amun-code/sources/operators.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) 2008-2024 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: OPERATORS
!!
!!  This module provides differential operators like gradient, divergence, or
!!  curl.
!!
!!*****************************************************************************
!
module operators

  implicit none

! interfaces for procedure pointers
!
  abstract interface
    subroutine derivative_1st_iface(d, h, u, v)
      integer                       , intent(in)  :: d
      real(kind=8)                  , intent(in)  :: h
      real(kind=8), dimension(:,:,:), intent(in)  :: u
      real(kind=8), dimension(:,:,:), intent(out) :: v
    end subroutine
    subroutine derivative_2nd_iface(d, h, u, v)
      integer                       , intent(in)  :: d
      real(kind=8)                  , intent(in)  :: h
      real(kind=8), dimension(:,:,:), intent(in)  :: u
      real(kind=8), dimension(:,:,:), intent(out) :: v
    end subroutine
  end interface

! procedure pointers for derivatives
!
  procedure(derivative_1st_iface), pointer, save :: derivative_1st => null()
  procedure(derivative_2nd_iface), pointer, save :: derivative_2nd => null()

  private

  public :: initialize_operators, finalize_operators
  public :: divergence, gradient, curl, laplace
  public :: derivative_1st, derivative_2nd

!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
  contains
!
!===============================================================================
!!
!!***  PUBLIC SUBROUTINES  *****************************************************
!!
!===============================================================================
!
!===============================================================================
!
! subroutine INITIALIZE_OPERATORS:
! -------------------------------
!
!   Subroutine initializes the module structures, pointers and variables.
!
!   Arguments:
!
!     order  - the order of the interpolation method;
!     status - return flag of the procedure execution status;
!
!===============================================================================
!
  subroutine initialize_operators(order, status)

    implicit none

    integer, intent(in)  :: order
    integer, intent(out) :: status

!-------------------------------------------------------------------------------
!
    status = 0

    select case(order)
    case(9)
      derivative_1st => derivative_1st_9o
      derivative_2nd => derivative_2nd_9o

    case(7)
      derivative_1st => derivative_1st_7o
      derivative_2nd => derivative_2nd_7o

    case(5)
      derivative_1st => derivative_1st_5o
      derivative_2nd => derivative_2nd_5o

    case default
      derivative_1st => derivative_1st_3o
      derivative_2nd => derivative_2nd_3o
    end select

!-------------------------------------------------------------------------------
!
  end subroutine initialize_operators
!
!===============================================================================
!
! subroutine FINALIZE_OPERATORS:
! -----------------------------
!
!   Subroutine releases the memory used by module variables and pointers.
!
!   Arguments:
!
!     status - return flag of the procedure execution status;
!
!===============================================================================
!
  subroutine finalize_operators(status)

    implicit none

    integer, intent(out) :: status

!-------------------------------------------------------------------------------
!
    status = 0

!-------------------------------------------------------------------------------
!
  end subroutine finalize_operators
!
!===============================================================================
!
! subroutine DIVERGENCE:
! ---------------------
!
!   Subroutine calculates the cell centered divergence of the input vector
!   field.
!
!      div(U) = ∇.([Ux, Uy, Uz]) = ∂x Ux + ∂y Uy + ∂z Uz
!
!   Arguments:
!
!     dh  - the spacial intervals in all direction;
!     u   - the input vector field;
!     v   - the output divergence field;
!
!===============================================================================
!
  subroutine divergence(dh, u, v)

    implicit none

    real(kind=8), dimension(3)      , intent(in)  :: dh
    real(kind=8), dimension(:,:,:,:), intent(in)  :: u
    real(kind=8), dimension(:,:,:)  , intent(out) :: v

    integer :: dir

    real(kind=8), dimension(size(u,2), size(u,3), size(u,4)) :: w

!-------------------------------------------------------------------------------
!
    v(:,:,:) = 0.0d+00

! iterate over directions and update divergence with directional derivatives
!
    do dir = 1, NDIMS

! calculate derivative along the current direction
!
      call derivative_1st(dir, dh(dir), u(dir,:,:,:), w(:,:,:))

! update the divergence array
!
      v(:,:,:) = v(:,:,:) + w(:,:,:)

    end do ! directions

!-------------------------------------------------------------------------------
!
  end subroutine divergence
!
!===============================================================================
!
! subroutine GRADIENT:
! -------------------
!
!   Subroutine calculates the cell centered gradient of the input scalar field.
!
!      grad(U) = ∇ U = [ ∂x U, ∂y U, ∂z U ]
!
!   Arguments:
!
!     dh  - the spacial intervals in all direction;
!     u   - the input scalar field;
!     v   - the output gradient vector field;
!
!===============================================================================
!
  subroutine gradient(dh, u, v)

    implicit none

    real(kind=8), dimension(3)      , intent(in)  :: dh
    real(kind=8), dimension(:,:,:)  , intent(in)  :: u
    real(kind=8), dimension(:,:,:,:), intent(out) :: v

    integer :: dir

!-------------------------------------------------------------------------------
!
    v(:,:,:,:) = 0.0d+00

! iterate over directions and calculate gradient components
!
    do dir = 1, NDIMS

