wrfv2_fire/chem/KPP/kpp/kpp-2.1/int/rosenbrock.f90

   1 MODULE KPP_ROOT_Integrator
   2
   3   IMPLICIT NONE
   4   PUBLIC
   5   SAVE
   6 !~~~>  Statistics on the work performed by the Rosenbrock method
   7   INTEGER :: Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng
   8   INTEGER, PARAMETER :: ifun=1, ijac=2, istp=3, iacc=4, &
   9     irej=5, idec=6, isol=7, isng=8, itexit=1, ihexit=2
  10
  11
  12   ! description of the error numbers IERR
  13   CHARACTER(LEN=50), PARAMETER, DIMENSION(-8:1) :: IERR_NAMES = (/ &
  14     'Matrix is repeatedly singular                     ', & ! -8
  15     'Step size too small                               ', & ! -7
  16     'No of steps exceeds maximum bound                 ', & ! -6
  17     'Improper tolerance values                         ', & ! -5
  18     'FacMin/FacMax/FacRej must be positive             ', & ! -4
  19     'Hmin/Hmax/Hstart must be positive                 ', & ! -3
  20     'Selected Rosenbrock method not implemented        ', & ! -2
  21     'Improper value for maximal no of steps            ', & ! -1
  22     '                                                  ', & !  0 (not used)
  23     'Success                                           ' /) !  1
  24
  25 CONTAINS
  26
  27 SUBROUTINE INTEGRATE( TIN, TOUT, &
  28   ICNTRL_U, RCNTRL_U, ISTATUS_U, RSTATUS_U, IERR_U )
  29
  30    USE KPP_ROOT_Parameters
  31    USE KPP_ROOT_Global
  32    IMPLICIT NONE
  33
  34    KPP_REAL, INTENT(IN) :: TIN  ! Start Time
  35    KPP_REAL, INTENT(IN) :: TOUT ! End Time
  36    ! Optional input parameters and statistics
  37    INTEGER,  INTENT(IN),  OPTIONAL :: ICNTRL_U(20)
  38    KPP_REAL, INTENT(IN),  OPTIONAL :: RCNTRL_U(20)
  39    INTEGER,  INTENT(OUT), OPTIONAL :: ISTATUS_U(20)
  40    KPP_REAL, INTENT(OUT), OPTIONAL :: RSTATUS_U(20)
  41    INTEGER,  INTENT(OUT), OPTIONAL :: IERR_U
  42
  43    INTEGER :: N_stp, N_acc, N_rej, N_sng
  44    SAVE N_stp, N_acc, N_rej, N_sng
  45    INTEGER :: i, IERR
  46    KPP_REAL :: RCNTRL(20), RSTATUS(20)
  47    INTEGER :: ICNTRL(20), ISTATUS(20)
  48
  49    ICNTRL(:)  = 0
  50    RCNTRL(:)  = 0.0_dp
  51    ISTATUS(:) = 0
  52    RSTATUS(:) = 0.0_dp
  53
  54    ! If optional parameters are given, and if they are >0,
  55    ! then they overwrite default settings.
  56    IF (PRESENT(ICNTRL_U)) THEN
  57      WHERE(ICNTRL_U(:) > 0) ICNTRL(:) = ICNTRL_U(:)
  58    END IF
  59    IF (PRESENT(RCNTRL_U)) THEN
  60      WHERE(RCNTRL_U(:) > 0) RCNTRL(:) = RCNTRL_U(:)
  61    END IF
  62
  63    CALL Rosenbrock(VAR,TIN,TOUT,   &
  64          ATOL,RTOL,                &
  65          RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR)
  66
  67    STEPMIN = RCNTRL(ihexit)
  68    ! if optional parameters are given for output they to return information
  69    IF (PRESENT(ISTATUS_U)) ISTATUS_U(:) = ISTATUS(:)
  70    IF (PRESENT(RSTATUS_U)) RSTATUS_U(:) = RSTATUS(:)
  71    IF (PRESENT(IERR_U))    IERR_U       = IERR
  72
  73 END SUBROUTINE INTEGRATE
  74
  75 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  76 SUBROUTINE Rosenbrock(Y,Tstart,Tend, &
  77            AbsTol,RelTol,            &
  78            RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR)
  79 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  80 !
  81 !    Solves the system y'=F(t,y) using a Rosenbrock method defined by:
  82 !
  83 !     G = 1/(H*gamma(1)) - Jac(t0,Y0)
  84 !     T_i = t0 + Alpha(i)*H
  85 !     Y_i = Y0 + \sum_{j=1}^{i-1} A(i,j)*K_j
  86 !     G * K_i = Fun( T_i, Y_i ) + \sum_{j=1}^S C(i,j)/H * K_j +
  87 !         gamma(i)*dF/dT(t0, Y0)
  88 !     Y1 = Y0 + \sum_{j=1}^S M(j)*K_j
  89 !
  90 !    For details on Rosenbrock methods and their implementation consult:
  91 !      E. Hairer and G. Wanner
  92 !      "Solving ODEs II. Stiff and differential-algebraic problems".
  93 !      Springer series in computational mathematics, Springer-Verlag, 1996.
  94 !    The codes contained in the book inspired this implementation.
  95 !
  96 !    (C)  Adrian Sandu, August 2004
  97 !    Virginia Polytechnic Institute and State University
  98 !    Contact: sandu@cs.vt.edu
  99 !    This implementation is part of KPP - the Kinetic PreProcessor
 100 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 101 !
 102 !~~~>   INPUT ARGUMENTS:
 103 !
 104 !-     Y(NVAR)    = vector of initial conditions (at T=Tstart)
 105 !-    [Tstart,Tend]  = time range of integration
 106 !     (if Tstart>Tend the integration is performed backwards in time)
 107 !-    RelTol, AbsTol = user precribed accuracy
 108 !- SUBROUTINE  Fun( T, Y, Ydot ) = ODE function,
 109 !                       returns Ydot = Y' = F(T,Y)
 110 !- SUBROUTINE  Jac( T, Y, Jcb ) = Jacobian of the ODE function,
 111 !                       returns Jcb = dFun/dY
 112 !-    ICNTRL(1:20)    = integer inputs parameters
 113 !-    RCNTRL(1:20)    = real inputs parameters
 114 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 115 !
 116 !~~~>     OUTPUT ARGUMENTS:
 117 !
 118 !-    Y(NVAR)    -> vector of final states (at T->Tend)
 119 !-    ISTATUS(1:20)   -> integer output parameters
 120 !-    RSTATUS(1:20)   -> real output parameters
 121 !-    IERR            -> job status upon return
 122 !                        success (positive value) or
 123 !                        failure (negative value)
 124 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 125 !
 126 !~~~>     INPUT PARAMETERS:
 127 !
 128 !    Note: For input parameters equal to zero the default values of the
 129 !       corresponding variables are used.
 130 !
 131 !    ICNTRL(1) = 1: F = F(y)   Independent of T (AUTONOMOUS)
 132 !              = 0: F = F(t,y) Depends on T (NON-AUTONOMOUS)
 133 !
 134 !    ICNTRL(2) = 0: AbsTol, RelTol are NVAR-dimensional vectors
 135 !              = 1: AbsTol, RelTol are scalars
 136 !
 137 !    ICNTRL(3)  -> selection of a particular Rosenbrock method
 138 !        = 0 :  default method is Rodas3
 139 !        = 1 :  method is  Ros2
 140 !        = 2 :  method is  Ros3
 141 !        = 3 :  method is  Ros4
 142 !        = 4 :  method is  Rodas3
 143 !        = 5:   method is  Rodas4
 144 !
