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[wrffire.git] / wrfv2_fire / chem / KPP / kpp / kpp-2.1 / int / rosenbrock_tlm.f90
blob50429ce116737ed2246013c1e718c1e02a47b8ea
1 MODULE KPP_ROOT_Integrator
2
3 USE KPP_ROOT_Precision
4 USE KPP_ROOT_Parameters
5 USE KPP_ROOT_Global
6 USE KPP_ROOT_LinearAlgebra
7 USE KPP_ROOT_Rates
8 USE KPP_ROOT_Function
9 USE KPP_ROOT_Jacobian
10 USE KPP_ROOT_Hessian
11 USE KPP_ROOT_Util
12
13 IMPLICIT NONE
14 PUBLIC
15 SAVE
16 !~~~> Statistics on the work performed by the Rosenbrock method
17 INTEGER :: Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng
18 INTEGER, PARAMETER :: ifun=1, ijac=2, istp=3, iacc=4, &
19 irej=5, idec=6, isol=7, isng=8, &
20 itexit=1,ihexit=2
21
22 KPP_REAL, PARAMETER :: ZERO = 0.0d0, ONE = 1.0d0
23
24
25 CONTAINS ! Functions in the module KPP_ROOT_Integrator
26
27 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
28 SUBROUTINE INTEGRATE_TLM( NTLM, Y, Y_tlm, TIN, TOUT, &
29 ICNTRL_U, RCNTRL_U, ISTATUS_U, RSTATUS_U )
30 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
31 IMPLICIT NONE
32
33 !~~~> Y - Concentrations
34 KPP_REAL :: Y(NVAR)
35 !~~~> NTLM - No. of sensitivity coefficients
36 INTEGER NTLM
37 !~~~> Y_tlm - Sensitivities of concentrations
38 ! Note: Y_tlm (1:NVAR,j) contains sensitivities of
39 ! Y(1:NVAR) w.r.t. the j-th parameter, j=1...NTLM
40 KPP_REAL :: Y_tlm(NVAR,NTLM)
41 KPP_REAL, INTENT(IN) :: TIN ! TIN - Start Time
42 KPP_REAL, INTENT(IN) :: TOUT ! TOUT - End Time
43 !~~~> Optional input parameters and statistics
44 INTEGER, INTENT(IN), OPTIONAL :: ICNTRL_U(20)
45 KPP_REAL, INTENT(IN), OPTIONAL :: RCNTRL_U(20)
46 INTEGER, INTENT(OUT), OPTIONAL :: ISTATUS_U(20)
47 KPP_REAL, INTENT(OUT), OPTIONAL :: RSTATUS_U(20)
48
49 INTEGER, SAVE :: N_stp, N_acc, N_rej, N_sng, IERR
50 KPP_REAL :: RCNTRL(20), RSTATUS(20)
51 INTEGER :: ICNTRL(20), ISTATUS(20)
52
53 ICNTRL(1:10) = 0
54 RCNTRL(1:10) = 0.0_dp
55 ISTATUS(1:10) = 0
56 RSTATUS(1:10) = 0.0_dp
57
58 ICNTRL(1) = 0 ! non-autonomous
59 ICNTRL(2) = 1 ! vector tolerances
60 RCNTRL(3) = STEPMIN ! starting step
61 ICNTRL(4) = 5 ! choice of the method
62
63 ! if optional parameters are given, and if they are >=0, then they overwrite default settings
64 IF (PRESENT(ICNTRL_U)) THEN
65 WHERE(ICNTRL_U(:) >= 0) ICNTRL(1:20) = ICNTRL_U(:)
66 ENDIF
67 IF (PRESENT(RCNTRL_U)) THEN
68 WHERE(RCNTRL_U(:) >= 0) RCNTRL(1:20) = RCNTRL_U(:)
69 ENDIF
70
71 CALL RosenbrockTLM(VAR, NTLM, Y_tlm, &
72 TIN,TOUT, &
73 ATOL,RTOL, &
74 FunTemplate,JacTemplate,HessTemplate, &
75 RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR)
76
77 ! N_stp = N_stp + ICNTRL(istp)
78 ! N_acc = N_acc + ICNTRL(iacc)
79 ! N_rej = N_rej + ICNTRL(irej)
80 ! N_sng = N_sng + ICNTRL(isng)
81 ! PRINT*,'Step=',N_stp,' Acc=',N_acc,' Rej=',N_rej, &
82 ! ' Singular=',N_sng
83
84 IF (IERR < 0) THEN
85 print *,'Rosenbrock: Unsucessful step at T=', &
86 TIN,' (IERR=',IERR,')'
87 END IF
88
89 STEPMIN = RCNTRL(ihexit)
90 ! if optional parameters are given for output they return information
91 IF (PRESENT(ISTATUS_U)) ISTATUS_U(:) = ISTATUS(1:20)
92 IF (PRESENT(RSTATUS_U)) RSTATUS_U(:) = RSTATUS(1:20)
93
94 END SUBROUTINE INTEGRATE_TLM
95
96
97 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
98 SUBROUTINE RosenbrockTLM(Y,NTLM,Y_tlm,&
99 Tstart,Tend, &
100 AbsTol,RelTol, &
101 RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR)
102 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
103 !
104 ! TLM = Tangent Linear Model of a Rosenbrock Method
105 !
106 ! Solves the system y'=F(t,y) using a Rosenbrock method defined by:
107 !
108 ! G = 1/(H*gamma(1)) - Jac(t0,Y0)
109 ! T_i = t0 + Alpha(i)*H
110 ! Y_i = Y0 + \sum_{j=1}^{i-1} A(i,j)*K_j
111 ! G * K_i = Fun( T_i, Y_i ) + \sum_{j=1}^S C(i,j)/H * K_j +
112 ! gamma(i)*dF/dT(t0, Y0)
113 ! Y1 = Y0 + \sum_{j=1}^S M(j)*K_j
114 !
115 ! For details on Rosenbrock methods and their implementation consult:
116 ! E. Hairer and G. Wanner
117 ! "Solving ODEs II. Stiff and differential-algebraic problems".
118 ! Springer series in computational mathematics, Springer-Verlag, 1996.
119 ! The codes contained in the book inspired this implementation.
120 !
121 ! (C) Adrian Sandu, August 2004
122 ! Virginia Polytechnic Institute and State University
123 ! Contact: sandu@cs.vt.edu
124 ! This implementation is part of KPP - the Kinetic PreProcessor
125 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
126 !
127 !~~~> INPUT ARGUMENTS:
128 !
129 !- Y(NVAR) -> vector of initial conditions (at T=Tstart)
130 ! NTLM -> dimension of linearized system,
131 ! i.e. the number of sensitivity coefficients
132 !- Y_tlm(NVAR*NTLM) -> vector of initial sensitivity conditions (at T=Tstart)
133 !- [Tstart,Tend] -> time range of integration
134 ! (if Tstart>Tend the integration is performed backwards in time)
135 !- RelTol, AbsTol -> user precribed accuracy
136 !- SUBROUTINE Fun( T, Y, Ydot ) -> ODE function,
137 ! returns Ydot = Y' = F(T,Y)
138 !- SUBROUTINE Jac( T, Y, Jcb ) -> Jacobian of the ODE function,
139 ! returns Jcb = dF/dY
140 !- ICNTRL(1:10) -> integer inputs parameters
141 !- RCNTRL(1:10) -> real inputs parameters
142 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
143 !
144 !~~~> OUTPUT ARGUMENTS:
145 !