! calculate derivative along the current direction
!
      call derivative_1st(dir, dh(dir), u(:,:,:), v(dir,:,:,:))

    end do ! directions

!-------------------------------------------------------------------------------
!
  end subroutine gradient
!
!===============================================================================
!
! subroutine CURL:
! ---------------
!
!   Subroutine calculates the cell centered curl of the input vector field.
!
!      curl(U) = ∇x([Ux, Uy, Uz])
!                              = [∂y Uz - ∂z Uy, ∂z Ux - ∂x Uz, ∂x Uy - ∂y Ux]
!
!   Arguments:
!
!     dh  - the spacial intervals in all direction;
!     u   - the input vector field;
!     v   - the output divergence field;
!
!===============================================================================
!
  subroutine curl(dh, u, v)

    implicit none

    real(kind=8), dimension(3)      , intent(in)  :: dh
    real(kind=8), dimension(:,:,:,:), intent(in)  :: u
    real(kind=8), dimension(:,:,:,:), intent(out) :: v

    real(kind=8), dimension(size(u,2), size(u,3), size(u,4)) :: w

!-------------------------------------------------------------------------------
!
! === calculate Vx component ===
!
! contribution from the Y derivative of Uz
!
    call derivative_1st(2, dh(2), u(3,:,:,:), w(:,:,:))

! update Vx
!
    v(1,:,:,:) = w(:,:,:)

#if NDIMS == 3
! contribution from the Z derivative of Uy
!
    call derivative_1st(3, dh(3), u(2,:,:,:), w(:,:,:))

! update Vx
!
    v(1,:,:,:) = v(1,:,:,:) - w(:,:,:)
#endif /* NDIMS == 3 */

! === calculate Vy component ===
!
#if NDIMS == 3
! contribution from the Z derivative of Ux
!
    call derivative_1st(3, dh(3), u(1,:,:,:), w(:,:,:))

! update Vy
!
    v(2,:,:,:) = w(:,:,:)

! contribution from the X derivative of Uz
!
    call derivative_1st(1, dh(1), u(3,:,:,:), w(:,:,:))

! update Vy
!
    v(2,:,:,:) = v(2,:,:,:) - w(:,:,:)
#else /* NDIMS == 3 */
! contribution from the X derivative of Az
!
    call derivative_1st(1, dh(1), u(3,:,:,:), w(:,:,:))

! update Vy
!
    v(2,:,:,:) = - w(:,:,:)
#endif /* NDIMS == 3 */

! === calculate Vz component ===
!
! contribution from the X derivative of Uy
!
    call derivative_1st(1, dh(1), u(2,:,:,:), w(:,:,:))

! update Vz
!
    v(3,:,:,:) = w(:,:,:)

! contribution from the Y derivative of Ux
!
    call derivative_1st(2, dh(2), u(1,:,:,:), w(:,:,:))

! update Vz
!
    v(3,:,:,:) = v(3,:,:,:) - w(:,:,:)

!-------------------------------------------------------------------------------
!
  end subroutine curl
!
!===============================================================================
!
! subroutine LAPLACE:
! ------------------
!
!   Subroutine calculates the Laplace operator of the input scalar field.
!
!      laplace(U) = Δ(U) = ∂²x U + ∂²y U + ∂²z U
!
!   Arguments:
!
!     dh  - the spacial intervals in all direction;
!     u   - the input scalar field U;
!     v   - the output field representing the laplacian of u;
!
!===============================================================================
!
  subroutine laplace(dh, u, v)

    implicit none

    real(kind=8), dimension(3)    , intent(in)  :: dh
    real(kind=8), dimension(:,:,:), intent(in)  :: u
    real(kind=8), dimension(:,:,:), intent(out) :: v

    real(kind=8), dimension(size(u,1), size(u,2), size(u,3)) :: w

!-------------------------------------------------------------------------------
!
! calculate the second derivative of U along the X direction
!
    call derivative_2nd(1, dh(1), u(:,:,:), w(:,:,:))

! update V
!
    v(:,:,:) = w(:,:,:)

! calculate the second derivative of U along the X direction
!
    call derivative_2nd(2, dh(2), u(:,:,:), w(:,:,:))

! update V
!
    v(:,:,:) = v(:,:,:) + w(:,:,:)

#if NDIMS == 3
! calculate the second derivative of U along the Z direction
!
    call derivative_2nd(3, dh(3), u(:,:,:), w(:,:,:))

! update V
!
    v(:,:,:) = v(:,:,:) + w(:,:,:)
#endif /* NDIMS == 3 */

!-------------------------------------------------------------------------------
!
  end subroutine laplace
!
!===============================================================================
!
! subroutine DERIVATIVE_1ST_3O:
! ----------------------------
!
!   Subroutine calculates the first order derivative of the input scalar field
!   along the given direction with the 3rd order approximation.
!
!   Arguments:
!
!     d - the direction of derivative;
!     h - the spacial interval;
!     u  - the input scalar field;
!     v  - the first derivative of the input field;
!
!===============================================================================
!
  subroutine derivative_1st_3o(d, h, u, v)

    implicit none

    integer                       , intent(in)  :: d
    real(kind=8)                  , intent(in)  :: h
    real(kind=8), dimension(:,:,:), intent(in)  :: u
    real(kind=8), dimension(:,:,:), intent(out) :: v

    integer :: m0, m1, m2

!-------------------------------------------------------------------------------
!
    m0 = size(u, d)
    m1 = m0 - 1
    m2 = m0 - 2

    select case(d)

    case(1)

      v(1   ,:,:) =           (u(2   ,:,:) - u(1   ,:,:)) / h
      v(2:m1,:,:) = 5.0d-01 * (u(3:m0,:,:) - u(1:m2,:,:)) / h
      v(  m0,:,:) =           (u(  m0,:,:) - u(  m1,:,:)) / h

    case(2)

      v(:,1   ,:) =           (u(:,2   ,:) - u(:,1   ,:)) / h
      v(:,2:m1,:) = 5.0d-01 * (u(:,3:m0,:) - u(:,1:m2,:)) / h
      v(:,  m0,:) =           (u(:,  m0,:) - u(:,  m1,:)) / h

