1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747 | c karline: changes to take into account zero-grad and conc boundary conditions
subroutine advection (N, Y, dt, h, hint, v, Bcup, Bcdown, &
& Yup, Ydown, VFint, VF, Aint, A, method, mode, split, &
& dY, cu, it)
c-----------------------------------------------------------------------------------------
c based on the advection scheme in the GOTM model, code from 2006-11-06
c fluxes are defined on the interfaces, in an upstream-biased way.
c slope-delimeters are applied to obtain monotonic and positive schemes
c also in the presence of large gradients.
c there are 5 different slope delimeters; first-order upstream,
c 3rd order upstream-biased polynomial, 3rd order with superbee delimiter,
c 3rd order with MUSCL limiter
c as described in Pietrzak 98
c Karline: made changes to make it work for negative ww...
c added volume fraction, surface area; these two properties will generally = 1.
cc-----------------------------------------------------------------------------------------
IMPLICIT NONE
c number of vertical layers, time step
INTEGER :: N
DOUBLE PRECISION :: dt
c layer thickness (m), distance from mid to mid of each layer
DOUBLE PRECISION :: h(N), hint(0:N)
c advection speed in the direction of the axis
DOUBLE PRECISION :: v(0:N), ww(0:N)
c volume fraction and surface at interface and in middle of layers
DOUBLE PRECISION :: VFint(0:N), Aint(0:N), VF(N), A(N)
c type of upper and lower Boundary Condition (only 1 and 2 used in R)
INTEGER :: Bcdown, Bcup
integer, parameter :: Flux = 1
integer, parameter :: Value = 2
integer, parameter :: ZeroGrad = 3
integer, parameter :: zeroDivergence = 4
c value of upper and lower bnd conc
DOUBLE PRECISION :: Ydown, Yup
c type of advection scheme, slope delimeters
INTEGER :: method
integer, parameter :: UPSTREAM =5
integer, parameter :: P2 =4
integer, parameter :: P2_PDM =3
integer, parameter :: Superbee =2
integer, parameter :: MUSCL =1
c advection mode: 0= non-conservative (e.g.water flow), 1= conservative, e.g. sinking
INTEGER :: mode
c splitting mode: >0 split, <= 0: do not split, one step and return)
INTEGER :: split
c concentration to be transported
DOUBLE PRECISION :: Y(N)
c rate of change due to advection
DOUBLE PRECISION :: dY(N)
DOUBLE PRECISION :: one6th=1.0d0/6.0d0
c maximal number of iterations
INTEGER, parameter :: itmax=10000
c LOCAL VARIABLES:
integer :: i,k,it, istart, istop
DOUBLE PRECISION :: x,r,Phi,limit=0.d0
DOUBLE PRECISION :: Yu,Yc,Yd
DOUBLE PRECISION :: c,cmax
DOUBLE PRECISION :: cu(0:N)
c-----------------------------------------------------------------------------------------
c copy of current value of state variables
do k =1,N
dy(k) = y(k)
enddo
istart = 1
istop = N-1
IF (Bcdown == Value) THEN
istop = N
ELSEIF (Bcdown == ZeroGrad) THEN
istop = N
Ydown = y(N)
ENDIF
IF (Bcup == Value) THEN
istart = 0
ELSEIF (Bcup == ZeroGrad) THEN
istart = 0
Yup = y(1)
ENDIF
c initialize maximum Courant number
cmax = 0.d0
c convert to per fraction (in case VFint != 1)
do k=0,N
ww(k) = v(k)*VFint(k)
enddo
IF (split .GT. 0) THEN
c compute maximum Courant number; estimate nr of iterations
do k=0,N
c = dabs(ww(k))*dt/hint(k)
if (c.gt.cmax) cmax=c
enddo
if (cmax . GT. 1) then
it = min(itmax, int(cmax)+1)
it = max(1,it)
else
it = 1
endif
ELSE
it = 1
ENDIF
c (time) splitting loop
do i=1,it
c initialize upstream interface fluxes with zero
cu = 0.d0
c spatial loop - karline : changed into 1:N-1
do k = istart, istop
c positive speed