146 !- Y(NVAR) -> vector of final states (at T->Tend)
147 !- Y_tlm(NVAR*NTLM) -> vector of final sensitivities (at T=Tend)
148 !- ICNTRL(11:20) -> integer output parameters
149 !- RCNTRL(11:20) -> real output parameters
150 !- IERR -> job status upon return
151 ! - succes (positive value) or failure (negative value) -
152 ! = 1 : Success
153 ! = -1 : Improper value for maximal no of steps
154 ! = -2 : Selected Rosenbrock method not implemented
155 ! = -3 : Hmin/Hmax/Hstart must be positive
156 ! = -4 : FacMin/FacMax/FacRej must be positive
157 ! = -5 : Improper tolerance values
158 ! = -6 : No of steps exceeds maximum bound
159 ! = -7 : Step size too small
160 ! = -8 : Matrix is repeatedly singular
161 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
162 !
163 !~~~> INPUT PARAMETERS:
164 !
165 ! Note: For input parameters equal to zero the default values of the
166 ! corresponding variables are used.
167 !
168 ! ICNTRL(1) = 1: F = F(y) Independent of T (AUTONOMOUS)
169 ! = 0: F = F(t,y) Depends on T (NON-AUTONOMOUS)
170 ! ICNTRL(2) = 0: AbsTol, RelTol are NVAR-dimensional vectors
171 ! = 1: AbsTol, RelTol are scalars
172 ! ICNTRL(3) -> maximum number of integration steps
173 ! For ICNTRL(3)=0) the default value of 100000 is used
174 !
175 ! ICNTRL(4) -> selection of a particular Rosenbrock method
176 ! = 0 : default method is Rodas3
177 ! = 1 : method is Ros2
178 ! = 2 : method is Ros3
179 ! = 3 : method is Ros4
180 ! = 4 : method is Rodas3
181 ! = 5: method is Rodas4
182 !
183 ! RCNTRL(1) -> Hmin, lower bound for the integration step size
184 ! It is strongly recommended to keep Hmin = ZERO
185 ! RCNTRL(2) -> Hmax, upper bound for the integration step size
186 ! RCNTRL(3) -> Hstart, starting value for the integration step size
187 !
188 ! RCNTRL(4) -> FacMin, lower bound on step decrease factor (default=0.2)
189 ! RCNTRL(5) -> FacMin,upper bound on step increase factor (default=6)
190 ! RCNTRL(6) -> FacRej, step decrease factor after multiple rejections
191 ! (default=0.1)
192 ! RCNTRL(7) -> FacSafe, by which the new step is slightly smaller
193 ! than the predicted value (default=0.9)
194 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
195 !
196 !~~~> OUTPUT PARAMETERS:
197 !
198 ! Note: each call to Rosenbrock adds the corrent no. of fcn calls
199 ! to previous value of ISTATUS(1), and similar for the other params.
200 ! Set ISTATUS(1:10) = 0 before call to avoid this accumulation.
201 !
202 ! ISTATUS(1) = No. of function calls
203 ! ISTATUS(2) = No. of jacobian calls
204 ! ISTATUS(3) = No. of steps
205 ! ISTATUS(4) = No. of accepted steps
206 ! ISTATUS(5) = No. of rejected steps (except at the beginning)
207 ! ISTATUS(6) = No. of LU decompositions
208 ! ISTATUS(7) = No. of forward/backward substitutions
209 ! ISTATUS(8) = No. of singular matrix decompositions
210 !
211 ! RSTATUS(1) -> Texit, the time corresponding to the
212 ! computed Y upon return
213 ! RSTATUS(2) -> Hexit, last accepted step before exit
214 ! For multiple restarts, use Hexit as Hstart in the following run
215 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
216
217 IMPLICIT NONE
218
219 !~~~> Arguments
220 KPP_REAL, INTENT(INOUT) :: Y(NVAR)
221 INTEGER, INTENT(IN) :: NTLM
222 KPP_REAL, INTENT(INOUT) :: Y_tlm(NVAR,NTLM)
223 KPP_REAL, INTENT(IN) :: Tstart, Tend
224 KPP_REAL, INTENT(IN) :: AbsTol(NVAR),RelTol(NVAR)
225 INTEGER, INTENT(IN) :: ICNTRL(10)
226 KPP_REAL, INTENT(IN) :: RCNTRL(10)
227 INTEGER, INTENT(INOUT) :: ISTATUS(10)
228 KPP_REAL, INTENT(INOUT) :: RSTATUS(10)
229 INTEGER, INTENT(OUT) :: IERR
230 !~~~> The method parameters
231 INTEGER, PARAMETER :: Smax = 6
232 INTEGER :: Method, ros_S
233 KPP_REAL, DIMENSION(Smax) :: ros_M, ros_E, ros_Alpha, ros_Gamma
234 KPP_REAL, DIMENSION(Smax*(Smax-1)/2) :: ros_A, ros_C
235 KPP_REAL :: ros_ELO
236 LOGICAL, DIMENSION(Smax) :: ros_NewF
237 CHARACTER(LEN=12) :: ros_Name
238 !~~~> Local variables
239 KPP_REAL :: Roundoff, FacMin, FacMax, FacRej, FacSafe
240 KPP_REAL :: Hmin, Hmax, Hstart, Hexit
241 KPP_REAL :: Texit
242 INTEGER :: i, UplimTol, Max_no_steps
243 LOGICAL :: Autonomous, VectorTol
244 !~~~> Parameters
245 KPP_REAL, PARAMETER :: ZERO = 0.0d0, ONE = 1.0d0
246 KPP_REAL, PARAMETER :: DeltaMin = 1.0d-5
247
248 !~~~> Initialize statistics
249 Nfun = ISTATUS(ifun)
250 Njac = ISTATUS(ijac)
251 Nstp = ISTATUS(istp)
252 Nacc = ISTATUS(iacc)
253 Nrej = ISTATUS(irej)
254 Ndec = ISTATUS(idec)
255 Nsol = ISTATUS(isol)
256 Nsng = ISTATUS(isng)
257
258 !~~~> Autonomous or time dependent ODE. Default is time dependent.
259 Autonomous = .NOT.(ICNTRL(1) == 0)
260
261 !~~~> For Scalar tolerances (ICNTRL(2).NE.0) the code uses AbsTol(1) and RelTol(1)
262 ! For Vector tolerances (ICNTRL(2) == 0) the code uses AbsTol(1:NVAR) and RelTol(1:NVAR)