#if NDIMS == 3
    case(3)

      v(:,:,1   ) =           (u(:,:,2   ) - u(:,:,1   )) / h
      v(:,:,2:m1) = 5.0d-01 * (u(:,:,3:m0) - u(:,:,1:m2)) / h
      v(:,:,  m0) =           (u(:,:,  m0) - u(:,:,  m1)) / h
#endif /* NDIMS == 3 */

    end select

!-------------------------------------------------------------------------------
!
  end subroutine derivative_1st_3o
!
!===============================================================================
!
! subroutine DERIVATIVE_1ST_5O:
! ----------------------------
!
!   Subroutine calculates the first order derivative of the input scalar field
!   along the given direction with the 5th order approximation.
!
!   Arguments:
!
!     d - the direction of derivative;
!     h - the spacial interval;
!     u  - the input scalar field;
!     v  - the first derivative of the input field;
!
!===============================================================================
!
  subroutine derivative_1st_5o(d, h, u, v)

    implicit none

    integer                       , intent(in)  :: d
    real(kind=8)                  , intent(in)  :: h
    real(kind=8), dimension(:,:,:), intent(in)  :: u
    real(kind=8), dimension(:,:,:), intent(out) :: v

    integer :: m0, m1, m2, m3, m4

!-------------------------------------------------------------------------------
!
    m0 = size(u, d)
    m1 = m0 - 1
    m2 = m0 - 2
    m3 = m0 - 3
    m4 = m0 - 4

    select case(d)

    case(1)

      v(1   ,:,:) =            (u(2   ,:,:) - u(1   ,:,:)) / h
      v(2   ,:,:) =  5.0d-01 * (u(3   ,:,:) - u(1   ,:,:)) / h
      v(3:m2,:,:) = (8.0d+00 * (u(4:m1,:,:) - u(2:m3,:,:))                     &
                             - (u(5:m0,:,:) - u(1:m4,:,:))) / (1.2d+01 * h)
      v(  m1,:,:) =  5.0d-01 * (u(  m0,:,:) - u(  m2,:,:)) / h
      v(  m0,:,:) =            (u(  m0,:,:) - u(  m1,:,:)) / h

    case(2)

      v(:,1   ,:) =            (u(:,2   ,:) - u(:,1   ,:)) / h
      v(:,2   ,:) =  5.0d-01 * (u(:,3   ,:) - u(:,1   ,:)) / h
      v(:,3:m2,:) = (8.0d+00 * (u(:,4:m1,:) - u(:,2:m3,:))                     &
                             - (u(:,5:m0,:) - u(:,1:m4,:))) / (1.2d+01 * h)
      v(:,  m1,:) =  5.0d-01 * (u(:,  m0,:) - u(:,  m2,:)) / h
      v(:,  m0,:) =            (u(:,  m0,:) - u(:,  m1,:)) / h

#if NDIMS == 3
    case(3)

      v(:,:,1   ) =            (u(:,:,2   ) - u(:,:,1   )) / h
      v(:,:,2   ) =  5.0d-01 * (u(:,:,3   ) - u(:,:,1   )) / h
      v(:,:,3:m2) = (8.0d+00 * (u(:,:,4:m1) - u(:,:,2:m3))                     &
                             - (u(:,:,5:m0) - u(:,:,1:m4))) / (1.2d+01 * h)
      v(:,:,  m1) =  5.0d-01 * (u(:,:,  m0) - u(:,:,  m2)) / h
      v(:,:,  m0) =            (u(:,:,  m0) - u(:,:,  m1)) / h
#endif /* NDIMS == 3 */

    end select

!-------------------------------------------------------------------------------
!
  end subroutine derivative_1st_5o
!
!===============================================================================
!
! subroutine DERIVATIVE_1ST_7O:
! ----------------------------
!
!   Subroutine calculates the first order derivative of the input scalar field
!   along the given direction with the 7th order approximation.
!
!   Arguments:
!
!     d - the direction of derivative;
!     h - the spacial interval;
!     u  - the input scalar field;
!     v  - the first derivative of the input field;
!
!===============================================================================
!
  subroutine derivative_1st_7o(d, h, u, v)

    implicit none

    integer                       , intent(in)  :: d
    real(kind=8)                  , intent(in)  :: h
    real(kind=8), dimension(:,:,:), intent(in)  :: u
    real(kind=8), dimension(:,:,:), intent(out) :: v

    integer :: m0, m1, m2, m3, m4, m5, m6

!-------------------------------------------------------------------------------
!
    m0 = size(u, d)
    m1 = m0 - 1
    m2 = m0 - 2
    m3 = m0 - 3
    m4 = m0 - 4
    m5 = m0 - 5
    m6 = m0 - 6

    select case(d)

    case(1)

      v(1   ,:,:) =            (u(2   ,:,:) - u(1   ,:,:)) / h
      v(2   ,:,:) =  5.0d-01 * (u(3   ,:,:) - u(1   ,:,:)) / h
      v(3   ,:,:) = (8.0d+00 * (u(4   ,:,:) - u(2   ,:,:))                     &
                             - (u(5   ,:,:) - u(1   ,:,:))) / (1.2d+01 * h)
      v(4:m3,:,:) = (4.5d+01 * (u(5:m2,:,:) - u(3:m4,:,:))                     &
                   - 9.0d+00 * (u(6:m1,:,:) - u(2:m5,:,:))                     &
                   +           (u(7:m0,:,:) - u(1:m6,:,:))) / (6.0d+01 * h)
      v(  m2,:,:) = (8.0d+00 * (u(  m1,:,:) - u(  m3,:,:))                     &
                             - (u(  m0,:,:) - u(  m4,:,:))) / (1.2d+01 * h)
      v(  m1,:,:) =  5.0d-01 * (u(  m0,:,:) - u(  m2,:,:)) / h
      v(  m0,:,:) =            (u(  m0,:,:) - u(  m1,:,:)) / h

    case(2)

