source: palm/trunk/SOURCE/tridia_solver_mod.f90

!> @file tridia_solver_mod.f90
!--------------------------------------------------------------------------------------------------!
! This file is part of the PALM model system.
!
! PALM 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.
!
! PALM 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 PALM. If not, see
! <http://www.gnu.org/licenses/>.
!
! Copyright 1997-2021 Leibniz Universitaet Hannover
!--------------------------------------------------------------------------------------------------!
!
!
! Current revisions:
! -----------------
! 
! 
! Former revisions:
! -----------------
! $Id: tridia_solver_mod.f90 4828 2021-01-05 11:21:41Z banzhafs $
! file re-formatted to follow the PALM coding standard
!
! 4360 2020-01-07 11:25:50Z suehring
! Added missing OpenMP directives
!
! 4182 2019-08-22 15:20:23Z scharf
! Corrected "Former revisions" section
!
! 3761 2019-02-25 15:31:42Z raasch
! OpenACC modification to prevent compiler warning about unused variable
!
! 3690 2019-01-22 22:56:42Z knoop
! OpenACC port for SPEC
!
! 1212 2013-08-15 08:46:27Z raasch
! Initial revision.
! Routines have been moved to seperate module from former file poisfft to here. The tridiagonal
! matrix coefficients of array tri are calculated only once at the beginning, i.e. routine split is
! called within tridia_init.
!
!
! Description:
! ------------
!> Solves the linear system of equations:
!>
!> -(4 pi^2(i^2/(dx^2*nnx^2)+j^2/(dy^2*nny^2))+ 1/(dzu(k)*dzw(k))+1/(dzu(k-1)*dzw(k)))*p(i,j,k)+
!> 1/(dzu(k)*dzw(k))*p(i,j,k+1)+1/(dzu(k-1)*dzw(k))*p(i,j,k-1)=d(i,j,k)
!>
!> by using the Thomas algorithm
!--------------------------------------------------------------------------------------------------!

#define __acc_fft_device ( defined( _OPENACC ) && ( defined ( __cuda_fft ) ) )

 MODULE tridia_solver


    USE basic_constants_and_equations_mod,                                                         &
        ONLY:  pi

    USE indices,                                                                                   &
        ONLY:  nx,                                                                                 &
               ny,                                                                                 &
               nz

    USE kinds

    USE transpose_indices,                                                                         &
        ONLY:  nxl_z,                                                                              &
               nyn_z,                                                                              &
               nxr_z,                                                                              &
               nys_z

    IMPLICIT NONE

    REAL(wp), DIMENSION(:,:), ALLOCATABLE ::  ddzuw  !<

    PRIVATE

    INTERFACE tridia_substi
       MODULE PROCEDURE tridia_substi
    END INTERFACE tridia_substi

    INTERFACE tridia_substi_overlap
       MODULE PROCEDURE tridia_substi_overlap
    END INTERFACE tridia_substi_overlap

    PUBLIC  tridia_substi, tridia_substi_overlap, tridia_init, tridia_1dd

 CONTAINS


!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> @todo Missing subroutine description.
!--------------------------------------------------------------------------------------------------!
 SUBROUTINE tridia_init

    USE arrays_3d,                                                                                 &
        ONLY:  ddzu_pres,                                                                          &
               ddzw,                                                                               &
               rho_air_zw

#if defined( _OPENACC )
       USE arrays_3d,                                                                              &
           ONLY:  tri
#endif

    IMPLICIT NONE

    INTEGER(iwp) ::  k  !<

    ALLOCATE( ddzuw(0:nz-1,3) )

    DO  k = 0, nz-1
       ddzuw(k,1) = ddzu_pres(k+1) * ddzw(k+1) * rho_air_zw(k)
       ddzuw(k,2) = ddzu_pres(k+2) * ddzw(k+1) * rho_air_zw(k+1)
       ddzuw(k,3) = -1.0_wp * ( ddzu_pres(k+2) * ddzw(k+1) * rho_air_zw(k+1) +                     &
                                ddzu_pres(k+1) * ddzw(k+1) * rho_air_zw(k) )
    ENDDO
!
!-- Calculate constant coefficients of the tridiagonal matrix
    CALL maketri
    CALL split