 145 !    ICNTRL(4)  -> maximum number of integration steps
 146 !        For ICNTRL(4)=0) the default value of 100000 is used
 147 !
 148 !    RCNTRL(1)  -> Hmin, lower bound for the integration step size
 149 !          It is strongly recommended to keep Hmin = ZERO
 150 !    RCNTRL(2)  -> Hmax, upper bound for the integration step size
 151 !    RCNTRL(3)  -> Hstart, starting value for the integration step size
 152 !
 153 !    RCNTRL(4)  -> FacMin, lower bound on step decrease factor (default=0.2)
 154 !    RCNTRL(5)  -> FacMax, upper bound on step increase factor (default=6)
 155 !    RCNTRL(6)  -> FacRej, step decrease factor after multiple rejections
 156 !            (default=0.1)
 157 !    RCNTRL(7)  -> FacSafe, by which the new step is slightly smaller
 158 !         than the predicted value  (default=0.9)
 159 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 160 !
 161 !~~~>     OUTPUT PARAMETERS:
 162 !
 163 !    Note: each call to Rosenbrock adds the current no. of fcn calls
 164 !      to previous value of ISTATUS(1), and similar for the other params.
 165 !      Set ISTATUS(1:20) = 0 before call to avoid this accumulation.
 166 !
 167 !    ISTATUS(1) = No. of function calls
 168 !    ISTATUS(2) = No. of jacobian calls
 169 !    ISTATUS(3) = No. of steps
 170 !    ISTATUS(4) = No. of accepted steps
 171 !    ISTATUS(5) = No. of rejected steps (except at the beginning)
 172 !    ISTATUS(6) = No. of LU decompositions
 173 !    ISTATUS(7) = No. of forward/backward substitutions
 174 !    ISTATUS(8) = No. of singular matrix decompositions
 175 !
 176 !    RSTATUS(1)  -> Texit, the time corresponding to the
 177 !                   computed Y upon return
 178 !    RSTATUS(2)  -> Hexit, last accepted step before exit
 179 !    For multiple restarts, use Hexit as Hstart in the following run
 180 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 181
 182   USE KPP_ROOT_Parameters
 183   USE KPP_ROOT_LinearAlgebra
 184   IMPLICIT NONE
 185
 186 !~~~>  Arguments
 187    KPP_REAL, INTENT(INOUT) :: Y(NVAR)
 188    KPP_REAL, INTENT(IN)   :: Tstart,Tend
 189    KPP_REAL, INTENT(IN)   :: AbsTol(NVAR),RelTol(NVAR)
 190    INTEGER, INTENT(IN)    :: ICNTRL(20)
 191    KPP_REAL, INTENT(IN)   :: RCNTRL(20)
 192    INTEGER, INTENT(INOUT) :: ISTATUS(20)
 193    KPP_REAL, INTENT(INOUT) :: RSTATUS(20)
 194    INTEGER, INTENT(OUT)   :: IERR
 195 !~~~>  The method parameters
 196    INTEGER, PARAMETER :: Smax = 6
 197    INTEGER  :: Method, ros_S
 198    KPP_REAL, DIMENSION(Smax) :: ros_M, ros_E, ros_Alpha, ros_Gamma
 199    KPP_REAL, DIMENSION(Smax*(Smax-1)/2) :: ros_A, ros_C
 200    KPP_REAL :: ros_ELO
 201    LOGICAL, DIMENSION(Smax) :: ros_NewF
 202    CHARACTER(LEN=12) :: ros_Name
 203 !~~~>  Local variables
 204    KPP_REAL :: Roundoff, FacMin, FacMax, FacRej, FacSafe
 205    KPP_REAL :: Hmin, Hmax, Hstart, Hexit
 206    KPP_REAL :: Texit
 207    INTEGER :: i, UplimTol, Max_no_steps
 208    LOGICAL :: Autonomous, VectorTol
 209 !~~~>   Parameters
 210    KPP_REAL, PARAMETER :: ZERO = 0.0_dp, ONE  = 1.0_dp
 211    KPP_REAL, PARAMETER :: DeltaMin = 1.0E-5_dp
 212
 213 !~~~>  Initialize statistics
 214    Nfun = ISTATUS(ifun)
 215    Njac = ISTATUS(ijac)
 216    Nstp = ISTATUS(istp)
 217    Nacc = ISTATUS(iacc)
 218    Nrej = ISTATUS(irej)
 219    Ndec = ISTATUS(idec)
 220    Nsol = ISTATUS(isol)
 221    Nsng = ISTATUS(isng)
 222
 223 !~~~>  Autonomous or time dependent ODE. Default is time dependent.
 224    Autonomous = .NOT.(ICNTRL(1) == 0)
 225
 226 !~~~>  For Scalar tolerances (ICNTRL(2).NE.0)  the code uses AbsTol(1) and RelTol(1)
 227 !   For Vector tolerances (ICNTRL(2) == 0) the code uses AbsTol(1:NVAR) and RelTol(1:NVAR)
 228    IF (ICNTRL(2) == 0) THEN
 229       VectorTol = .TRUE.
 230          UplimTol  = NVAR
 231    ELSE
 232       VectorTol = .FALSE.
 233          UplimTol  = 1
 234    END IF
 235
 236 !~~~>  The particular Rosenbrock method chosen
 237    IF (ICNTRL(3) == 0) THEN
 238       Method = 4
 239    ELSEIF ( (ICNTRL(3) >= 1).AND.(ICNTRL(3) <= 5) ) THEN
 240       Method = ICNTRL(3)
 241    ELSE
 242       PRINT * , 'User-selected Rosenbrock method: ICNTRL(3)=', Method
 243       CALL ros_ErrorMsg(-2,Tstart,ZERO,IERR)
 244       RETURN
 245    END IF
 246
 247 !~~~>   The maximum number of steps admitted
 248    IF (ICNTRL(4) == 0) THEN
 249       Max_no_steps = 100000
 250    ELSEIF (ICNTRL(4) > 0) THEN
 251       Max_no_steps=ICNTRL(4)
 252    ELSE
 253       PRINT * ,'User-selected max no. of steps: ICNTRL(4)=',ICNTRL(4)
 254       CALL ros_ErrorMsg(-1,Tstart,ZERO,IERR)
 255       RETURN
 256    END IF
 257
 258 !~~~>  Unit roundoff (1+Roundoff>1)
 259    Roundoff = WLAMCH('E')
 260
 261 !~~~>  Lower bound on the step size: (positive value)
 262    IF (RCNTRL(1) == ZERO) THEN
 263       Hmin = ZERO
 264    ELSEIF (RCNTRL(1) > ZERO) THEN
 265       Hmin = RCNTRL(1)
 266    ELSE
 267       PRINT * , 'User-selected Hmin: RCNTRL(1)=', RCNTRL(1)
 268       CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR)
 269       RETURN
 270    END IF
 271 !~~~>  Upper bound on the step size: (positive value)
 272    IF (RCNTRL(2) == ZERO) THEN