      v(:,1   ,:) =            (u(:,2   ,:) - u(:,1   ,:)) / h
      v(:,2   ,:) =  5.0d-01 * (u(:,3   ,:) - u(:,1   ,:)) / h
      v(:,3   ,:) = (8.0d+00 * (u(:,4   ,:) - u(:,2   ,:))                     &
                             - (u(:,5   ,:) - u(:,1   ,:))) / (1.2d+01 * h)
      v(:,4:m3,:) = (4.5d+01 * (u(:,5:m2,:) - u(:,3:m4,:))                     &
                   - 9.0d+00 * (u(:,6:m1,:) - u(:,2:m5,:))                     &
                   +           (u(:,7:m0,:) - u(:,1:m6,:))) / (6.0d+01 * h)
      v(:,  m2,:) = (8.0d+00 * (u(:,  m1,:) - u(:,  m3,:))                     &
                             - (u(:,  m0,:) - u(:,  m4,:))) / (1.2d+01 * h)
      v(:,  m1,:) =  5.0d-01 * (u(:,  m0,:) - u(:,  m2,:)) / h
      v(:,  m0,:) =            (u(:,  m0,:) - u(:,  m1,:)) / h

#if NDIMS == 3
    case(3)

      v(:,:,1   ) =            (u(:,:,2   ) - u(:,:,1   )) / h
      v(:,:,2   ) =  5.0d-01 * (u(:,:,3   ) - u(:,:,1   )) / h
      v(:,:,3   ) = (8.0d+00 * (u(:,:,4   ) - u(:,:,2   ))                     &
                             - (u(:,:,5   ) - u(:,:,1   ))) / (1.2d+01 * h)
      v(:,:,4:m3) = (4.5d+01 * (u(:,:,5:m2) - u(:,:,3:m4))                     &
                   - 9.0d+00 * (u(:,:,6:m1) - u(:,:,2:m5))                     &
                   +           (u(:,:,7:m0) - u(:,:,1:m6))) / (6.0d+01 * h)
      v(:,:,  m2) = (8.0d+00 * (u(:,:,  m1) - u(:,:,  m3))                     &
                             - (u(:,:,  m0) - u(:,:,  m4))) / (1.2d+01 * h)
      v(:,:,  m1) =  5.0d-01 * (u(:,:,  m0) - u(:,:,  m2)) / h
      v(:,:,  m0) =            (u(:,:,  m0) - u(:,:,  m1)) / h
#endif /* NDIMS == 3 */

    end select

!-------------------------------------------------------------------------------
!
  end subroutine derivative_1st_7o
!
!===============================================================================
!
! subroutine DERIVATIVE_1ST_9O:
! ----------------------------
!
!   Subroutine calculates the first order derivative of the input scalar field
!   along the given direction with the 9th order approximation.
!
!   Arguments:
!
!     d - the direction of derivative;
!     h - the spacial interval;
!     u  - the input scalar field;
!     v  - the first derivative of the input field;
!
!===============================================================================
!
  subroutine derivative_1st_9o(d, h, u, v)

    implicit none

    integer                       , intent(in)  :: d
    real(kind=8)                  , intent(in)  :: h
    real(kind=8), dimension(:,:,:), intent(in)  :: u
    real(kind=8), dimension(:,:,:), intent(out) :: v

    integer :: m0, m1, m2, m3, m4, m5, m6, m7, m8

!-------------------------------------------------------------------------------
!
    m0 = size(u, d)
    m1 = m0 - 1
    m2 = m0 - 2
    m3 = m0 - 3
    m4 = m0 - 4
    m5 = m0 - 5
    m6 = m0 - 6
    m7 = m0 - 7
    m8 = m0 - 8

    select case(d)

    case(1)

      v(1   ,:,:) =             (u(2   ,:,:) - u(1   ,:,:)) / h
      v(2   ,:,:) =  5.00d-01 * (u(3   ,:,:) - u(1   ,:,:)) / h
      v(3   ,:,:) = (8.00d+00 * (u(4   ,:,:) - u(2   ,:,:))                    &
                   -            (u(5   ,:,:) - u(1   ,:,:))) / (1.2d+01 * h)
      v(4   ,:,:) = (4.50d+01 * (u(5   ,:,:) - u(3   ,:,:))                    &
                   - 9.00d+00 * (u(6   ,:,:) - u(2   ,:,:))                    &
                   +            (u(7   ,:,:) - u(1   ,:,:))) / (6.0d+01 * h)
      v(5:m4,:,:) = (6.72d+02 * (u(6:m3,:,:) - u(4:m5,:,:))                    &
                   - 1.68d+02 * (u(7:m2,:,:) - u(3:m6,:,:))                    &
                   + 3.20d+01 * (u(8:m1,:,:) - u(2:m7,:,:))                    &
                   - 3.00d+00 * (u(9:m0,:,:) - u(1:m8,:,:))) / (8.4d+02 * h)
      v(  m3,:,:) = (4.50d+01 * (u(  m2,:,:) - u(  m4,:,:))                    &
                   - 9.00d+00 * (u(  m1,:,:) - u(  m5,:,:))                    &
                   +            (u(  m0,:,:) - u(  m6,:,:))) / (6.0d+01 * h)
      v(  m2,:,:) = (8.00d+00 * (u(  m1,:,:) - u(  m3,:,:))                    &
                   -            (u(  m0,:,:) - u(  m4,:,:))) / (1.2d+01 * h)
      v(  m1,:,:) =  5.00d-01 * (u(  m0,:,:) - u(  m2,:,:)) / h
      v(  m0,:,:) =             (u(  m0,:,:) - u(  m1,:,:)) / h

    case(2)

