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

   1 MODULE KPP_ROOT_Integrator
   2
   3  USE KPP_ROOT_Parameters
   4  USE KPP_ROOT_Precision
   5  USE KPP_ROOT_JacobianSP
   6
   7   IMPLICIT NONE
   8
   9
  10
  11   INTEGER, PARAMETER :: ifun=1, ijac=2, istp=3, iacc=4, &
  12     irej=5, idec=6, isol=7, isng=8, itexit=1, ihexit=2
  13
  14
  15   ! description of the error numbers IERR
  16   CHARACTER(LEN=50), PARAMETER, DIMENSION(-8:1) :: IERR_NAMES = (/ &
  17     'Matrix is repeatedly singular                     ', & ! -8
  18     'Step size too small                               ', & ! -7
  19     'No of steps exceeds maximum bound                 ', & ! -6
  20     'Improper tolerance values                         ', & ! -5
  21     'FacMin/FacMax/FacRej must be positive             ', & ! -4
  22     'Hmin/Hmax/Hstart must be positive                 ', & ! -3
  23     'Selected Rosenbrock method not implemented        ', & ! -2
  24     'Improper value for maximal no of steps            ', & ! -1
  25     '                                                  ', & !  0 (not used)
  26     'Success                                           ' /) !  1
  27
  28 CONTAINS
  29
  30 SUBROUTINE  KPP_ROOT_INTEGRATE( TIN, TOUT, &
  31   FIX, VAR,  RCONST, ATOL, RTOL, IRR_WRK,  &
  32   ICNTRL_U, RCNTRL_U, ISTATUS_U, RSTATUS_U, IERR_U  )
  33
  34    USE KPP_ROOT_Parameters
  35 !!   USE KPP_ROOT_Global
  36    IMPLICIT NONE
  37    KPP_REAL, INTENT(INOUT), DIMENSION(NFIX) :: FIX
  38    KPP_REAL, INTENT(INOUT), DIMENSION(NVAR) :: VAR
  39    KPP_REAL, INTENT(INOUT) :: IRR_WRK(NREACT)
  40    KPP_REAL, INTENT(IN), DIMENSION(NSPEC) :: ATOL, RTOL
  41    KPP_REAL, INTENT(IN), DIMENSION(NREACT) :: RCONST
  42    KPP_REAL, INTENT(IN) :: TIN  ! Start Time
  43    KPP_REAL, INTENT(IN) :: TOUT ! End Time
  44    ! Optional input parameters and statistics
  45    INTEGER,  INTENT(IN),  OPTIONAL :: ICNTRL_U(20)
  46    KPP_REAL, INTENT(IN),  OPTIONAL :: RCNTRL_U(20)
  47    INTEGER,  INTENT(OUT), OPTIONAL :: ISTATUS_U(20)
  48    KPP_REAL, INTENT(OUT), OPTIONAL :: RSTATUS_U(20)
  49    INTEGER,  INTENT(OUT), OPTIONAL :: IERR_U
  50
  51    KPP_REAL  :: STEPMIN
  52
  53
  54    INTEGER :: N_stp, N_acc, N_rej, N_sng
  55    SAVE N_stp, N_acc, N_rej, N_sng
  56    INTEGER :: i, IERR
  57    KPP_REAL :: RCNTRL(20), RSTATUS(20)
  58    INTEGER :: ICNTRL(20), ISTATUS(20)
  59
  60
  61    ICNTRL(:)  = 0
  62    RCNTRL(:)  = 0.0_dp
  63    ISTATUS(:) = 0
  64    RSTATUS(:) = 0.0_dp
  65
  66    ! If optional parameters are given, and if they are >0,
  67    ! then they overwrite default settings.
  68    IF (PRESENT(ICNTRL_U)) THEN
  69      WHERE(ICNTRL_U(:) > 0) ICNTRL(:) = ICNTRL_U(:)
  70    END IF
  71    IF (PRESENT(RCNTRL_U)) THEN
  72      WHERE(RCNTRL_U(:) > 0) RCNTRL(:) = RCNTRL_U(:)
  73    END IF
  74
  75    CALL KPP_ROOT_Rosenbrock(VAR, FIX, RCONST, TIN,TOUT,   &
  76          ATOL,RTOL,               &
  77          RCNTRL,ICNTRL,RSTATUS,ISTATUS,IRR_WRK,IERR)
  78
  79    STEPMIN = RCNTRL(ihexit)
  80    ! if optional parameters are given for output they to return information
  81    IF (PRESENT(ISTATUS_U)) ISTATUS_U(:) = ISTATUS(:)
  82    IF (PRESENT(RSTATUS_U)) RSTATUS_U(:) = RSTATUS(:)
  83    IF (PRESENT(IERR_U))    IERR_U       = IERR
  84
  85 END SUBROUTINE  KPP_ROOT_INTEGRATE
  86
  87 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  88 SUBROUTINE  KPP_ROOT_Rosenbrock(Y, FIX, RCONST, Tstart,Tend, &
  89            AbsTol,RelTol,            &
  90            RCNTRL,ICNTRL,RSTATUS,ISTATUS,IRR_WRK,IERR)
  91 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  92 !
  93 !    Solves the system y'=F(t,y) using a Rosenbrock method defined by:
  94 !
  95 !     G = 1/(H*gamma(1)) - Jac(t0,Y0)
  96 !     T_i = t0 + Alpha(i)*H
  97 !     Y_i = Y0 + \sum_{j=1}^{i-1} A(i,j)*K_j
  98 !     G * K_i = Fun( T_i, Y_i ) + \sum_{j=1}^S C(i,j)/H * K_j +
  99 !         gamma(i)*dF/dT(t0, Y0)
 100 !     Y1 = Y0 + \sum_{j=1}^S M(j)*K_j
 101 !
 102 !    For details on Rosenbrock methods and their implementation consult:
 103 !      E. Hairer and G. Wanner
 104 !      "Solving ODEs II. Stiff and differential-algebraic problems".
 105 !      Springer series in computational mathematics, Springer-Verlag, 1996.
 106 !    The codes contained in the book inspired this implementation.
 107 !
 108 !    (C)  Adrian Sandu, August 2004
 109 !    Virginia Polytechnic Institute and State University
 110 !    Contact: sandu@cs.vt.edu
 111 !    This implementation is part of KPP - the Kinetic PreProcessor
 112 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 113 !
 114 !~~~>   INPUT ARGUMENTS:
 115 !
 116 !-     Y(NVAR)    = vector of initial conditions (at T=Tstart)
 117 !-    [Tstart,Tend]  = time range of integration
 118 !     (if Tstart>Tend the integration is performed backwards in time)
 119 !-    RelTol, AbsTol = user precribed accuracy
 120 !- SUBROUTINE  Fun( T, Y, Ydot ) = ODE function,
 121 !                       returns Ydot = Y' = F(T,Y)
 122 !- SUBROUTINE  Jac( T, Y, Jcb ) = Jacobian of the ODE function,
 123 !                       returns Jcb = dFun/dY
 124 !-    ICNTRL(1:20)    = integer inputs parameters
 125 !-    RCNTRL(1:20)    = real inputs parameters
 126 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 127 !
 128 !~~~>     OUTPUT ARGUMENTS:
 129 !
 130 !-    Y(NVAR)    -> vector of final states (at T->Tend)
 131 !-    ISTATUS(1:20)   -> integer output parameters
 132 !-    RSTATUS(1:20)   -> real output parameters
 133 !-    IERR            -> job status upon return
 134 !                        success (positive value) or
 135 !                        failure (negative value)
 136 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 137 !
 138 !~~~>     INPUT PARAMETERS:
 139 !
 140 !    Note: For input parameters equal to zero the default values of the
 141 !       corresponding variables are used.
 142 !
 143 !    ICNTRL(1) = 1: F = F(y)   Independent of T (AUTONOMOUS)
 144 !              = 0: F = F(t,y) Depends on T (NON-AUTONOMOUS)
 145 !
 146 !    ICNTRL(2) = 0: AbsTol, RelTol are NVAR-dimensional vectors
 147 !              = 1: AbsTol, RelTol are scalars
 148 !
 149 !    ICNTRL(3)  -> selection of a particular Rosenbrock method
 150 !        = 0 :  default method is Rodas3
 151 !        = 1 :  method is  Ros2
 152 !        = 2 :  method is  Ros3
 153 !        = 3 :  method is  Ros4
 154 !        = 4 :  method is  Rodas3
 155 !        = 5:   method is  Rodas4
 156 !
