)
#endif /* NDIMS == 3 */
            do i = il, iu
              im1 = max( 1, i - 1)
              ip1 = min(nn, i + 1)

! make normal derivatives zero
!
              qn(1:nv,i,j,k) = qn(1:nv,i,ne,k)

! decay density and pressure to their limits
!
              qn(idn,i,j,k) = fl * qn(idn,i,ne,k) + fr * dens
              if (ipr > 0) qn(ipr,i,j,k) = fl * qn(ipr,i,ne,k) + fr * pres

! decay magnetic field to its limit
!
              qn(ibx,i,j,k) = fl * qn(ibx,i,ne,k) + fr * bamp
              qn(ibz,i,j,k) = fl * qn(ibz,i,ne,k) + fr * bgui

! update By from div(B)=0
!
              qn(iby,i,j,k) = qn(iby,i,jm2,k)                              &
                           + (qn(ibx,im1,jm1,k) - qn(ibx,ip1,jm1,k)) * dyx
#if NDIMS == 3
              qn(iby,i,j,k) = qn(iby,i,j  ,k)                              &
                           + (qn(ibz,i,jm1,km1) - qn(ibz,i,jm1,kp1)) * dyz
#endif /* NDIMS == 3 */
              qn(ibp,i,j,k) = 0.0d+00
            end do ! i = il, iu
#if NDIMS == 3
          end do ! k = kl, ku
#endif /* NDIMS == 3 */
        end do ! j = neu, nn
      end if ! jc = 1
    else ! ibx > 0
      if (jc == 1) then
        do j = nbl, 1, -1
#if NDIMS == 3
          qn(1:nv,il:iu,j,kl:ku) = qn(1:nv,il:iu,nb,kl:ku)
          qn(ivy ,il:iu,j,kl:ku) = min(0.0d+00, qn(ivy,il:iu,nb,kl:ku))
#else /* NDIMS == 3 */
          qn(1:nv,il:iu,j,  :  ) = qn(1:nv,il:iu,nb,  :  )
          qn(ivy ,il:iu,j,  :  ) = min(0.0d+00, qn(ivy,il:iu,nb,  :  ))
#endif /* NDIMS == 3 */
        end do ! j = nbl, 1, -1
      else
        do j = neu, nn
#if NDIMS == 3
          qn(1:nv,il:iu,j,kl:ku) = qn(1:nv,il:iu,ne,kl:ku)
          qn(ivy ,il:iu,j,kl:ku) = max(0.0d+00, qn(ivy,il:iu,ne,kl:ku))
#else /* NDIMS == 3 */
          qn(1:nv,il:iu,j,  :  ) = qn(1:nv,il:iu,ne,  :  )
          qn(ivy ,il:iu,j,  :  ) = max(0.0d+00, qn(ivy,il:iu,ne,  :  ))
#endif /* NDIMS == 3 */
        end do ! j = neu, nn
      end if
    end if ! ibx > 0

#ifdef PROFILE
! stop accounting time for the boundary update
!
    call stop_timer(imb)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine boundary_user_y
!
!===============================================================================
!
! subroutine BOUNDARY_USER_Z:
! --------------------------
!
!   Subroutine updates ghost zones within the specific region along
!   the Z direction.
!
!   Arguments:
!
!     kc      - the block side along the Z direction for the ghost zone update;
!     il, iu  - the cell index limits for the X direction;
!     jl, ju  - the cell index limits for the Y direction;
!     t, dt   - time and time increment;
!     x, y, z - the block coordinates;
!     qn      - the array of variables to update;
!
!===============================================================================
!
  subroutine boundary_user_z(kc, il, iu, jl, ju, t, dt, x, y, z, qn)

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    integer                         , intent(in)    :: kc
    integer                         , intent(in)    :: il, iu
    integer                         , intent(in)    :: jl, ju
    real(kind=8)                    , intent(in)    :: t, dt
    real(kind=8), dimension(:)      , intent(in)    :: x
    real(kind=8), dimension(:)      , intent(in)    :: y
    real(kind=8), dimension(:)      , intent(in)    :: z
    real(kind=8), dimension(:,:,:,:), intent(inout) :: qn
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! start accounting time for the boundary update
!
    call start_timer(imb)
#endif /* PROFILE */

#ifdef PROFILE
! stop accounting time for the boundary update
!
    call stop_timer(imb)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine boundary_user_z

!===============================================================================
!
end module user_problem