 273       Hmax = ABS(Tend-Tstart)
 274    ELSEIF (RCNTRL(2) > ZERO) THEN
 275       Hmax = MIN(ABS(RCNTRL(2)),ABS(Tend-Tstart))
 276    ELSE
 277       PRINT * , 'User-selected Hmax: RCNTRL(2)=', RCNTRL(2)
 278       CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR)
 279       RETURN
 280    END IF
 281 !~~~>  Starting step size: (positive value)
 282    IF (RCNTRL(3) == ZERO) THEN
 283       Hstart = MAX(Hmin,DeltaMin)
 284    ELSEIF (RCNTRL(3) > ZERO) THEN
 285       Hstart = MIN(ABS(RCNTRL(3)),ABS(Tend-Tstart))
 286    ELSE
 287       PRINT * , 'User-selected Hstart: RCNTRL(3)=', RCNTRL(3)
 288       CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR)
 289       RETURN
 290    END IF
 291 !~~~>  Step size can be changed s.t.  FacMin < Hnew/Hexit < FacMax
 292    IF (RCNTRL(4) == ZERO) THEN
 293       FacMin = 0.2_dp
 294    ELSEIF (RCNTRL(4) > ZERO) THEN
 295       FacMin = RCNTRL(4)
 296    ELSE
 297       PRINT * , 'User-selected FacMin: RCNTRL(4)=', RCNTRL(4)
 298       CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR)
 299       RETURN
 300    END IF
 301    IF (RCNTRL(5) == ZERO) THEN
 302       FacMax = 6.0_dp
 303    ELSEIF (RCNTRL(5) > ZERO) THEN
 304       FacMax = RCNTRL(5)
 305    ELSE
 306       PRINT * , 'User-selected FacMax: RCNTRL(5)=', RCNTRL(5)
 307       CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR)
 308       RETURN
 309    END IF
 310 !~~~>   FacRej: Factor to decrease step after 2 succesive rejections
 311    IF (RCNTRL(6) == ZERO) THEN
 312       FacRej = 0.1_dp
 313    ELSEIF (RCNTRL(6) > ZERO) THEN
 314       FacRej = RCNTRL(6)
 315    ELSE
 316       PRINT * , 'User-selected FacRej: RCNTRL(6)=', RCNTRL(6)
 317       CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR)
 318       RETURN
 319    END IF
 320 !~~~>   FacSafe: Safety Factor in the computation of new step size
 321    IF (RCNTRL(7) == ZERO) THEN
 322       FacSafe = 0.9_dp
 323    ELSEIF (RCNTRL(7) > ZERO) THEN
 324       FacSafe = RCNTRL(7)
 325    ELSE
 326       PRINT * , 'User-selected FacSafe: RCNTRL(7)=', RCNTRL(7)
 327       CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR)
 328       RETURN
 329    END IF
 330 !~~~>  Check if tolerances are reasonable
 331     DO i=1,UplimTol
 332       IF ( (AbsTol(i) <= ZERO) .OR. (RelTol(i) <= 10.0_dp*Roundoff) &
 333          .OR. (RelTol(i) >= 1.0_dp) ) THEN
 334         PRINT * , ' AbsTol(',i,') = ',AbsTol(i)
 335         PRINT * , ' RelTol(',i,') = ',RelTol(i)
 336         CALL ros_ErrorMsg(-5,Tstart,ZERO,IERR)
 337         RETURN
 338       END IF
 339     END DO
 340
 341
 342 !~~~>   Initialize the particular Rosenbrock method
 343    SELECT CASE (Method)
 344      CASE (1)
 345        CALL Ros2(ros_S, ros_A, ros_C, ros_M, ros_E,   &
 346           ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 347      CASE (2)
 348        CALL Ros3(ros_S, ros_A, ros_C, ros_M, ros_E,   &
 349           ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 350      CASE (3)
 351        CALL Ros4(ros_S, ros_A, ros_C, ros_M, ros_E,   &
 352           ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 353      CASE (4)
 354        CALL Rodas3(ros_S, ros_A, ros_C, ros_M, ros_E, &
 355           ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 356      CASE (5)
 357        CALL Rodas4(ros_S, ros_A, ros_C, ros_M, ros_E, &
 358           ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 359      CASE DEFAULT
 360        PRINT * , 'Unknown Rosenbrock method: ICNTRL(4)=', Method
 361        CALL ros_ErrorMsg(-2,Tstart,ZERO,IERR)
 362        RETURN
 363    END SELECT
 364
 365 !~~~>  CALL Rosenbrock method
 366    CALL ros_Integrator(Y,Tstart,Tend,Texit,      &
 367         AbsTol, RelTol,                          &
 368 !  Rosenbrock method coefficients
 369         ros_S, ros_M, ros_E, ros_A, ros_C,       &
 370         ros_Alpha, ros_Gamma, ros_ELO, ros_NewF, &
 371 !  Integration parameters
 372         Autonomous, VectorTol, Max_no_steps,     &
 373         Roundoff, Hmin, Hmax, Hstart, Hexit,     &
 374         FacMin, FacMax, FacRej, FacSafe,         &
 375 !  Error indicator
 376         IERR)
 377
 378
 379 !~~~>  Collect run statistics
 380    ISTATUS(ifun) = Nfun
 381    ISTATUS(ijac) = Njac
 382    ISTATUS(istp) = Nstp
 383    ISTATUS(iacc) = Nacc
 384    ISTATUS(irej) = Nrej
 385    ISTATUS(idec) = Ndec
 386    ISTATUS(isol) = Nsol
 387    ISTATUS(isng) = Nsng
 388 !~~~> Last T and H
 389    RSTATUS(itexit) = Texit
 390    RSTATUS(ihexit) = Hexit
 391
 392 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 393 CONTAINS !  SUBROUTINES internal to Rosenbrock
 394 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 395
 396 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 397  SUBROUTINE ros_ErrorMsg(Code,T,H,IERR)
 398 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 399 !    Handles all error messages
 400 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 401
 402    KPP_REAL, INTENT(IN) :: T, H
 403    INTEGER, INTENT(IN)  :: Code
 404    INTEGER, INTENT(OUT) :: IERR
 405
 406    IERR = Code
 407    PRINT * , &
 408      'Forced exit from Rosenbrock due to the following error:'
