[2696] | 1 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~! |
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| 2 | ! RungeKuttaTLM - Tangent Linear Model of Fully Implicit 3-stage RK: ! |
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| 3 | ! * Radau-2A quadrature (order 5) ! |
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| 4 | ! * Radau-1A quadrature (order 5) ! |
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| 5 | ! * Lobatto-3C quadrature (order 4) ! |
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| 6 | ! * Gauss quadrature (order 6) ! |
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| 7 | ! By default the code employs the KPP sparse linear algebra routines ! |
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| 8 | ! Compile with -DFULL_ALGEBRA to use full linear algebra (LAPACK) ! |
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| 9 | ! ! |
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| 10 | ! (C) Adrian Sandu, August 2005 ! |
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| 11 | ! Virginia Polytechnic Institute and State University ! |
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| 12 | ! Contact: sandu@cs.vt.edu ! |
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| 13 | ! Revised by Philipp Miehe and Adrian Sandu, May 2006 ! |
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| 14 | ! This implementation is part of KPP - the Kinetic PreProcessor ! |
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| 15 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~! |
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| 16 | |
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| 17 | MODULE KPP_ROOT_Integrator |
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| 18 | |
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| 19 | USE KPP_ROOT_Precision |
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| 20 | USE KPP_ROOT_Parameters, ONLY: NVAR, NSPEC, NFIX, LU_NONZERO |
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| 21 | USE KPP_ROOT_Global, ONLY: FIX, RCONST, TIME |
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| 22 | USE KPP_ROOT_Jacobian |
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| 23 | USE KPP_ROOT_LinearAlgebra |
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| 24 | |
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| 25 | IMPLICIT NONE |
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| 26 | PUBLIC |
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| 27 | SAVE |
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| 28 | |
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| 29 | !~~~> Statistics on the work performed by the Runge-Kutta method |
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| 30 | INTEGER, PARAMETER :: Nfun=1, Njac=2, Nstp=3, Nacc=4, & |
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| 31 | Nrej=5, Ndec=6, Nsol=7, Nsng=8, Ntexit=1, Nhacc=2, Nhnew=3 |
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| 32 | |
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| 33 | CONTAINS |
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| 34 | |
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| 35 | ! ************************************************************************** |
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| 36 | |
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| 37 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 38 | SUBROUTINE INTEGRATE_TLM( NTLM, Y, Y_tlm, TIN, TOUT, ATOL_tlm, RTOL_tlm, & |
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| 39 | ICNTRL_U, RCNTRL_U, ISTATUS_U, RSTATUS_U, IERR_U ) |
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| 40 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 41 | |
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| 42 | USE KPP_ROOT_Parameters, ONLY: NVAR |
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| 43 | USE KPP_ROOT_Global, ONLY: ATOL,RTOL,VAR |
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| 44 | |
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| 45 | IMPLICIT NONE |
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| 46 | |
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| 47 | !~~~> Y - Concentrations |
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| 48 | KPP_REAL :: Y(NVAR) |
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| 49 | !~~~> NTLM - No. of sensitivity coefficients |
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| 50 | INTEGER NTLM |
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| 51 | !~~~> Y_tlm - Sensitivities of concentrations |
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| 52 | ! Note: Y_tlm (1:NVAR,j) contains sensitivities of |
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| 53 | ! Y(1:NVAR) w.r.t. the j-th parameter, j=1...NTLM |
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| 54 | KPP_REAL :: Y_tlm(NVAR,NTLM) |
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| 55 | KPP_REAL :: TIN ! TIN - Start Time |
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| 56 | KPP_REAL :: TOUT ! TOUT - End Time |
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| 57 | KPP_REAL, INTENT(IN), OPTIONAL :: RTOL_tlm(NVAR,NTLM),ATOL_tlm(NVAR,NTLM) |
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| 58 | INTEGER, INTENT(IN), OPTIONAL :: ICNTRL_U(20) |
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| 59 | KPP_REAL, INTENT(IN), OPTIONAL :: RCNTRL_U(20) |
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| 60 | INTEGER, INTENT(OUT), OPTIONAL :: ISTATUS_U(20) |
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| 61 | KPP_REAL, INTENT(OUT), OPTIONAL :: RSTATUS_U(20) |
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| 62 | INTEGER, INTENT(OUT), OPTIONAL :: IERR_U |
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| 63 | |
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| 64 | INTEGER :: IERR |
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| 65 | |
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| 66 | KPP_REAL :: RCNTRL(20), RSTATUS(20), T1, T2 |
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| 67 | INTEGER :: ICNTRL(20), ISTATUS(20) |
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| 68 | INTEGER, SAVE :: Ntotal = 0 |