 157 !    ICNTRL(4)  -> maximum number of integration steps
 158 !        For ICNTRL(4)=0) the default value of 100000 is used
 159 !
 160 !    RCNTRL(1)  -> Hmin, lower bound for the integration step size
 161 !          It is strongly recommended to keep Hmin = ZERO
 162 !    RCNTRL(2)  -> Hmax, upper bound for the integration step size
 163 !    RCNTRL(3)  -> Hstart, starting value for the integration step size
 164 !
 165 !    RCNTRL(4)  -> FacMin, lower bound on step decrease factor (default=0.2)
 166 !    RCNTRL(5)  -> FacMax, upper bound on step increase factor (default=6)
 167 !    RCNTRL(6)  -> FacRej, step decrease factor after multiple rejections
 168 !            (default=0.1)
 169 !    RCNTRL(7)  -> FacSafe, by which the new step is slightly smaller
 170 !         than the predicted value  (default=0.9)
 171 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 172 !
 173 !~~~>     OUTPUT PARAMETERS:
 174 !
 175 !    Note: each call to Rosenbrock adds the current no. of fcn calls
 176 !      to previous value of ISTATUS(1), and similar for the other params.
 177 !      Set ISTATUS(1:20) = 0 before call to avoid this accumulation.
 178 !
 179 !    ISTATUS(1) = No. of function calls
 180 !    ISTATUS(2) = No. of jacobian calls
 181 !    ISTATUS(3) = No. of steps
 182 !    ISTATUS(4) = No. of accepted steps
 183 !    ISTATUS(5) = No. of rejected steps (except at the beginning)
 184 !    ISTATUS(6) = No. of LU decompositions
 185 !    ISTATUS(7) = No. of forward/backward substitutions
 186 !    ISTATUS(8) = No. of singular matrix decompositions
 187 !
 188 !    RSTATUS(1)  -> Texit, the time corresponding to the
 189 !                   computed Y upon return
 190 !    RSTATUS(2)  -> Hexit, last accepted step before exit
 191 !    For multiple restarts, use Hexit as Hstart in the following run
 192 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 193
 194   USE KPP_ROOT_Parameters
 195 !!  USE KPP_ROOT_LinearAlgebra
 196   IMPLICIT NONE
 197
 198 !~~~>  Arguments
 199    KPP_REAL, INTENT(INOUT) :: Y(NVAR)
 200    KPP_REAL, INTENT(INOUT) :: IRR_WRK(NREACT)
 201    KPP_REAL, INTENT(IN), DIMENSION(NFIX) :: FIX
 202    KPP_REAL, INTENT(IN), DIMENSION(NREACT) :: RCONST
 203    KPP_REAL, INTENT(IN)   :: Tstart,Tend
 204    KPP_REAL, INTENT(IN)   :: AbsTol(NVAR),RelTol(NVAR)
 205    INTEGER, INTENT(IN)    :: ICNTRL(20)
 206    KPP_REAL, INTENT(IN)   :: RCNTRL(20)
 207    INTEGER, INTENT(INOUT) :: ISTATUS(20)
 208    KPP_REAL, INTENT(INOUT) :: RSTATUS(20)
 209    INTEGER, INTENT(OUT)   :: IERR
 210 !~~~>  The method parameters
 211    INTEGER, PARAMETER :: Smax = 6
 212    INTEGER  :: Method, ros_S
 213    KPP_REAL, DIMENSION(Smax) :: ros_M, ros_E, ros_Alpha, ros_Gamma
 214    KPP_REAL, DIMENSION(Smax*(Smax-1)/2) :: ros_A, ros_C
 215    KPP_REAL :: ros_ELO
 216    LOGICAL, DIMENSION(Smax) :: ros_NewF
 217    CHARACTER(LEN=12) :: ros_Name
 218
 219 !~~~>  Statistics on the work performed by the Rosenbrock method
 220   INTEGER :: Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng
 221
 222
 223 !~~~>  Local variables
 224    KPP_REAL :: Roundoff, FacMin, FacMax, FacRej, FacSafe
 225    KPP_REAL :: Hmin, Hmax, Hstart, Hexit
 226    KPP_REAL :: Texit
 227    INTEGER :: i, UplimTol, Max_no_steps
 228    LOGICAL :: Autonomous, VectorTol
 229 !~~~>   Parameters
 230    KPP_REAL, PARAMETER :: ZERO = 0.0_dp, ONE  = 1.0_dp
 231    KPP_REAL, PARAMETER :: DeltaMin = 1.0E-5_dp
 232
 233 !~~~>  Initialize statistics
 234    Nfun = ISTATUS(ifun)
 235    Njac = ISTATUS(ijac)
 236    Nstp = ISTATUS(istp)
 237    Nacc = ISTATUS(iacc)
 238    Nrej = ISTATUS(irej)
 239    Ndec = ISTATUS(idec)
 240    Nsol = ISTATUS(isol)
 241    Nsng = ISTATUS(isng)
 242
 243 !~~~>  Autonomous or time dependent ODE. Default is time dependent.
 244    Autonomous = .NOT.(ICNTRL(1) == 0)
 245
 246
 247
 248
 249
 250 !~~~>  For Scalar tolerances (ICNTRL(2).NE.0)  the code uses AbsTol(1) and RelTol(1)
 251 !   For Vector tolerances (ICNTRL(2) == 0) the code uses AbsTol(1:NVAR) and RelTol(1:NVAR)
 252    IF (ICNTRL(2) == 0) THEN
 253       VectorTol = .TRUE.
 254          UplimTol  = NVAR
 255    ELSE
 256       VectorTol = .FALSE.
 257          UplimTol  = 1
 258    END IF
 259
 260 !~~~>  The particular Rosenbrock method chosen
 261    IF (ICNTRL(3) == 0) THEN
 262       Method = 4
 263    ELSEIF ( (ICNTRL(3) >= 1).AND.(ICNTRL(3) <= 5) ) THEN
 264       Method = ICNTRL(3)
 265    ELSE
 266       PRINT * , 'User-selected Rosenbrock method: ICNTRL(3)=', Method
 267       CALL KPP_ROOT_ros_ErrorMsg(-2,Tstart,ZERO,IERR)
 268       RETURN
 269    END IF
 270
 271 !~~~>   The maximum number of steps admitted
 272    IF (ICNTRL(4) == 0) THEN
 273       Max_no_steps = 100000
 274    ELSEIF (ICNTRL(4) > 0) THEN
 275       Max_no_steps=ICNTRL(4)
 276    ELSE
 277       PRINT * ,'User-selected max no. of steps: ICNTRL(4)=',ICNTRL(4)
 278       CALL KPP_ROOT_ros_ErrorMsg(-1,Tstart,ZERO,IERR)
 279       RETURN
 280    END IF
 281
 282 !~~~>  Unit roundoff (1+Roundoff>1)
 283    Roundoff = KPP_ROOT_WLAMCH('E')
 284
 285 !~~~>  Lower bound on the step size: (positive value)
 286    IF (RCNTRL(1) == ZERO) THEN
 287       Hmin = ZERO
 288    ELSEIF (RCNTRL(1) > ZERO) THEN
 289       Hmin = RCNTRL(1)
 290    ELSE
 291       PRINT * , 'User-selected Hmin: RCNTRL(1)=', RCNTRL(1)
 292       CALL KPP_ROOT_ros_ErrorMsg(-3,Tstart,ZERO,IERR)
 293       RETURN
 294    END IF
 295 !~~~>  Upper bound on the step size: (positive value)
 296    IF (RCNTRL(2) == ZERO) THEN
 297       Hmax = ABS(Tend-Tstart)
 298    ELSEIF (RCNTRL(2) > ZERO) THEN
 299       Hmax = MIN(ABS(RCNTRL(2)),ABS(Tend-Tstart))
 300    ELSE