263 IF (ICNTRL(2) == 0) THEN
264 VectorTol = .TRUE.
265 UplimTol = NVAR
266 ELSE
267 VectorTol = .FALSE.
268 UplimTol = 1
269 END IF
270
271 !~~~> The maximum number of steps admitted
272 IF (ICNTRL(3) == 0) THEN
273 Max_no_steps = 100000
274 ELSEIF (Max_no_steps > 0) THEN
275 Max_no_steps=ICNTRL(3)
276 ELSE
277 PRINT * ,'User-selected max no. of steps: ICNTRL(3)=',ICNTRL(3)
278 CALL ros_ErrorMsg(-1,Tstart,ZERO,IERR)
279 RETURN
280 END IF
281
282 !~~~> The particular Rosenbrock method chosen
283 IF (ICNTRL(4) == 0) THEN
284 Method = 3
285 ELSEIF ( (ICNTRL(4) >= 1).AND.(ICNTRL(4) <= 5) ) THEN
286 Method = ICNTRL(4)
287 ELSE
288 PRINT * , 'User-selected Rosenbrock method: ICNTRL(4)=', Method
289 CALL ros_ErrorMsg(-2,Tstart,ZERO,IERR)
290 RETURN
291 END IF
292
293 !~~~> Unit roundoff (1+Roundoff>1)
294 Roundoff = WLAMCH('E')
295
296 !~~~> Lower bound on the step size: (positive value)
297 IF (RCNTRL(1) == ZERO) THEN
298 Hmin = ZERO
299 ELSEIF (RCNTRL(1) > ZERO) THEN
300 Hmin = RCNTRL(1)
301 ELSE
302 PRINT * , 'User-selected Hmin: RCNTRL(1)=', RCNTRL(1)
303 CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR)
304 RETURN
305 END IF
306 !~~~> Upper bound on the step size: (positive value)
307 IF (RCNTRL(2) == ZERO) THEN
308 Hmax = ABS(Tend-Tstart)
309 ELSEIF (RCNTRL(2) > ZERO) THEN
310 Hmax = MIN(ABS(RCNTRL(2)),ABS(Tend-Tstart))
311 ELSE
312 PRINT * , 'User-selected Hmax: RCNTRL(2)=', RCNTRL(2)
313 CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR)
314 RETURN
315 END IF
316 !~~~> Starting step size: (positive value)
317 IF (RCNTRL(3) == ZERO) THEN
318 Hstart = MAX(Hmin,DeltaMin)
319 ELSEIF (RCNTRL(3) > ZERO) THEN
320 Hstart = MIN(ABS(RCNTRL(3)),ABS(Tend-Tstart))
321 ELSE
322 PRINT * , 'User-selected Hstart: RCNTRL(3)=', RCNTRL(3)
323 CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR)
324 RETURN
325 END IF
326 !~~~> Step size can be changed s.t. FacMin < Hnew/Hexit < FacMax
327 IF (RCNTRL(4) == ZERO) THEN
328 FacMin = 0.2d0
329 ELSEIF (RCNTRL(4) > ZERO) THEN
330 FacMin = RCNTRL(4)
331 ELSE
332 PRINT * , 'User-selected FacMin: RCNTRL(4)=', RCNTRL(4)
333 CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR)
334 RETURN
335 END IF
336 IF (RCNTRL(5) == ZERO) THEN
337 FacMax = 6.0d0
338 ELSEIF (RCNTRL(5) > ZERO) THEN
339 FacMax = RCNTRL(5)
340 ELSE
341 PRINT * , 'User-selected FacMax: RCNTRL(5)=', RCNTRL(5)
342 CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR)
343 RETURN
344 END IF
345 !~~~> FacRej: Factor to decrease step after 2 succesive rejections
346 IF (RCNTRL(6) == ZERO) THEN
347 FacRej = 0.1d0
348 ELSEIF (RCNTRL(6) > ZERO) THEN
349 FacRej = RCNTRL(6)
350 ELSE
351 PRINT * , 'User-selected FacRej: RCNTRL(6)=', RCNTRL(6)
352 CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR)
353 RETURN
354 END IF
355 !~~~> FacSafe: Safety Factor in the computation of new step size
356 IF (RCNTRL(7) == ZERO) THEN
357 FacSafe = 0.9d0
358 ELSEIF (RCNTRL(7) > ZERO) THEN
359 FacSafe = RCNTRL(7)
360 ELSE
361 PRINT * , 'User-selected FacSafe: RCNTRL(7)=', RCNTRL(7)
362 CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR)
363 RETURN
364 END IF
365 !~~~> Check if tolerances are reasonable
366 DO i=1,UplimTol
367 IF ( (AbsTol(i) <= ZERO) .OR. (RelTol(i) <= 10.d0*Roundoff) &
368 .OR. (RelTol(i) >= 1.0d0) ) THEN
369 PRINT * , ' AbsTol(',i,') = ',AbsTol(i)
370 PRINT * , ' RelTol(',i,') = ',RelTol(i)
371 CALL ros_ErrorMsg(-5,Tstart,ZERO,IERR)
372 RETURN
373 END IF
374 END DO
375
376
377 !~~~> Initialize the particular Rosenbrock method
378 SELECT CASE (Method)
379 CASE (1)
380 CALL Ros2(ros_S, ros_A, ros_C, ros_M, ros_E, &
381 ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
382 CASE (2)
383 CALL Ros3(ros_S, ros_A, ros_C, ros_M, ros_E, &
384 ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
385 CASE (3)
386 CALL Ros4(ros_S, ros_A, ros_C, ros_M, ros_E, &
387 ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
388 CASE (4)
389 CALL Rodas3(ros_S, ros_A, ros_C, ros_M, ros_E, &
390 ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
391 CASE (5)
392 CALL Rodas4(ros_S, ros_A, ros_C, ros_M, ros_E, &
393 ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
394 CASE DEFAULT
395 PRINT * , 'Unknown Rosenbrock method: ICNTRL(4)=', Method
396 CALL ros_ErrorMsg(-2,Tstart,ZERO,IERR)
397 RETURN
398 END SELECT
399
400 !~~~> CALL Rosenbrock method
401 CALL ros_TLM_Int(Y, NTLM, Y_tlm, &
402 Tstart, Tend, Texit, &
403 AbsTol, RelTol, &
404 ! Rosenbrock method coefficients
405 ros_S, ros_M, ros_E, ros_A, ros_C, &
406 ros_Alpha, ros_Gamma, ros_ELO, ros_NewF, &
407 ! Integration parameters
408 Autonomous, VectorTol, Max_no_steps, &
409 Roundoff, Hmin, Hmax, Hstart, Hexit, &
410 FacMin, FacMax, FacRej, FacSafe, &
411 ! Error indicator
412 IERR)
413
414
415 !~~~> Collect run statistics
416 ISTATUS(ifun) = Nfun
417 ISTATUS(ijac) = Njac
418 ISTATUS(istp) = Nstp
419 ISTATUS(iacc) = Nacc
420 ISTATUS(irej) = Nrej
421 ISTATUS(idec) = Ndec
422 ISTATUS(isol) = Nsol
423 ISTATUS(isng) = Nsng
424 !~~~> Last T and H
425 RSTATUS(itexit) = Texit
426 RSTATUS(ihexit) = Hexit
427
428
429 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
430 END SUBROUTINE RosenbrockTLM
431 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
432
433
434 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
435 SUBROUTINE ros_ErrorMsg(Code,T,H,IERR)
436 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
437 ! Handles all error messages
438 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
439
440 KPP_REAL, INTENT(IN) :: T, H
441 INTEGER, INTENT(IN) :: Code
442 INTEGER, INTENT(OUT) :: IERR
443
444 IERR = Code
445 PRINT * , &
446 'Forced exit from Rosenbrock due to the following error:'
447
448 SELECT CASE (Code)
449 CASE (-1)
450 PRINT * , '--> Improper value for maximal no of steps'
451 CASE (-2)
452 PRINT * , '--> Selected Rosenbrock method not implemented'
453 CASE (-3)
454 PRINT * , '--> Hmin/Hmax/Hstart must be positive'
455 CASE (-4)