#if __acc_fft_device
    !$ACC ENTER DATA &
    !$ACC COPYIN(ddzuw(0:nz-1,1:3)) &
    !$ACC COPYIN(tri(nxl_z:nxr_z,nys_z:nyn_z,0:nz-1,1:2))
#endif

 END SUBROUTINE tridia_init


!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Computes the i- and j-dependent component of the matrix.
!> Provide the constant coefficients of the tridiagonal matrix for solution of the Poisson equation
!> in Fourier space. The coefficients are computed following the method of Schmidt et al.
!> (DFVLR-Mitteilung 84-15), which departs from Stephan Siano's original version by discretizing the
!> Poisson equation, before it is Fourier-transformed.
!--------------------------------------------------------------------------------------------------!
 SUBROUTINE maketri


    USE arrays_3d,                                                                                 &
        ONLY:  tric,                                                                               &
               rho_air

    USE control_parameters,                                                                        &
        ONLY:  ibc_p_b,                                                                            &
               ibc_p_t

    USE grid_variables,                                                                            &
        ONLY:  dx,                                                                                 &
               dy


    IMPLICIT NONE

    INTEGER(iwp) ::  i     !<
    INTEGER(iwp) ::  j     !<
    INTEGER(iwp) ::  k     !<
    INTEGER(iwp) ::  nnxh  !<
    INTEGER(iwp) ::  nnyh  !<

    REAL(wp)    ::  ll(nxl_z:nxr_z,nys_z:nyn_z)  !<


    nnxh = ( nx + 1 ) / 2
    nnyh = ( ny + 1 ) / 2

    DO  j = nys_z, nyn_z
       DO  i = nxl_z, nxr_z
          IF ( j >= 0  .AND.  j <= nnyh )  THEN
             IF ( i >= 0  .AND.  i <= nnxh )  THEN
                ll(i,j) = 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * i ) /                           &
                                     REAL( nx+1, KIND=wp ) ) ) / ( dx * dx ) +                     &
                          2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * j ) /                           &
                                     REAL( ny+1, KIND=wp ) ) ) / ( dy * dy )
             ELSE
                ll(i,j) = 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * ( nx+1-i ) ) /                  &
                                     REAL( nx+1, KIND=wp ) ) ) / ( dx * dx ) +                     &
                          2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * j ) /                           &
                                     REAL( ny+1, KIND=wp ) ) ) / ( dy * dy )
             ENDIF
          ELSE
             IF ( i >= 0  .AND.  i <= nnxh )  THEN
                ll(i,j) = 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * i ) /                           &
                                     REAL( nx+1, KIND=wp ) ) ) / ( dx * dx ) +                     &
                          2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * ( ny+1-j ) ) /                  &
                                     REAL( ny+1, KIND=wp ) ) ) / ( dy * dy )
             ELSE
                ll(i,j) = 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * ( nx+1-i ) ) /                  &
                                     REAL( nx+1, KIND=wp ) ) ) / ( dx * dx ) +                     &
                          2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * ( ny+1-j ) ) /                  &
                                     REAL( ny+1, KIND=wp ) ) ) / ( dy * dy )
             ENDIF
          ENDIF
       ENDDO
    ENDDO

    DO  k = 0, nz-1
       DO  j = nys_z, nyn_z
          DO  i = nxl_z, nxr_z
             tric(i,j,k) = ddzuw(k,3) - ll(i,j) * rho_air(k+1)
          ENDDO
       ENDDO
    ENDDO