 409    IF ((Code>=-8).AND.(Code<=-1)) THEN
 410      PRINT *, IERR_NAMES(Code)
 411    ELSE
 412      PRINT *, 'Unknown Error code: ', Code
 413    ENDIF
 414
 415    PRINT *, "T=", T, "and H=", H
 416
 417  END SUBROUTINE ros_ErrorMsg
 418
 419 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 420  SUBROUTINE ros_Integrator (Y, Tstart, Tend, T,     &
 421         AbsTol, RelTol,                          &
 422 !~~~> Rosenbrock method coefficients
 423         ros_S, ros_M, ros_E, ros_A, ros_C,       &
 424         ros_Alpha, ros_Gamma, ros_ELO, ros_NewF, &
 425 !~~~> Integration parameters
 426         Autonomous, VectorTol, Max_no_steps,     &
 427         Roundoff, Hmin, Hmax, Hstart, Hexit,     &
 428         FacMin, FacMax, FacRej, FacSafe,         &
 429 !~~~> Error indicator
 430         IERR )
 431 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 432 !   Template for the implementation of a generic Rosenbrock method
 433 !      defined by ros_S (no of stages)
 434 !      and its coefficients ros_{A,C,M,E,Alpha,Gamma}
 435 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 436
 437   IMPLICIT NONE
 438
 439 !~~~> Input: the initial condition at Tstart; Output: the solution at T
 440    KPP_REAL, INTENT(INOUT) :: Y(NVAR)
 441 !~~~> Input: integration interval
 442    KPP_REAL, INTENT(IN) :: Tstart,Tend
 443 !~~~> Output: time at which the solution is returned (T=Tend if success)
 444    KPP_REAL, INTENT(OUT) ::  T
 445 !~~~> Input: tolerances
 446    KPP_REAL, INTENT(IN) ::  AbsTol(NVAR), RelTol(NVAR)
 447 !~~~> Input: The Rosenbrock method parameters
 448    INTEGER, INTENT(IN) ::  ros_S
 449    KPP_REAL, INTENT(IN) :: ros_M(ros_S), ros_E(ros_S),  &
 450        ros_Alpha(ros_S), ros_A(ros_S*(ros_S-1)/2), &
 451        ros_Gamma(ros_S), ros_C(ros_S*(ros_S-1)/2), ros_ELO
 452    LOGICAL, INTENT(IN) :: ros_NewF(ros_S)
 453 !~~~> Input: integration parameters
 454    LOGICAL, INTENT(IN) :: Autonomous, VectorTol
 455    KPP_REAL, INTENT(IN) :: Hstart, Hmin, Hmax
 456    INTEGER, INTENT(IN) :: Max_no_steps
 457    KPP_REAL, INTENT(IN) :: Roundoff, FacMin, FacMax, FacRej, FacSafe
 458 !~~~> Output: last accepted step
 459    KPP_REAL, INTENT(OUT) :: Hexit
 460 !~~~> Output: Error indicator
 461    INTEGER, INTENT(OUT) :: IERR
 462 ! ~~~~ Local variables
 463    KPP_REAL :: Ynew(NVAR), Fcn0(NVAR), Fcn(NVAR)
 464    KPP_REAL :: K(NVAR*ros_S), dFdT(NVAR)
 465 #ifdef FULL_ALGEBRA
 466    KPP_REAL :: Jac0(NVAR,NVAR), Ghimj(NVAR,NVAR)
 467 #else
 468    KPP_REAL :: Jac0(LU_NONZERO), Ghimj(LU_NONZERO)
 469 #endif
 470    KPP_REAL :: H, Hnew, HC, HG, Fac, Tau
 471    KPP_REAL :: Err, Yerr(NVAR)
 472    INTEGER :: Pivot(NVAR), Direction, ioffset, j, istage
 473    LOGICAL :: RejectLastH, RejectMoreH, Singular
 474 !~~~>  Local parameters
 475    KPP_REAL, PARAMETER :: ZERO = 0.0_dp, ONE  = 1.0_dp
 476    KPP_REAL, PARAMETER :: DeltaMin = 1.0E-5_dp
 477 !~~~>  Locally called functions
 478 !    KPP_REAL WLAMCH
 479 !    EXTERNAL WLAMCH
 480 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 481
 482
 483 !~~~>  Initial preparations
 484    T = Tstart
 485    Hexit = 0.0_dp
 486    H = MIN(Hstart,Hmax)
 487    IF (ABS(H) <= 10.0_dp*Roundoff) H = DeltaMin
 488
 489    IF (Tend  >=  Tstart) THEN
 490      Direction = +1
 491    ELSE
 492      Direction = -1
 493    END IF
 494
 495    RejectLastH=.FALSE.
 496    RejectMoreH=.FALSE.
 497
 498 !~~~> Time loop begins below
 499
 500 TimeLoop: DO WHILE ( (Direction > 0).AND.((T-Tend)+Roundoff <= ZERO) &
 501        .OR. (Direction < 0).AND.((Tend-T)+Roundoff <= ZERO) )
 502
 503    IF ( Nstp > Max_no_steps ) THEN  ! Too many steps
 504       CALL ros_ErrorMsg(-6,T,H,IERR)
 505       RETURN
 506    END IF
 507    IF ( ((T+0.1_dp*H) == T).OR.(H <= Roundoff) ) THEN  ! Step size too small
 508       CALL ros_ErrorMsg(-7,T,H,IERR)
 509       RETURN
 510    END IF
 511
 512 !~~~>  Limit H if necessary to avoid going beyond Tend
 513    Hexit = H
 514    H = MIN(H,ABS(Tend-T))
 515
 516 !~~~>   Compute the function at current time
 517    CALL FunTemplate(T,Y,Fcn0)
 518
 519 !~~~>  Compute the function derivative with respect to T
 520    IF (.NOT.Autonomous) THEN
 521       CALL ros_FunTimeDerivative ( T, Roundoff, Y, &
 522                 Fcn0, dFdT )
 523    END IF
 524
 525 !~~~>   Compute the Jacobian at current time
 526    CALL JacTemplate(T,Y,Jac0)
 527
 528 !~~~>  Repeat step calculation until current step accepted
 529 UntilAccepted: DO
 530
 531    CALL ros_PrepareMatrix(H,Direction,ros_Gamma(1), &
 532           Jac0,Ghimj,Pivot,Singular)
 533    IF (Singular) THEN ! More than 5 consecutive failed decompositions
 534        CALL ros_ErrorMsg(-8,T,H,IERR)
 535        RETURN
 536    END IF
 537
 538 !~~~>   Compute the stages
 539 Stage: DO istage = 1, ros_S
 540
 541       ! Current istage offset. Current istage vector is K(ioffset+1:ioffset+NVAR)
 542        ioffset = NVAR*(istage-1)
 543
 544       ! For the 1st istage the function has been computed previously
 545        IF ( istage == 1 ) THEN
 546          CALL WCOPY(NVAR,Fcn0,1,Fcn,1)
 547       ! istage>1 and a new function evaluation is needed at the current istage
