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source: palm/trunk/SOURCE/tridia_solver_mod.f90 @ 2405

Last change on this file since 2405 was 2119, checked in by raasch, 8 years ago
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1	!> @file tridia_solver_mod.f90
2	!------------------------------------------------------------------------------!
3	! This file is part of PALM.
4	!
5	! PALM is free software: you can redistribute it and/or modify it under the
6	! terms of the GNU General Public License as published by the Free Software
7	! Foundation, either version 3 of the License, or (at your option) any later
8	! version.
9	!
10	! PALM is distributed in the hope that it will be useful, but WITHOUT ANY
11	! WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
12	! A PARTICULAR PURPOSE. See the GNU General Public License for more details.
13	!
14	! You should have received a copy of the GNU General Public License along with
15	! PALM. If not, see <http://www.gnu.org/licenses/>.
16	!
17	! Copyright 1997-2017 Leibniz Universitaet Hannover
18	!------------------------------------------------------------------------------!
19	!
20	! Current revisions:
21	! ------------------
22	!
23	!
24	! Former revisions:
25	! -----------------
26	! $Id: tridia_solver_mod.f90 2119 2017-01-17 16:51:50Z raasch $
27	!
28	! 2118 2017-01-17 16:38:49Z raasch
29	! OpenACC directives removed
30	!
31	! 2037 2016-10-26 11:15:40Z knoop
32	! Anelastic approximation implemented
33	!
34	! 2000 2016-08-20 18:09:15Z knoop
35	! Forced header and separation lines into 80 columns
36	!
37	! 1850 2016-04-08 13:29:27Z maronga
38	! Module renamed
39	!
40	!
41	! 1815 2016-04-06 13:49:59Z raasch
42	! cpp-switch intel11 removed
43	!
44	! 1808 2016-04-05 19:44:00Z raasch
45	! test output removed
46	!
47	! 1804 2016-04-05 16:30:18Z maronga
48	! Removed code for parameter file check (__check)
49	!
50	! 1682 2015-10-07 23:56:08Z knoop
51	! Code annotations made doxygen readable
52	!
53	! 1406 2014-05-16 13:47:01Z raasch
54	! bugfix for pgi 14.4: declare create moved after array declaration
55	!
56	! 1342 2014-03-26 17:04:47Z kanani
57	! REAL constants defined as wp-kind
58	!
59	! 1322 2014-03-20 16:38:49Z raasch
60	! REAL functions provided with KIND-attribute
61	!
62	! 1320 2014-03-20 08:40:49Z raasch
63	! ONLY-attribute added to USE-statements,
64	! kind-parameters added to all INTEGER and REAL declaration statements,
65	! kinds are defined in new module kinds,
66	! old module precision_kind is removed,
67	! revision history before 2012 removed,
68	! comment fields (!:) to be used for variable explanations added to
69	! all variable declaration statements
70	!
71	! 1257 2013-11-08 15:18:40Z raasch
72	! openacc loop and loop vector clauses removed, declare create moved after
73	! the FORTRAN declaration statement
74	!
75	! 1221 2013-09-10 08:59:13Z raasch
76	! dummy argument tri in 1d-routines replaced by tri_for_1d because of name
77	! conflict with arry tri in module arrays_3d
78	!
79	! 1216 2013-08-26 09:31:42Z raasch
80	! +tridia_substi_overlap for handling overlapping fft / transposition
81	!
82	! 1212 2013-08-15 08:46:27Z raasch
83	! Initial revision.
84	! Routines have been moved to seperate module from former file poisfft to here.
85	! The tridiagonal matrix coefficients of array tri are calculated only once at
86	! the beginning, i.e. routine split is called within tridia_init.
87	!
88	!
89	! Description:
90	! ------------
91	!> solves the linear system of equations:
92	!>
93	!> -(4 pi^2(i^2/(dx^2nnx^2)+j^2/(dy^2nny^2))+
94	!> 1/(dzu(k)dzw(k))+1/(dzu(k-1)dzw(k)))*p(i,j,k)+
95	!> 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)
96	!>
97	!> by using the Thomas algorithm
98	!------------------------------------------------------------------------------!