    IF ( ibc_p_b == 1 )  THEN
       DO  j = nys_z, nyn_z
          DO  i = nxl_z, nxr_z
             tric(i,j,0) = tric(i,j,0) + ddzuw(0,1)
          ENDDO
       ENDDO
    ENDIF
    IF ( ibc_p_t == 1 )  THEN
       DO  j = nys_z, nyn_z
          DO  i = nxl_z, nxr_z
             tric(i,j,nz-1) = tric(i,j,nz-1) + ddzuw(nz-1,2)
          ENDDO
       ENDDO
    ENDIF

 END SUBROUTINE maketri


!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Substitution (Forward and Backward) (Thomas algorithm).
!--------------------------------------------------------------------------------------------------!
 SUBROUTINE tridia_substi( ar )


    USE arrays_3d,                                                                                 &
        ONLY:  tri

    USE control_parameters,                                                                        &
        ONLY:  ibc_p_b,                                                                            &
               ibc_p_t

    IMPLICIT NONE

    INTEGER(iwp) ::  i !<
    INTEGER(iwp) ::  j !<
    INTEGER(iwp) ::  k !<

    REAL(wp)     ::  ar(nxl_z:nxr_z,nys_z:nyn_z,1:nz)  !<

    REAL(wp), DIMENSION(nxl_z:nxr_z,nys_z:nyn_z,0:nz-1) ::  ar1  !<
#if __acc_fft_device
    !$ACC DECLARE CREATE(ar1)
#endif

    !$OMP PARALLEL PRIVATE(i,j,k)

!
!-- Forward substitution
#if __acc_fft_device
    !$ACC PARALLEL PRESENT(ar, ar1, tri) PRIVATE(i,j,k)
#endif
    DO  k = 0, nz - 1
#if __acc_fft_device
       !$ACC LOOP COLLAPSE(2)
#endif
       !$OMP DO
       DO  j = nys_z, nyn_z
          DO  i = nxl_z, nxr_z

             IF ( k == 0 )  THEN
                ar1(i,j,k) = ar(i,j,k+1)
             ELSE
                ar1(i,j,k) = ar(i,j,k+1) - tri(i,j,k,2) * ar1(i,j,k-1)
             ENDIF

          ENDDO
       ENDDO
    ENDDO
#if __acc_fft_device
    !$ACC END PARALLEL
#endif

!
!-- Backward substitution
!-- Note, the 1.0E-20 in the denominator is due to avoid divisions by zero appearing if the
!-- pressure bc is set to neumann at the top of the model domain.
#if __acc_fft_device
    !$ACC PARALLEL PRESENT(ar, ar1, ddzuw, tri) PRIVATE(i,j,k)
#endif
    DO  k = nz-1, 0, -1
#if __acc_fft_device
       !$ACC LOOP COLLAPSE(2)
#endif
       !$OMP DO
       DO  j = nys_z, nyn_z
          DO  i = nxl_z, nxr_z

             IF ( k == nz-1 )  THEN
                ar(i,j,k+1) = ar1(i,j,k) / ( tri(i,j,k,1) + 1.0E-20_wp )
             ELSE
                ar(i,j,k+1) = ( ar1(i,j,k) - ddzuw(k,2) * ar(i,j,k+2) ) / tri(i,j,k,1)
             ENDIF
          ENDDO
       ENDDO
    ENDDO
#if __acc_fft_device
    !$ACC END PARALLEL
#endif

    !$OMP END PARALLEL

!
!-- Indices i=0, j=0 correspond to horizontally averaged pressure. The respective values of ar
!-- should be zero at all k-levels if acceleration of horizontally averaged vertical velocity
!-- is zero.
    IF ( ibc_p_b == 1  .AND.  ibc_p_t == 1 )  THEN
       IF ( nys_z == 0  .AND.  nxl_z == 0 )  THEN
#if __acc_fft_device
             !$ACC PARALLEL LOOP PRESENT(ar)
#endif
          DO  k = 1, nz
             ar(nxl_z,nys_z,k) = 0.0_wp
          ENDDO
       ENDIF
    ENDIF