 548        ELSEIF ( ros_NewF(istage) ) THEN
 549          CALL WCOPY(NVAR,Y,1,Ynew,1)
 550          DO j = 1, istage-1
 551            CALL WAXPY(NVAR,ros_A((istage-1)*(istage-2)/2+j), &
 552             K(NVAR*(j-1)+1),1,Ynew,1)
 553          END DO
 554          Tau = T + ros_Alpha(istage)*Direction*H
 555          CALL FunTemplate(Tau,Ynew,Fcn)
 556        END IF ! if istage == 1 elseif ros_NewF(istage)
 557        CALL WCOPY(NVAR,Fcn,1,K(ioffset+1),1)
 558        DO j = 1, istage-1
 559          HC = ros_C((istage-1)*(istage-2)/2+j)/(Direction*H)
 560          CALL WAXPY(NVAR,HC,K(NVAR*(j-1)+1),1,K(ioffset+1),1)
 561        END DO
 562        IF ((.NOT. Autonomous).AND.(ros_Gamma(istage).NE.ZERO)) THEN
 563          HG = Direction*H*ros_Gamma(istage)
 564          CALL WAXPY(NVAR,HG,dFdT,1,K(ioffset+1),1)
 565        END IF
 566        CALL ros_Solve(Ghimj, Pivot, K(ioffset+1))
 567
 568    END DO Stage
 569
 570
 571 !~~~>  Compute the new solution
 572    CALL WCOPY(NVAR,Y,1,Ynew,1)
 573    DO j=1,ros_S
 574          CALL WAXPY(NVAR,ros_M(j),K(NVAR*(j-1)+1),1,Ynew,1)
 575    END DO
 576
 577 !~~~>  Compute the error estimation
 578    CALL WSCAL(NVAR,ZERO,Yerr,1)
 579    DO j=1,ros_S
 580         CALL WAXPY(NVAR,ros_E(j),K(NVAR*(j-1)+1),1,Yerr,1)
 581    END DO
 582    Err = ros_ErrorNorm ( Y, Ynew, Yerr, AbsTol, RelTol, VectorTol )
 583
 584 !~~~> New step size is bounded by FacMin <= Hnew/H <= FacMax
 585    Fac  = MIN(FacMax,MAX(FacMin,FacSafe/Err**(ONE/ros_ELO)))
 586    Hnew = H*Fac
 587
 588 !~~~>  Check the error magnitude and adjust step size
 589    Nstp = Nstp+1
 590    IF ( (Err <= ONE).OR.(H <= Hmin) ) THEN  !~~~> Accept step
 591       Nacc = Nacc+1
 592       CALL WCOPY(NVAR,Ynew,1,Y,1)
 593       T = T + Direction*H
 594       Hnew = MAX(Hmin,MIN(Hnew,Hmax))
 595       IF (RejectLastH) THEN  ! No step size increase after a rejected step
 596          Hnew = MIN(Hnew,H)
 597       END IF
 598       RejectLastH = .FALSE.
 599       RejectMoreH = .FALSE.
 600       H = Hnew
 601       EXIT UntilAccepted ! EXIT THE LOOP: WHILE STEP NOT ACCEPTED
 602    ELSE           !~~~> Reject step
 603       IF (RejectMoreH) THEN
 604          Hnew = H*FacRej
 605       END IF
 606       RejectMoreH = RejectLastH
 607       RejectLastH = .TRUE.
 608       H = Hnew
 609       IF (Nacc >= 1) THEN
 610          Nrej = Nrej+1
 611       END IF
 612    END IF ! Err <= 1
 613
 614    END DO UntilAccepted
 615
 616    END DO TimeLoop
 617
 618 !~~~> Succesful exit
 619    IERR = 1  !~~~> The integration was successful
 620
 621   END SUBROUTINE ros_Integrator
 622
 623
 624 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 625   KPP_REAL FUNCTION ros_ErrorNorm ( Y, Ynew, Yerr, &
 626                AbsTol, RelTol, VectorTol )
 627 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 628 !~~~> Computes the "scaled norm" of the error vector Yerr
 629 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 630    IMPLICIT NONE
 631
 632 ! Input arguments
 633    KPP_REAL, INTENT(IN) :: Y(NVAR), Ynew(NVAR), &
 634           Yerr(NVAR), AbsTol(NVAR), RelTol(NVAR)
 635    LOGICAL, INTENT(IN) ::  VectorTol
 636 ! Local variables
 637    KPP_REAL :: Err, Scale, Ymax
 638    INTEGER  :: i
 639    KPP_REAL, PARAMETER :: ZERO = 0.0_dp
 640
 641    Err = ZERO
 642    DO i=1,NVAR
 643      Ymax = MAX(ABS(Y(i)),ABS(Ynew(i)))
 644      IF (VectorTol) THEN
 645        Scale = AbsTol(i)+RelTol(i)*Ymax
 646      ELSE
 647        Scale = AbsTol(1)+RelTol(1)*Ymax
 648      END IF
 649      Err = Err+(Yerr(i)/Scale)**2
 650    END DO
 651    Err  = SQRT(Err/NVAR)
 652
 653    ros_ErrorNorm = Err
 654
 655   END FUNCTION ros_ErrorNorm
 656
 657
 658 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 659   SUBROUTINE ros_FunTimeDerivative ( T, Roundoff, Y, &
 660                 Fcn0, dFdT )
 661 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 662 !~~~> The time partial derivative of the function by finite differences
 663 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 664    IMPLICIT NONE
 665
 666 !~~~> Input arguments
 667    KPP_REAL, INTENT(IN) :: T, Roundoff, Y(NVAR), Fcn0(NVAR)
 668 !~~~> Output arguments
 669    KPP_REAL, INTENT(OUT) :: dFdT(NVAR)
 670 !~~~> Local variables
 671    KPP_REAL :: Delta
 672    KPP_REAL, PARAMETER :: ONE = 1.0_dp, DeltaMin = 1.0E-6_dp
 673
 674    Delta = SQRT(Roundoff)*MAX(DeltaMin,ABS(T))
 675    CALL FunTemplate(T+Delta,Y,dFdT)
 676    CALL WAXPY(NVAR,(-ONE),Fcn0,1,dFdT,1)
 677    CALL WSCAL(NVAR,(ONE/Delta),dFdT,1)
 678
 679   END SUBROUTINE ros_FunTimeDerivative
 680
 681
 682 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 683   SUBROUTINE ros_PrepareMatrix ( H, Direction, gam, &
 684              Jac0, Ghimj, Pivot, Singular )
 685 ! --- --- --- --- --- --- --- --- --- --- --- --- ---
 686 !  Prepares the LHS matrix for stage calculations
 687 !  1.  Construct Ghimj = 1/(H*ham) - Jac0
 688 !      "(Gamma H) Inverse Minus Jacobian"
 689 !  2.  Repeat LU decomposition of Ghimj until successful.
 690 !       -half the step size if LU decomposition fails and retry
 691 !       -exit after 5 consecutive fails
 692 ! --- --- --- --- --- --- --- --- --- --- --- --- ---
 693    IMPLICIT NONE
 694
 695 !~~~> Input arguments
 696 #ifdef FULL_ALGEBRA
 697    KPP_REAL, INTENT(IN) ::  Jac0(NVAR,NVAR)
 698 #else
 699    KPP_REAL, INTENT(IN) ::  Jac0(LU_NONZERO)
 700 #endif
 701    KPP_REAL, INTENT(IN) ::  gam
 702    INTEGER, INTENT(IN) ::  Direction
 703 !~~~> Output arguments
 704 #ifdef FULL_ALGEBRA
 705    KPP_REAL, INTENT(OUT) :: Ghimj(NVAR,NVAR)
 706 #else
 707    KPP_REAL, INTENT(OUT) :: Ghimj(LU_NONZERO)
 708 #endif
 709    LOGICAL, INTENT(OUT) ::  Singular
 710    INTEGER, INTENT(OUT) ::  Pivot(NVAR)
 711 !~~~> Inout arguments
 712    KPP_REAL, INTENT(INOUT) :: H   ! step size is decreased when LU fails
 713 !~~~> Local variables
 714    INTEGER  :: i, ising, Nconsecutive
 715    KPP_REAL :: ghinv
 716    KPP_REAL, PARAMETER :: ONE  = 1.0_dp, HALF = 0.5_dp
 717
 718    Nconsecutive = 0
 719    Singular = .TRUE.
 720
 721    DO WHILE (Singular)
 722
 723 !~~~>    Construct Ghimj = 1/(H*gam) - Jac0
 724 #ifdef FULL_ALGEBRA
 725      CALL WCOPY(NVAR*NVAR,Jac0,1,Ghimj,1)
 726      CALL WSCAL(NVAR*NVAR,(-ONE),Ghimj,1)
 727      ghinv = ONE/(Direction*H*gam)
 728      DO i=1,NVAR
 729        Ghimj(i,i) = Ghimj(i,i)+ghinv
 730      END DO
 731 #else
 732      CALL WCOPY(LU_NONZERO,Jac0,1,Ghimj,1)
 733      CALL WSCAL(LU_NONZERO,(-ONE),Ghimj,1)
 734      ghinv = ONE/(Direction*H*gam)