456 PRINT * , '--> FacMin/FacMax/FacRej must be positive'
457 CASE (-5)
458 PRINT * , '--> Improper tolerance values'
459 CASE (-6)
460 PRINT * , '--> No of steps exceeds maximum bound'
461 CASE (-7)
462 PRINT * , '--> Step size too small: T + 10*H = T', &
463 ' or H < Roundoff'
464 CASE (-8)
465 PRINT * , '--> Matrix is repeatedly singular'
466 CASE DEFAULT
467 PRINT *, 'Unknown Error code: ', Code
468 END SELECT
469
470 PRINT *, "T=", T, "and H=", H
471
472 END SUBROUTINE ros_ErrorMsg
473
474
475
476 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
477 SUBROUTINE ros_TLM_Int (Y, NTLM, Y_tlm, &
478 Tstart, Tend, T, &
479 AbsTol, RelTol, &
480 !~~~> Rosenbrock method coefficients
481 ros_S, ros_M, ros_E, ros_A, ros_C, &
482 ros_Alpha, ros_Gamma, ros_ELO, ros_NewF, &
483 !~~~> Integration parameters
484 Autonomous, VectorTol, Max_no_steps, &
485 Roundoff, Hmin, Hmax, Hstart, Hexit, &
486 FacMin, FacMax, FacRej, FacSafe, &
487 !~~~> Error indicator
488 IERR )
489 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
490 ! Template for the implementation of a generic Rosenbrock method
491 ! defined by ros_S (no of stages)
492 ! and its coefficients ros_{A,C,M,E,Alpha,Gamma}
493 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
494
495 IMPLICIT NONE
496
497 !~~~> Input: the initial condition at Tstart; Output: the solution at T
498 KPP_REAL, INTENT(INOUT) :: Y(NVAR)
499 !~~~> Input: Number of sensitivity coefficients
500 INTEGER, INTENT(IN) :: NTLM
501 !~~~> Input: the initial sensitivites at Tstart; Output: the sensitivities at T
502 KPP_REAL, INTENT(INOUT) :: Y_tlm(NVAR,NTLM)
503 !~~~> Input: integration interval
504 KPP_REAL, INTENT(IN) :: Tstart,Tend
505 !~~~> Output: time at which the solution is returned (T=Tend if success)
506 KPP_REAL, INTENT(OUT) :: T
507 !~~~> Input: tolerances
508 KPP_REAL, INTENT(IN) :: AbsTol(NVAR), RelTol(NVAR)
509 !~~~> Input: The Rosenbrock method parameters
510 INTEGER, INTENT(IN) :: ros_S
511 KPP_REAL, INTENT(IN) :: ros_M(ros_S), ros_E(ros_S), &
512 ros_Alpha(ros_S), ros_A(ros_S*(ros_S-1)/2), &
513 ros_Gamma(ros_S), ros_C(ros_S*(ros_S-1)/2), ros_ELO
514 LOGICAL, INTENT(IN) :: ros_NewF(ros_S)
515 !~~~> Input: integration parameters
516 LOGICAL, INTENT(IN) :: Autonomous, VectorTol
517 KPP_REAL, INTENT(IN) :: Hstart, Hmin, Hmax
518 INTEGER, INTENT(IN) :: Max_no_steps
519 KPP_REAL, INTENT(IN) :: Roundoff, FacMin, FacMax, FacRej, FacSafe
520 !~~~> Output: last accepted step
521 KPP_REAL, INTENT(OUT) :: Hexit
522 !~~~> Output: Error indicator
523 INTEGER, INTENT(OUT) :: IERR
524 ! ~~~~ Local variables
525 KPP_REAL :: Ynew(NVAR), Fcn0(NVAR), Fcn(NVAR)
526 KPP_REAL :: K(NVAR*ros_S)
527 KPP_REAL :: Ynew_tlm(NVAR,NTLM), Fcn0_tlm(NVAR,NTLM), Fcn_tlm(NVAR,NTLM)
528 KPP_REAL :: K_tlm(NVAR*ros_S,NTLM)
529 KPP_REAL :: Hes0(NHESS)
530 KPP_REAL :: dFdT(NVAR), dJdT(LU_NONZERO)
531 KPP_REAL :: Jac0(LU_NONZERO), Jac(LU_NONZERO), Ghimj(LU_NONZERO)
532 KPP_REAL :: H, Hnew, HC, HG, Fac, Tau
533 KPP_REAL :: Err, Yerr(NVAR)
534 INTEGER :: Pivot(NVAR), Direction, ioffset, i, j, istage, itlm
535 LOGICAL :: RejectLastH, RejectMoreH, Singular
536 !~~~> Local parameters
537 KPP_REAL, PARAMETER :: ZERO = 0.0d0, ONE = 1.0d0
538 KPP_REAL, PARAMETER :: DeltaMin = 1.0d-5
539 !~~~> Locally called functions
540 ! KPP_REAL WLAMCH
541 ! EXTERNAL WLAMCH
542 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
543
544
545 !~~~> Initial preparations
546 T = Tstart
547 Hexit = 0.0_dp
548 H = MIN(Hstart,Hmax)
549 IF (ABS(H) <= 10.D0*Roundoff) H = DeltaMin
550
551 IF (Tend >= Tstart) THEN
552 Direction = +1
553 ELSE
554 Direction = -1
555 END IF
556
557 RejectLastH=.FALSE.
558 RejectMoreH=.FALSE.
559
560 !~~~> Time loop begins below
561
562 TimeLoop: DO WHILE ( (Direction > 0).AND.((T-Tend)+Roundoff <= ZERO) &
563 .OR. (Direction < 0).AND.((Tend-T)+Roundoff <= ZERO) )
564
565 IF ( Nstp > Max_no_steps ) THEN ! Too many steps
566 CALL ros_ErrorMsg(-6,T,H,IERR)
567 RETURN
568 END IF
569 IF ( ((T+0.1d0*H) == T).OR.(H <= Roundoff) ) THEN ! Step size too small
570 CALL ros_ErrorMsg(-7,T,H,IERR)
571 RETURN
572 END IF
573
574 !~~~> Limit H if necessary to avoid going beyond Tend
575 Hexit = H
576 H = MIN(H,ABS(Tend-T))
577
578 !~~~> Compute the function at current time
579 CALL FunTemplate(T,Y,Fcn0)
580
581 !~~~> Compute the Jacobian at current time
582 CALL JacTemplate(T,Y,Jac0)
583
584 !~~~> Compute the Hessian at current time
585 CALL HessTemplate(T,Y,Hes0)
586
587 !~~~> Compute the TLM function at current time
588 DO itlm = 1, NTLM
589 CALL Jac_SP_Vec ( Jac0, Y_tlm(1,itlm), Fcn0_tlm(1,itlm) )
590 END DO
591
592 !~~~> Compute the function and Jacobian derivatives with respect to T
593 IF (.NOT.Autonomous) THEN
594 CALL ros_FunTimeDerivative ( T, Roundoff, Y, Fcn0, dFdT )
595 CALL ros_JacTimeDerivative ( T, Roundoff, Y, Jac0, dJdT )
596 END IF
597
598 !~~~> Repeat step calculation until current step accepted
599 UntilAccepted: DO
600
601 CALL ros_PrepareMatrix(H,Direction,ros_Gamma(1),&
602 Jac0,Ghimj,Pivot,Singular)
603 IF (Singular) THEN ! More than 5 consecutive failed decompositions
604 CALL ros_ErrorMsg(-8,T,H,IERR)
605 RETURN
606 END IF
607
608 !~~~> Compute the stages
609 Stage: DO istage = 1, ros_S
610
611 ! Current istage offset. Current istage vector is K(ioffset+1:ioffset+NVAR)
612 ioffset = NVAR*(istage-1)
613
614 ! For the 1st istage the function has been computed previously
615 IF ( istage == 1 ) THEN
616 CALL WCOPY(NVAR,Fcn0,1,Fcn,1)
617 CALL WCOPY(NVAR*NTLM,Fcn0_tlm,1,Fcn_tlm,1)
618 ! istage>1 and a new function evaluation is needed at the current istage
619 ELSEIF ( ros_NewF(istage) ) THEN
620 CALL WCOPY(NVAR,Y,1,Ynew,1)
621 CALL WCOPY(NVAR*NTLM,Y_tlm,1,Ynew_tlm,1)
622 DO j = 1, istage-1
623 CALL WAXPY(NVAR,ros_A((istage-1)*(istage-2)/2+j), &
624 K(NVAR*(j-1)+1),1,Ynew,1)
625 DO itlm=1,NTLM
626 CALL WAXPY(NVAR,ros_A((istage-1)*(istage-2)/2+j), &
627 K_tlm(NVAR*(j-1)+1,itlm),1,Ynew_tlm(1,itlm),1)
628 END DO
629 END DO