 END SUBROUTINE tridia_substi


!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Substitution (Forward and Backward) (Thomas algorithm).
!--------------------------------------------------------------------------------------------------!
 SUBROUTINE tridia_substi_overlap( ar, jj )


    USE arrays_3d,                                                                                 &
        ONLY:  tri

    USE control_parameters,                                                                        &
        ONLY:  ibc_p_b,                                                                            &
               ibc_p_t

    IMPLICIT NONE

    INTEGER(iwp) ::  i  !<
    INTEGER(iwp) ::  j  !<
    INTEGER(iwp) ::  jj !<
    INTEGER(iwp) ::  k  !<

    REAL(wp)     ::  ar(nxl_z:nxr_z,nys_z:nyn_z,1:nz)  !<

    REAL(wp), DIMENSION(nxl_z:nxr_z,nys_z:nyn_z,0:nz-1) ::  ar1  !<

!
!-- Forward substitution
    DO  k = 0, nz - 1
       DO  j = nys_z, nyn_z
          DO  i = nxl_z, nxr_z

             IF ( k == 0 )  THEN
                ar1(i,j,k) = ar(i,j,k+1)
             ELSE
                ar1(i,j,k) = ar(i,j,k+1) - tri(i,jj,k,2) * ar1(i,j,k-1)
             ENDIF

          ENDDO
       ENDDO
    ENDDO

!
!-- Backward substitution
!-- Note, the 1.0E-20 in the denominator is due to avoid divisions by zero appearing if the
!-- pressure bc is set to neumann at the top of the model domain.
    DO  k = nz-1, 0, -1
       DO  j = nys_z, nyn_z
          DO  i = nxl_z, nxr_z

             IF ( k == nz-1 )  THEN
                ar(i,j,k+1) = ar1(i,j,k) / ( tri(i,jj,k,1) + 1.0E-20_wp )
             ELSE
                ar(i,j,k+1) = ( ar1(i,j,k) - ddzuw(k,2) * ar(i,j,k+2) ) / tri(i,jj,k,1)
             ENDIF
          ENDDO
       ENDDO
    ENDDO

!
!-- Indices i=0, j=0 correspond to horizontally averaged pressure. The respective values of ar
!-- should be zero at all k-levels if acceleration of horizontally averaged vertical velocity
!-- is zero.
    IF ( ibc_p_b == 1  .AND.  ibc_p_t == 1 )  THEN
       IF ( nys_z == 0  .AND.  nxl_z == 0 )  THEN
          DO  k = 1, nz
             ar(nxl_z,nys_z,k) = 0.0_wp
          ENDDO
       ENDIF
    ENDIF

 END SUBROUTINE tridia_substi_overlap


!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Splitting of the tridiagonal matrix (Thomas algorithm).
!--------------------------------------------------------------------------------------------------!
 SUBROUTINE split


    USE arrays_3d,                                                                                 &
        ONLY:  tri,                                                                                &
               tric

    IMPLICIT NONE

    INTEGER(iwp) ::  i  !<
    INTEGER(iwp) ::  j  !<
    INTEGER(iwp) ::  k  !<
!
!   Splitting
    DO  j = nys_z, nyn_z
       DO  i = nxl_z, nxr_z
          tri(i,j,0,1) = tric(i,j,0)
       ENDDO
    ENDDO

    DO  k = 1, nz-1
       DO  j = nys_z, nyn_z
          DO  i = nxl_z, nxr_z
             tri(i,j,k,2) = ddzuw(k,1) / tri(i,j,k-1,1)
             tri(i,j,k,1) = tric(i,j,k) - ddzuw(k-1,2) * tri(i,j,k,2)
          ENDDO
       ENDDO
    ENDDO

 END SUBROUTINE split


!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Solves the linear system of equations for a 1d-decomposition along x (see tridia).
!>
!> @attention When using intel compilers older than 12.0, array tri must be passed as an argument to
!>            the contained subroutines. Otherwise address faults will occur. This feature can be
!>            activated with cpp-switch __intel11. On NEC, tri should not be passed
!>            (except for routine substi_1dd) because this causes very bad performance.
!--------------------------------------------------------------------------------------------------!