 735      DO i=1,NVAR
 736        Ghimj(LU_DIAG(i)) = Ghimj(LU_DIAG(i))+ghinv
 737      END DO
 738 #endif
 739 !~~~>    Compute LU decomposition
 740      CALL ros_Decomp( Ghimj, Pivot, ising )
 741      IF (ising == 0) THEN
 742 !~~~>    If successful done
 743         Singular = .FALSE.
 744      ELSE ! ising .ne. 0
 745 !~~~>    If unsuccessful half the step size; if 5 consecutive fails then return
 746         Nsng = Nsng+1
 747         Nconsecutive = Nconsecutive+1
 748         Singular = .TRUE.
 749         PRINT*,'Warning: LU Decomposition returned ising = ',ising
 750         IF (Nconsecutive <= 5) THEN ! Less than 5 consecutive failed decompositions
 751            H = H*HALF
 752         ELSE  ! More than 5 consecutive failed decompositions
 753            RETURN
 754         END IF  ! Nconsecutive
 755       END IF    ! ising
 756
 757    END DO ! WHILE Singular
 758
 759   END SUBROUTINE ros_PrepareMatrix
 760
 761
 762 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 763   SUBROUTINE ros_Decomp( A, Pivot, ising )
 764 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 765 !  Template for the LU decomposition
 766 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 767    IMPLICIT NONE
 768 !~~~> Inout variables
 769    KPP_REAL, INTENT(INOUT) :: A(LU_NONZERO)
 770 !~~~> Output variables
 771    INTEGER, INTENT(OUT) :: Pivot(NVAR), ising
 772
 773 #ifdef FULL_ALGEBRA
 774    CALL  DGETRF( NVAR, NVAR, A, NVAR, Pivot, ising )
 775 #else
 776    CALL KppDecomp ( A, ising )
 777    Pivot(1) = 1
 778 #endif
 779    Ndec = Ndec + 1
 780
 781   END SUBROUTINE ros_Decomp
 782
 783
 784 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 785   SUBROUTINE ros_Solve( A, Pivot, b )
 786 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 787 !  Template for the forward/backward substitution (using pre-computed LU decomposition)
 788 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 789    IMPLICIT NONE
 790 !~~~> Input variables
 791    KPP_REAL, INTENT(IN) :: A(LU_NONZERO)
 792    INTEGER, INTENT(IN) :: Pivot(NVAR)
 793 !~~~> InOut variables
 794    KPP_REAL, INTENT(INOUT) :: b(NVAR)
 795
 796 #ifdef FULL_ALGEBRA
 797    CALL  DGETRS( 'N', NVAR , 1, A, NVAR, Pivot, b, NVAR, 0 )
 798 #else
 799    CALL KppSolve( A, b )
 800 #endif
 801
 802    Nsol = Nsol+1
 803
 804   END SUBROUTINE ros_Solve
 805
 806
 807
 808 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 809   SUBROUTINE Ros2 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
 810             ros_Gamma,ros_NewF,ros_ELO,ros_Name)
 811 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 812 ! --- AN L-STABLE METHOD, 2 stages, order 2
 813 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 814
 815   IMPLICIT NONE
 816
 817    INTEGER, PARAMETER :: S=2
 818    INTEGER, INTENT(OUT) ::  ros_S
 819    KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
 820    KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
 821    KPP_REAL, INTENT(OUT) :: ros_ELO
 822    LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
 823    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
 824    DOUBLE PRECISION g
 825
 826     g = 1.0_dp + 1.0_dp/SQRT(2.0_dp)
 827
 828 !~~~> Name of the method
 829     ros_Name = 'ROS-2'
 830 !~~~> Number of stages
 831     ros_S = S
 832
 833 !~~~> The coefficient matrices A and C are strictly lower triangular.
 834 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
 835 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
 836 !   The general mapping formula is:
 837 !       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
 838 !       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
 839
 840     ros_A(1) = (1.0_dp)/g
 841     ros_C(1) = (-2.0_dp)/g
 842 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
 843 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
 844     ros_NewF(1) = .TRUE.
 845     ros_NewF(2) = .TRUE.
 846 !~~~> M_i = Coefficients for new step solution
 847     ros_M(1)= (3.0_dp)/(2.0_dp*g)
 848     ros_M(2)= (1.0_dp)/(2.0_dp*g)
 849 ! E_i = Coefficients for error estimator
 850     ros_E(1) = 1.0_dp/(2.0_dp*g)
 851     ros_E(2) = 1.0_dp/(2.0_dp*g)
 852 !~~~> ros_ELO = estimator of local order - the minimum between the
 853 !    main and the embedded scheme orders plus one
 854     ros_ELO = 2.0_dp
 855 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
 856     ros_Alpha(1) = 0.0_dp
 857     ros_Alpha(2) = 1.0_dp
 858 !~~~> Gamma_i = \sum_j  gamma_{i,j}
 859     ros_Gamma(1) = g
 860     ros_Gamma(2) =-g
 861
 862  END SUBROUTINE Ros2
 863
 864
 865 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 866   SUBROUTINE Ros3 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
 867            ros_Gamma,ros_NewF,ros_ELO,ros_Name)
 868 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 869 ! --- AN L-STABLE METHOD, 3 stages, order 3, 2 function evaluations
 870 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 871
 872   IMPLICIT NONE
 873
 874    INTEGER, PARAMETER :: S=3
 875    INTEGER, INTENT(OUT) ::  ros_S
 876    KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
 877    KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
 878    KPP_REAL, INTENT(OUT) :: ros_ELO
 879    LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
 880    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
 881
 882 !~~~> Name of the method
 883    ros_Name = 'ROS-3'
 884 !~~~> Number of stages
 885    ros_S = S
 886
 887 !~~~> The coefficient matrices A and C are strictly lower triangular.
 888 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
 889 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
 890 !   The general mapping formula is:
 891 !       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
 892 !       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
 893
 894    ros_A(1)= 1.0_dp
 895    ros_A(2)= 1.0_dp
 896    ros_A(3)= 0.0_dp
 897
 898    ros_C(1) = -0.10156171083877702091975600115545E+01_dp
 899    ros_C(2) =  0.40759956452537699824805835358067E+01_dp
 900    ros_C(3) =  0.92076794298330791242156818474003E+01_dp
 901 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
 902 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
 903    ros_NewF(1) = .TRUE.
 904    ros_NewF(2) = .TRUE.
 905    ros_NewF(3) = .FALSE.
 906 !~~~> M_i = Coefficients for new step solution
 907    ros_M(1) =  0.1E+01_dp
 908    ros_M(2) =  0.61697947043828245592553615689730E+01_dp
 909    ros_M(3) = -0.42772256543218573326238373806514E+00_dp
 910 ! E_i = Coefficients for error estimator
 911    ros_E(1) =  0.5E+00_dp
 912    ros_E(2) = -0.29079558716805469821718236208017E+01_dp
 913    ros_E(3) =  0.22354069897811569627360909276199E+00_dp
 914 !~~~> ros_ELO = estimator of local order - the minimum between the
 915 !    main and the embedded scheme orders plus 1
 916    ros_ELO = 3.0_dp
 917 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
 918    ros_Alpha(1)= 0.0E+00_dp
 919    ros_Alpha(2)= 0.43586652150845899941601945119356E+00_dp
 920    ros_Alpha(3)= 0.43586652150845899941601945119356E+00_dp
 921 !~~~> Gamma_i = \sum_j  gamma_{i,j}
 922    ros_Gamma(1)= 0.43586652150845899941601945119356E+00_dp
 923    ros_Gamma(2)= 0.24291996454816804366592249683314E+00_dp
 924    ros_Gamma(3)= 0.21851380027664058511513169485832E+01_dp
 925
 926   END SUBROUTINE Ros3
 927
 928 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 929
 930
 931 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 932   SUBROUTINE Ros4 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
 933            ros_Gamma,ros_NewF,ros_ELO,ros_Name)
 934 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 935 !     L-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 4 STAGES
 936 !     L-STABLE EMBEDDED ROSENBROCK METHOD OF ORDER 3
 937 !