99	MODULE tridia_solver
100
101
102	USE indices, &
103	ONLY: nx, ny, nz
104
105	USE kinds
106
107	USE transpose_indices, &
108	ONLY: nxl_z, nyn_z, nxr_z, nys_z
109
110	IMPLICIT NONE
111
112	REAL(wp), DIMENSION(:,:), ALLOCATABLE :: ddzuw !<
113
114	PRIVATE
115
116	INTERFACE tridia_substi
117	MODULE PROCEDURE tridia_substi
118	END INTERFACE tridia_substi
119
120	INTERFACE tridia_substi_overlap
121	MODULE PROCEDURE tridia_substi_overlap
122	END INTERFACE tridia_substi_overlap
123
124	PUBLIC tridia_substi, tridia_substi_overlap, tridia_init, tridia_1dd
125
126	CONTAINS
127
128
129	!------------------------------------------------------------------------------!
130	! Description:
131	! ------------
132	!> @todo Missing subroutine description.
133	!------------------------------------------------------------------------------!
134	SUBROUTINE tridia_init
135
136	USE arrays_3d, &
137	ONLY: ddzu_pres, ddzw, rho_air_zw
138
139	USE kinds
140
141	IMPLICIT NONE
142
143	INTEGER(iwp) :: k !<
144
145	ALLOCATE( ddzuw(0:nz-1,3) )
146
147	DO k = 0, nz-1
148	ddzuw(k,1) = ddzu_pres(k+1) * ddzw(k+1) * rho_air_zw(k)
149	ddzuw(k,2) = ddzu_pres(k+2) * ddzw(k+1) * rho_air_zw(k+1)
150	ddzuw(k,3) = -1.0_wp * &
151	( ddzu_pres(k+2) * ddzw(k+1) * rho_air_zw(k+1) + &
152	ddzu_pres(k+1) * ddzw(k+1) * rho_air_zw(k) )
153	ENDDO
154	!
155	!-- Calculate constant coefficients of the tridiagonal matrix
156	CALL maketri
157	CALL split
158
159	END SUBROUTINE tridia_init
160
161
162	!------------------------------------------------------------------------------!
163	! Description:
164	! ------------
165	!> Computes the i- and j-dependent component of the matrix
166	!> Provide the constant coefficients of the tridiagonal matrix for solution
167	!> of the Poisson equation in Fourier space.
168	!> The coefficients are computed following the method of
169	!> Schmidt et al. (DFVLR-Mitteilung 84-15), which departs from Stephan
170	!> Siano's original version by discretizing the Poisson equation,
171	!> before it is Fourier-transformed.
172	!------------------------------------------------------------------------------!
173	SUBROUTINE maketri
174
175
176	USE arrays_3d, &
177	ONLY: tric, rho_air
178
179	USE constants, &
180	ONLY: pi
181
182	USE control_parameters, &
183	ONLY: ibc_p_b, ibc_p_t
184
185	USE grid_variables, &
186	ONLY: dx, dy
187
188
189	USE kinds
190
191	IMPLICIT NONE
192
193	INTEGER(iwp) :: i !<
194	INTEGER(iwp) :: j !<
195	INTEGER(iwp) :: k !<
196	INTEGER(iwp) :: nnxh !<
197	INTEGER(iwp) :: nnyh !<
198
199	REAL(wp) :: ll(nxl_z:nxr_z,nys_z:nyn_z) !<
200
201
202	nnxh = ( nx + 1 ) / 2
203	nnyh = ( ny + 1 ) / 2
204
205	DO j = nys_z, nyn_z
206	DO i = nxl_z, nxr_z
207	IF ( j >= 0 .AND. j <= nnyh ) THEN
208	IF ( i >= 0 .AND. i <= nnxh ) THEN
209	ll(i,j) = 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * i ) / &
210	REAL( nx+1, KIND=wp ) ) ) / ( dx * dx ) + &
211	2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * j ) / &
212	REAL( ny+1, KIND=wp ) ) ) / ( dy * dy )
213	ELSE
214	ll(i,j) = 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * ( nx+1-i ) ) / &
215	REAL( nx+1, KIND=wp ) ) ) / ( dx * dx ) + &