 SUBROUTINE tridia_1dd( ddx2, ddy2, nx, ny, j, ar, tri_for_1d )


    USE arrays_3d,                                                                                 &
        ONLY:  ddzu_pres,                                                                          &
               ddzw,                                                                               &
               rho_air,                                                                            &
               rho_air_zw

    USE control_parameters,                                                                        &
        ONLY:  ibc_p_b,                                                                            &
               ibc_p_t

    IMPLICIT NONE

    INTEGER(iwp) ::  i     !<
    INTEGER(iwp) ::  j     !<
    INTEGER(iwp) ::  k     !<
    INTEGER(iwp) ::  nnyh  !<
    INTEGER(iwp) ::  nx    !<
    INTEGER(iwp) ::  ny    !<

    REAL(wp)     ::  ddx2  !<
    REAL(wp)     ::  ddy2  !<

    REAL(wp), DIMENSION(0:nx,1:nz)     ::  ar          !<
    REAL(wp), DIMENSION(5,0:nx,0:nz-1) ::  tri_for_1d  !<


    nnyh = ( ny + 1 ) / 2

!
!-- Define constant elements of the tridiagonal matrix. The compiler on SX6 does loop exchange.
!-- If 0:nx is a high power of 2, the exchanged loops create bank conflicts. The following directive
!-- prohibits loop exchange and the loops perform much better.
!CDIR NOLOOPCHG
    DO  k = 0, nz-1
       DO  i = 0,nx
          tri_for_1d(2,i,k) = ddzu_pres(k+1) * ddzw(k+1) * rho_air_zw(k)
          tri_for_1d(3,i,k) = ddzu_pres(k+2) * ddzw(k+1) * rho_air_zw(k+1)
       ENDDO
    ENDDO

    IF ( j <= nnyh )  THEN
       CALL maketri_1dd( j )
    ELSE
       CALL maketri_1dd( ny+1-j )
    ENDIF

    CALL split_1dd
    CALL substi_1dd( ar, tri_for_1d )

 CONTAINS


!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Computes the i- and j-dependent component of the matrix.
!--------------------------------------------------------------------------------------------------!
 SUBROUTINE maketri_1dd( j )

    IMPLICIT NONE

    INTEGER(iwp) ::  i    !<
    INTEGER(iwp) ::  j    !<
    INTEGER(iwp) ::  k    !<
    INTEGER(iwp) ::  nnxh !<

    REAL(wp)     ::  a !<
    REAL(wp)     ::  c !<

    REAL(wp), DIMENSION(0:nx) ::  l !<


    nnxh = ( nx + 1 ) / 2
!
!-- Provide the tridiagonal matrix for solution of the Poisson equation in Fourier space.
!-- The coefficients are computed following the method of Schmidt et al. (DFVLR-Mitteilung 84-15),
!-- which departs from Stephan Siano's original version by discretizing the Poisson equation,
!-- before it is Fourier-transformed.
    DO  i = 0, nx
       IF ( i >= 0 .AND. i <= nnxh ) THEN
          l(i) = 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * i ) /                                    &
                            REAL( nx+1, KIND=wp ) ) ) * ddx2 +                                     &
                 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * j ) /                                    &
                            REAL( ny+1, KIND=wp ) ) ) * ddy2
       ELSE
          l(i) = 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * ( nx+1-i ) ) /                           &
                            REAL( nx+1, KIND=wp ) ) ) * ddx2 +                                     &
                 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * j ) /                                    &
                            REAL( ny+1, KIND=wp ) ) ) * ddy2
       ENDIF
    ENDDO