 938 !      E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL
 939 !      EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS.
 940 !      SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS,
 941 !      SPRINGER-VERLAG (1990)
 942 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 943
 944   IMPLICIT NONE
 945
 946    INTEGER, PARAMETER :: S=4
 947    INTEGER, INTENT(OUT) ::  ros_S
 948    KPP_REAL, DIMENSION(4), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
 949    KPP_REAL, DIMENSION(6), INTENT(OUT) :: ros_A, ros_C
 950    KPP_REAL, INTENT(OUT) :: ros_ELO
 951    LOGICAL, DIMENSION(4), INTENT(OUT) :: ros_NewF
 952    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
 953    DOUBLE PRECISION g
 954
 955 !~~~> Name of the method
 956    ros_Name = 'ROS-4'
 957 !~~~> Number of stages
 958    ros_S = S
 959
 960 !~~~> The coefficient matrices A and C are strictly lower triangular.
 961 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
 962 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
 963 !   The general mapping formula is:
 964 !       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
 965 !       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
 966
 967    ros_A(1) = 0.2000000000000000E+01_dp
 968    ros_A(2) = 0.1867943637803922E+01_dp
 969    ros_A(3) = 0.2344449711399156E+00_dp
 970    ros_A(4) = ros_A(2)
 971    ros_A(5) = ros_A(3)
 972    ros_A(6) = 0.0_dp
 973
 974    ros_C(1) =-0.7137615036412310E+01_dp
 975    ros_C(2) = 0.2580708087951457E+01_dp
 976    ros_C(3) = 0.6515950076447975E+00_dp
 977    ros_C(4) =-0.2137148994382534E+01_dp
 978    ros_C(5) =-0.3214669691237626E+00_dp
 979    ros_C(6) =-0.6949742501781779E+00_dp
 980 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
 981 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
 982    ros_NewF(1)  = .TRUE.
 983    ros_NewF(2)  = .TRUE.
 984    ros_NewF(3)  = .TRUE.
 985    ros_NewF(4)  = .FALSE.
 986 !~~~> M_i = Coefficients for new step solution
 987    ros_M(1) = 0.2255570073418735E+01_dp
 988    ros_M(2) = 0.2870493262186792E+00_dp
 989    ros_M(3) = 0.4353179431840180E+00_dp
 990    ros_M(4) = 0.1093502252409163E+01_dp
 991 !~~~> E_i  = Coefficients for error estimator
 992    ros_E(1) =-0.2815431932141155E+00_dp
 993    ros_E(2) =-0.7276199124938920E-01_dp
 994    ros_E(3) =-0.1082196201495311E+00_dp
 995    ros_E(4) =-0.1093502252409163E+01_dp
 996 !~~~> ros_ELO  = estimator of local order - the minimum between the
 997 !    main and the embedded scheme orders plus 1
 998    ros_ELO  = 4.0_dp
 999 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
1000    ros_Alpha(1) = 0.0_dp
1001    ros_Alpha(2) = 0.1145640000000000E+01_dp
1002    ros_Alpha(3) = 0.6552168638155900E+00_dp
1003    ros_Alpha(4) = ros_Alpha(3)
1004 !~~~> Gamma_i = \sum_j  gamma_{i,j}
1005    ros_Gamma(1) = 0.5728200000000000E+00_dp
1006    ros_Gamma(2) =-0.1769193891319233E+01_dp
1007    ros_Gamma(3) = 0.7592633437920482E+00_dp
1008    ros_Gamma(4) =-0.1049021087100450E+00_dp
1009
1010   END SUBROUTINE Ros4
1011
1012 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1013   SUBROUTINE Rodas3 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
1014             ros_Gamma,ros_NewF,ros_ELO,ros_Name)
1015 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1016 ! --- A STIFFLY-STABLE METHOD, 4 stages, order 3
1017 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1018
1019   IMPLICIT NONE
1020
1021    INTEGER, PARAMETER :: S=4
1022    INTEGER, INTENT(OUT) ::  ros_S
1023    KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
1024    KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
1025    KPP_REAL, INTENT(OUT) :: ros_ELO
1026    LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
1027    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
1028    DOUBLE PRECISION g
1029
1030 !~~~> Name of the method
1031    ros_Name = 'RODAS-3'
1032 !~~~> Number of stages
1033    ros_S = S
1034
1035 !~~~> The coefficient matrices A and C are strictly lower triangular.
1036 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
1037 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
1038 !   The general mapping formula is:
1039 !       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
1040 !       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
1041
1042    ros_A(1) = 0.0E+00_dp
1043    ros_A(2) = 2.0E+00_dp
1044    ros_A(3) = 0.0E+00_dp
1045    ros_A(4) = 2.0E+00_dp
1046    ros_A(5) = 0.0E+00_dp
1047    ros_A(6) = 1.0E+00_dp
1048
1049    ros_C(1) = 4.0E+00_dp
1050    ros_C(2) = 1.0E+00_dp
1051    ros_C(3) =-1.0E+00_dp
1052    ros_C(4) = 1.0E+00_dp
1053    ros_C(5) =-1.0E+00_dp
1054    ros_C(6) =-(8.0E+00_dp/3.0E+00_dp)
1055
1056 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
1057 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
1058    ros_NewF(1)  = .TRUE.
1059    ros_NewF(2)  = .FALSE.
1060    ros_NewF(3)  = .TRUE.
1061    ros_NewF(4)  = .TRUE.
1062 !~~~> M_i = Coefficients for new step solution
1063    ros_M(1) = 2.0E+00_dp
1064    ros_M(2) = 0.0E+00_dp
1065    ros_M(3) = 1.0E+00_dp
1066    ros_M(4) = 1.0E+00_dp
1067 !~~~> E_i  = Coefficients for error estimator
1068    ros_E(1) = 0.0E+00_dp
1069    ros_E(2) = 0.0E+00_dp
1070    ros_E(3) = 0.0E+00_dp
1071    ros_E(4) = 1.0E+00_dp
1072 !~~~> ros_ELO  = estimator of local order - the minimum between the
1073 !    main and the embedded scheme orders plus 1
1074    ros_ELO  = 3.0E+00_dp
1075 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
1076    ros_Alpha(1) = 0.0E+00_dp
1077    ros_Alpha(2) = 0.0E+00_dp
1078    ros_Alpha(3) = 1.0E+00_dp
1079    ros_Alpha(4) = 1.0E+00_dp
1080 !~~~> Gamma_i = \sum_j  gamma_{i,j}
1081    ros_Gamma(1) = 0.5E+00_dp
1082    ros_Gamma(2) = 1.5E+00_dp
1083    ros_Gamma(3) = 0.0E+00_dp
1084    ros_Gamma(4) = 0.0E+00_dp
1085
1086   END SUBROUTINE Rodas3
1087
1088 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1089   SUBROUTINE Rodas4 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
1090              ros_Gamma,ros_NewF,ros_ELO,ros_Name)
1091 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1092 !     STIFFLY-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 6 STAGES
1093 !
1094 !      E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL
1095 !      EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS.