216	2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * j ) / &
217	REAL( ny+1, KIND=wp ) ) ) / ( dy * dy )
218	ENDIF
219	ELSE
220	IF ( i >= 0 .AND. i <= nnxh ) THEN
221	ll(i,j) = 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * i ) / &
222	REAL( nx+1, KIND=wp ) ) ) / ( dx * dx ) + &
223	2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * ( ny+1-j ) ) / &
224	REAL( ny+1, KIND=wp ) ) ) / ( dy * dy )
225	ELSE
226	ll(i,j) = 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * ( nx+1-i ) ) / &
227	REAL( nx+1, KIND=wp ) ) ) / ( dx * dx ) + &
228	2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * ( ny+1-j ) ) / &
229	REAL( ny+1, KIND=wp ) ) ) / ( dy * dy )
230	ENDIF
231	ENDIF
232	ENDDO
233	ENDDO
234
235	DO k = 0, nz-1
236	DO j = nys_z, nyn_z
237	DO i = nxl_z, nxr_z
238	tric(i,j,k) = ddzuw(k,3) - ll(i,j) * rho_air(k+1)
239	ENDDO
240	ENDDO
241	ENDDO
242
243	IF ( ibc_p_b == 1 ) THEN
244	DO j = nys_z, nyn_z
245	DO i = nxl_z, nxr_z
246	tric(i,j,0) = tric(i,j,0) + ddzuw(0,1)
247	ENDDO
248	ENDDO
249	ENDIF
250	IF ( ibc_p_t == 1 ) THEN
251	DO j = nys_z, nyn_z
252	DO i = nxl_z, nxr_z
253	tric(i,j,nz-1) = tric(i,j,nz-1) + ddzuw(nz-1,2)
254	ENDDO
255	ENDDO
256	ENDIF
257
258	END SUBROUTINE maketri
259
260
261	!------------------------------------------------------------------------------!
262	! Description:
263	! ------------
264	!> Substitution (Forward and Backward) (Thomas algorithm)
265	!------------------------------------------------------------------------------!
266	SUBROUTINE tridia_substi( ar )
267
268
269	USE arrays_3d, &
270	ONLY: tri
271
272	USE control_parameters, &
273	ONLY: ibc_p_b, ibc_p_t
274
275	USE kinds
276
277	IMPLICIT NONE
278
279	INTEGER(iwp) :: i !<
280	INTEGER(iwp) :: j !<
281	INTEGER(iwp) :: k !<
282
283	REAL(wp) :: ar(nxl_z:nxr_z,nys_z:nyn_z,1:nz) !<
284
285	REAL(wp), DIMENSION(nxl_z:nxr_z,nys_z:nyn_z,0:nz-1) :: ar1 !<
286
287	!
288	!-- Forward substitution
289	DO k = 0, nz - 1
290	DO j = nys_z, nyn_z
291	DO i = nxl_z, nxr_z
292
293	IF ( k == 0 ) THEN
294	ar1(i,j,k) = ar(i,j,k+1)
295	ELSE
296	ar1(i,j,k) = ar(i,j,k+1) - tri(i,j,k,2) * ar1(i,j,k-1)
297	ENDIF
298
299	ENDDO
300	ENDDO
301	ENDDO
302
303	!
304	!-- Backward substitution
305	!-- Note, the 1.0E-20 in the denominator is due to avoid divisions
306	!-- by zero appearing if the pressure bc is set to neumann at the top of
307	!-- the model domain.
308	DO k = nz-1, 0, -1
309	DO j = nys_z, nyn_z
310	DO i = nxl_z, nxr_z
311
312	IF ( k == nz-1 ) THEN
313	ar(i,j,k+1) = ar1(i,j,k) / ( tri(i,j,k,1) + 1.0E-20_wp )
314	ELSE
315	ar(i,j,k+1) = ( ar1(i,j,k) - ddzuw(k,2) * ar(i,j,k+2) ) &
316	/ tri(i,j,k,1)
317	ENDIF
318	ENDDO
319	ENDDO
320	ENDDO
321
322	!
323	!-- Indices i=0, j=0 correspond to horizontally averaged pressure.
324	!-- The respective values of ar should be zero at all k-levels if
325	!-- acceleration of horizontally averaged vertical velocity is zero.
326	IF ( ibc_p_b == 1 .AND. ibc_p_t == 1 ) THEN
327	IF ( nys_z == 0 .AND. nxl_z == 0 ) THEN
328	DO k = 1, nz
329	ar(nxl_z,nys_z,k) = 0.0_wp
330	ENDDO
331	ENDIF
332	ENDIF
333
334	END SUBROUTINE tridia_substi
335
336
337	!------------------------------------------------------------------------------!