630 Tau = T + ros_Alpha(istage)*Direction*H
631 CALL FunTemplate(Tau,Ynew,Fcn)
632 CALL JacTemplate(Tau,Ynew,Jac)
633 DO itlm=1,NTLM
634 CALL Jac_SP_Vec ( Jac, Ynew_tlm(1,itlm), Fcn_tlm(1,itlm) )
635 END DO
636 END IF ! if istage == 1 elseif ros_NewF(istage)
637 CALL WCOPY(NVAR,Fcn,1,K(ioffset+1),1)
638 DO itlm=1,NTLM
639 CALL WCOPY(NVAR,Fcn_tlm(1,itlm),1,K_tlm(ioffset+1,itlm),1)
640 END DO
641 DO j = 1, istage-1
642 HC = ros_C((istage-1)*(istage-2)/2+j)/(Direction*H)
643 CALL WAXPY(NVAR,HC,K(NVAR*(j-1)+1),1,K(ioffset+1),1)
644 DO itlm=1,NTLM
645 CALL WAXPY(NVAR,HC,K_tlm(NVAR*(j-1)+1,itlm),1,K_tlm(ioffset+1,itlm),1)
646 END DO
647 END DO
648 IF ((.NOT. Autonomous).AND.(ros_Gamma(istage).NE.ZERO)) THEN
649 HG = Direction*H*ros_Gamma(istage)
650 CALL WAXPY(NVAR,HG,dFdT,1,K(ioffset+1),1)
651 DO itlm=1,NTLM
652 CALL Jac_SP_Vec ( dJdT, Ynew_tlm(1,itlm), Fcn_tlm(1,itlm) )
653 CALL WAXPY(NVAR,HG,Fcn_tlm(1,itlm),1,K_tlm(ioffset+1,itlm),1)
654 END DO
655 END IF
656 CALL ros_Solve(Ghimj, Pivot, K(ioffset+1))
657 DO itlm=1,NTLM
658 CALL Hess_Vec ( Hes0, K(ioffset+1), Y_tlm(1,itlm), Fcn_tlm(1,itlm) )
659 CALL WAXPY(NVAR,ONE,Fcn_tlm(1,itlm),1,K_tlm(ioffset+1,itlm),1)
660 CALL ros_Solve(Ghimj, Pivot, K_tlm(ioffset+1,itlm))
661 END DO
662
663 END DO Stage
664
665
666 !~~~> Compute the new solution
667 CALL WCOPY(NVAR,Y,1,Ynew,1)
668 DO j=1,ros_S
669 CALL WAXPY(NVAR,ros_M(j),K(NVAR*(j-1)+1),1,Ynew,1)
670 END DO
671 DO itlm=1,NTLM
672 CALL WCOPY(NVAR,Y_tlm(1,itlm),1,Ynew_tlm(1,itlm),1)
673 DO j=1,ros_S
674 CALL WAXPY(NVAR,ros_M(j),K_tlm(NVAR*(j-1)+1,itlm),1,Ynew_tlm(1,itlm),1)
675 END DO
676 END DO
677
678 !~~~> Compute the error estimation
679 CALL WSCAL(NVAR,ZERO,Yerr,1)
680 DO j=1,ros_S
681 CALL WAXPY(NVAR,ros_E(j),K(NVAR*(j-1)+1),1,Yerr,1)
682 END DO
683 Err = ros_ErrorNorm ( Y, Ynew, Yerr, AbsTol, RelTol, VectorTol )
684
685 !~~~> New step size is bounded by FacMin <= Hnew/H <= FacMax
686 Fac = MIN(FacMax,MAX(FacMin,FacSafe/Err**(ONE/ros_ELO)))
687 Hnew = H*Fac
688
689 !~~~> Check the error magnitude and adjust step size
690 Nstp = Nstp+1
691 IF ( (Err <= ONE).OR.(H <= Hmin) ) THEN !~~~> Accept step
692 Nacc = Nacc+1
693 CALL WCOPY(NVAR,Ynew,1,Y,1)
694 CALL WCOPY(NVAR*NTLM,Ynew_tlm,1,Y_tlm,1)
695 T = T + Direction*H
696 Hnew = MAX(Hmin,MIN(Hnew,Hmax))
697 IF (RejectLastH) THEN ! No step size increase after a rejected step
698 Hnew = MIN(Hnew,H)
699 END IF
700 RejectLastH = .FALSE.
701 RejectMoreH = .FALSE.
702 H = Hnew
703 EXIT UntilAccepted ! EXIT THE LOOP: WHILE STEP NOT ACCEPTED
704 ELSE !~~~> Reject step
705 IF (RejectMoreH) THEN
706 Hnew = H*FacRej
707 END IF
708 RejectMoreH = RejectLastH
709 RejectLastH = .TRUE.
710 H = Hnew
711 IF (Nacc >= 1) THEN
712 Nrej = Nrej+1
713 END IF
714 END IF ! Err <= 1
715
716 END DO UntilAccepted
717
718 END DO TimeLoop
719
720 !~~~> Succesful exit
721 IERR = 1 !~~~> The integration was successful
722
723 END SUBROUTINE ros_TLM_Int
724
725
726 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
727 KPP_REAL FUNCTION ros_ErrorNorm ( Y, Ynew, Yerr, &
728 AbsTol, RelTol, VectorTol )
729 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
730 !~~~> Computes the "scaled norm" of the error vector Yerr
731 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
732 IMPLICIT NONE
733
734 ! Input arguments
735 KPP_REAL, INTENT(IN) :: Y(NVAR), Ynew(NVAR), &
736 Yerr(NVAR), AbsTol(NVAR), RelTol(NVAR)
737 LOGICAL, INTENT(IN) :: VectorTol
738 ! Local variables
739 KPP_REAL :: Err, Scale, Ymax
740 INTEGER :: i
741 KPP_REAL, PARAMETER :: ZERO = 0.0d0
742
743 Err = ZERO
744 DO i=1,NVAR
745 Ymax = MAX(ABS(Y(i)),ABS(Ynew(i)))
746 IF (VectorTol) THEN
747 Scale = AbsTol(i)+RelTol(i)*Ymax
748 ELSE
749 Scale = AbsTol(1)+RelTol(1)*Ymax
750 END IF
751 Err = Err+(Yerr(i)/Scale)**2
752 END DO
753 Err = SQRT(Err/NVAR)
754
755 ros_ErrorNorm = Err
756
757 END FUNCTION ros_ErrorNorm
758
759
760 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
761 SUBROUTINE ros_FunTimeDerivative ( T, Roundoff, Y, &
762 Fcn0, dFdT )
763 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
764 !~~~> The time partial derivative of the function by finite differences
765 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
766 IMPLICIT NONE
767
768 !~~~> Input arguments
769 KPP_REAL, INTENT(IN) :: T, Roundoff, Y(NVAR), Fcn0(NVAR)
770 !~~~> Output arguments
771 KPP_REAL, INTENT(OUT) :: dFdT(NVAR)
772 !~~~> Local variables
773 KPP_REAL :: Delta
774 KPP_REAL, PARAMETER :: ONE = 1.0d0, DeltaMin = 1.0d-6
775
776 Delta = SQRT(Roundoff)*MAX(DeltaMin,ABS(T))
777 CALL FunTemplate(T+Delta,Y,dFdT)
778 CALL WAXPY(NVAR,(-ONE),Fcn0,1,dFdT,1)
779 CALL WSCAL(NVAR,(ONE/Delta),dFdT,1)
780
781 END SUBROUTINE ros_FunTimeDerivative
782
783
784 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
785 SUBROUTINE ros_JacTimeDerivative ( T, Roundoff, Y, &
786 Jac0, dJdT )
787 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
788 !~~~> The time partial derivative of the Jacobian by finite differences
789 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
790 IMPLICIT NONE
791
792 !~~~> Input arguments
793 KPP_REAL, INTENT(IN) :: T, Roundoff, Y(NVAR), Jac0(LU_NONZERO)
794 !~~~> Output arguments
795 KPP_REAL, INTENT(OUT) :: dJdT(LU_NONZERO)
796 !~~~> Local variables
797 KPP_REAL Delta
798 KPP_REAL, PARAMETER :: ONE = 1.0d0, DeltaMin = 1.0d-6
799
800 Delta = SQRT(Roundoff)*MAX(DeltaMin,ABS(T))
801 CALL JacTemplate(T+Delta,Y,dJdT)
802 CALL WAXPY(LU_NONZERO,(-ONE),Jac0,1,dJdT,1)
803 CALL WSCAL(LU_NONZERO,(ONE/Delta),dJdT,1)
804
805 END SUBROUTINE ros_JacTimeDerivative
806
807
808 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
809 SUBROUTINE ros_PrepareMatrix ( H, Direction, gam, &
810 Jac0, Ghimj, Pivot, Singular )
811 ! --- --- --- --- --- --- --- --- --- --- --- --- ---
812 ! Prepares the LHS matrix for stage calculations
813 ! 1. Construct Ghimj = 1/(H*ham) - Jac0
814 ! "(Gamma H) Inverse Minus Jacobian"
815 ! 2. Repeat LU decomposition of Ghimj until successful.