if (ww(k) .gt. 0.d0) then
if(k > 0) THEN
c=ww(k)/dble(it)*dt/hint(k-1) ! courant number
else
c=ww(k)/dble(it)*dt/hint(k) ! courant number
endif
if (k .gt. 1) then
Yu=Y(k-1) ! upstream value
else
Yu=Yup
end if
if (k .gt. 0) then
Yc = Y(k) ! central value
else
Yc = Yup
end if
if (k .lt. N) then
Yd = Y(k+1) ! downstream value
else
Yd = Ydown
endif
c negative speed
else
c=-ww(k)/dble(it)*dt/hint(k) ! courant number
if (k .gt. 0) then
Yd = Y(k) ! downstream value
else
Yd = Yup
end if
if (k .lt. N) then
Yc = Y(k+1) ! central value
else
Yc = Ydown
endif
if (k .lt. N-1) then
Yu = Y(k+2) ! upstream value
else
Yu = Yup
endif
end if
if (abs(Yd - Yc) .gt. 1e-10) then ! slope ratio
r=(Yc - Yu)/(Yd - Yc)
else
r=(Yc - Yu)*1.e10
end if
c limit the flux according to different suggestions
select case (method)
case (UPSTREAM)
limit = 0.d0
case ((P2),(P2_PDM))
c the flux-factor phi
x = one6th*(1.-2.0*c)
Phi = (0.5+x)+(0.5-x)*r
if (method.eq.P2) then
limit=Phi
else
limit=max(0.d0,min(Phi,2./(1.d0-c),2.*r/(c+1.e-10)))
end if
case (Superbee)
limit=max(0.d0, min(1.d0, 2.0*r), min(r,2.*1.d0) )
case (MUSCL)
limit=max(0.d0,min(2.*1.d0,2.0*r,0.5*(1.d0+r)))
case default
c call rerror( 'unkown advection method') ! should not happen
end select
c compute the limited flux
cu(k) = ww(k) *(Yc+0.5d0*limit*(1.-c)*(Yd-Yc))
end do
c downstream boundary conditions
select case (Bcdown)
case (flux)
cu(N) = Ydown ! flux OUT of the domain is positive
case (zeroDivergence)
cu(N) = cu(N-1)
c case default
c call rwarn('unkown downstream boundary condition type')
end select
c upstream boundary conditions
select case (Bcup)
case (flux)
cu(0) = Yup ! flux into the domain is positive
case (zeroDivergence)
cu(0) = cu(1)
c case default
c call rwarn('unkown upstream boundary condition type')
end select
c the advection step
if (mode.eq.0) then ! conservative - KARLINE CHECK...
do k=1,N
Y(k)=Y(k)-1.d0/dble(it)*dt*((Aint(k)*cu(k)- &
& Aint(k-1)*cu(k-1))/ h(k)/A(k)/VF(k) &
& -Y(k)*(ww(k)-ww(k-1))/h(k)/A(k)/VF(k))
enddo
else ! non-conservative
do k=1,N
Y(k)=Y(k)-1.d0/dble(it)*dt*((Aint(k)*cu(k)- &
& Aint(k-1)*cu(k-1))/h(k)/A(k)/VF(k))
enddo
end if
end do ! end of the iteration loop
c rate of change due to advection
do k =1,N
dy(k) = (y(k)-dy(k))/dt
enddo
c Still to do: integrate fluxes in time (now cu = cu of last step )
c flux = 0. ! at start
c do k=1,N
c flux(k)=flux(k)+1.d0/dble(it)*dt*cu(k)
return
end subroutine advection
!-----------------------------------------------------------------------
! Copyright by the GOTM-team under the GNU Public License - www.gnu.org
! ... extensively modified by Karline Soetaert
!-----------------------------------------------------------------------
c karline: original routine, differs with respect to boundary conditions
subroutine advection_ori(N, Y, dt, h, hint, v, Bcup, Bcdown, &
& Yup, Ydown, VFint, VF, Aint, A, method, mode, split, &
& dY, cu, it)
c-----------------------------------------------------------------------------------------
c based on the advection scheme in the GOTM model, code from 2006-11-06
c fluxes are defined on the interfaces, in an upstream-biased way.
c slope-delimeters are applied to obtain monotonic and positive schemes
c also in the presence of large gradients.
c there are 5 different slope delimeters; first-order upstream,
c 3rd order upstream-biased polynomial, 3rd order with superbee delimiter,
c 3rd order with MUSCL limiter
c as described in Pietrzak 98
c Karline: made changes to make it work for negative ww...