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| 69 | |
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| 70 | ICNTRL(1:20) = 0 |
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| 71 | RCNTRL(1:20) = 0.0_dp |
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| 72 | |
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| 73 | !~~~> fine-tune the integrator: |
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| 74 | ICNTRL(2) = 0 ! Tolerances: 0=vector, 1=scalar |
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| 75 | ICNTRL(5) = 8 ! Max no. of Newton iterations |
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| 76 | ICNTRL(6) = 0 ! Starting values for Newton are: 0=interpolated, 1=zero |
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| 77 | ICNTRL(7) = 1 ! TLM solution: 0=iterations, 1=direct |
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| 78 | ICNTRL(9) = 0 ! TLM Newton iteration error: 0=fwd Newton, 1=TLM Newton error est |
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| 79 | ICNTRL(10) = 1 ! FWD error estimation: 0=classic, 1=SDIRK |
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| 80 | ICNTRL(11) = 1 ! Step size ontroller: 1=Gustaffson, 2=classic |
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| 81 | ICNTRL(12) = 0 ! Trunc. error estimate: 0=fwd only, 1=fwd and TLM |
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| 82 | |
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| 83 | !~~~> if optional parameters are given, and if they are >0, |
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| 84 | ! then use them to overwrite default settings |
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| 85 | IF (PRESENT(ICNTRL_U)) THEN |
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| 86 | WHERE(ICNTRL_U(:) > 0) ICNTRL(:) = ICNTRL_U(:) |
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| 87 | END IF |
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| 88 | IF (PRESENT(RCNTRL_U)) THEN |
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| 89 | WHERE(RCNTRL_U(:) > 0) RCNTRL(:) = RCNTRL_U(:) |
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| 90 | END IF |
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| 91 | |
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| 92 | T1 = TIN; T2 = TOUT |
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| 93 | CALL RungeKuttaTLM( NVAR, NTLM, Y, Y_tlm, T1, T2, RTOL, ATOL, & |
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| 94 | RTOL_tlm, ATOL_tlm, & |
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| 95 | RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR ) |
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| 96 | |
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| 97 | Ntotal = Ntotal + ISTATUS(Nstp) |
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| 98 | PRINT*,'NSTEPS=',ISTATUS(Nstp),' (',Ntotal,')',' O3=', VAR(ind_O3) |
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| 99 | |
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| 100 | ! if optional parameters are given for output |
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| 101 | ! use them to store information in them |
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| 102 | IF (PRESENT(ISTATUS_U)) ISTATUS_U(:) = ISTATUS(:) |
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| 103 | IF (PRESENT(RSTATUS_U)) RSTATUS_U(:) = RSTATUS(:) |
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| 104 | IF (PRESENT(IERR_U)) IERR_U = IERR |
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| 105 | |
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| 106 | IF (IERR < 0) THEN |
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| 107 | PRINT *,'Runge-Kutta-TLM: Unsuccessful exit at T=', TIN,' (IERR=',IERR,')' |
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| 108 | ENDIF |
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| 109 | |
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| 110 | END SUBROUTINE INTEGRATE_TLM |
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| 111 | |
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| 112 | |
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| 113 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 114 | SUBROUTINE RungeKuttaTLM( N, NTLM, Y, Y_tlm, T, Tend,RelTol,AbsTol, & |
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| 115 | RelTol_tlm,AbsTol_tlm,RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR ) |
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| 116 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 117 | ! |
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| 118 | ! This implementation is based on the book and the code Radau5: |
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| 119 | ! |
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| 120 | ! E. HAIRER AND G. WANNER |
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| 121 | ! "SOLVING ORDINARY DIFFERENTIAL EQUATIONS II. |
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| 122 | ! STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS." |
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| 123 | ! SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS 14, |
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| 124 | ! SPRINGER-VERLAG (1991) |
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| 125 | ! |
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| 126 | ! UNIVERSITE DE GENEVE, DEPT. DE MATHEMATIQUES |
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| 127 | ! CH-1211 GENEVE 24, SWITZERLAND |
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| 128 | ! E-MAIL: HAIRER@DIVSUN.UNIGE.CH, WANNER@DIVSUN.UNIGE.CH |
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| 129 | ! |
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| 130 | ! Methods: |
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| 131 | ! * Radau-2A quadrature (order 5) |
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| 132 | ! * Radau-1A quadrature (order 5) |
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| 133 | ! * Lobatto-3C quadrature (order 4) |
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| 134 | ! * Gauss quadrature (order 6) |
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| 135 | ! |