338	! Description:
339	! ------------
340	!> Substitution (Forward and Backward) (Thomas algorithm)
341	!------------------------------------------------------------------------------!
342	SUBROUTINE tridia_substi_overlap( ar, jj )
343
344
345	USE arrays_3d, &
346	ONLY: tri
347
348	USE control_parameters, &
349	ONLY: ibc_p_b, ibc_p_t
350
351	USE kinds
352
353	IMPLICIT NONE
354
355	INTEGER(iwp) :: i !<
356	INTEGER(iwp) :: j !<
357	INTEGER(iwp) :: jj !<
358	INTEGER(iwp) :: k !<
359
360	REAL(wp) :: ar(nxl_z:nxr_z,nys_z:nyn_z,1:nz) !<
361
362	REAL(wp), DIMENSION(nxl_z:nxr_z,nys_z:nyn_z,0:nz-1) :: ar1 !<
363
364	!
365	!-- Forward substitution
366	DO k = 0, nz - 1
367	DO j = nys_z, nyn_z
368	DO i = nxl_z, nxr_z
369
370	IF ( k == 0 ) THEN
371	ar1(i,j,k) = ar(i,j,k+1)
372	ELSE
373	ar1(i,j,k) = ar(i,j,k+1) - tri(i,jj,k,2) * ar1(i,j,k-1)
374	ENDIF
375
376	ENDDO
377	ENDDO
378	ENDDO
379
380	!
381	!-- Backward substitution
382	!-- Note, the 1.0E-20 in the denominator is due to avoid divisions
383	!-- by zero appearing if the pressure bc is set to neumann at the top of
384	!-- the model domain.
385	DO k = nz-1, 0, -1
386	DO j = nys_z, nyn_z
387	DO i = nxl_z, nxr_z
388
389	IF ( k == nz-1 ) THEN
390	ar(i,j,k+1) = ar1(i,j,k) / ( tri(i,jj,k,1) + 1.0E-20_wp )
391	ELSE
392	ar(i,j,k+1) = ( ar1(i,j,k) - ddzuw(k,2) * ar(i,j,k+2) ) &
393	/ tri(i,jj,k,1)
394	ENDIF
395	ENDDO
396	ENDDO
397	ENDDO
398
399	!
400	!-- Indices i=0, j=0 correspond to horizontally averaged pressure.
401	!-- The respective values of ar should be zero at all k-levels if
402	!-- acceleration of horizontally averaged vertical velocity is zero.
403	IF ( ibc_p_b == 1 .AND. ibc_p_t == 1 ) THEN
404	IF ( nys_z == 0 .AND. nxl_z == 0 ) THEN
405	DO k = 1, nz
406	ar(nxl_z,nys_z,k) = 0.0_wp
407	ENDDO
408	ENDIF
409	ENDIF
410
411	END SUBROUTINE tridia_substi_overlap
412
413
414	!------------------------------------------------------------------------------!
415	! Description:
416	! ------------
417	!> Splitting of the tridiagonal matrix (Thomas algorithm)
418	!------------------------------------------------------------------------------!
419	SUBROUTINE split
420
421
422	USE arrays_3d, &
423	ONLY: tri, tric
424
425	USE kinds
426
427	IMPLICIT NONE
428
429	INTEGER(iwp) :: i !<
430	INTEGER(iwp) :: j !<
431	INTEGER(iwp) :: k !<
432	!
433	!-- Splitting
434	DO j = nys_z, nyn_z
435	DO i = nxl_z, nxr_z
436	tri(i,j,0,1) = tric(i,j,0)
437	ENDDO
438	ENDDO
439
440	DO k = 1, nz-1
441	DO j = nys_z, nyn_z
442	DO i = nxl_z, nxr_z
443	tri(i,j,k,2) = ddzuw(k,1) / tri(i,j,k-1,1)
444	tri(i,j,k,1) = tric(i,j,k) - ddzuw(k-1,2) * tri(i,j,k,2)
445	ENDDO
446	ENDDO
447	ENDDO
448
449	END SUBROUTINE split
450
451
452	!------------------------------------------------------------------------------!