 301       PRINT * , 'User-selected Hmax: RCNTRL(2)=', RCNTRL(2)
 302       CALL KPP_ROOT_ros_ErrorMsg(-3,Tstart,ZERO,IERR)
 303       RETURN
 304    END IF
 305 !~~~>  Starting step size: (positive value)
 306    IF (RCNTRL(3) == ZERO) THEN
 307       Hstart = MAX(Hmin,DeltaMin)
 308    ELSEIF (RCNTRL(3) > ZERO) THEN
 309       Hstart = MIN(ABS(RCNTRL(3)),ABS(Tend-Tstart))
 310    ELSE
 311       PRINT * , 'User-selected Hstart: RCNTRL(3)=', RCNTRL(3)
 312       CALL KPP_ROOT_ros_ErrorMsg(-3,Tstart,ZERO,IERR)
 313       RETURN
 314    END IF
 315 !~~~>  Step size can be changed s.t.  FacMin < Hnew/Hexit < FacMax
 316    IF (RCNTRL(4) == ZERO) THEN
 317       FacMin = 0.2_dp
 318    ELSEIF (RCNTRL(4) > ZERO) THEN
 319       FacMin = RCNTRL(4)
 320    ELSE
 321       PRINT * , 'User-selected FacMin: RCNTRL(4)=', RCNTRL(4)
 322       CALL KPP_ROOT_ros_ErrorMsg(-4,Tstart,ZERO,IERR)
 323       RETURN
 324    END IF
 325    IF (RCNTRL(5) == ZERO) THEN
 326       FacMax = 6.0_dp
 327    ELSEIF (RCNTRL(5) > ZERO) THEN
 328       FacMax = RCNTRL(5)
 329    ELSE
 330       PRINT * , 'User-selected FacMax: RCNTRL(5)=', RCNTRL(5)
 331       CALL KPP_ROOT_ros_ErrorMsg(-4,Tstart,ZERO,IERR)
 332       RETURN
 333    END IF
 334 !~~~>   FacRej: Factor to decrease step after 2 succesive rejections
 335    IF (RCNTRL(6) == ZERO) THEN
 336       FacRej = 0.1_dp
 337    ELSEIF (RCNTRL(6) > ZERO) THEN
 338       FacRej = RCNTRL(6)
 339    ELSE
 340       PRINT * , 'User-selected FacRej: RCNTRL(6)=', RCNTRL(6)
 341       CALL KPP_ROOT_ros_ErrorMsg(-4,Tstart,ZERO,IERR)
 342       RETURN
 343    END IF
 344 !~~~>   FacSafe: Safety Factor in the computation of new step size
 345    IF (RCNTRL(7) == ZERO) THEN
 346       FacSafe = 0.9_dp
 347    ELSEIF (RCNTRL(7) > ZERO) THEN
 348       FacSafe = RCNTRL(7)
 349    ELSE
 350       PRINT * , 'User-selected FacSafe: RCNTRL(7)=', RCNTRL(7)
 351       CALL KPP_ROOT_ros_ErrorMsg(-4,Tstart,ZERO,IERR)
 352       RETURN
 353    END IF
 354 !~~~>  Check if tolerances are reasonable
 355     DO i=1,UplimTol
 356       IF ( (AbsTol(i) <= ZERO) .OR. (RelTol(i) <= 10.0_dp*Roundoff) &
 357          .OR. (RelTol(i) >= 1.0_dp) ) THEN
 358         PRINT * , ' AbsTol(',i,') = ',AbsTol(i)
 359         PRINT * , ' RelTol(',i,') = ',RelTol(i)
 360         CALL KPP_ROOT_ros_ErrorMsg(-5,Tstart,ZERO,IERR)
 361         RETURN
 362       END IF
 363     END DO
 364
 365
 366 !~~~>   Initialize the particular Rosenbrock method
 367    SELECT CASE (Method)
 368      CASE (1)
 369        CALL KPP_ROOT_Ros2(ros_S, ros_A, ros_C, ros_M, ros_E,   &
 370           ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 371      CASE (2)
 372        CALL KPP_ROOT_Ros3(ros_S, ros_A, ros_C, ros_M, ros_E,   &
 373           ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 374      CASE (3)
 375        CALL KPP_ROOT_Ros4(ros_S, ros_A, ros_C, ros_M, ros_E,   &
 376           ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 377      CASE (4)
 378        CALL KPP_ROOT_Rodas3(ros_S, ros_A, ros_C, ros_M, ros_E, &
 379           ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 380      CASE (5)
 381        CALL KPP_ROOT_Rodas4(ros_S, ros_A, ros_C, ros_M, ros_E, &
 382           ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name)
 383      CASE DEFAULT
 384        PRINT * , 'Unknown Rosenbrock method: ICNTRL(4)=', Method
 385        CALL KPP_ROOT_ros_ErrorMsg(-2,Tstart,ZERO,IERR)
 386        RETURN
 387    END SELECT
 388
 389 !~~~>  CALL Rosenbrock method
 390    CALL KPP_ROOT_ros_Integrator(Y,Tstart,Tend,Texit,      &
 391         AbsTol, RelTol,                          &
 392 !  Rosenbrock method coefficients
 393         ros_S, ros_M, ros_E, ros_A, ros_C,       &
 394         ros_Alpha, ros_Gamma, ros_ELO, ros_NewF, &
 395 !  Integration parameters
 396         Autonomous, VectorTol, Max_no_steps,     &
 397         Roundoff, Hmin, Hmax, Hstart, Hexit,     &
 398         FacMin, FacMax, FacRej, FacSafe,         &
 399 !  Error indicator
 400         IRR_WRK,IERR,                            &
 401 !  Statistics on the work performed by the Rosenbrock method
 402          Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng,&
 403 !~~~>
 404          RCONST, FIX &
 405 )
 406
 407
 408 !~~~>  Collect run statistics
 409    ISTATUS(ifun) = Nfun
 410    ISTATUS(ijac) = Njac
 411    ISTATUS(istp) = Nstp
 412    ISTATUS(iacc) = Nacc
 413    ISTATUS(irej) = Nrej
 414    ISTATUS(idec) = Ndec
 415    ISTATUS(isol) = Nsol
 416    ISTATUS(isng) = Nsng
 417 !~~~> Last T and H
 418    RSTATUS(itexit) = Texit
 419    RSTATUS(ihexit) = Hexit
 420
 421 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 422 CONTAINS !  SUBROUTINES internal to Rosenbrock
 423 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 424
 425 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 426  SUBROUTINE  KPP_ROOT_ros_ErrorMsg(Code,T,H,IERR)
 427 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 428 !    Handles all error messages
 429 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 430    USE KPP_ROOT_Precision
 431
 432    KPP_REAL, INTENT(IN) :: T, H
 433    INTEGER, INTENT(IN)  :: Code
 434    INTEGER, INTENT(OUT) :: IERR
 435
 436    IERR = Code
 437    PRINT * , &
 438      'Forced exit from Rosenbrock due to the following error:'
 439    IF ((Code>=-8).AND.(Code<=-1)) THEN
 440      PRINT *, IERR_NAMES(Code)
 441    ELSE
 442      PRINT *, 'Unknown Error code: ', Code
 443    ENDIF
 444
 445    PRINT *, "T=", T, "and H=", H
 446
 447  END SUBROUTINE  KPP_ROOT_ros_ErrorMsg
 448
 449 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 450  SUBROUTINE  KPP_ROOT_ros_Integrator (Y, Tstart, Tend, T,     &
