text: C     --------------------------------------------------------------------
  C     SUNDIALS Copyright Start
  C     Copyright (c) 2002-2020, Lawrence Livermore National Security
  C     and Southern Methodist University.
  C     All rights reserved.
  C
  C     See the top-level LICENSE and NOTICE files for details.
  C
  C     SPDX-License-Identifier: BSD-3-Clause
  C     SUNDIALS Copyright End
  C     --------------------------------------------------------------------
  C     FCVODE Example Problem: 2D kinetics-transport, precond. Krylov
  C     solver. 
  C     
  C     An ODE system is generated from the following 2-species diurnal
  C     kinetics advection-diffusion PDE system in 2 space dimensions:
  C     
  C     dc(i)/dt = Kh*(d/dx)**2 c(i) + V*dc(i)/dx + (d/dy)(Kv(y)*dc(i)/dy)
  C                           + Ri(c1,c2,t)      for i = 1,2,   where
  C     R1(c1,c2,t) = -q1*c1*c3 - q2*c1*c2 + 2*q3(t)*c3 + q4(t)*c2 ,
  C     R2(c1,c2,t) =  q1*c1*c3 - q2*c1*c2 - q4(t)*c2 ,
  C     Kv(y) = Kv0*exp(y/5) ,
  C     Kh, V, Kv0, q1, q2, and c3 are constants, and q3(t) and q4(t)
  C     vary diurnally.
  C
  C     The problem is posed on the square
  C     0 .le. x .le. 20,    30 .le. y .le. 50   (all in km),
  C     with homogeneous Neumann boundary conditions, and for time t
  C     in 0 .le. t .le. 86400 sec (1 day).
  C
  C     The PDE system is treated by central differences on a uniform
  C     10 x 10 mesh, with simple polynomial initial profiles.
  C     The problem is solved with CVODE, with the BDF/GMRES method and
  C     using the FCVBP banded preconditioner.
  C     
  C     The second and third dimensions of U here must match the values
  C     of MX and MY, for consistency with the output statements below.
  C     --------------------------------------------------------------------
  C
        IMPLICIT NONE
  C
        INTEGER*4 MX, MY
        PARAMETER (MX=10, MY=10)
  C
        INTEGER*4 LNST, LNFE, LNSETUP, LNNI, LNCF, LNPE, LNLI, LNPS
        INTEGER*4 LNCFL, LH, LQ, METH, IATOL, ITASK
        INTEGER*4 LNETF, IER, MAXL, JPRETYPE, IGSTYPE, JOUT
        INTEGER*4 LLENRW, LLENIW, LLENRWLS, LLENIWLS
  C The following declaration specification should match C type long int.
        INTEGER*8 NEQ, IOUT(25), IPAR(4)
        INTEGER*4 NST, NFE, NPSET, NPE, NPS, NNI
        INTEGER*4 NLI, NCFN, NCFL, NETF
        INTEGER*4 LENRW, LENIW, LENRWLS, LENIWLS
  C The following declaration specification should match C type long int.
        INTEGER*8 MU, ML, LENRWBP, LENIWBP, NFEBP
        DOUBLE PRECISION ATOL, AVDIM, DELT, FLOOR, RTOL, T, TOUT, TWOHR
        DOUBLE PRECISION ROUT(10), U(2,MX,MY), RPAR(12)
  C
        DATA TWOHR/7200.0D0/, RTOL/1.0D-5/, FLOOR/100.0D0/,
       1     JPRETYPE/1/, IGSTYPE/1/, MAXL/0/, DELT/0.0D0/
        DATA LLENRW/1/, LLENIW/2/, LNST/3/, LNFE/4/, LNETF/5/,  LNCF/6/,
       1     LNNI/7/, LNSETUP/8/, LQ/9/, LLENRWLS/13/, LLENIWLS/14/,
       1     LNPE/20/, LNLI/22/, LNPS/21/, LNCFL/23/
        DATA LH/2/
  C
  C     Load problem constants into IPAR, RPAR, and set initial values
        CALL INITKX(MX, MY, U, IPAR, RPAR)
  C
  C     Set other input arguments.