453	! Description:
454	! ------------
455	!> Solves the linear system of equations for a 1d-decomposition along x (see
456	!> tridia)
457	!>
458	!> @attention when using the intel compilers older than 12.0, array tri must
459	!> be passed as an argument to the contained subroutines. Otherwise
460	!> addres faults will occur. This feature can be activated with
461	!> cpp-switch __intel11
462	!> On NEC, tri should not be passed (except for routine substi_1dd)
463	!> because this causes very bad performance.
464	!------------------------------------------------------------------------------!
465
466	SUBROUTINE tridia_1dd( ddx2, ddy2, nx, ny, j, ar, tri_for_1d )
467
468
469	USE arrays_3d, &
470	ONLY: ddzu_pres, ddzw, rho_air, rho_air_zw
471
472	USE control_parameters, &
473	ONLY: ibc_p_b, ibc_p_t
474
475	USE kinds
476
477	IMPLICIT NONE
478
479	INTEGER(iwp) :: i !<
480	INTEGER(iwp) :: j !<
481	INTEGER(iwp) :: k !<
482	INTEGER(iwp) :: nnyh !<
483	INTEGER(iwp) :: nx !<
484	INTEGER(iwp) :: ny !<
485	INTEGER(iwp) :: omp_get_thread_num !<
486	INTEGER(iwp) :: tn !<
487
488	REAL(wp) :: ddx2 !<
489	REAL(wp) :: ddy2 !<
490
491	REAL(wp), DIMENSION(0:nx,1:nz) :: ar !<
492	REAL(wp), DIMENSION(5,0:nx,0:nz-1) :: tri_for_1d !<
493
494
495	nnyh = ( ny + 1 ) / 2
496
497	!
498	!-- Define constant elements of the tridiagonal matrix.
499	!-- The compiler on SX6 does loop exchange. If 0:nx is a high power of 2,
500	!-- the exchanged loops create bank conflicts. The following directive
501	!-- prohibits loop exchange and the loops perform much better.
502	!CDIR NOLOOPCHG
503	DO k = 0, nz-1
504	DO i = 0,nx
505	tri_for_1d(2,i,k) = ddzu_pres(k+1) * ddzw(k+1) * rho_air_zw(k)
506	tri_for_1d(3,i,k) = ddzu_pres(k+2) * ddzw(k+1) * rho_air_zw(k+1)
507	ENDDO
508	ENDDO
509
510	IF ( j <= nnyh ) THEN
511	CALL maketri_1dd( j )
512	ELSE
513	CALL maketri_1dd( ny+1-j )
514	ENDIF
515
516	CALL split_1dd
517	CALL substi_1dd( ar, tri_for_1d )
518
519	CONTAINS
520
521
522	!------------------------------------------------------------------------------!
523	! Description:
524	! ------------
525	!> computes the i- and j-dependent component of the matrix
526	!------------------------------------------------------------------------------!
527	SUBROUTINE maketri_1dd( j )
528
529	USE constants, &
530	ONLY: pi
531
532	USE kinds
533
534	IMPLICIT NONE
535
536	INTEGER(iwp) :: i !<
537	INTEGER(iwp) :: j !<
538	INTEGER(iwp) :: k !<
539	INTEGER(iwp) :: nnxh !<
540
541	REAL(wp) :: a !<
542	REAL(wp) :: c !<
543
544	REAL(wp), DIMENSION(0:nx) :: l !<
545
546
547	nnxh = ( nx + 1 ) / 2
548	!