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| 136 | ! (C) Adrian Sandu, August 2005 |
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| 137 | ! Virginia Polytechnic Institute and State University |
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| 138 | ! Contact: sandu@cs.vt.edu |
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| 139 | ! Revised by Philipp Miehe and Adrian Sandu, May 2006 |
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| 140 | ! This implementation is part of KPP - the Kinetic PreProcessor |
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| 141 | ! |
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| 142 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 143 | ! |
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| 144 | !~~~> INPUT ARGUMENTS: |
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| 145 | ! ---------------- |
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| 146 | ! |
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| 147 | ! Note: For input parameters equal to zero the default values of the |
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| 148 | ! corresponding variables are used. |
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| 149 | ! |
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| 150 | ! N Dimension of the system |
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| 151 | ! T Initial time value |
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| 152 | ! |
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| 153 | ! Tend Final T value (Tend-T may be positive or negative) |
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| 154 | ! |
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| 155 | ! Y(N) Initial values for Y |
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| 156 | ! |
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| 157 | ! RelTol,AbsTol Relative and absolute error tolerances. |
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| 158 | ! for ICNTRL(2) = 0: AbsTol, RelTol are N-dimensional vectors |
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| 159 | ! = 1: AbsTol, RelTol are scalars |
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| 160 | ! |
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| 161 | !~~~> Integer input parameters: |
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| 162 | ! |
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| 163 | ! ICNTRL(1) = not used |
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| 164 | ! |
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| 165 | ! ICNTRL(2) = 0: AbsTol, RelTol are NVAR-dimensional vectors |
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| 166 | ! = 1: AbsTol, RelTol are scalars |
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| 167 | ! |
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| 168 | ! ICNTRL(3) = RK method selection |
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| 169 | ! = 1: Radau-2A (the default) |
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| 170 | ! = 2: Lobatto-3C |
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| 171 | ! = 3: Gauss |
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| 172 | ! = 4: Radau-1A |
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| 173 | ! |
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| 174 | ! ICNTRL(4) -> maximum number of integration steps |
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| 175 | ! For ICNTRL(4)=0 the default value of 100000 is used |
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| 176 | ! |
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| 177 | ! ICNTRL(5) -> maximum number of Newton iterations |
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| 178 | ! For ICNTRL(5)=0 the default value of 8 is used |
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| 179 | ! |
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| 180 | ! ICNTRL(6) -> starting values of Newton iterations: |
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| 181 | ! ICNTRL(6)=0 : starting values are obtained from |
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| 182 | ! the extrapolated collocation solution |
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| 183 | ! (the default) |
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| 184 | ! ICNTRL(6)=1 : starting values are zero |
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| 185 | ! |
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| 186 | ! ICNTRL(7) -> method to solve TLM equations: |
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| 187 | ! ICNTRL(7)=0 : modified Newton re-using LU (the default) |
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| 188 | ! ICNTRL(7)=1 : direct solution (additional one *big* LU factorization) |
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| 189 | ! |
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| 190 | ! ICNTRL(9) -> switch for TLM Newton iteration error estimation strategy |
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| 191 | ! ICNTRL(9) = 0: base number of iterations as forward solution |
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| 192 | ! ICNTRL(9) = 1: use RTOL_tlm and ATOL_tlm to calculate |
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| 193 | ! error estimation for TLM at Newton stages |
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| 194 | ! |
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| 195 | ! ICNTRL(10) -> switch for error estimation strategy |
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| 196 | ! ICNTRL(10) = 0: one additional stage: at c=0, |
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| 197 | ! see Hairer (default) |
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| 198 | ! ICNTRL(10) = 1: two additional stages: one at c=0 |
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| 199 | ! and one SDIRK at c=1, stiffly accurate |
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| 200 | ! |
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| 201 | ! ICNTRL(11) -> switch for step size strategy |
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| 202 | ! ICNTRL(11)=1: mod. predictive controller (Gustafsson, default) |