        NEQ = 2*MX*MY
        T = 0.0D0
        METH = 2
        IATOL = 1
        ATOL = RTOL * FLOOR
        ITASK = 1
  C
        WRITE(6,10) NEQ
   10   FORMAT('Krylov example problem:'//
       1       ' Kinetics-transport, NEQ = ', I4/)
  C     
  C     Initialize vector specification
        CALL FNVINITS(1, NEQ, IER)
        IF (IER .NE. 0) THEN
          WRITE(6,20) IER
   20     FORMAT(///' SUNDIALS_ERROR: FNVINITS returned IER = ', I5)
          STOP
        ENDIF
  C
  C     Initialize SPGMR linear solver module
        call FSUNSPGMRINIT(1, JPRETYPE, MAXL, IER)
        IF (IER .NE. 0) THEN
          WRITE(6,25) IER
   25     FORMAT(///' SUNDIALS_ERROR: FSUNSPGMRINIT IER = ', I5)
          STOP
        ENDIF
        call FSUNSPGMRSETGSTYPE(1, IGSTYPE, IER)
        IF (IER .NE. 0) THEN
          WRITE(6,27) IER
   27     FORMAT(///' SUNDIALS_ERROR: FSUNSPGMRSETGSTYPE IER = ', I5)
          STOP
        ENDIF
  C     
  C     Initialize CVODE
        CALL FCVMALLOC(T, U, METH, IATOL, RTOL, ATOL,
       1               IOUT, ROUT, IPAR, RPAR, IER)
        IF (IER .NE. 0) THEN
          WRITE(6,30) IER
   30     FORMAT(///' SUNDIALS_ERROR: FCVMALLOC returned IER = ', I5)
          STOP
        ENDIF
  C
  C     attach linear solver module to CVLs interface
        CALL FCVLSINIT(IER)
        IF (IER .NE. 0) THEN
          WRITE(6,40) IER
   40     FORMAT(///' SUNDIALS_ERROR: FCVLSINIT returned IER = ',I5)
          CALL FCVFREE
          STOP
        ENDIF
  C     
  C     Initialize band preconditioner
        MU = 2
        ML = 2
        CALL FCVBPINIT(NEQ, MU, ML, IER) 
        IF (IER .NE. 0) THEN
           WRITE(6,45) IER
   45      FORMAT(///' SUNDIALS_ERROR: FCVBPINIT returned IER = ', I5)
           CALL FCVFREE
           STOP
        ENDIF
  C     
  C     Loop over output points, call FCVODE, print sample solution values.
        TOUT = TWOHR
        DO 70 JOUT = 1, 12
  C
           CALL FCVODE(TOUT, T, U, ITASK, IER)
  C     
           WRITE(6,50) T, IOUT(LNST), IOUT(LQ), ROUT(LH)
   50      FORMAT(/' t = ', E14.6, 5X, 'no. steps = ', I5,
       1           '   order = ', I3, '   stepsize = ', E14.6)
           WRITE(6,55) U(1,1,1), U(1,5,5), U(1,10,10),
       1               U(2,1,1), U(2,5,5), U(2,10,10)
   55      FORMAT('  c1 (bot.left/middle/top rt.) = ', 3E14.6/
       1          '  c2 (bot.left/middle/top rt.) = ', 3E14.6)
  C
           IF (IER .NE. 0) THEN
              WRITE(6,60) IER, IOUT(15)
   60         FORMAT(///' SUNDIALS_ERROR: FCVODE returned IER = ', I5, /,
       1             '                 Linear Solver returned IER = ', I5)
              CALL FCVFREE
              STOP
           ENDIF
  C     
           TOUT = TOUT + TWOHR
   70   CONTINUE
        
  C     Print final statistics.