      v(:,1   ,:) =             (u(:,2   ,:) - u(:,1   ,:)) / h
      v(:,2   ,:) =  5.00d-01 * (u(:,3   ,:) - u(:,1   ,:)) / h
      v(:,3   ,:) = (8.00d+00 * (u(:,4   ,:) - u(:,2   ,:))                    &
                   -            (u(:,5   ,:) - u(:,1   ,:))) / (1.2d+01 * h)
      v(:,4   ,:) = (4.50d+01 * (u(:,5   ,:) - u(:,3   ,:))                    &
                   - 9.00d+00 * (u(:,6   ,:) - u(:,2   ,:))                    &
                   +            (u(:,7   ,:) - u(:,1   ,:))) / (6.0d+01 * h)
      v(:,5:m4,:) = (6.72d+02 * (u(:,6:m3,:) - u(:,4:m5,:))                    &
                   - 1.68d+02 * (u(:,7:m2,:) - u(:,3:m6,:))                    &
                   + 3.20d+01 * (u(:,8:m1,:) - u(:,2:m7,:))                    &
                   - 3.00d+00 * (u(:,9:m0,:) - u(:,1:m8,:))) / (8.4d+02 * h)
      v(:,  m3,:) = (4.50d+01 * (u(:,  m2,:) - u(:,  m4,:))                    &
                   - 9.00d+00 * (u(:,  m1,:) - u(:,  m5,:))                    &
                   +            (u(:,  m0,:) - u(:,  m6,:))) / (6.0d+01 * h)
      v(:,  m2,:) = (8.00d+00 * (u(:,  m1,:) - u(:,  m3,:))                    &
                   -            (u(:,  m0,:) - u(:,  m4,:))) / (1.2d+01 * h)
      v(:,  m1,:) =  5.00d-01 * (u(:,  m0,:) - u(:,  m2,:)) / h
      v(:,  m0,:) =             (u(:,  m0,:) - u(:,  m1,:)) / h

#if NDIMS == 3
    case(3)

      v(:,:,1   ) =             (u(:,:,2   ) - u(:,:,1   )) / h
      v(:,:,2   ) =  5.00d-01 * (u(:,:,3   ) - u(:,:,1   )) / h
      v(:,:,3   ) = (8.00d+00 * (u(:,:,4   ) - u(:,:,2   ))                    &
                   -            (u(:,:,5   ) - u(:,:,1   ))) / (1.2d+01 * h)
      v(:,:,4   ) = (4.50d+01 * (u(:,:,5   ) - u(:,:,3   ))                    &
                   - 9.00d+00 * (u(:,:,6   ) - u(:,:,2   ))                    &
                   +            (u(:,:,7   ) - u(:,:,1   ))) / (6.0d+01 * h)
      v(:,:,5:m4) = (6.72d+02 * (u(:,:,6:m3) - u(:,:,4:m5))                    &
                   - 1.68d+02 * (u(:,:,7:m2) - u(:,:,3:m6))                    &
                   + 3.20d+01 * (u(:,:,8:m1) - u(:,:,2:m7))                    &
                   - 3.00d+00 * (u(:,:,9:m0) - u(:,:,1:m8))) / (8.4d+02 * h)
      v(:,:,  m3) = (4.50d+01 * (u(:,:,  m2) - u(:,:,  m4))                    &
                   - 9.00d+00 * (u(:,:,  m1) - u(:,:,  m5))                    &
                   +            (u(:,:,  m0) - u(:,:,  m6))) / (6.0d+01 * h)
      v(:,:,  m2) = (8.00d+00 * (u(:,:,  m1) - u(:,:,  m3))                    &
                   -            (u(:,:,  m0) - u(:,:,  m4))) / (1.2d+01 * h)
      v(:,:,  m1) =  5.00d-01 * (u(:,:,  m0) - u(:,:,  m2)) / h
      v(:,:,  m0) =             (u(:,:,  m0) - u(:,:,  m1)) / h
#endif /* NDIMS == 3 */

    end select

!-------------------------------------------------------------------------------
!
  end subroutine derivative_1st_9o
!
!===============================================================================
!
! subroutine DERIVATIVE_2ND_3O:
! ----------------------------
!
!   Subroutine calculates the second order derivative of the input scalar field
!   along the given direction with the 3rd order approximation.
!
!   Arguments:
!
!     d - the direction of derivative;
!     h - the spacial interval;
!     u - the input scalar field;
!     v - the output scalar field representing the second derivative of u;
!
!===============================================================================
!
  subroutine derivative_2nd_3o(d, h, u, v)

    implicit none

    integer                       , intent(in)  :: d
    real(kind=8)                  , intent(in)  :: h
    real(kind=8), dimension(:,:,:), intent(in)  :: u
    real(kind=8), dimension(:,:,:), intent(out) :: v

    integer      :: m0, m1, m2
    real(kind=8) :: h2

!-------------------------------------------------------------------------------
!
    m0 = size(u, d)
    m1 = m0 - 1
    m2 = m0 - 2

    h2 = h * h

    select case(d)

    case(1)

      v(1   ,:,:) = 0.0d+00
      v(2:m1,:,:) = ((u(3:m0,:,:) + u(1:m2,:,:)) - 2.0d+00 * u(2:m1,:,:)) / h2
      v(  m0,:,:) = 0.0d+00

    case(2)

      v(:,1   ,:) = 0.0d+00
      v(:,2:m1,:) = ((u(:,3:m0,:) + u(:,1:m2,:)) - 2.0d+00 * u(:,2:m1,:)) / h2
      v(:,  m0,:) = 0.0d+00