 451         AbsTol, RelTol,                          &
 452 !~~~> Rosenbrock method coefficients
 453         ros_S, ros_M, ros_E, ros_A, ros_C,       &
 454         ros_Alpha, ros_Gamma, ros_ELO, ros_NewF, &
 455 !~~~> Integration parameters
 456         Autonomous, VectorTol, Max_no_steps,     &
 457         Roundoff, Hmin, Hmax, Hstart, Hexit,     &
 458         FacMin, FacMax, FacRej, FacSafe,         &
 459 !~~~> Error indicator
 460         IRR_WRK,IERR,                            &
 461 !~~~>   Statistics on the work performed by the Rosenbrock method
 462         Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng, &
 463 !~~~>
 464         RCONST, FIX )
 465 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 466 !   Template for the implementation of a generic Rosenbrock method
 467 !      defined by ros_S (no of stages)
 468 !      and its coefficients ros_{A,C,M,E,Alpha,Gamma}
 469 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 470
 471   IMPLICIT NONE
 472
 473 !~~~> Input: the initial condition at Tstart; Output: the solution at T
 474    KPP_REAL, INTENT(INOUT) :: Y(NVAR)
 475 !~~~> Output: the reaction rates
 476    KPP_REAL, INTENT(INOUT) :: IRR_WRK(NREACT)
 477 !~~~> Input: integration interval
 478    KPP_REAL, INTENT(IN) :: Tstart,Tend
 479 !~~~> Output: time at which the solution is returned (T=Tend if success)
 480    KPP_REAL, INTENT(OUT) ::  T
 481 !~~~> Input: tolerances
 482    KPP_REAL, INTENT(IN) ::  AbsTol(NVAR), RelTol(NVAR)
 483 !~~~> Input: The Rosenbrock method parameters
 484    INTEGER, INTENT(IN) ::  ros_S
 485    KPP_REAL, INTENT(IN) :: ros_M(ros_S), ros_E(ros_S),  &
 486        ros_Alpha(ros_S), ros_A(ros_S*(ros_S-1)/2), &
 487        ros_Gamma(ros_S), ros_C(ros_S*(ros_S-1)/2), ros_ELO
 488    LOGICAL, INTENT(IN) :: ros_NewF(ros_S)
 489 !~~~> Input: integration parameters
 490    LOGICAL, INTENT(IN) :: Autonomous, VectorTol
 491    KPP_REAL, INTENT(IN) :: Hstart, Hmin, Hmax
 492    INTEGER, INTENT(IN) :: Max_no_steps
 493    KPP_REAL, INTENT(IN) :: Roundoff, FacMin, FacMax, FacRej, FacSafe
 494 !~~~> Output: last accepted step
 495    KPP_REAL, INTENT(OUT) :: Hexit
 496 !~~~> Output: Error indicator
 497    INTEGER, INTENT(OUT) :: IERR
 498 !~~~> Input
 499    KPP_REAL, INTENT(IN), DIMENSION(NFIX) :: FIX
 500 !~~~> Input
 501    KPP_REAL, INTENT(IN), DIMENSION(NREACT) :: RCONST
 502
 503 !~~~>  Statistics on the work performed by the Rosenbrock method
 504   INTEGER, INTENT(INOUT)  :: Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng
 505
 506 ! ~~~~ Local variables
 507    KPP_REAL :: Ynew(NVAR), Fcn0(NVAR), Fcn(NVAR)
 508    KPP_REAL :: K(NVAR*ros_S), dFdT(NVAR)
 509 #ifdef FULL_ALGEBRA
 510    KPP_REAL :: Jac0(NVAR,NVAR), Ghimj(NVAR,NVAR)
 511 #else
 512    KPP_REAL :: Jac0(LU_NONZERO), Ghimj(LU_NONZERO)
 513 #endif
 514    KPP_REAL :: H, Hnew, HC, HG, Fac, Tau
 515    KPP_REAL :: Err, Yerr(NVAR)
 516    INTEGER :: Pivot(NVAR), Direction, ioffset, j, istage
 517    LOGICAL :: RejectLastH, RejectMoreH, Singular
 518 !~~~>  Local parameters
 519    KPP_REAL, PARAMETER :: ZERO = 0.0_dp, ONE  = 1.0_dp
 520    KPP_REAL, PARAMETER :: DeltaMin = 1.0E-5_dp
 521 !~~~>  Locally called functions
 522 !    KPP_REAL WLAMCH
 523 !    EXTERNAL WLAMCH
 524 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 525
 526
 527 !~~~>  Initial preparations
 528    T = Tstart
 529    Hexit = 0.0_dp
 530    H = MIN(Hstart,Hmax)
 531    IF (ABS(H) <= 10.0_dp*Roundoff) H = DeltaMin
 532
 533    IF (Tend  >=  Tstart) THEN
 534      Direction = +1
 535    ELSE
 536      Direction = -1
 537    END IF
 538
 539    RejectLastH=.FALSE.
 540    RejectMoreH=.FALSE.
 541
 542 !~~~> Time loop begins below
 543
 544 TimeLoop: DO WHILE ( (Direction > 0).AND.((T-Tend)+Roundoff <= ZERO) &
 545        .OR. (Direction < 0).AND.((Tend-T)+Roundoff <= ZERO) )
 546
 547    IF ( Nstp > Max_no_steps ) THEN  ! Too many steps
 548       CALL KPP_ROOT_ros_ErrorMsg(-6,T,H,IERR)
 549       RETURN
 550    END IF
 551    IF ( ((T+0.1_dp*H) == T).OR.(H <= Roundoff) ) THEN  ! Step size too small
 552       CALL KPP_ROOT_ros_ErrorMsg(-7,T,H,IERR)
 553       RETURN
 554    END IF
 555
 556 !~~~>  Limit H if necessary to avoid going beyond Tend
 557    Hexit = H
 558    H = MIN(H,ABS(Tend-T))
 559
 560 !~~~>   Compute the function at current time
 561    CALL KPP_ROOT_FunTemplate(T,Y,Fcn0, RCONST, FIX, Nfun)
 562    IF( T == Tstart ) THEN
 563      CALL KPP_ROOT_IRRFun( Y, FIX, RCONST, IRR_WRK )
 564    ENDIF
 565
 566 !~~~>  Compute the function derivative with respect to T
 567    IF (.NOT.Autonomous) THEN
 568       CALL KPP_ROOT_ros_FunTimeDeriv ( T, Roundoff, Y, &
 569                 Fcn0, dFdT, RCONST, FIX, Nfun )
 570    END IF
 571
 572 !~~~>   Compute the Jacobian at current time
 573    CALL KPP_ROOT_JacTemplate(T,Y,Jac0, FIX, Njac, RCONST)
 574
 575 !~~~>  Repeat step calculation until current step accepted
 576 UntilAccepted: DO
 577
 578    CALL KPP_ROOT_ros_PrepareMatrix(H,Direction,ros_Gamma(1), &
 579           Jac0,Ghimj,Pivot,Singular, Ndec,  Nsng )
 580    IF (Singular) THEN ! More than 5 consecutive failed decompositions
 581        CALL KPP_ROOT_ros_ErrorMsg(-8,T,H,IERR)
 582        RETURN
 583    END IF
 584
 585 !~~~>   Compute the stages
 586 Stage: DO istage = 1, ros_S
 587
 588       ! Current istage offset. Current istage vector is K(ioffset+1:ioffset+NVAR)
 589        ioffset = NVAR*(istage-1)
 590
 591       ! For the 1st istage the function has been computed previously
 592        IF ( istage == 1 ) THEN
 593          CALL KPP_ROOT_WCOPY(NVAR,Fcn0,1,Fcn,1)
 594       ! istage>1 and a new function evaluation is needed at the current istage