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| 203 | ! ICNTRL(11)=2: classical step size control |
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| 204 | ! the choice 1 seems to produce safer results; |
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| 205 | ! for simple problems, the choice 2 produces |
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| 206 | ! often slightly faster runs |
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| 207 | ! |
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| 208 | ! ICNTRL(12) -> switch for TLM truncation error control |
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| 209 | ! ICNTRL(12) = 0: TLM error is not used |
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| 210 | ! ICNTRL(12) = 1: TLM error is computed and used |
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| 211 | ! |
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| 212 | ! |
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| 213 | !~~~> Real input parameters: |
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| 214 | ! |
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| 215 | ! RCNTRL(1) -> Hmin, lower bound for the integration step size |
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| 216 | ! (highly recommended to keep Hmin = ZERO, the default) |
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| 217 | ! |
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| 218 | ! RCNTRL(2) -> Hmax, upper bound for the integration step size |
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| 219 | ! |
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| 220 | ! RCNTRL(3) -> Hstart, the starting step size |
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| 221 | ! |
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| 222 | ! RCNTRL(4) -> FacMin, lower bound on step decrease factor (default=0.2) |
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| 223 | ! |
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| 224 | ! RCNTRL(5) -> FacMax, upper bound on step increase factor (default=6) |
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| 225 | ! |
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| 226 | ! RCNTRL(6) -> FacRej, step decrease factor after multiple rejections |
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| 227 | ! (default=0.1) |
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| 228 | ! |
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| 229 | ! RCNTRL(7) -> FacSafe, by which the new step is slightly smaller |
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| 230 | ! than the predicted value (default=0.9) |
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| 231 | ! |
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| 232 | ! RCNTRL(8) -> ThetaMin. If Newton convergence rate smaller |
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| 233 | ! than ThetaMin the Jacobian is not recomputed; |
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| 234 | ! (default=0.001) |
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| 235 | ! |
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| 236 | ! RCNTRL(9) -> NewtonTol, stopping criterion for Newton's method |
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| 237 | ! (default=0.03) |
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| 238 | ! |
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| 239 | ! RCNTRL(10) -> Qmin |
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| 240 | ! |
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| 241 | ! RCNTRL(11) -> Qmax. If Qmin < Hnew/Hold < Qmax, then the |
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| 242 | ! step size is kept constant and the LU factorization |
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| 243 | ! reused (default Qmin=1, Qmax=1.2) |
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| 244 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 245 | ! |
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| 246 | ! |
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| 247 | ! OUTPUT ARGUMENTS: |
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| 248 | ! ----------------- |
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| 249 | ! |
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| 250 | ! T -> T value for which the solution has been computed |
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| 251 | ! (after successful return T=Tend). |
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| 252 | ! |
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| 253 | ! Y(N) -> Numerical solution at T |
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| 254 | ! |
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| 255 | ! IERR -> Reports on successfulness upon return: |
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| 256 | ! = 1 for success |
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| 257 | ! < 0 for error (value equals error code) |
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| 258 | ! |
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| 259 | ! ISTATUS(1) -> No. of function calls |
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| 260 | ! ISTATUS(2) -> No. of jacobian calls |
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| 261 | ! ISTATUS(3) -> No. of steps |
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| 262 | ! ISTATUS(4) -> No. of accepted steps |
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| 263 | ! ISTATUS(5) -> No. of rejected steps (except at very beginning) |
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| 264 | ! ISTATUS(6) -> No. of LU decompositions |
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| 265 | ! ISTATUS(7) -> No. of forward/backward substitutions |
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| 266 | ! ISTATUS(8) -> No. of singular matrix decompositions |
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| 267 | ! |
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| 268 | ! RSTATUS(1) -> Texit, the time corresponding to the |
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| 269 | ! computed Y upon return |
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| 270 | ! RSTATUS(2) -> Hexit, last accepted step before exit |
