!> @file tridia_solver_mod.f90 !------------------------------------------------------------------------------! ! This file is part of PALM. ! ! 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 . ! ! Copyright 1997-2017 Leibniz Universitaet Hannover !------------------------------------------------------------------------------! ! ! Current revisions: ! ------------------ ! ! ! Former revisions: ! ----------------- ! $Id: tridia_solver_mod.f90 2119 2017-01-17 16:51:50Z witha $ ! ! 2118 2017-01-17 16:38:49Z raasch ! OpenACC directives removed ! ! 2037 2016-10-26 11:15:40Z knoop ! Anelastic approximation implemented ! ! 2000 2016-08-20 18:09:15Z knoop ! Forced header and separation lines into 80 columns ! ! 1850 2016-04-08 13:29:27Z maronga ! Module renamed ! ! ! 1815 2016-04-06 13:49:59Z raasch ! cpp-switch intel11 removed ! ! 1808 2016-04-05 19:44:00Z raasch ! test output removed ! ! 1804 2016-04-05 16:30:18Z maronga ! Removed code for parameter file check (__check) ! ! 1682 2015-10-07 23:56:08Z knoop ! Code annotations made doxygen readable ! ! 1406 2014-05-16 13:47:01Z raasch ! bugfix for pgi 14.4: declare create moved after array declaration ! ! 1342 2014-03-26 17:04:47Z kanani ! REAL constants defined as wp-kind ! ! 1322 2014-03-20 16:38:49Z raasch ! REAL functions provided with KIND-attribute ! ! 1320 2014-03-20 08:40:49Z raasch ! ONLY-attribute added to USE-statements, ! kind-parameters added to all INTEGER and REAL declaration statements, ! kinds are defined in new module kinds, ! old module precision_kind is removed, ! revision history before 2012 removed, ! comment fields (!:) to be used for variable explanations added to ! all variable declaration statements ! ! 1257 2013-11-08 15:18:40Z raasch ! openacc loop and loop vector clauses removed, declare create moved after ! the FORTRAN declaration statement ! ! 1221 2013-09-10 08:59:13Z raasch ! dummy argument tri in 1d-routines replaced by tri_for_1d because of name ! conflict with arry tri in module arrays_3d ! ! 1216 2013-08-26 09:31:42Z raasch ! +tridia_substi_overlap for handling overlapping fft / transposition ! ! 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 !------------------------------------------------------------------------------! MODULE tridia_solver 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 USE kinds 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 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 constants, & ONLY: pi USE control_parameters, & ONLY: ibc_p_b, ibc_p_t USE grid_variables, & ONLY: dx, dy USE kinds 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 USE kinds 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 !< ! !-- 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,j,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,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 ! !-- 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 !------------------------------------------------------------------------------! ! 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 USE kinds 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 USE kinds 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 the intel compilers older than 12.0, array tri must !> be passed as an argument to the contained subroutines. Otherwise !> addres 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 USE kinds IMPLICIT NONE INTEGER(iwp) :: i !< INTEGER(iwp) :: j !< INTEGER(iwp) :: k !< INTEGER(iwp) :: nnyh !< INTEGER(iwp) :: nx !< INTEGER(iwp) :: ny !< INTEGER(iwp) :: omp_get_thread_num !< INTEGER(iwp) :: tn !< 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 ) USE constants, & ONLY: pi USE kinds 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