chem/KPP/kpp/kpp-2.1/int/rosenbrock_tlm.f90

   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