c added volume fraction, surface area; these two properties will generally = 1.
cc-----------------------------------------------------------------------------------------
IMPLICIT NONE
c number of vertical layers, time step
INTEGER :: N
DOUBLE PRECISION :: dt
c layer thickness (m), distance from mid to mid of each layer
DOUBLE PRECISION :: h(N), hint(0:N)
c advection speed in the direction of the axis
DOUBLE PRECISION :: v(0:N), ww(0:N)
c volume fraction and surface at interface and in middle of layers
DOUBLE PRECISION :: VFint(0:N), Aint(0:N), VF(N), A(N)
c type of upper and lower Boundary Condition (only 1 and 2 used in R)
INTEGER :: Bcdown, Bcup
integer, parameter :: Flux = 1
integer, parameter :: Value = 2
integer, parameter :: ZeroGrad = 3
integer, parameter :: zeroDivergence = 4
c value of upper and lower bnd conc
DOUBLE PRECISION :: Ydown, Yup
c type of advection scheme, slope delimeters
INTEGER :: method
integer, parameter :: UPSTREAM =5
integer, parameter :: P2 =4
integer, parameter :: P2_PDM =3
integer, parameter :: Superbee =2
integer, parameter :: MUSCL =1
c advection mode: 0= non-conservative (e.g.water flow), 1= conservative, e.g. sinking
INTEGER :: mode
c splitting mode: >0 split, <= 0: do not split, one step and return)
INTEGER :: split
c concentration to be transported
DOUBLE PRECISION :: Y(N)
c rate of change due to advection
DOUBLE PRECISION :: dY(N)
DOUBLE PRECISION :: one6th=1.0d0/6.0d0
c maximal number of iterations
INTEGER, parameter :: itmax=10000
c LOCAL VARIABLES:
integer :: i,k,it
DOUBLE PRECISION :: x,r,Phi,limit=0.d0
DOUBLE PRECISION :: Yu,Yc,Yd
DOUBLE PRECISION :: c,cmax
DOUBLE PRECISION :: cu(0:N)
c-----------------------------------------------------------------------------------------
c copy of current value of state variables
do k =1,N
dy(k) = y(k)
enddo
c initialize maximum Courant number
cmax = 0.d0
c convert to per fraction (in case VFint != 1)
do k=0,N
ww(k) = v(k)*VFint(k)
enddo
IF (split .GT. 0) THEN
c compute maximum Courant number; estimate nr of iterations
do k=0,N
c = dabs(ww(k))*dt/hint(k)
if (c.gt.cmax) cmax=c
enddo
if (cmax . GT. 1) then
it = min(itmax,int(cmax)+1) ! WAS: min(itmax,int(cmax)+1)
it = max(1,it)
else
it = 1
endif
ELSE
it = 1
ENDIF
c (time) splitting loop
do i=1,it
c initialize upstream interface fluxes with zero
cu = 0.d0
c spatial loop - karline : changed into 1:N-1
do k = 1, N-1
c positive speed
if (ww(k) .gt. 0.d0) then
c=ww(k)/dble(it)*dt/hint(k-1) ! courant number
if (k .gt. 1) then
Yu=Y(k-1) ! upstream value
else
Yu=Y(k)
end if
Yc=Y(k ) ! central value
Yd=Y(k+1) ! downstream value
c negative speed
else
c=-ww(k)/dble(it)*dt/hint(k) ! courant number
Yd = Y(k) ! downstream value
Yc = Y(k+1) ! central value
if (k .lt. N-1) then
Yu = Y(k+2)
else
Yu = Y(N)
endif
end if
if (abs(Yd-Yc) .gt. 1e-10) then ! slope ratio
r=(Yc-Yu)/(Yd-Yc)
else
r=(Yc-Yu)*1.e10
end if
c limit the flux according to different suggestions
select case (method)
case (UPSTREAM)
limit=0.d0
case ((P2),(P2_PDM))
c the flux-factor phi
x = one6th*(1.-2.0*c)
Phi = (0.5+x)+(0.5-x)*r
if (method.eq.P2) then
limit=Phi
else
limit=max(0.d0,min(Phi,2./(1.d0-c),2.*r/(c+1.e-10)))
end if
case (Superbee)