#if NDIMS == 3
    case(3)

      v(:,:,1   ) = 0.0d+00
      v(:,:,2:m1) = ((u(:,:,3:m0) + u(:,:,1:m2)) - 2.0d+00 * u(:,:,2:m1)) / h2
      v(:,:,  m0) = 0.0d+00
#endif /* NDIMS == 3 */

    end select

!-------------------------------------------------------------------------------
!
  end subroutine derivative_2nd_3o
!
!===============================================================================
!
! subroutine DERIVATIVE_2ND_5O:
! ----------------------------
!
!   Subroutine calculates the second order derivative of the input scalar field
!   along the given direction with the 5th order approximation.
!
!   Arguments:
!
!     d - the direction of derivative;
!     h - the spacial interval;
!     u - the input scalar field;
!     v - the output scalar field representing the second derivative of u;
!
!===============================================================================
!
  subroutine derivative_2nd_5o(d, h, u, v)

    implicit none

    integer                       , intent(in)  :: d
    real(kind=8)                  , intent(in)  :: h
    real(kind=8), dimension(:,:,:), intent(in)  :: u
    real(kind=8), dimension(:,:,:), intent(out) :: v

    integer      :: m0, m1, m2, m3, m4
    real(kind=8) :: h2

!-------------------------------------------------------------------------------
!
    m0 = size(u, d)
    m1 = m0 - 1
    m2 = m0 - 2
    m3 = m0 - 3
    m4 = m0 - 4

    h2 = h * h

    select case(d)

    case(1)

      v(1   ,:,:) = 0.0d+00
      v(2   ,:,:) = ((u(3   ,:,:) + u(1   ,:,:)) - 2.0d+00 * u(2   ,:,:)) / h2
      v(3:m2,:,:) = (1.6d+01 * (u(4:m1,:,:) + u(2:m3,:,:))                     &
                   -           (u(5:m0,:,:) + u(1:m4,:,:))                     &
                   - 3.0d+01 *  u(3:m2,:,:)) / (1.2d+01 * h2)
      v(  m1,:,:) = ((u(  m0,:,:) + u(  m2,:,:)) - 2.0d+00 * u(  m1,:,:)) / h2
      v(  m0,:,:) = 0.0d+00

    case(2)

      v(:,1   ,:) = 0.0d+00
      v(:,2   ,:) = ((u(:,3   ,:) + u(:,1   ,:)) - 2.0d+00 * u(:,2   ,:)) / h2
      v(:,3:m2,:) = (1.6d+01 * (u(:,4:m1,:) + u(:,2:m3,:))                     &
                   -           (u(:,5:m0,:) + u(:,1:m4,:))                     &
                   - 3.0d+01 *  u(:,3:m2,:)) / (1.2d+01 * h2)
      v(:,  m1,:) = ((u(:,  m0,:) + u(:,  m2,:)) - 2.0d+00 * u(:,  m1,:)) / h2
      v(:,  m0,:) = 0.0d+00

#if NDIMS == 3
    case(3)

      v(:,:,1   ) = 0.0d+00
      v(:,:,2   ) = ((u(:,:,3   ) + u(:,:,1   )) - 2.0d+00 * u(:,:,2   )) / h2
      v(:,:,3:m2) = (1.6d+01 * (u(:,:,4:m1) + u(:,:,2:m3))                     &
                   -           (u(:,:,5:m0) + u(:,:,1:m4))                     &
                   - 3.0d+01 *  u(:,:,3:m2)) / (1.2d+01 * h2)
      v(:,:,  m1) = ((u(:,:,  m0) + u(:,:,  m2)) - 2.0d+00 * u(:,:,  m1)) / h2
      v(:,:,  m0) = 0.0d+00
#endif /* NDIMS == 3 */

    end select

!-------------------------------------------------------------------------------
!
  end subroutine derivative_2nd_5o
!
!===============================================================================
!
! subroutine DERIVATIVE_2ND_7O:
! ----------------------------
!
!   Subroutine calculates the second order derivative of the input scalar field
!   along the given direction with the 7th order approximation.
!
!   Arguments:
!
!     d - the direction of derivative;
!     h - the spacial interval;
!     u - the input scalar field;
!     v - the output scalar field representing the second derivative of u;
!
!===============================================================================
!
  subroutine derivative_2nd_7o(d, h, u, v)

    implicit none

    integer                       , intent(in)  :: d
    real(kind=8)                  , intent(in)  :: h
    real(kind=8), dimension(:,:,:), intent(in)  :: u
    real(kind=8), dimension(:,:,:), intent(out) :: v

    integer      :: m0, m1, m2, m3, m4, m5, m6
    real(kind=8) :: h2

!-------------------------------------------------------------------------------
!
    m0 = size(u, d)
    m1 = m0 - 1
    m2 = m0 - 2
    m3 = m0 - 3
    m4 = m0 - 4
    m5 = m0 - 5
    m6 = m0 - 6

    h2 = h * h

    select case(d)

    case(1)

      v(1   ,:,:) = 0.0d+00
      v(2   ,:,:) = ((u(3   ,:,:) + u(1   ,:,:)) - 2.0d+00 * u(2   ,:,:)) / h2
      v(3   ,:,:) = (1.60d+01 * (u(4   ,:,:) + u(2   ,:,:))                    &
                   -            (u(5   ,:,:) + u(1   ,:,:))                    &
                   - 3.00d+01 *  u(3   ,:,:)) / (1.2d+01 * h2)
      v(4:m3,:,:) = (1.35d+02 * (u(5:m2,:,:) + u(3:m4,:,:))                    &
                   - 1.35d+01 * (u(6:m1,:,:) + u(2:m5,:,:))                    &
                   +            (u(7:m0,:,:) + u(1:m6,:,:))                    &
                   - 2.45d+02 *  u(4:m3,:,:)) / (9.0d+01 * h2)
      v(  m2,:,:) = (1.60d+01 * (u(  m1,:,:) + u(  m3,:,:))                    &
                   -            (u(  m0,:,:) + u(  m4,:,:))                    &
                   - 3.00d+01 *  u(  m2,:,:)) / (1.2d+01 * h2)
      v(  m1,:,:) = ((u(  m0,:,:) + u(  m2,:,:)) - 2.0d+00 * u(  m1,:,:)) / h2
      v(  m0,:,:) = 0.0d+00