549	!-- Provide the tridiagonal matrix for solution of the Poisson equation in
550	!-- Fourier space. The coefficients are computed following the method of
551	!-- Schmidt et al. (DFVLR-Mitteilung 84-15), which departs from Stephan
552	!-- Siano's original version by discretizing the Poisson equation,
553	!-- before it is Fourier-transformed
554	DO i = 0, nx
555	IF ( i >= 0 .AND. i <= nnxh ) THEN
556	l(i) = 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * i ) / &
557	REAL( nx+1, KIND=wp ) ) ) * ddx2 + &
558	2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * j ) / &
559	REAL( ny+1, KIND=wp ) ) ) * ddy2
560	ELSE
561	l(i) = 2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * ( nx+1-i ) ) / &
562	REAL( nx+1, KIND=wp ) ) ) * ddx2 + &
563	2.0_wp * ( 1.0_wp - COS( ( 2.0_wp * pi * j ) / &
564	REAL( ny+1, KIND=wp ) ) ) * ddy2
565	ENDIF
566	ENDDO
567
568	DO k = 0, nz-1
569	DO i = 0, nx
570	a = -1.0_wp * ddzu_pres(k+2) * ddzw(k+1) * rho_air_zw(k+1)
571	c = -1.0_wp * ddzu_pres(k+1) * ddzw(k+1) * rho_air_zw(k)
572	tri_for_1d(1,i,k) = a + c - l(i) * rho_air(k+1)
573	ENDDO
574	ENDDO
575	IF ( ibc_p_b == 1 ) THEN
576	DO i = 0, nx
577	tri_for_1d(1,i,0) = tri_for_1d(1,i,0) + tri_for_1d(2,i,0)
578	ENDDO
579	ENDIF
580	IF ( ibc_p_t == 1 ) THEN
581	DO i = 0, nx
582	tri_for_1d(1,i,nz-1) = tri_for_1d(1,i,nz-1) + tri_for_1d(3,i,nz-1)
583	ENDDO
584	ENDIF
585
586	END SUBROUTINE maketri_1dd
587
588
589	!------------------------------------------------------------------------------!
590	! Description:
591	! ------------
592	!> Splitting of the tridiagonal matrix (Thomas algorithm)
593	!------------------------------------------------------------------------------!
594	SUBROUTINE split_1dd
595
596	IMPLICIT NONE
597
598	INTEGER(iwp) :: i !<
599	INTEGER(iwp) :: k !<
600
601
602	!
603	!-- Splitting
604	DO i = 0, nx
605	tri_for_1d(4,i,0) = tri_for_1d(1,i,0)
606	ENDDO
607	DO k = 1, nz-1
608	DO i = 0, nx
609	tri_for_1d(5,i,k) = tri_for_1d(2,i,k) / tri_for_1d(4,i,k-1)
610	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)
611	ENDDO
612	ENDDO
613
614	END SUBROUTINE split_1dd
615
616
617	!------------------------------------------------------------------------------!
618	! Description:
619	! ------------
620	!> Substitution (Forward and Backward) (Thomas algorithm)
621	!------------------------------------------------------------------------------!
622	SUBROUTINE substi_1dd( ar, tri_for_1d )
623
624
625	IMPLICIT NONE
626
627	INTEGER(iwp) :: i !<
628	INTEGER(iwp) :: k !<
629
630	REAL(wp), DIMENSION(0:nx,nz) :: ar !<
631	REAL(wp), DIMENSION(0:nx,0:nz-1) :: ar1 !<
632	REAL(wp), DIMENSION(5,0:nx,0:nz-1) :: tri_for_1d !<
633
634	!
635	!-- Forward substitution
636	DO i = 0, nx
637	ar1(i,0) = ar(i,1)
638	ENDDO
639	DO k = 1, nz-1
640	DO i = 0, nx
641	ar1(i,k) = ar(i,k+1) - tri_for_1d(5,i,k) * ar1(i,k-1)
642	ENDDO
643	ENDDO
644
645	!
646	!-- Backward substitution
647	!-- Note, the add of 1.0E-20 in the denominator is due to avoid divisions
648	!-- by zero appearing if the pressure bc is set to neumann at the top of
649	!-- the model domain.
650	DO i = 0, nx
651	ar(i,nz) = ar1(i,nz-1) / ( tri_for_1d(4,i,nz-1) + 1.0E-20_wp )
652	ENDDO
653	DO k = nz-2, 0, -1
654	DO i = 0, nx
655	ar(i,k+1) = ( ar1(i,k) - tri_for_1d(3,i,k) * ar(i,k+2) ) &
656	/ tri_for_1d(4,i,k)
657	ENDDO
658	ENDDO
659
660	!
661	!-- Indices i=0, j=0 correspond to horizontally averaged pressure.
662	!-- The respective values of ar should be zero at all k-levels if
663	!-- acceleration of horizontally averaged vertical velocity is zero.
664	IF ( ibc_p_b == 1 .AND. ibc_p_t == 1 ) THEN
665	IF ( j == 0 ) THEN
666	DO k = 1, nz
667	ar(0,k) = 0.0_wp
668	ENDDO
669	ENDIF
670	ENDIF
671
672	END SUBROUTINE substi_1dd
673
674	END SUBROUTINE tridia_1dd
675
676
677	END MODULE tridia_solver

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