limit=max(0.d0, min(1.d0, 2.0*r), min(r,2.*1.d0) )
case (MUSCL)
limit=max(0.d0,min(2.*1.d0,2.0*r,0.5*(1.d0+r)))
case default
c call rerror( 'unkown advection method') ! should not happen
end select
c compute the limited flux
cu(k) = ww(k) *(Yc+0.5d0*limit*(1.-c)*(Yd-Yc))
end do
c downstream boundary conditions
select case (Bcdown)
case (flux)
cu(N) = Ydown ! flux OUT of the domain is positive
case (value)
if (ww(N).lt. 0.d0) then
cu(N) = ww(N)*Ydown
else
cu(N) = ww(N)*Y(N)
end if
case (zerograd)
cu(N) = ww(N)*Y(N)
case (zeroDivergence)
cu(N) = cu(N-1)
case default
call rwarn('unkown downstream boundary condition type')
end select
c upstream boundary conditions
select case (Bcup)
case (flux)
cu(0) = Yup ! flux into the domain is positive
case (value)
if(ww(0) .gt. 0.d0) then
cu(0) = ww(0)*Yup
else
cu(0) = ww(0)*Y(1)
end if
case (ZeroGrad)
cu(0) = ww(0)*Y(1)
case (zeroDivergence)
cu(0) = cu(1)
case default
call rwarn('unkown upstream boundary condition type')
end select
c the advection step
if (mode.eq.0) then ! conservative - KARLINE CHECK...
do k=1,N
Y(k)=Y(k)-1.d0/dble(it)*dt*((Aint(k)*cu(k)- &
& Aint(k-1)*cu(k-1))/ h(k)/A(k)/VF(k) &
& -Y(k)*(ww(k)-ww(k-1))/h(k)/A(k)/VF(k))
enddo
else ! non-conservative
do k=1,N
Y(k)=Y(k)-1.d0/dble(it)*dt*((Aint(k)*cu(k)- &
& Aint(k-1)*cu(k-1))/h(k)/A(k)/VF(k))
enddo
end if
end do ! end of the iteration loop
c rate of change due to advection
do k =1,N
dy(k) = (y(k)-dy(k))/dt
enddo
c Still to do: integrate fluxes in time (now cu = cu of last step )
c flux = 0. ! at start
c do k=1,N
c flux(k)=flux(k)+1.d0/dble(it)*dt*cu(k)
return
end subroutine advection_ori
!-----------------------------------------------------------------------
! Copyright by the GOTM-team under the GNU Public License - www.gnu.org
! ... extensively modified by Karline Soetaert
!-----------------------------------------------------------------------
subroutine advectvol(N, Y, dt, V, Vint, flow, Bcup, Bcdown, &
& Yup, Ydown, method,mode,dY, cu, it)
c-----------------------------------------------------------------------------------------
c Similar as above, but for volumetric transport:
c use flow = v*A rather than v
c volume = h*A rather than h
c hint -> c(volume(1),volume) ... for now ... not so important
c-----------------------------------------------------------------------------------------
IMPLICIT NONE
c number of vertical layers, time step
INTEGER :: N
DOUBLE PRECISION :: dt
c layer thickness (m), distance from mid to mid
DOUBLE PRECISION :: V(N), Vint(0:N)
c vertical advection speed
DOUBLE PRECISION :: flow(0:N)
c type of upper and lower Boundary Condition (1 and 2 used)
INTEGER :: Bcdown, Bcup
integer, parameter :: Flux =1
integer, parameter :: Value =2
integer, parameter :: ZeroGrad =3 ! not used
integer, parameter :: zeroDivergence =4 ! not used
c value of upper and lower bnd conc
DOUBLE PRECISION :: Ydown, Yup
c type of advection scheme, slope delimeters
INTEGER :: method
integer, parameter :: UPSTREAM =5
integer, parameter :: P2 =4
integer, parameter :: P2_PDM =3
integer, parameter :: Superbee =2
integer, parameter :: MUSCL =1
c advection mode 0: non-conservative (e.g. water flow), 1: conservative, eg.g. sinking
INTEGER :: mode