    case(2)

      v(:,1   ,:) = 0.0d+00
      v(:,2   ,:) = ((u(:,3   ,:) + u(:,1   ,:)) - 2.0d+00 * u(:,2   ,:)) / h2
      v(:,3   ,:) = (1.60d+01 * (u(:,4   ,:) + u(:,2   ,:))                    &
                   -            (u(:,5   ,:) + u(:,1   ,:))                    &
                   - 3.00d+01 *  u(:,3   ,:)) / (1.2d+01 * h2)
      v(:,4:m3,:) = (1.35d+02 * (u(:,5:m2,:) + u(:,3:m4,:))                    &
                   - 1.35d+01 * (u(:,6:m1,:) + u(:,2:m5,:))                    &
                   +            (u(:,7:m0,:) + u(:,1:m6,:))                    &
                   - 2.45d+02 *  u(:,4:m3,:)) / (9.0d+01 * h2)
      v(:,  m2,:) = (1.60d+01 * (u(:,  m1,:) + u(:,  m3,:))                    &
                   -            (u(:,  m0,:) + u(:,  m4,:))                    &
                   - 3.00d+01 *  u(:,  m2,:)) / (1.2d+01 * h2)
      v(:,  m1,:) = ((u(:,  m0,:) + u(:,  m2,:)) - 2.0d+00 * u(:,  m1,:)) / h2
      v(:,  m0,:) = 0.0d+00

#if NDIMS == 3
    case(3)

      v(:,:,1   ) = 0.0d+00
      v(:,:,2   ) = ((u(:,:,3   ) + u(:,:,1   )) - 2.0d+00 * u(:,:,2   )) / h2
      v(:,:,3   ) = (1.60d+01 * (u(:,:,4   ) + u(:,:,2   ))                    &
                   -            (u(:,:,5   ) + u(:,:,1   ))                    &
                   - 3.00d+01 *  u(:,:,3   )) / (1.2d+01 * h2)
      v(:,:,4:m3) = (1.35d+02 * (u(:,:,5:m2) + u(:,:,3:m4))                    &
                   - 1.35d+01 * (u(:,:,6:m1) + u(:,:,2:m5))                    &
                   +            (u(:,:,7:m0) + u(:,:,1:m6))                    &
                   - 2.45d+02 *  u(:,:,4:m3)) / (9.0d+01 * h2)
      v(:,:,  m2) = (1.60d+01 * (u(:,:,  m1) + u(:,:,  m3))                    &
                   -            (u(:,:,  m0) + u(:,:,  m4))                    &
                   - 3.00d+01 *  u(:,:,  m2)) / (1.2d+01 * h2)
      v(:,:,  m1) = ((u(:,:,  m0) + u(:,:,  m2)) - 2.0d+00 * u(:,:,  m1)) / h2
      v(:,:,  m0) = 0.0d+00
#endif /* NDIMS == 3 */

    end select

!-------------------------------------------------------------------------------
!
  end subroutine derivative_2nd_7o
!
!===============================================================================
!
! subroutine DERIVATIVE_2ND_9O:
! ----------------------------
!
!   Subroutine calculates the second order derivative of the input scalar field
!   along the given direction with the 9th order approximation.
!
!   Arguments:
!
!     d - the direction of derivative;
!     h - the spacial interval;
!     u - the input scalar field;
!     v - the output scalar field representing the second derivative of u;
!
!===============================================================================
!
  subroutine derivative_2nd_9o(d, h, u, v)

    implicit none

    integer                       , intent(in)  :: d
    real(kind=8)                  , intent(in)  :: h
    real(kind=8), dimension(:,:,:), intent(in)  :: u
    real(kind=8), dimension(:,:,:), intent(out) :: v

    integer      :: m0, m1, m2, m3, m4, m5, m6, m7, m8
    real(kind=8) :: h2

!-------------------------------------------------------------------------------
!
    m0 = size(u, d)
    m1 = m0 - 1
    m2 = m0 - 2
    m3 = m0 - 3
    m4 = m0 - 4
    m5 = m0 - 5
    m6 = m0 - 6
    m7 = m0 - 7
    m8 = m0 - 8

    h2 = h * h

    select case(d)

    case(1)

      v(1   ,:,:) = 0.0d+00
      v(2   ,:,:) = ((u(3   ,:,:) + u(1   ,:,:)) - 2.0d+00 * u(2   ,:,:)) / h2
      v(3   ,:,:) = (1.600d+01 * (u(4   ,:,:) + u(2   ,:,:))                   &
                   -             (u(5   ,:,:) + u(1   ,:,:))                   &
                   - 3.000d+01 *  u(3   ,:,:)) / (1.2d+01 * h2)
      v(4   ,:,:) = (1.350d+02 * (u(5   ,:,:) + u(3   ,:,:))                   &
                   - 1.350d+01 * (u(6   ,:,:) + u(2   ,:,:))                   &
                   +             (u(7   ,:,:) + u(1   ,:,:))                   &
                   - 2.450d+02 *  u(4   ,:,:)) / (9.0d+01 * h2)
      v(5:m4,:,:) = (8.064d+03 * (u(6:m3,:,:) + u(4:m5,:,:))                   &
                   - 1.008d+03 * (u(7:m2,:,:) + u(3:m6,:,:))                   &
                   + 1.280d+02 * (u(8:m1,:,:) + u(2:m7,:,:))                   &
                   - 9.000d+00 * (u(9:m0,:,:) + u(1:m8,:,:))                   &
                   - 1.435d+04 *  u(5:m4,:,:)) / (5.04d+03 * h2)
      v(  m3,:,:) = (1.350d+02 * (u(  m2,:,:) + u(  m4,:,:))                   &
                   - 1.350d+01 * (u(  m1,:,:) + u(  m5,:,:))                   &
                   +             (u(  m0,:,:) + u(  m6,:,:))                   &
                   - 2.450d+02 *  u(  m3,:,:)) / (9.0d+01 * h2)
      v(  m2,:,:) = (1.600d+01 * (u(  m1,:,:) + u(  m3,:,:))                   &
                   -             (u(  m0,:,:) + u(  m4,:,:))                   &
                   - 3.000d+01 *  u(  m2,:,:)) / (1.2d+01 * h2)
      v(  m1,:,:) = ((u(  m0,:,:) + u(  m2,:,:)) - 2.0d+00 * u(  m1,:,:)) / h2
      v(  m0,:,:) = 0.0d+00