 595        ELSEIF ( ros_NewF(istage) ) THEN
 596          CALL KPP_ROOT_WCOPY(NVAR,Y,1,Ynew,1)
 597          DO j = 1, istage-1
 598            CALL KPP_ROOT_WAXPY(NVAR,ros_A((istage-1)*(istage-2)/2+j), &
 599             K(NVAR*(j-1)+1),1,Ynew,1)
 600          END DO
 601          Tau = T + ros_Alpha(istage)*Direction*H
 602          CALL KPP_ROOT_FunTemplate(Tau,Ynew,Fcn, RCONST, FIX, Nfun)
 603        END IF ! if istage == 1 elseif ros_NewF(istage)
 604        CALL KPP_ROOT_WCOPY(NVAR,Fcn,1,K(ioffset+1),1)
 605        DO j = 1, istage-1
 606          HC = ros_C((istage-1)*(istage-2)/2+j)/(Direction*H)
 607          CALL KPP_ROOT_WAXPY(NVAR,HC,K(NVAR*(j-1)+1),1,K(ioffset+1),1)
 608        END DO
 609        IF ((.NOT. Autonomous).AND.(ros_Gamma(istage).NE.ZERO)) THEN
 610          HG = Direction*H*ros_Gamma(istage)
 611          CALL KPP_ROOT_WAXPY(NVAR,HG,dFdT,1,K(ioffset+1),1)
 612        END IF
 613        CALL KPP_ROOT_ros_Solve(Ghimj, Pivot, K(ioffset+1), Nsol)
 614
 615    END DO Stage
 616
 617
 618 !~~~>  Compute the new solution
 619    CALL KPP_ROOT_WCOPY(NVAR,Y,1,Ynew,1)
 620    DO j=1,ros_S
 621          CALL KPP_ROOT_WAXPY(NVAR,ros_M(j),K(NVAR*(j-1)+1),1,Ynew,1)
 622    END DO
 623
 624 !~~~>  Compute the error estimation
 625    CALL KPP_ROOT_WSCAL(NVAR,ZERO,Yerr,1)
 626    DO j=1,ros_S
 627         CALL KPP_ROOT_WAXPY(NVAR,ros_E(j),K(NVAR*(j-1)+1),1,Yerr,1)
 628    END DO
 629    Err = KPP_ROOT_ros_ErrorNorm ( Y, Ynew, Yerr, AbsTol, RelTol, VectorTol )
 630
 631 !~~~> New step size is bounded by FacMin <= Hnew/H <= FacMax
 632    Fac  = MIN(FacMax,MAX(FacMin,FacSafe/Err**(ONE/ros_ELO)))
 633    Hnew = H*Fac
 634
 635 !~~~>  Check the error magnitude and adjust step size
 636    Nstp = Nstp+1
 637    IF ( (Err <= ONE).OR.(H <= Hmin) ) THEN  !~~~> Accept step
 638       Nacc = Nacc+1
 639       CALL KPP_ROOT_WCOPY(NVAR,Ynew,1,Y,1)
 640       T = T + Direction*H
 641       Hnew = MAX(Hmin,MIN(Hnew,Hmax))
 642       IF (RejectLastH) THEN  ! No step size increase after a rejected step
 643          Hnew = MIN(Hnew,H)
 644       END IF
 645       RejectLastH = .FALSE.
 646       RejectMoreH = .FALSE.
 647       H = Hnew
 648       EXIT UntilAccepted ! EXIT THE LOOP: WHILE STEP NOT ACCEPTED
 649    ELSE           !~~~> Reject step
 650       IF (RejectMoreH) THEN
 651          Hnew = H*FacRej
 652       END IF
 653       RejectMoreH = RejectLastH
 654       RejectLastH = .TRUE.
 655       H = Hnew
 656       IF (Nacc >= 1) THEN
 657          Nrej = Nrej+1
 658       END IF
 659    END IF ! Err <= 1
 660
 661    END DO UntilAccepted
 662
 663    END DO TimeLoop
 664
 665 !~~~> Succesful exit
 666    IERR = 1  !~~~> The integration was successful
 667
 668   END SUBROUTINE  KPP_ROOT_ros_Integrator
 669
 670
 671 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 672   KPP_REAL FUNCTION  KPP_ROOT_ros_ErrorNorm ( Y, Ynew, Yerr, &
 673                AbsTol, RelTol, VectorTol )
 674 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 675 !~~~> Computes the "scaled norm" of the error vector Yerr
 676 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 677    IMPLICIT NONE
 678
 679 ! Input arguments
 680    KPP_REAL, INTENT(IN) :: Y(NVAR), Ynew(NVAR), &
 681           Yerr(NVAR), AbsTol(NVAR), RelTol(NVAR)
 682    LOGICAL, INTENT(IN) ::  VectorTol
 683 ! Local variables
 684    KPP_REAL :: Err, Scale, Ymax
 685    INTEGER  :: i
 686    KPP_REAL, PARAMETER :: ZERO = 0.0_dp
 687
 688    Err = ZERO
 689    DO i=1,NVAR
 690      Ymax = MAX(ABS(Y(i)),ABS(Ynew(i)))
 691      IF (VectorTol) THEN
 692        Scale = AbsTol(i)+RelTol(i)*Ymax
 693      ELSE
 694        Scale = AbsTol(1)+RelTol(1)*Ymax
 695      END IF
 696      Err = Err+(Yerr(i)/Scale)**2
 697    END DO
 698    Err  = SQRT(Err/NVAR)
 699
 700     KPP_ROOT_ros_ErrorNorm = Err
 701
 702   END FUNCTION  KPP_ROOT_ros_ErrorNorm
 703
 704
 705 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 706   SUBROUTINE KPP_ROOT_ros_FunTimeDeriv ( T, Roundoff, Y, &
 707                 Fcn0, dFdT, RCONST, FIX, Nfun )
 708 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 709 !~~~> The time partial derivative of the function by finite differences
 710 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 711    IMPLICIT NONE
 712
 713 !~~~> Input arguments
 714    KPP_REAL, INTENT(IN) :: T, Roundoff, Y(NVAR), Fcn0(NVAR)
 715    KPP_REAL, INTENT(IN) :: RCONST(NREACT), FIX(NFIX)
 716 !~~~> Output arguments
 717    KPP_REAL, INTENT(OUT) :: dFdT(NVAR)
 718 !~~~> InOut args
 719    INTEGER, INTENT(INOUT) ::Nfun
 720 !~~~> Local variables
 721    KPP_REAL :: Delta
 722    KPP_REAL, PARAMETER :: ONE = 1.0_dp, DeltaMin = 1.0E-6_dp
 723
 724    Delta = SQRT(Roundoff)*MAX(DeltaMin,ABS(T))
 725    CALL KPP_ROOT_FunTemplate(T+Delta,Y,dFdT, RCONST, FIX, Nfun)
 726    CALL KPP_ROOT_WAXPY(NVAR,(-ONE),Fcn0,1,dFdT,1)
 727    CALL KPP_ROOT_WSCAL(NVAR,(ONE/Delta),dFdT,1)
 728
 729   END SUBROUTINE  KPP_ROOT_ros_FunTimeDeriv
 730
 731
 732 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 733   SUBROUTINE  KPP_ROOT_ros_PrepareMatrix ( H, Direction, gam, &
 734              Jac0, Ghimj, Pivot, Singular, Ndec,  Nsng  )
 735 ! --- --- --- --- --- --- --- --- --- --- --- --- ---
 736 !  Prepares the LHS matrix for stage calculations
 737 !  1.  Construct Ghimj = 1/(H*ham) - Jac0
 738 !      "(Gamma H) Inverse Minus Jacobian"
 739 !  2.  Repeat LU decomposition of Ghimj until successful.
 740 !       -half the step size if LU decomposition fails and retry
 741 !       -exit after 5 consecutive fails
 742 ! --- --- --- --- --- --- --- --- --- --- --- --- ---
 743    IMPLICIT NONE
 744
 745 !~~~> Input arguments
 746 #ifdef FULL_ALGEBRA
 747    KPP_REAL, INTENT(IN) ::  Jac0(NVAR,NVAR)
 748 #else
 749    KPP_REAL, INTENT(IN) ::  Jac0(LU_NONZERO)
 750 #endif
 751    KPP_REAL, INTENT(IN) ::  gam
 752    INTEGER, INTENT(IN) ::  Direction
 753 !~~~> Output arguments
 754 #ifdef FULL_ALGEBRA
 755    KPP_REAL, INTENT(OUT) :: Ghimj(NVAR,NVAR)
 756 #else
 757    KPP_REAL, INTENT(OUT) :: Ghimj(LU_NONZERO)
 758 #endif
 759    LOGICAL, INTENT(OUT) ::  Singular
 760    INTEGER, INTENT(OUT) ::  Pivot(NVAR)
 761 !~~~> Inout arguments
 762    KPP_REAL, INTENT(INOUT) :: H   ! step size is decreased when LU fails
 763    INTEGER, INTENT(INOUT) ::  Ndec, Nsng
 764 !~~~> Local variables
 765    INTEGER  :: i, ising, Nconsecutive
 766    KPP_REAL :: ghinv
 767    KPP_REAL, PARAMETER :: ONE  = 1.0_dp, HALF = 0.5_dp
 768
 769    Nconsecutive = 0
 770    Singular = .TRUE.
 771
 772    DO WHILE (Singular)
 773
 774 !~~~>    Construct Ghimj = 1/(H*gam) - Jac0
 775 #ifdef FULL_ALGEBRA
 776      CALL KPP_ROOT_WCOPY(NVAR*NVAR,Jac0,1,Ghimj,1)
 777      CALL KPP_ROOT_WSCAL(NVAR*NVAR,(-ONE),Ghimj,1)
 778      ghinv = ONE/(Direction*H*gam)
 779      DO i=1,NVAR
 780        Ghimj(i,i) = Ghimj(i,i)+ghinv
 781      END DO
 782 #else
 783      CALL KPP_ROOT_WCOPY(LU_NONZERO,Jac0,1,Ghimj,1)
 784      CALL KPP_ROOT_WSCAL(LU_NONZERO,(-ONE),Ghimj,1)
 785      ghinv = ONE/(Direction*H*gam)