816 ! -half the step size if LU decomposition fails and retry
817 ! -exit after 5 consecutive fails
818 ! --- --- --- --- --- --- --- --- --- --- --- --- ---
819 IMPLICIT NONE
820
821 !~~~> Input arguments
822 KPP_REAL, INTENT(IN) :: gam, Jac0(LU_NONZERO)
823 INTEGER, INTENT(IN) :: Direction
824 !~~~> Output arguments
825 KPP_REAL, INTENT(OUT) :: Ghimj(LU_NONZERO)
826 LOGICAL, INTENT(OUT) :: Singular
827 INTEGER, INTENT(OUT) :: Pivot(NVAR)
828 !~~~> Inout arguments
829 KPP_REAL, INTENT(INOUT) :: H ! step size is decreased when LU fails
830 !~~~> Local variables
831 INTEGER :: i, ising, Nconsecutive
832 KPP_REAL :: ghinv
833 KPP_REAL, PARAMETER :: ONE = 1.0d0, HALF = 0.5d0
834
835 Nconsecutive = 0
836 Singular = .TRUE.
837
838 DO WHILE (Singular)
839
840 !~~~> Construct Ghimj = 1/(H*ham) - Jac0
841 CALL WCOPY(LU_NONZERO,Jac0,1,Ghimj,1)
842 CALL WSCAL(LU_NONZERO,(-ONE),Ghimj,1)
843 ghinv = ONE/(Direction*H*gam)
844 DO i=1,NVAR
845 Ghimj(LU_DIAG(i)) = Ghimj(LU_DIAG(i))+ghinv
846 END DO
847 !~~~> Compute LU decomposition
848 CALL ros_Decomp( Ghimj, Pivot, ising )
849 IF (ising == 0) THEN
850 !~~~> If successful done
851 Singular = .FALSE.
852 ELSE ! ising .ne. 0
853 !~~~> If unsuccessful half the step size; if 5 consecutive fails then return
854 Nsng = Nsng+1
855 Nconsecutive = Nconsecutive+1
856 Singular = .TRUE.
857 PRINT*,'Warning: LU Decomposition returned ising = ',ising
858 IF (Nconsecutive <= 5) THEN ! Less than 5 consecutive failed decompositions
859 H = H*HALF
860 ELSE ! More than 5 consecutive failed decompositions
861 RETURN
862 END IF ! Nconsecutive
863 END IF ! ising
864
865 END DO ! WHILE Singular
866
867 END SUBROUTINE ros_PrepareMatrix
868
869
870 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
871 SUBROUTINE ros_Decomp( A, Pivot, ising )
872 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
873 ! Template for the LU decomposition
874 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
875 IMPLICIT NONE
876 !~~~> Inout variables
877 KPP_REAL, INTENT(INOUT) :: A(LU_NONZERO)
878 !~~~> Output variables
879 INTEGER, INTENT(OUT) :: Pivot(NVAR), ising
880
881 CALL KppDecomp ( A, ising )
882 !~~~> Note: for a full matrix use Lapack:
883 ! CALL DGETRF( NVAR, NVAR, A, NVAR, Pivot, ising )
884 Pivot(1) = 1
885
886 Ndec = Ndec + 1
887
888 END SUBROUTINE ros_Decomp
889
890
891 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
892 SUBROUTINE ros_Solve( A, Pivot, b )
893 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
894 ! Template for the forward/backward substitution (using pre-computed LU decomposition)
895 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
896 IMPLICIT NONE
897 !~~~> Input variables
898 KPP_REAL, INTENT(IN) :: A(LU_NONZERO)
899 INTEGER, INTENT(IN) :: Pivot(NVAR)
900 !~~~> InOut variables
901 KPP_REAL, INTENT(INOUT) :: b(NVAR)
902
903 CALL KppSolve( A, b )
904 !~~~> Note: for a full matrix use Lapack:
905 ! NRHS = 1
906 ! CALL DGETRS( 'N', NVAR , NRHS, A, NVAR, Pivot, b, NVAR, INFO )
907
908 Nsol = Nsol+1
909
910 END SUBROUTINE ros_Solve
911
912
913
914 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
915 SUBROUTINE Ros2 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
916 ros_Gamma,ros_NewF,ros_ELO,ros_Name)
917 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
918 ! --- AN L-STABLE METHOD, 2 stages, order 2
919 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
920
921 IMPLICIT NONE
922
923 INTEGER, PARAMETER :: S = 2
924 INTEGER, INTENT(OUT) :: ros_S
925 KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
926 KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
927 KPP_REAL, INTENT(OUT) :: ros_ELO
928 LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
929 CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
930 DOUBLE PRECISION g
931
932 g = 1.0d0 + 1.0d0/SQRT(2.0d0)
933
934 !~~~> Name of the method
935 ros_Name = 'ROS-2'
936 !~~~> Number of stages
937 ros_S = S
938
939 !~~~> The coefficient matrices A and C are strictly lower triangular.
940 ! The lower triangular (subdiagonal) elements are stored in row-wise order:
941 ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
942 ! The general mapping formula is:
943 ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
944 ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
945
946 ros_A(1) = (1.d0)/g
947 ros_C(1) = (-2.d0)/g
948 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
949 ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
950 ros_NewF(1) = .TRUE.
951 ros_NewF(2) = .TRUE.
952 !~~~> M_i = Coefficients for new step solution
953 ros_M(1)= (3.d0)/(2.d0*g)
954 ros_M(2)= (1.d0)/(2.d0*g)
955 ! E_i = Coefficients for error estimator
956 ros_E(1) = 1.d0/(2.d0*g)
957 ros_E(2) = 1.d0/(2.d0*g)
958 !~~~> ros_ELO = estimator of local order - the minimum between the
959 ! main and the embedded scheme orders plus one
960 ros_ELO = 2.0d0
961 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
962 ros_Alpha(1) = 0.0d0
963 ros_Alpha(2) = 1.0d0
964 !~~~> Gamma_i = \sum_j gamma_{i,j}
965 ros_Gamma(1) = g
966 ros_Gamma(2) =-g
967
968 END SUBROUTINE Ros2
969
970
971 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
972 SUBROUTINE Ros3 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
973 ros_Gamma,ros_NewF,ros_ELO,ros_Name)
974 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
975 ! --- AN L-STABLE METHOD, 3 stages, order 3, 2 function evaluations
976 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
977
978 IMPLICIT NONE
979
980 INTEGER, PARAMETER :: S = 3
981 INTEGER, INTENT(OUT) :: ros_S
982 KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
983 KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
984 KPP_REAL, INTENT(OUT) :: ros_ELO
985 LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
986 CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
987
988 !~~~> Name of the method
989 ros_Name = 'ROS-3'
990 !~~~> Number of stages
991 ros_S = S
992
993 !~~~> The coefficient matrices A and C are strictly lower triangular.