        NST = IOUT(LNST)
        NFE = IOUT(LNFE)
        NPSET = IOUT(LNSETUP)
        NPE = IOUT(LNPE)
        NPS = IOUT(LNPS)
        NNI = IOUT(LNNI)
        NLI = IOUT(LNLI)
        AVDIM = DBLE(NLI) / DBLE(NNI)
        NCFN = IOUT(LNCF)
        NCFL = IOUT(LNCFL)
        NETF = IOUT(LNETF)
        LENRW = IOUT(LLENRW)
        LENIW = IOUT(LLENIW)
        LENRWLS = IOUT(LLENRWLS)
        LENIWLS = IOUT(LLENIWLS)
        WRITE(6,80) NST, NFE, NPSET, NPE, NPS, NNI, NLI, AVDIM, NCFN,
       1     NCFL, NETF, LENRW, LENIW, LENRWLS, LENIWLS
   80   FORMAT(//'Final statistics:'//
       &   ' number of steps        = ', I5, 4X,
       &   ' number of f evals.     = ', I5/
       &   ' number of prec. setups = ', I5/
       &   ' number of prec. evals. = ', I5, 4X,
       &   ' number of prec. solves = ', I5/
       &   ' number of nonl. iters. = ', I5, 4X,
       &   ' number of lin. iters.  = ', I5/
       &   ' average Krylov subspace dimension (NLI/NNI) = ', E14.6/
       &   ' number of conv. failures.. nonlinear =', I3,
       &   ' linear = ', I3/
       &   ' number of error test failures = ', I3/
       &   ' main solver real/int workspace sizes   = ',2I7/
       &   ' linear solver real/int workspace sizes = ',2I5)
        CALL FCVBPOPT(LENRWBP, LENIWBP, NFEBP)
        WRITE(6,82) LENRWBP, LENIWBP, NFEBP
   82   FORMAT('In CVBANDPRE:'/
       &        ' real/int workspace sizes = ', 2I7/
       &        ' number of f evaluations  = ', I5)
  C     
        CALL FCVFREE
  C
        STOP
        END
  
  C     ----------------------------------------------------------------
  
        SUBROUTINE INITKX(MX, MY, U0, IPAR, RPAR)
  C     Routine to set problem constants and initial values
  C
        IMPLICIT NONE
  C
        INTEGER*4 MX, MY
  C The following declaration specification should match C type long int.
        INTEGER*8 IPAR(*)
        DOUBLE PRECISION RPAR(*)
  C
        INTEGER*8 MM, JY, JX, NEQ
        DOUBLE PRECISION U0
        DIMENSION U0(2,MX,MY)
        DOUBLE PRECISION Q1, Q2, Q3, Q4, A3, A4, OM, C3, DY, HDCO
        DOUBLE PRECISION VDCO, HACO, X, Y
        DOUBLE PRECISION CX, CY, DKH, DKV0, DX, HALFDA, PI, VEL
  C
        DATA DKH/4.0D-6/, VEL/0.001D0/, DKV0/1.0D-8/, HALFDA/4.32D4/,
       1     PI/3.1415926535898D0/
  C
  C     Problem constants
        MM = MX * MY
        NEQ = 2 * MM
        Q1 = 1.63D-16
        Q2 = 4.66D-16
        Q3 = 0.0D0
        Q4 = 0.0D0
        A3 = 22.62D0
        A4 = 7.601D0
        OM = PI / HALFDA
        C3 = 3.7D16
        DX = 20.0D0 / (MX - 1.0D0)
        DY = 20.0D0 / (MY - 1.0D0)
        HDCO = DKH / DX**2
        HACO = VEL / (2.0D0 * DX)
        VDCO = (1.0D0 / DY**2) * DKV0
  C
  C     Load constants in IPAR and RPAR
        IPAR(1) = MX
        IPAR(2) = MY
        IPAR(3) = MM
        IPAR(4) = NEQ
  C
        RPAR(1)  = Q1
        RPAR(2)  = Q2
        RPAR(3)  = Q3
        RPAR(4)  = Q4
        RPAR(5)  = A3
        RPAR(6)  = A4
        RPAR(7)  = OM
        RPAR(8)  = C3
        RPAR(9)  = DY
        RPAR(10) = HDCO
        RPAR(11) = VDCO
        RPAR(12) = HACO
  C
  C     Set initial profiles.
        DO 20 JY = 1, MY
          Y = 30.0D0 + (JY - 1.0D0) * DY
          CY = (0.1D0 * (Y - 40.0D0))**2
          CY = 1.0D0 - CY + 0.5D0 * CY**2
          DO 10 JX = 1, MX
            X = (JX - 1.0D0) * DX
            CX = (0.1D0 * (X - 10.0D0))**2
            CX = 1.0D0 - CX + 0.5D0 * CX**2
            U0(1,JX,JY) = 1.0D6 * CX * CY
            U0(2,JX,JY) = 1.0D12 * CX * CY
   10       CONTINUE
   20     CONTINUE
  C
        RETURN
        END
  
  C     ----------------------------------------------------------------
  
        SUBROUTINE FCVFUN(T, U, UDOT, IPAR, RPAR, IER)
  C     Routine for right-hand side function f
  C
        IMPLICIT NONE
  C
  C The following declaration specification should match C type long int.