 786      DO i=1,NVAR
 787        Ghimj(LU_DIAG(i)) = Ghimj(LU_DIAG(i))+ghinv
 788      END DO
 789 #endif
 790 !~~~>    Compute LU decomposition
 791      CALL KPP_ROOT_ros_Decomp( Ghimj, Pivot, ising, Ndec )
 792      IF (ising == 0) THEN
 793 !~~~>    If successful done
 794         Singular = .FALSE.
 795      ELSE ! ising .ne. 0
 796 !~~~>    If unsuccessful half the step size; if 5 consecutive fails then return
 797         Nsng = Nsng+1
 798         Nconsecutive = Nconsecutive+1
 799         Singular = .TRUE.
 800         PRINT*,'Warning: LU Decomposition returned ising = ',ising
 801         IF (Nconsecutive <= 5) THEN ! Less than 5 consecutive failed decompositions
 802            H = H*HALF
 803         ELSE  ! More than 5 consecutive failed decompositions
 804            RETURN
 805         END IF  ! Nconsecutive
 806       END IF    ! ising
 807
 808    END DO ! WHILE Singular
 809
 810   END SUBROUTINE  KPP_ROOT_ros_PrepareMatrix
 811
 812
 813 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 814   SUBROUTINE  KPP_ROOT_ros_Decomp( A, Pivot, ising, Ndec )
 815 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 816 !  Template for the LU decomposition
 817 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 818    IMPLICIT NONE
 819 !~~~> Inout variables
 820    KPP_REAL, INTENT(INOUT) :: A(LU_NONZERO)
 821 !~~~> Output variables
 822    INTEGER, INTENT(OUT) :: Pivot(NVAR), ising
 823    INTEGER, INTENT(INOUT) :: Ndec
 824
 825 #ifdef FULL_ALGEBRA
 826    CALL  KPP_ROOT_DGETRF( NVAR, NVAR, A, NVAR, Pivot, ising )
 827 #else
 828    CALL KPP_ROOT_KppDecomp ( A, ising )
 829    Pivot(1) = 1
 830 #endif
 831    Ndec = Ndec + 1
 832
 833   END SUBROUTINE  KPP_ROOT_ros_Decomp
 834
 835
 836 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 837   SUBROUTINE  KPP_ROOT_ros_Solve( A, Pivot, b, Nsol )
 838 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 839 !  Template for the forward/backward substitution (using pre-computed LU decomposition)
 840 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 841    IMPLICIT NONE
 842 !~~~> Input variables
 843    KPP_REAL, INTENT(IN) :: A(LU_NONZERO)
 844    INTEGER, INTENT(IN) :: Pivot(NVAR)
 845 !~~~~>  InOut args
 846    INTEGER, INTENT(INOUT) :: nsol
 847 !~~~> InOut variables
 848    KPP_REAL, INTENT(INOUT) :: b(NVAR)
 849
 850
 851 #ifdef FULL_ALGEBRA
 852    CALL  KPP_ROOT_DGETRS( 'N', NVAR , 1, A, NVAR, Pivot, b, NVAR, 0 )
 853 #else
 854    CALL KPP_ROOT_KppSolve( A, b )
 855 #endif
 856
 857    Nsol = Nsol+1
 858
 859   END SUBROUTINE  KPP_ROOT_ros_Solve
 860
 861
 862
 863 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 864   SUBROUTINE  KPP_ROOT_Ros2 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
 865             ros_Gamma,ros_NewF,ros_ELO,ros_Name)
 866 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 867 ! --- AN L-STABLE METHOD, 2 stages, order 2
 868 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 869
 870   IMPLICIT NONE
 871
 872    INTEGER, PARAMETER :: S=2
 873    INTEGER, INTENT(OUT) ::  ros_S
 874    KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
 875    KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
 876    KPP_REAL, INTENT(OUT) :: ros_ELO
 877    LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
 878    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
 879 !cms   DOUBLE PRECISION g
 880     KPP_REAL :: g
 881
 882     g = 1.0_dp + 1.0_dp/SQRT(2.0_dp)
 883
 884 !~~~> Name of the method
 885     ros_Name = 'ROS-2'
 886 !~~~> Number of stages
 887     ros_S = S
 888
 889 !~~~> The coefficient matrices A and C are strictly lower triangular.
 890 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
 891 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
 892 !   The general mapping formula is:
 893 !       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
 894 !       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
 895
 896     ros_A(1) = (1.0_dp)/g
 897     ros_C(1) = (-2.0_dp)/g
 898 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
 899 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
 900     ros_NewF(1) = .TRUE.
 901     ros_NewF(2) = .TRUE.
 902 !~~~> M_i = Coefficients for new step solution
 903     ros_M(1)= (3.0_dp)/(2.0_dp*g)
 904     ros_M(2)= (1.0_dp)/(2.0_dp*g)
 905 ! E_i = Coefficients for error estimator
 906     ros_E(1) = 1.0_dp/(2.0_dp*g)
 907     ros_E(2) = 1.0_dp/(2.0_dp*g)
 908 !~~~> ros_ELO = estimator of local order - the minimum between the
 909 !    main and the embedded scheme orders plus one
 910     ros_ELO = 2.0_dp
 911 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
 912     ros_Alpha(1) = 0.0_dp
 913     ros_Alpha(2) = 1.0_dp
 914 !~~~> Gamma_i = \sum_j  gamma_{i,j}
 915     ros_Gamma(1) = g
 916     ros_Gamma(2) =-g
 917
 918  END SUBROUTINE  KPP_ROOT_Ros2
 919
 920
 921 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 922   SUBROUTINE  KPP_ROOT_Ros3 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
 923            ros_Gamma,ros_NewF,ros_ELO,ros_Name)
 924 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 925 ! --- AN L-STABLE METHOD, 3 stages, order 3, 2 function evaluations
 926 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 927
 928   IMPLICIT NONE
 929
 930    INTEGER, PARAMETER :: S=3
 931    INTEGER, INTENT(OUT) ::  ros_S
 932    KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
 933    KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
 934    KPP_REAL, INTENT(OUT) :: ros_ELO
 935    LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
 936    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
 937
 938 !~~~> Name of the method
 939    ros_Name = 'ROS-3'
 940 !~~~> Number of stages
 941    ros_S = S
 942
 943 !~~~> The coefficient matrices A and C are strictly lower triangular.
 944 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
 945 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
 946 !   The general mapping formula is:
 947 !       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
 948 !       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
 949
 950    ros_A(1)= 1.0_dp
 951    ros_A(2)= 1.0_dp
 952    ros_A(3)= 0.0_dp
 953
 954    ros_C(1) = -0.10156171083877702091975600115545E+01_dp
 955    ros_C(2) =  0.40759956452537699824805835358067E+01_dp
 956    ros_C(3) =  0.92076794298330791242156818474003E+01_dp
 957 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
 958 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
 959    ros_NewF(1) = .TRUE.
 960    ros_NewF(2) = .TRUE.
 961    ros_NewF(3) = .FALSE.
 962 !~~~> M_i = Coefficients for new step solution
 963    ros_M(1) =  0.1E+01_dp
 964    ros_M(2) =  0.61697947043828245592553615689730E+01_dp
 965    ros_M(3) = -0.42772256543218573326238373806514E+00_dp
 966 ! E_i = Coefficients for error estimator
 967    ros_E(1) =  0.5E+00_dp
 968    ros_E(2) = -0.29079558716805469821718236208017E+01_dp
 969    ros_E(3) =  0.22354069897811569627360909276199E+00_dp
 970 !~~~> ros_ELO = estimator of local order - the minimum between the
 971 !    main and the embedded scheme orders plus 1
 972    ros_ELO = 3.0_dp
 973 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
 974    ros_Alpha(1)= 0.0E+00_dp
 975    ros_Alpha(2)= 0.43586652150845899941601945119356E+00_dp
 976    ros_Alpha(3)= 0.43586652150845899941601945119356E+00_dp
 977 !~~~> Gamma_i = \sum_j  gamma_{i,j}
 978    ros_Gamma(1)= 0.43586652150845899941601945119356E+00_dp
 979    ros_Gamma(2)= 0.24291996454816804366592249683314E+00_dp
 980    ros_Gamma(3)= 0.21851380027664058511513169485832E+01_dp
 981
 982   END SUBROUTINE  KPP_ROOT_Ros3
 983
 984 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 985
 986
 987 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 988   SUBROUTINE  KPP_ROOT_Ros4 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
 989            ros_Gamma,ros_NewF,ros_ELO,ros_Name)
 990 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 991 !     L-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 4 STAGES
 992 !     L-STABLE EMBEDDED ROSENBROCK METHOD OF ORDER 3
 993 !