c concentration to be transported
DOUBLE PRECISION :: Y(N)
c rate of change due to advection
DOUBLE PRECISION :: dY(N)
DOUBLE PRECISION :: one6th=1.0d0/6.0d0
INTEGER, parameter :: itmax=100
c LOCAL VARIABLES:
integer :: i,k,it
DOUBLE PRECISION :: x,r,Phi,limit=0.d0
DOUBLE PRECISION :: Yu,Yc,Yd
DOUBLE PRECISION :: c,cmax
DOUBLE PRECISION :: cu(0:N)
c-----------------------------------------------------------------------------------------
c initialize maximum Courant number
cmax = 0.d0
c copy of current value of state variables
do k =1,N
dy(k) = y(k)
enddo
c compute maximum Courant number; estimate nr of iterations
do k=0,N
c=dabs(flow(k))*dt/Vint(k)
if (c.gt.cmax) cmax=c
enddo
it=min(itmax,int(cmax)+1)
c (time) splitting loop
do i=1,it
c initialize interface fluxes with zero
cu = 0.d0
c spatial loop
do k=1,N-1
c positive speed
if (flow(k) .gt. 0.d0) then
c=flow(k)/dble(it)*dt/Vint(k-1) ! courant number
if (k .gt. 1) then
Yu=Y(k-1) ! upstream value
else
Yu=Y(k)
end if
Yc=Y(k ) ! central value
Yd=Y(k+1) ! downstream value
c negative speed
else
c=-flow(k)/dble(it)*dt/Vint(k) ! courant number
if (k .lt. N-1) then
Yu=Y(k+2) ! upstream value
else
Yu=Y(N)
end if
Yc=Y(k+1) ! central value
Yd=Y(k)
end if
if (abs(Yd-Yc) .gt. 1e-10) then ! slope ratio
r=(Yc-Yu)/(Yd-Yc)
else
r=(Yc-Yu)*1.e10
end if
c limit the flux according to different suggestions, phi = flux-factor
select case (method)
case (UPSTREAM)
limit=0.d0
case ((P2),(P2_PDM))
c - for quickest
x = one6th*(1.d0-2.d0*c)
Phi = (0.5d0+x)+(0.5d0-x)*r
if (method.eq.P2) then
limit=Phi
else
limit=max(0.d0,min(Phi,2./(1.-c),2.*r/(c+1.e-10)))
end if
case (Superbee)
limit=max(0.d0, min(1.d0, 2.0*r), min(r,2.*1.d0) )
case (MUSCL)
limit=max(0.d0,min(2.*1.d0,2.0*r,0.5*(1.0+r)))
case default
c call rerror( 'unkown advection method')
end select
c compute the limited flux
cu(k) = flow(k) *(Yc+0.5d0*limit*(1.-c)*(Yd-Yc))
end do
c downstream boundary conditions
select case (Bcdown)
case (flux)
cu(N) = Ydown ! flux OUT of the domain is positive
case (value)
if (flow(N).lt. 0.d0) then
cu(N) = flow(N)*Ydown
else
cu(N) = flow(N)*Y(N)
end if
case (ZeroGrad)
cu(N) = flow(N)*Y(N)
case (zeroDivergence)
cu(N) = cu(N-1)
case default
call rwarn('unkown downstream boundary condition type')
end select
c upstream boundary conditions
select case (Bcup)
case (flux)
cu(0) = Yup ! flux into the domain is positive
case (value)
if(flow(0) .gt. 0.d0) then ! Karline: CHECK!
cu(0) = flow(0)*Yup
else
cu(0) = flow(0)*Y(1)
end if
case (ZeroGrad)
cu(0) = flow(0)*Y(1)
case (zeroDivergence)
cu(0) = cu(1)
case default
call rwarn('unkown upstream boundary condition type')
end select
c the advection step
if (mode.eq.0) then ! non-conservative - KARLINE CHECK...
do k=1,N
Y(k)=Y(k)-1.d0/dble(it)*dt*((cu(k)- cu(k-1))/ V(k) &
& -Y(k)*(flow(k)-flow(k-1))/V(k))
enddo
else ! conservative - this is actually used
do k=1,N
Y(k)=Y(k)-1.d0/dble(it)*dt*((cu(k)- cu(k-1))/V(k))
enddo
end if
end do ! end of the iteration loop
c rate of change due to advection
do k =1,N
dy(k) = (y(k)-dy(k))/dt
enddo
c Still to do: integrate fluxes in time (now cu = cu of last step )
c flux = 0. ! at start
c do k=1,N
c flux(k)=flux(k)+1.d0/real(it)*dt*cu(k)
return
end subroutine advectvol
|