1096 !      SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS,
1097 !      SPRINGER-VERLAG (1996)
1098 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1099
1100   IMPLICIT NONE
1101
1102    INTEGER, PARAMETER :: S=6
1103    INTEGER, INTENT(OUT) ::  ros_S
1104    KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
1105    KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
1106    KPP_REAL, INTENT(OUT) :: ros_ELO
1107    LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
1108    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
1109    DOUBLE PRECISION g
1110
1111 !~~~> Name of the method
1112     ros_Name = 'RODAS-4'
1113 !~~~> Number of stages
1114     ros_S = 6
1115
1116 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
1117     ros_Alpha(1) = 0.000_dp
1118     ros_Alpha(2) = 0.386_dp
1119     ros_Alpha(3) = 0.210_dp
1120     ros_Alpha(4) = 0.630_dp
1121     ros_Alpha(5) = 1.000_dp
1122     ros_Alpha(6) = 1.000_dp
1123
1124 !~~~> Gamma_i = \sum_j  gamma_{i,j}
1125     ros_Gamma(1) = 0.2500000000000000E+00_dp
1126     ros_Gamma(2) =-0.1043000000000000E+00_dp
1127     ros_Gamma(3) = 0.1035000000000000E+00_dp
1128     ros_Gamma(4) =-0.3620000000000023E-01_dp
1129     ros_Gamma(5) = 0.0_dp
1130     ros_Gamma(6) = 0.0_dp
1131
1132 !~~~> The coefficient matrices A and C are strictly lower triangular.
1133 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
1134 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
1135 !   The general mapping formula is:  A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
1136 !                  C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
1137
1138     ros_A(1) = 0.1544000000000000E+01_dp
1139     ros_A(2) = 0.9466785280815826E+00_dp
1140     ros_A(3) = 0.2557011698983284E+00_dp
1141     ros_A(4) = 0.3314825187068521E+01_dp
1142     ros_A(5) = 0.2896124015972201E+01_dp
1143     ros_A(6) = 0.9986419139977817E+00_dp
1144     ros_A(7) = 0.1221224509226641E+01_dp
1145     ros_A(8) = 0.6019134481288629E+01_dp
1146     ros_A(9) = 0.1253708332932087E+02_dp
1147     ros_A(10) =-0.6878860361058950E+00_dp
1148     ros_A(11) = ros_A(7)
1149     ros_A(12) = ros_A(8)
1150     ros_A(13) = ros_A(9)
1151     ros_A(14) = ros_A(10)
1152     ros_A(15) = 1.0E+00_dp
1153
1154     ros_C(1) =-0.5668800000000000E+01_dp
1155     ros_C(2) =-0.2430093356833875E+01_dp
1156     ros_C(3) =-0.2063599157091915E+00_dp
1157     ros_C(4) =-0.1073529058151375E+00_dp
1158     ros_C(5) =-0.9594562251023355E+01_dp
1159     ros_C(6) =-0.2047028614809616E+02_dp
1160     ros_C(7) = 0.7496443313967647E+01_dp
1161     ros_C(8) =-0.1024680431464352E+02_dp
1162     ros_C(9) =-0.3399990352819905E+02_dp
1163     ros_C(10) = 0.1170890893206160E+02_dp
1164     ros_C(11) = 0.8083246795921522E+01_dp
1165     ros_C(12) =-0.7981132988064893E+01_dp
1166     ros_C(13) =-0.3152159432874371E+02_dp
1167     ros_C(14) = 0.1631930543123136E+02_dp
1168     ros_C(15) =-0.6058818238834054E+01_dp
1169
1170 !~~~> M_i = Coefficients for new step solution
1171     ros_M(1) = ros_A(7)
1172     ros_M(2) = ros_A(8)
1173     ros_M(3) = ros_A(9)
1174     ros_M(4) = ros_A(10)
1175     ros_M(5) = 1.0E+00_dp
1176     ros_M(6) = 1.0E+00_dp
1177
1178 !~~~> E_i  = Coefficients for error estimator
1179     ros_E(1) = 0.0E+00_dp
1180     ros_E(2) = 0.0E+00_dp
1181     ros_E(3) = 0.0E+00_dp
1182     ros_E(4) = 0.0E+00_dp
1183     ros_E(5) = 0.0E+00_dp
1184     ros_E(6) = 1.0E+00_dp
1185
1186 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
1187 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
1188     ros_NewF(1) = .TRUE.
1189     ros_NewF(2) = .TRUE.
1190     ros_NewF(3) = .TRUE.
1191     ros_NewF(4) = .TRUE.
1192     ros_NewF(5) = .TRUE.
1193     ros_NewF(6) = .TRUE.
1194
1195 !~~~> ros_ELO  = estimator of local order - the minimum between the
1196 !        main and the embedded scheme orders plus 1
1197     ros_ELO = 4.0_dp
1198
1199   END SUBROUTINE Rodas4
1200
1201 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1202 !   End of the set of internal Rosenbrock subroutines
1203 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1204 END SUBROUTINE Rosenbrock
1205 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1206
1207
1208 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1209 SUBROUTINE FunTemplate( T, Y, Ydot )
1210 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1211 !  Template for the ODE function call.
1212 !  Updates the rate coefficients (and possibly the fixed species) at each call
1213 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1214  USE KPP_ROOT_Parameters
1215  USE KPP_ROOT_Global
1216  USE KPP_ROOT_Function
1217  USE KPP_ROOT_Rates
1218 !~~~> Input variables
1219    KPP_REAL :: T, Y(NVAR)
1220 !~~~> Output variables
1221    KPP_REAL :: Ydot(NVAR)
1222 !~~~> Local variables
1223    KPP_REAL :: Told
1224
1225    Told = TIME
1226    TIME = T
1227    CALL Update_SUN()
1228    CALL Update_RCONST()
1229    CALL Fun( Y, FIX, RCONST, Ydot )
1230    TIME = Told
1231
1232    Nfun = Nfun+1
1233
1234 END SUBROUTINE FunTemplate
1235
1236
1237 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1238 SUBROUTINE JacTemplate( T, Y, Jcb )
1239 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1240 !  Template for the ODE Jacobian call.
1241 !  Updates the rate coefficients (and possibly the fixed species) at each call
1242 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1243  USE KPP_ROOT_Parameters
1244  USE KPP_ROOT_Global
1245  USE KPP_ROOT_Jacobian
1246  USE KPP_ROOT_LinearAlgebra
1247  USE KPP_ROOT_Rates
1248 !~~~> Input variables
1249     KPP_REAL :: T, Y(NVAR)
1250 !~~~> Output variables
1251 #ifdef FULL_ALGEBRA
1252     KPP_REAL :: JV(LU_NONZERO), Jcb(NVAR,NVAR)
1253 #else
1254     KPP_REAL :: Jcb(LU_NONZERO)
1255 #endif
1256 !~~~> Local variables
1257     KPP_REAL :: Told
1258 #ifdef FULL_ALGEBRA
1259     INTEGER :: i, j
1260 #endif
1261
1262     Told = TIME
1263     TIME = T
1264     CALL Update_SUN()
1265     CALL Update_RCONST()
1266 #ifdef FULL_ALGEBRA
1267     CALL Jac_SP(Y, FIX, RCONST, JV)
1268     DO j=1,NVAR
1269       DO i=1,NVAR
1270          Jcb(i,j) = 0.0d0
1271       END DO
1272     END DO
1273     DO i=1,LU_NONZERO
1274        Jcb(LU_IROW(i),LU_ICOL(i)) = JV(i)
1275     END DO
1276 #else
1277     CALL Jac_SP( Y, FIX, RCONST, Jcb )
1278 #endif
1279     TIME = Told
1280
1281     Njac = Njac+1
1282
1283 END SUBROUTINE JacTemplate
1284
1285 END MODULE KPP_ROOT_Integrator