 994 !      E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL
 995 !      EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS.
 996 !      SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS,
 997 !      SPRINGER-VERLAG (1990)
 998 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 999
1000   IMPLICIT NONE
1001
1002    INTEGER, PARAMETER :: S=4
1003    INTEGER, INTENT(OUT) ::  ros_S
1004    KPP_REAL, DIMENSION(4), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
1005    KPP_REAL, DIMENSION(6), INTENT(OUT) :: ros_A, ros_C
1006    KPP_REAL, INTENT(OUT) :: ros_ELO
1007    LOGICAL, DIMENSION(4), INTENT(OUT) :: ros_NewF
1008    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
1009 !cms   DOUBLE PRECISION g
1010    KPP_REAL :: g
1011
1012
1013 !~~~> Name of the method
1014    ros_Name = 'ROS-4'
1015 !~~~> Number of stages
1016    ros_S = S
1017
1018 !~~~> The coefficient matrices A and C are strictly lower triangular.
1019 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
1020 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
1021 !   The general mapping formula is:
1022 !       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
1023 !       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
1024
1025    ros_A(1) = 0.2000000000000000E+01_dp
1026    ros_A(2) = 0.1867943637803922E+01_dp
1027    ros_A(3) = 0.2344449711399156E+00_dp
1028    ros_A(4) = ros_A(2)
1029    ros_A(5) = ros_A(3)
1030    ros_A(6) = 0.0_dp
1031
1032    ros_C(1) =-0.7137615036412310E+01_dp
1033    ros_C(2) = 0.2580708087951457E+01_dp
1034    ros_C(3) = 0.6515950076447975E+00_dp
1035    ros_C(4) =-0.2137148994382534E+01_dp
1036    ros_C(5) =-0.3214669691237626E+00_dp
1037    ros_C(6) =-0.6949742501781779E+00_dp
1038 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
1039 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
1040    ros_NewF(1)  = .TRUE.
1041    ros_NewF(2)  = .TRUE.
1042    ros_NewF(3)  = .TRUE.
1043    ros_NewF(4)  = .FALSE.
1044 !~~~> M_i = Coefficients for new step solution
1045    ros_M(1) = 0.2255570073418735E+01_dp
1046    ros_M(2) = 0.2870493262186792E+00_dp
1047    ros_M(3) = 0.4353179431840180E+00_dp
1048    ros_M(4) = 0.1093502252409163E+01_dp
1049 !~~~> E_i  = Coefficients for error estimator
1050    ros_E(1) =-0.2815431932141155E+00_dp
1051    ros_E(2) =-0.7276199124938920E-01_dp
1052    ros_E(3) =-0.1082196201495311E+00_dp
1053    ros_E(4) =-0.1093502252409163E+01_dp
1054 !~~~> ros_ELO  = estimator of local order - the minimum between the
1055 !    main and the embedded scheme orders plus 1
1056    ros_ELO  = 4.0_dp
1057 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
1058    ros_Alpha(1) = 0.0_dp
1059    ros_Alpha(2) = 0.1145640000000000E+01_dp
1060    ros_Alpha(3) = 0.6552168638155900E+00_dp
1061    ros_Alpha(4) = ros_Alpha(3)
1062 !~~~> Gamma_i = \sum_j  gamma_{i,j}
1063    ros_Gamma(1) = 0.5728200000000000E+00_dp
1064    ros_Gamma(2) =-0.1769193891319233E+01_dp
1065    ros_Gamma(3) = 0.7592633437920482E+00_dp
1066    ros_Gamma(4) =-0.1049021087100450E+00_dp
1067
1068   END SUBROUTINE  KPP_ROOT_Ros4
1069
1070 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1071   SUBROUTINE  KPP_ROOT_Rodas3 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
1072             ros_Gamma,ros_NewF,ros_ELO,ros_Name)
1073 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1074 ! --- A STIFFLY-STABLE METHOD, 4 stages, order 3
1075 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1076
1077   IMPLICIT NONE
1078
1079    INTEGER, PARAMETER :: S=4
1080    INTEGER, INTENT(OUT) ::  ros_S
1081    KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
1082    KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
1083    KPP_REAL, INTENT(OUT) :: ros_ELO
1084    LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
1085    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
1086 !cms   DOUBLE PRECISION g
1087    KPP_REAL :: g
1088
1089 !~~~> Name of the method
1090    ros_Name = 'RODAS-3'
1091 !~~~> Number of stages
1092    ros_S = S
1093
1094 !~~~> The coefficient matrices A and C are strictly lower triangular.
1095 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
1096 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
1097 !   The general mapping formula is:
1098 !       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
1099 !       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
1100
1101    ros_A(1) = 0.0E+00_dp
1102    ros_A(2) = 2.0E+00_dp
1103    ros_A(3) = 0.0E+00_dp
1104    ros_A(4) = 2.0E+00_dp
1105    ros_A(5) = 0.0E+00_dp
1106    ros_A(6) = 1.0E+00_dp
1107
1108    ros_C(1) = 4.0E+00_dp
1109    ros_C(2) = 1.0E+00_dp
1110    ros_C(3) =-1.0E+00_dp
1111    ros_C(4) = 1.0E+00_dp
1112    ros_C(5) =-1.0E+00_dp
1113    ros_C(6) =-(8.0E+00_dp/3.0E+00_dp)
1114
1115 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
1116 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
1117    ros_NewF(1)  = .TRUE.
1118    ros_NewF(2)  = .FALSE.
1119    ros_NewF(3)  = .TRUE.
1120    ros_NewF(4)  = .TRUE.
1121 !~~~> M_i = Coefficients for new step solution
1122    ros_M(1) = 2.0E+00_dp
1123    ros_M(2) = 0.0E+00_dp
1124    ros_M(3) = 1.0E+00_dp
1125    ros_M(4) = 1.0E+00_dp
1126 !~~~> E_i  = Coefficients for error estimator
1127    ros_E(1) = 0.0E+00_dp
1128    ros_E(2) = 0.0E+00_dp
1129    ros_E(3) = 0.0E+00_dp
1130    ros_E(4) = 1.0E+00_dp
1131 !~~~> ros_ELO  = estimator of local order - the minimum between the
1132 !    main and the embedded scheme orders plus 1
1133    ros_ELO  = 3.0E+00_dp
1134 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
1135    ros_Alpha(1) = 0.0E+00_dp
1136    ros_Alpha(2) = 0.0E+00_dp
1137    ros_Alpha(3) = 1.0E+00_dp
1138    ros_Alpha(4) = 1.0E+00_dp
1139 !~~~> Gamma_i = \sum_j  gamma_{i,j}
1140    ros_Gamma(1) = 0.5E+00_dp
1141    ros_Gamma(2) = 1.5E+00_dp
1142    ros_Gamma(3) = 0.0E+00_dp
1143    ros_Gamma(4) = 0.0E+00_dp
1144
1145   END SUBROUTINE  KPP_ROOT_Rodas3
1146
1147 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1148   SUBROUTINE  KPP_ROOT_Rodas4 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,&
1149              ros_Gamma,ros_NewF,ros_ELO,ros_Name)
1150 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1151 !     STIFFLY-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 6 STAGES
1152 !
1153 !      E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL
1154 !      EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS.