    DO  k = 0, nz-1
       DO  i = 0, nx
          a = -1.0_wp * ddzu_pres(k+2) * ddzw(k+1) * rho_air_zw(k+1)
          c = -1.0_wp * ddzu_pres(k+1) * ddzw(k+1) * rho_air_zw(k)
          tri_for_1d(1,i,k) = a + c - l(i) * rho_air(k+1)
       ENDDO
    ENDDO
    IF ( ibc_p_b == 1 )  THEN
       DO  i = 0, nx
          tri_for_1d(1,i,0) = tri_for_1d(1,i,0) + tri_for_1d(2,i,0)
       ENDDO
    ENDIF
    IF ( ibc_p_t == 1 )  THEN
       DO  i = 0, nx
          tri_for_1d(1,i,nz-1) = tri_for_1d(1,i,nz-1) + tri_for_1d(3,i,nz-1)
       ENDDO
    ENDIF

 END SUBROUTINE maketri_1dd


!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Splitting of the tridiagonal matrix (Thomas algorithm).
!--------------------------------------------------------------------------------------------------!
 SUBROUTINE split_1dd

    IMPLICIT NONE

    INTEGER(iwp) ::  i  !<
    INTEGER(iwp) ::  k  !<


!
!-- Splitting
    DO  i = 0, nx
       tri_for_1d(4,i,0) = tri_for_1d(1,i,0)
    ENDDO
    DO  k = 1, nz-1
       DO  i = 0, nx
          tri_for_1d(5,i,k) = tri_for_1d(2,i,k) / tri_for_1d(4,i,k-1)
          tri_for_1d(4,i,k) = tri_for_1d(1,i,k) - tri_for_1d(3,i,k-1) * tri_for_1d(5,i,k)
       ENDDO
    ENDDO

 END SUBROUTINE split_1dd


!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Substitution (Forward and Backward) (Thomas algorithm).
!--------------------------------------------------------------------------------------------------!
 SUBROUTINE substi_1dd( ar, tri_for_1d )


    IMPLICIT NONE

    INTEGER(iwp) ::  i  !<
    INTEGER(iwp) ::  k  !<

    REAL(wp), DIMENSION(0:nx,nz)       ::  ar          !<
    REAL(wp), DIMENSION(0:nx,0:nz-1)   ::  ar1         !<
    REAL(wp), DIMENSION(5,0:nx,0:nz-1) ::  tri_for_1d  !<

!
!-- Forward substitution
    DO  i = 0, nx
       ar1(i,0) = ar(i,1)
    ENDDO
    DO  k = 1, nz-1
       DO  i = 0, nx
          ar1(i,k) = ar(i,k+1) - tri_for_1d(5,i,k) * ar1(i,k-1)
       ENDDO
    ENDDO

!
!-- Backward substitution
!-- Note, the add of 1.0E-20 in the denominator is due to avoid divisions by zero appearing if the
!-- pressure bc is set to neumann at the top of the model domain.
    DO  i = 0, nx
       ar(i,nz) = ar1(i,nz-1) / ( tri_for_1d(4,i,nz-1) + 1.0E-20_wp )
    ENDDO
    DO  k = nz-2, 0, -1
       DO  i = 0, nx
          ar(i,k+1) = ( ar1(i,k) - tri_for_1d(3,i,k) * ar(i,k+2) ) / tri_for_1d(4,i,k)
       ENDDO
    ENDDO

!
!-- Indices i=0, j=0 correspond to horizontally averaged pressure. The respective values of ar
!-- should be zero at all k-levels if acceleration of horizontally averaged vertical velocity is
!-- zero.
    IF ( ibc_p_b == 1  .AND.  ibc_p_t == 1 )  THEN
       IF ( j == 0 )  THEN
          DO  k = 1, nz
             ar(0,k) = 0.0_wp
          ENDDO
       ENDIF
    ENDIF

 END SUBROUTINE substi_1dd

 END SUBROUTINE tridia_1dd


 END MODULE tridia_solver