994 ! The lower triangular (subdiagonal) elements are stored in row-wise order:
995 ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
996 ! The general mapping formula is:
997 ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
998 ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
999
1000 ros_A(1)= 1.d0
1001 ros_A(2)= 1.d0
1002 ros_A(3)= 0.d0
1003
1004 ros_C(1) = -0.10156171083877702091975600115545d+01
1005 ros_C(2) = 0.40759956452537699824805835358067d+01
1006 ros_C(3) = 0.92076794298330791242156818474003d+01
1007 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
1008 ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
1009 ros_NewF(1) = .TRUE.
1010 ros_NewF(2) = .TRUE.
1011 ros_NewF(3) = .FALSE.
1012 !~~~> M_i = Coefficients for new step solution
1013 ros_M(1) = 0.1d+01
1014 ros_M(2) = 0.61697947043828245592553615689730d+01
1015 ros_M(3) = -0.42772256543218573326238373806514d+00
1016 ! E_i = Coefficients for error estimator
1017 ros_E(1) = 0.5d+00
1018 ros_E(2) = -0.29079558716805469821718236208017d+01
1019 ros_E(3) = 0.22354069897811569627360909276199d+00
1020 !~~~> ros_ELO = estimator of local order - the minimum between the
1021 ! main and the embedded scheme orders plus 1
1022 ros_ELO = 3.0d0
1023 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
1024 ros_Alpha(1)= 0.0d+00
1025 ros_Alpha(2)= 0.43586652150845899941601945119356d+00
1026 ros_Alpha(3)= 0.43586652150845899941601945119356d+00
1027 !~~~> Gamma_i = \sum_j gamma_{i,j}
1028 ros_Gamma(1)= 0.43586652150845899941601945119356d+00
1029 ros_Gamma(2)= 0.24291996454816804366592249683314d+00
1030 ros_Gamma(3)= 0.21851380027664058511513169485832d+01
1031
1032 END SUBROUTINE Ros3
1033
1034 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1035
1036
1037 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1038 SUBROUTINE Ros4 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
1039 ros_Gamma,ros_NewF,ros_ELO,ros_Name)
1040 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1041 ! L-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 4 STAGES
1042 ! L-STABLE EMBEDDED ROSENBROCK METHOD OF ORDER 3
1043 !
1044 ! E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL
1045 ! EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS.
1046 ! SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS,
1047 ! SPRINGER-VERLAG (1990)
1048 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1049
1050 IMPLICIT NONE
1051
1052 INTEGER, PARAMETER :: S = 4
1053 INTEGER, INTENT(OUT) :: ros_S
1054 KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
1055 KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
1056 KPP_REAL, INTENT(OUT) :: ros_ELO
1057 LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
1058 CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
1059 DOUBLE PRECISION g
1060
1061 !~~~> Name of the method
1062 ros_Name = 'ROS-4'
1063 !~~~> Number of stages
1064 ros_S = S
1065
1066 !~~~> The coefficient matrices A and C are strictly lower triangular.
1067 ! The lower triangular (subdiagonal) elements are stored in row-wise order:
1068 ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
1069 ! The general mapping formula is:
1070 ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
1071 ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
1072
1073 ros_A(1) = 0.2000000000000000d+01
1074 ros_A(2) = 0.1867943637803922d+01
1075 ros_A(3) = 0.2344449711399156d+00
1076 ros_A(4) = ros_A(2)
1077 ros_A(5) = ros_A(3)
1078 ros_A(6) = 0.0D0
1079
1080 ros_C(1) =-0.7137615036412310d+01
1081 ros_C(2) = 0.2580708087951457d+01
1082 ros_C(3) = 0.6515950076447975d+00
1083 ros_C(4) =-0.2137148994382534d+01
1084 ros_C(5) =-0.3214669691237626d+00
1085 ros_C(6) =-0.6949742501781779d+00
1086 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
1087 ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
1088 ros_NewF(1) = .TRUE.
1089 ros_NewF(2) = .TRUE.
1090 ros_NewF(3) = .TRUE.
1091 ros_NewF(4) = .FALSE.
1092 !~~~> M_i = Coefficients for new step solution
1093 ros_M(1) = 0.2255570073418735d+01
1094 ros_M(2) = 0.2870493262186792d+00
1095 ros_M(3) = 0.4353179431840180d+00
1096 ros_M(4) = 0.1093502252409163d+01
1097 !~~~> E_i = Coefficients for error estimator
1098 ros_E(1) =-0.2815431932141155d+00
1099 ros_E(2) =-0.7276199124938920d-01
1100 ros_E(3) =-0.1082196201495311d+00
1101 ros_E(4) =-0.1093502252409163d+01
1102 !~~~> ros_ELO = estimator of local order - the minimum between the
1103 ! main and the embedded scheme orders plus 1
1104 ros_ELO = 4.0d0
1105 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
1106 ros_Alpha(1) = 0.D0
1107 ros_Alpha(2) = 0.1145640000000000d+01
1108 ros_Alpha(3) = 0.6552168638155900d+00
1109 ros_Alpha(4) = ros_Alpha(3)
1110 !~~~> Gamma_i = \sum_j gamma_{i,j}
1111 ros_Gamma(1) = 0.5728200000000000d+00
1112 ros_Gamma(2) =-0.1769193891319233d+01
1113 ros_Gamma(3) = 0.7592633437920482d+00
1114 ros_Gamma(4) =-0.1049021087100450d+00
1115
1116 END SUBROUTINE Ros4
1117
1118 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1119 SUBROUTINE Rodas3 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
1120 ros_Gamma,ros_NewF,ros_ELO,ros_Name)
1121 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1122 ! --- A STIFFLY-STABLE METHOD, 4 stages, order 3
1123 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1124
1125 IMPLICIT NONE
1126
1127 INTEGER, PARAMETER :: S = 4
1128 INTEGER, INTENT(OUT) :: ros_S
1129 KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
1130 KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
1131 KPP_REAL, INTENT(OUT) :: ros_ELO
1132 LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
1133 CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
1134 DOUBLE PRECISION g
1135
1136 !~~~> Name of the method
1137 ros_Name = 'RODAS-3'
1138 !~~~> Number of stages
1139 ros_S = S
1140
1141 !~~~> The coefficient matrices A and C are strictly lower triangular.
1142 ! The lower triangular (subdiagonal) elements are stored in row-wise order:
1143 ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
1144 ! The general mapping formula is:
1145 ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
1146 ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
1147
1148 ros_A(1) = 0.0d+00
1149 ros_A(2) = 2.0d+00
1150 ros_A(3) = 0.0d+00
1151 ros_A(4) = 2.0d+00
1152 ros_A(5) = 0.0d+00
1153 ros_A(6) = 1.0d+00
1154
1155 ros_C(1) = 4.0d+00
1156 ros_C(2) = 1.0d+00
1157 ros_C(3) =-1.0d+00
1158 ros_C(4) = 1.0d+00
1159 ros_C(5) =-1.0d+00
1160 ros_C(6) =-(8.0d+00/3.0d+00)
1161
1162 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
1163 ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
1164 ros_NewF(1) = .TRUE.
1165 ros_NewF(2) = .FALSE.
1166 ros_NewF(3) = .TRUE.
1167 ros_NewF(4) = .TRUE.
1168 !~~~> M_i = Coefficients for new step solution
1169 ros_M(1) = 2.0d+00
1170 ros_M(2) = 0.0d+00
1171 ros_M(3) = 1.0d+00
1172 ros_M(4) = 1.0d+00
1173 !~~~> E_i = Coefficients for error estimator
1174 ros_E(1) = 0.0d+00
1175 ros_E(2) = 0.0d+00
1176 ros_E(3) = 0.0d+00
1177 ros_E(4) = 1.0d+00
1178 !~~~> ros_ELO = estimator of local order - the minimum between the
1179 ! main and the embedded scheme orders plus 1
1180 ros_ELO = 3.0d+00
1181 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
1182 ros_Alpha(1) = 0.0d+00
1183 ros_Alpha(2) = 0.0d+00
1184 ros_Alpha(3) = 1.0d+00
1185 ros_Alpha(4) = 1.0d+00
1186 !~~~> Gamma_i = \sum_j gamma_{i,j}
1187 ros_Gamma(1) = 0.5d+00
1188 ros_Gamma(2) = 1.5d+00
1189 ros_Gamma(3) = 0.0d+00
1190 ros_Gamma(4) = 0.0d+00
1191
1192 END SUBROUTINE Rodas3
1193
1194 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1195 SUBROUTINE Rodas4 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
1196 ros_Gamma,ros_NewF,ros_ELO,ros_Name)
1197 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1198 ! STIFFLY-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 6 STAGES
1199 !