1155 !      SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS,
1156 !      SPRINGER-VERLAG (1996)
1157 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1158
1159   IMPLICIT NONE
1160
1161    INTEGER, PARAMETER :: S=6
1162    INTEGER, INTENT(OUT) ::  ros_S
1163    KPP_REAL, DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma
1164    KPP_REAL, DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C
1165    KPP_REAL, INTENT(OUT) :: ros_ELO
1166    LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF
1167    CHARACTER(LEN=12), INTENT(OUT) :: ros_Name
1168 !cms   DOUBLE PRECISION g
1169     KPP_REAL :: g
1170
1171 !~~~> Name of the method
1172     ros_Name = 'RODAS-4'
1173 !~~~> Number of stages
1174     ros_S = 6
1175
1176 !~~~> Y_stage_i ~ Y( T + H*Alpha_i )
1177     ros_Alpha(1) = 0.000_dp
1178     ros_Alpha(2) = 0.386_dp
1179     ros_Alpha(3) = 0.210_dp
1180     ros_Alpha(4) = 0.630_dp
1181     ros_Alpha(5) = 1.000_dp
1182     ros_Alpha(6) = 1.000_dp
1183
1184 !~~~> Gamma_i = \sum_j  gamma_{i,j}
1185     ros_Gamma(1) = 0.2500000000000000E+00_dp
1186     ros_Gamma(2) =-0.1043000000000000E+00_dp
1187     ros_Gamma(3) = 0.1035000000000000E+00_dp
1188     ros_Gamma(4) =-0.3620000000000023E-01_dp
1189     ros_Gamma(5) = 0.0_dp
1190     ros_Gamma(6) = 0.0_dp
1191
1192 !~~~> The coefficient matrices A and C are strictly lower triangular.
1193 !   The lower triangular (subdiagonal) elements are stored in row-wise order:
1194 !   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
1195 !   The general mapping formula is:  A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
1196 !                  C(i,j) = ros_C( (i-1)*(i-2)/2 + j )
1197
1198     ros_A(1) = 0.1544000000000000E+01_dp
1199     ros_A(2) = 0.9466785280815826E+00_dp
1200     ros_A(3) = 0.2557011698983284E+00_dp
1201     ros_A(4) = 0.3314825187068521E+01_dp
1202     ros_A(5) = 0.2896124015972201E+01_dp
1203     ros_A(6) = 0.9986419139977817E+00_dp
1204     ros_A(7) = 0.1221224509226641E+01_dp
1205     ros_A(8) = 0.6019134481288629E+01_dp
1206     ros_A(9) = 0.1253708332932087E+02_dp
1207     ros_A(10) =-0.6878860361058950E+00_dp
1208     ros_A(11) = ros_A(7)
1209     ros_A(12) = ros_A(8)
1210     ros_A(13) = ros_A(9)
1211     ros_A(14) = ros_A(10)
1212     ros_A(15) = 1.0E+00_dp
1213
1214     ros_C(1) =-0.5668800000000000E+01_dp
1215     ros_C(2) =-0.2430093356833875E+01_dp
1216     ros_C(3) =-0.2063599157091915E+00_dp
1217     ros_C(4) =-0.1073529058151375E+00_dp
1218     ros_C(5) =-0.9594562251023355E+01_dp
1219     ros_C(6) =-0.2047028614809616E+02_dp
1220     ros_C(7) = 0.7496443313967647E+01_dp
1221     ros_C(8) =-0.1024680431464352E+02_dp
1222     ros_C(9) =-0.3399990352819905E+02_dp
1223     ros_C(10) = 0.1170890893206160E+02_dp
1224     ros_C(11) = 0.8083246795921522E+01_dp
1225     ros_C(12) =-0.7981132988064893E+01_dp
1226     ros_C(13) =-0.3152159432874371E+02_dp
1227     ros_C(14) = 0.1631930543123136E+02_dp
1228     ros_C(15) =-0.6058818238834054E+01_dp
1229
1230 !~~~> M_i = Coefficients for new step solution
1231     ros_M(1) = ros_A(7)
1232     ros_M(2) = ros_A(8)
1233     ros_M(3) = ros_A(9)
1234     ros_M(4) = ros_A(10)
1235     ros_M(5) = 1.0E+00_dp
1236     ros_M(6) = 1.0E+00_dp
1237
1238 !~~~> E_i  = Coefficients for error estimator
1239     ros_E(1) = 0.0E+00_dp
1240     ros_E(2) = 0.0E+00_dp
1241     ros_E(3) = 0.0E+00_dp
1242     ros_E(4) = 0.0E+00_dp
1243     ros_E(5) = 0.0E+00_dp
1244     ros_E(6) = 1.0E+00_dp
1245
1246 !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
1247 !   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
1248     ros_NewF(1) = .TRUE.
1249     ros_NewF(2) = .TRUE.
1250     ros_NewF(3) = .TRUE.
1251     ros_NewF(4) = .TRUE.
1252     ros_NewF(5) = .TRUE.
1253     ros_NewF(6) = .TRUE.
1254
1255 !~~~> ros_ELO  = estimator of local order - the minimum between the
1256 !        main and the embedded scheme orders plus 1
1257     ros_ELO = 4.0_dp
1258
1259   END SUBROUTINE  KPP_ROOT_Rodas4
1260
1261 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1262 !   End of the set of internal Rosenbrock subroutines
1263 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1264 END SUBROUTINE  KPP_ROOT_Rosenbrock
1265 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1266
1267
1268 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1269 SUBROUTINE  KPP_ROOT_FunTemplate( T, Y, Ydot, RCONST, FIX, Nfun )
1270 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1271 !  Template for the ODE function call.
1272 !  Updates the rate coefficients (and possibly the fixed species) at each call
1273 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1274    USE KPP_ROOT_Parameters
1275 !! USE KPP_ROOT_Global
1276 !! USE KPP_ROOT_Function
1277 !! USE KPP_ROOT_Rates
1278 !~~~> Input variables
1279    KPP_REAL :: T, Y(NVAR)
1280    KPP_REAL :: RCONST(NREACT)
1281    KPP_REAL :: FIX(NFIX)
1282 !~~~> Output variables
1283    KPP_REAL :: Ydot(NVAR)
1284    INTEGER :: Nfun
1285
1286
1287 !~~~> Local variables
1288 !!   KPP_REAL :: Told
1289
1290 !!   Told = TIME
1291 !!   TIME = T
1292 !!   CALL Update_SUN()
1293 !!   CALL Update_RCONST()
1294    CALL KPP_ROOT_Fun( Y, FIX, RCONST, Ydot )
1295 !!   TIME = Told
1296
1297    Nfun = Nfun+1
1298
1299 END SUBROUTINE  KPP_ROOT_FunTemplate
1300
1301
1302 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1303 SUBROUTINE  KPP_ROOT_JacTemplate( T, Y, Jcb, FIX, Njac, RCONST )
1304 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1305 !  Template for the ODE Jacobian call.
1306 !  Updates the rate coefficients (and possibly the fixed species) at each call
1307 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1308  USE KPP_ROOT_Parameters
1309  !!USE KPP_ROOT_Global
1310  USE KPP_ROOT_Jacobian
1311 !! USE KPP_ROOT_LinearAlgebra
1312 !! USE KPP_ROOT_Rates
1313 !~~~> Input variables
1314     KPP_REAL :: T, Y(NVAR)
1315     KPP_REAL :: FIX(NFIX)
1316     KPP_REAL :: RCONST(NREACT)
1317
1318     INTEGER :: Njac
1319
1320 !~~~> Output variables
1321 #ifdef FULL_ALGEBRA
1322     KPP_REAL :: JV(LU_NONZERO), Jcb(NVAR,NVAR)
1323 #else
1324     KPP_REAL :: Jcb(LU_NONZERO)
1325 #endif
1326 !~~~> Local variables
1327     KPP_REAL :: Told
1328 #ifdef FULL_ALGEBRA
1329     INTEGER :: i, j
1330 #endif
1331
1332 !!    Told = TIME
1333 !!    TIME = T
1334 !!    CALL Update_SUN()
1335 !!    CALL Update_RCONST()
1336 #ifdef FULL_ALGEBRA
1337     CALL KPP_ROOT_Jac_SP(Y, FIX, RCONST, JV)
1338     DO j=1,NVAR
1339       DO i=1,NVAR
1340          Jcb(i,j) = 0.0d0
1341       END DO
1342     END DO
1343     DO i=1,LU_NONZERO
1344        Jcb(LU_IROW(i),LU_ICOL(i)) = JV(i)
1345     END DO
1346 #else
1347     CALL KPP_ROOT_Jac_SP( Y, FIX, RCONST, Jcb )
1348 #endif
1349 !!    TIME = Told
1350
1351     Njac = Njac+1
1352
1353 END SUBROUTINE  KPP_ROOT_JacTemplate
1354
1355 !!!END MODULE KPP_ROOT_Integrator