        INTEGER*8 IPAR(*)
        INTEGER*4 IER
        DOUBLE PRECISION T, U(2,*), UDOT(2,*), RPAR(*)
  C
        INTEGER*4 ILEFT, IRIGHT
        INTEGER*8 MX, MY, MM, JY, JX, IBLOK0, IDN, IUP, IBLOK
        DOUBLE PRECISION Q1,Q2,Q3,Q4, A3, A4, OM, C3, DY, HDCO, VDCO, HACO
        DOUBLE PRECISION C1, C2, C1DN, C2DN, C1UP, C2UP, C1LT, C2LT
        DOUBLE PRECISION C1RT, C2RT, CYDN, CYUP, HORD1, HORD2, HORAD1
        DOUBLE PRECISION HORAD2, QQ1, QQ2, QQ3, QQ4, RKIN1, RKIN2, S
        DOUBLE PRECISION VERTD1, VERTD2, YDN, YUP
  C
  C     Extract constants from IPAR and RPAR
        MX = IPAR(1)
        MY = IPAR(2)
        MM = IPAR(3)
  C
        Q1 = RPAR(1)
        Q2 = RPAR(2)
        Q3 = RPAR(3)
        Q4 = RPAR(4)
        A3 = RPAR(5)
        A4 = RPAR(6)
        OM = RPAR(7)
        C3 = RPAR(8)
        DY = RPAR(9)
        HDCO = RPAR(10)
        VDCO = RPAR(11)
        HACO = RPAR(12)
  C
  C     Set diurnal rate coefficients.
        S = SIN(OM * T)
        IF (S .GT. 0.0D0) THEN
           Q3 = EXP(-A3 / S)
           Q4 = EXP(-A4 / S)
        ELSE
           Q3 = 0.0D0
           Q4 = 0.0D0
        ENDIF
  C     
  C     Loop over all grid points.
        DO 20 JY = 1, MY
           YDN = 30.0D0 + (JY - 1.5D0) * DY
           YUP = YDN + DY
           CYDN = VDCO * EXP(0.2D0 * YDN)
           CYUP = VDCO * EXP(0.2D0 * YUP)
           IBLOK0 = (JY - 1) * MX
           IDN = -MX
           IF (JY .EQ. 1) IDN = MX
           IUP = MX
           IF (JY .EQ. MY) IUP = -MX
           DO 10 JX = 1, MX
              IBLOK = IBLOK0 + JX
              C1 = U(1,IBLOK)
              C2 = U(2,IBLOK)
  C     Set kinetic rate terms.
              QQ1 = Q1 * C1 * C3
              QQ2 = Q2 * C1 * C2
              QQ3 = Q3 * C3
              QQ4 = Q4 * C2
              RKIN1 = -QQ1 - QQ2 + 2.0D0 * QQ3 + QQ4
              RKIN2 = QQ1 - QQ2 - QQ4
  C     Set vertical diffusion terms.
              C1DN = U(1,IBLOK + IDN)
              C2DN = U(2,IBLOK + IDN)
              C1UP = U(1,IBLOK + IUP)
              C2UP = U(2,IBLOK + IUP)
              VERTD1 = CYUP * (C1UP - C1) - CYDN * (C1 - C1DN)
              VERTD2 = CYUP * (C2UP - C2) - CYDN * (C2 - C2DN)
  C     Set horizontal diffusion and advection terms.
              ILEFT = -1
              IF (JX .EQ. 1) ILEFT = 1
              IRIGHT = 1
              IF (JX .EQ. MX) IRIGHT = -1
              C1LT = U(1,IBLOK + ILEFT)
              C2LT = U(2,IBLOK + ILEFT)
              C1RT = U(1,IBLOK + IRIGHT)
              C2RT = U(2,IBLOK + IRIGHT)
              HORD1 = HDCO * (C1RT - 2.0D0 * C1 + C1LT)
              HORD2 = HDCO * (C2RT - 2.0D0 * C2 + C2LT)
              HORAD1 = HACO * (C1RT - C1LT)
              HORAD2 = HACO * (C2RT - C2LT)
  C     Load all terms into UDOT.
              UDOT(1,IBLOK) = VERTD1 + HORD1 + HORAD1 + RKIN1
              UDOT(2,IBLOK) = VERTD2 + HORD2 + HORAD2 + RKIN2
   10      CONTINUE
   20   CONTINUE
  C
        IER = 0
  C
        RETURN
        END
  