1200 ! E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL
1201 ! EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS.
1202 ! SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS,
1203 ! SPRINGER-VERLAG (1996)
1204 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1205
1206 IMPLICIT NONE
1207
1208 INTEGER, PARAMETER :: S = 6
1209 INTEGER, INTENT(OUT) :: ros_S
1210 KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
1211 KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
1212 KPP_REAL, INTENT(OUT) :: ros_ELO
1213 LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
1214 CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
1215 DOUBLE PRECISION g
1216
1217 !~~~> Name of the method
1218 ros_Name = 'RODAS-4'
1219 !~~~> Number of stages
1220 ros_S = S
1221
1222 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
1223 ros_Alpha(1) = 0.000d0
1224 ros_Alpha(2) = 0.386d0
1225 ros_Alpha(3) = 0.210d0
1226 ros_Alpha(4) = 0.630d0
1227 ros_Alpha(5) = 1.000d0
1228 ros_Alpha(6) = 1.000d0
1229
1230 !~~~> Gamma_i = \sum_j gamma_{i,j}
1231 ros_Gamma(1) = 0.2500000000000000d+00
1232 ros_Gamma(2) =-0.1043000000000000d+00
1233 ros_Gamma(3) = 0.1035000000000000d+00
1234 ros_Gamma(4) =-0.3620000000000023d-01
1235 ros_Gamma(5) = 0.0d0
1236 ros_Gamma(6) = 0.0d0
1237
1238 !~~~> The coefficient matrices A and C are strictly lower triangular.
1239 ! The lower triangular (subdiagonal) elements are stored in row-wise order:
1240 ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
1241 ! The general mapping formula is: A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
1242 ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
1243
1244 ros_A(1) = 0.1544000000000000d+01
1245 ros_A(2) = 0.9466785280815826d+00
1246 ros_A(3) = 0.2557011698983284d+00
1247 ros_A(4) = 0.3314825187068521d+01
1248 ros_A(5) = 0.2896124015972201d+01
1249 ros_A(6) = 0.9986419139977817d+00
1250 ros_A(7) = 0.1221224509226641d+01
1251 ros_A(8) = 0.6019134481288629d+01
1252 ros_A(9) = 0.1253708332932087d+02
1253 ros_A(10) =-0.6878860361058950d+00
1254 ros_A(11) = ros_A(7)
1255 ros_A(12) = ros_A(8)
1256 ros_A(13) = ros_A(9)
1257 ros_A(14) = ros_A(10)
1258 ros_A(15) = 1.0d+00
1259
1260 ros_C(1) =-0.5668800000000000d+01
1261 ros_C(2) =-0.2430093356833875d+01
1262 ros_C(3) =-0.2063599157091915d+00
1263 ros_C(4) =-0.1073529058151375d+00
1264 ros_C(5) =-0.9594562251023355d+01
1265 ros_C(6) =-0.2047028614809616d+02
1266 ros_C(7) = 0.7496443313967647d+01
1267 ros_C(8) =-0.1024680431464352d+02
1268 ros_C(9) =-0.3399990352819905d+02
1269 ros_C(10) = 0.1170890893206160d+02
1270 ros_C(11) = 0.8083246795921522d+01
1271 ros_C(12) =-0.7981132988064893d+01
1272 ros_C(13) =-0.3152159432874371d+02
1273 ros_C(14) = 0.1631930543123136d+02
1274 ros_C(15) =-0.6058818238834054d+01
1275
1276 !~~~> M_i = Coefficients for new step solution
1277 ros_M(1) = ros_A(7)
1278 ros_M(2) = ros_A(8)
1279 ros_M(3) = ros_A(9)
1280 ros_M(4) = ros_A(10)
1281 ros_M(5) = 1.0d+00
1282 ros_M(6) = 1.0d+00
1283
1284 !~~~> E_i = Coefficients for error estimator
1285 ros_E(1) = 0.0d+00
1286 ros_E(2) = 0.0d+00
1287 ros_E(3) = 0.0d+00
1288 ros_E(4) = 0.0d+00
1289 ros_E(5) = 0.0d+00
1290 ros_E(6) = 1.0d+00
1291
1292 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
1293 ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
1294 ros_NewF(1) = .TRUE.
1295 ros_NewF(2) = .TRUE.
1296 ros_NewF(3) = .TRUE.
1297 ros_NewF(4) = .TRUE.
1298 ros_NewF(5) = .TRUE.
1299 ros_NewF(6) = .TRUE.
1300
1301 !~~~> ros_ELO = estimator of local order - the minimum between the
1302 ! main and the embedded scheme orders plus 1
1303 ros_ELO = 4.0d0
1304
1305 END SUBROUTINE Rodas4
1306
1307
1308
1309 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1310 SUBROUTINE FunTemplate( T, Y, Ydot )
1311 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1312 ! Template for the ODE function call.
1313 ! Updates the rate coefficients (and possibly the fixed species) at each call
1314 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1315
1316 IMPLICIT NONE
1317 !~~~> Input variables
1318 KPP_REAL T, Y(NVAR)
1319 !~~~> Output variables
1320 KPP_REAL Ydot(NVAR)
1321 !~~~> Local variables
1322 KPP_REAL Told
1323
1324 Told = TIME
1325 TIME = T
1326 CALL Update_SUN()
1327 CALL Update_RCONST()
1328 CALL Fun( Y, FIX, RCONST, Ydot )
1329 TIME = Told
1330
1331 Nfun = Nfun+1
1332
1333 END SUBROUTINE FunTemplate
1334
1335
1336 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1337 SUBROUTINE JacTemplate( T, Y, Jcb )
1338 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1339 ! Template for the ODE Jacobian call.
1340 ! Updates the rate coefficients (and possibly the fixed species) at each call
1341 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1342 IMPLICIT NONE
1343
1344 !~~~> Input variables
1345 KPP_REAL T, Y(NVAR)
1346 !~~~> Output variables
1347 KPP_REAL Jcb(LU_NONZERO)
1348 !~~~> Local variables
1349 KPP_REAL Told
1350
1351 Told = TIME
1352 TIME = T
1353 CALL Update_SUN()
1354 CALL Update_RCONST()
1355 CALL Jac_SP( Y, FIX, RCONST, Jcb )
1356 TIME = Told
1357
1358 Njac = Njac+1
1359
1360 END SUBROUTINE JacTemplate
1361
1362
1363
1364 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1365 SUBROUTINE HessTemplate( T, Y, Hes )
1366 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1367 ! Template for the ODE Hessian call.
1368 ! Updates the rate coefficients (and possibly the fixed species) at each call
1369 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1370 IMPLICIT NONE
1371
1372 !~~~> Input variables
1373 KPP_REAL T, Y(NVAR)
1374 !~~~> Output variables
1375 KPP_REAL Hes(NHESS)
1376 !~~~> Local variables
1377 KPP_REAL Told
1378
1379 Told = TIME
1380 TIME = T
1381 CALL Update_SUN()
1382 CALL Update_RCONST()
1383 CALL Hessian( Y, FIX, RCONST, Hes )
1384 TIME = Told
1385
1386 END SUBROUTINE HessTemplate
1387
1388 END MODULE KPP_ROOT_Integrator
1389
1390
A coupled weather-fire forecasting model built on top of Weather Research and Forecasting (WRF).