    case(2)

      v(:,1   ,:) = 0.0d+00
      v(:,2   ,:) = ((u(:,3   ,:) + u(:,1   ,:)) - 2.0d+00 * u(:,2   ,:)) / h2
      v(:,3   ,:) = (1.600d+01 * (u(:,4   ,:) + u(:,2   ,:))                   &
                   -             (u(:,5   ,:) + u(:,1   ,:))                   &
                   - 3.000d+01 *  u(:,3   ,:)) / (1.2d+01 * h2)
      v(:,4   ,:) = (1.350d+02 * (u(:,5   ,:) + u(:,3   ,:))                   &
                   - 1.350d+01 * (u(:,6   ,:) + u(:,2   ,:))                   &
                   +             (u(:,7   ,:) + u(:,1   ,:))                   &
                   - 2.450d+02 *  u(:,4   ,:)) / (9.0d+01 * h2)
      v(:,5:m4,:) = (8.064d+03 * (u(:,6:m3,:) + u(:,4:m5,:))                   &
                   - 1.008d+03 * (u(:,7:m2,:) + u(:,3:m6,:))                   &
                   + 1.280d+02 * (u(:,8:m1,:) + u(:,2:m7,:))                   &
                   - 9.000d+00 * (u(:,9:m0,:) + u(:,1:m8,:))                   &
                   - 1.435d+04 *  u(:,5:m4,:)) / (5.04d+03 * h2)
      v(:,  m3,:) = (1.350d+02 * (u(:,  m2,:) + u(:,  m4,:))                   &
                   - 1.350d+01 * (u(:,  m1,:) + u(:,  m5,:))                   &
                   +             (u(:,  m0,:) + u(:,  m6,:))                   &
                   - 2.450d+02 *  u(:,  m3,:)) / (9.0d+01 * h2)
      v(:,  m2,:) = (1.600d+01 * (u(:,  m1,:) + u(:,  m3,:))                   &
                   -             (u(:,  m0,:) + u(:,  m4,:))                   &
                   - 3.000d+01 *  u(:,  m2,:)) / (1.2d+01 * h2)
      v(:,  m1,:) = ((u(:,  m0,:) + u(:,  m2,:)) - 2.0d+00 * u(:,  m1,:)) / h2
      v(:,  m0,:) = 0.0d+00

#if NDIMS == 3
    case(3)

      v(:,:,1   ) = 0.0d+00
      v(:,:,2   ) = ((u(:,:,3   ) + u(:,:,1   )) - 2.0d+00 * u(:,:,2   )) / h2
      v(:,:,3   ) = (1.600d+01 * (u(:,:,4   ) + u(:,:,2   ))                   &
                   -             (u(:,:,5   ) + u(:,:,1   ))                   &
                   - 3.000d+01 *  u(:,:,3   )) / (1.2d+01 * h2)
      v(:,:,4   ) = (1.350d+02 * (u(:,:,5   ) + u(:,:,3   ))                   &
                   - 1.350d+01 * (u(:,:,6   ) + u(:,:,2   ))                   &
                   +             (u(:,:,7   ) + u(:,:,1   ))                   &
                   - 2.450d+02 *  u(:,:,4   )) / (9.0d+01 * h2)
      v(:,:,5:m4) = (8.064d+03 * (u(:,:,6:m3) + u(:,:,4:m5))                   &
                   - 1.008d+03 * (u(:,:,7:m2) + u(:,:,3:m6))                   &
                   + 1.280d+02 * (u(:,:,8:m1) + u(:,:,2:m7))                   &
                   - 9.000d+00 * (u(:,:,9:m0) + u(:,:,1:m8))                   &
                   - 1.435d+04 *  u(:,:,5:m4)) / (5.04d+03 * h2)
      v(:,:,  m3) = (1.350d+02 * (u(:,:,  m2) + u(:,:,  m4))                   &
                   - 1.350d+01 * (u(:,:,  m1) + u(:,:,  m5))                   &
                   +             (u(:,:,  m0) + u(:,:,  m6))                   &
                   - 2.450d+02 *  u(:,:,  m3)) / (9.0d+01 * h2)
      v(:,:,  m2) = (1.600d+01 * (u(:,:,  m1) + u(:,:,  m3))                   &
                   -             (u(:,:,  m0) + u(:,:,  m4))                   &
                   - 3.000d+01 *  u(:,:,  m2)) / (1.2d+01 * h2)
      v(:,:,  m1) = ((u(:,:,  m0) + u(:,:,  m2)) - 2.0d+00 * u(:,:,  m1)) / h2
      v(:,:,  m0) = 0.0d+00
#endif /* NDIMS == 3 */

    end select

!-------------------------------------------------------------------------------
!
  end subroutine derivative_2nd_9o

!===============================================================================
!
end module operators