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| 271 | ! RSTATUS(3) -> Hnew, last predicted step (not yet taken) |
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| 272 | ! For multiple restarts, use Hnew as Hstart |
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| 273 | ! in the subsequent run |
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| 274 | ! |
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| 275 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 276 | |
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| 277 | IMPLICIT NONE |
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| 278 | |
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| 279 | INTEGER :: N, NTLM |
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| 280 | KPP_REAL :: Y(N),Y_tlm(N,NTLM) |
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| 281 | KPP_REAL :: AbsTol(N),RelTol(N),RCNTRL(20),RSTATUS(20) |
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| 282 | KPP_REAL :: AbsTol_tlm(N,NTLM),RelTol_tlm(N,NTLM) |
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| 283 | INTEGER :: ICNTRL(20), ISTATUS(20) |
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| 284 | LOGICAL :: StartNewton, Gustafsson, SdirkError, TLMNewtonEst, & |
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| 285 | TLMtruncErr, TLMDirect |
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| 286 | INTEGER :: IERR, ITOL |
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| 287 | KPP_REAL :: T,Tend |
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| 288 | |
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| 289 | !~~~> Control arguments |
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| 290 | INTEGER :: Max_no_steps, NewtonMaxit, rkMethod |
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| 291 | KPP_REAL :: Hmin,Hmax,Hstart,Qmin,Qmax |
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| 292 | KPP_REAL :: Roundoff, ThetaMin, NewtonTol |
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| 293 | KPP_REAL :: FacSafe,FacMin,FacMax,FacRej |
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| 294 | ! Runge-Kutta method parameters |
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| 295 | INTEGER, PARAMETER :: R2A=1, R1A=2, L3C=3, GAU=4, L3A=5 |
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| 296 | KPP_REAL :: rkT(3,3), rkTinv(3,3), rkTinvAinv(3,3), rkAinvT(3,3), & |
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| 297 | rkA(3,3), rkB(3), rkC(3), rkD(0:3), rkE(0:3), & |
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| 298 | rkBgam(0:4), rkBhat(0:4), rkTheta(0:3), & |
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| 299 | rkGamma, rkAlpha, rkBeta, rkELO |
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| 300 | !~~~> Local variables |
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| 301 | INTEGER :: i |
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| 302 | KPP_REAL, PARAMETER :: ZERO = 0.0d0, ONE = 1.0d0 |
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| 303 | |
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| 304 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 305 | ! SETTING THE PARAMETERS |
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| 306 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 307 | IERR = 0 |
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| 308 | ISTATUS(1:20) = 0 |
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| 309 | RSTATUS(1:20) = ZERO |
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| 310 | |
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| 311 | !~~~> ICNTRL(1) - autonomous system - not used |
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| 312 | !~~~> ITOL: 1 for vector and 0 for scalar AbsTol/RelTol |
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| 313 | IF (ICNTRL(2) == 0) THEN |
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| 314 | ITOL = 1 |
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| 315 | ELSE |
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| 316 | ITOL = 0 |
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| 317 | END IF |
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| 318 | !~~~> Error control selection |
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| 319 | IF (ICNTRL(10) == 0) THEN |
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| 320 | SdirkError = .FALSE. |
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| 321 | ELSE |
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| 322 | SdirkError = .TRUE. |
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| 323 | END IF |
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| 324 | !~~~> Method selection |
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| 325 | SELECT CASE (ICNTRL(3)) |
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| 326 | CASE (0,1) |
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| 327 | CALL Radau2A_Coefficients |
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| 328 | CASE (2) |
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| 329 | CALL Lobatto3C_Coefficients |
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| 330 | CASE (3) |
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| 331 | CALL Gauss_Coefficients |
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| 332 | CASE (4) |
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| 333 | CALL Radau1A_Coefficients |
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| 334 | CASE DEFAULT |
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| 335 | WRITE(6,*) 'ICNTRL(3)=',ICNTRL(3) |
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| 336 | CALL RK_ErrorMsg(-13,T,ZERO,IERR) |
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| 337 | END SELECT |
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| 338 | !~~~> Max_no_steps: the maximal number of time steps |
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| 339 | IF (ICNTRL(4) == 0) THEN |
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| 340 | Max_no_steps = 200000 |
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| 341 | ELSE |
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| 342 | Max_no_steps=ICNTRL(4) |
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| 343 | IF (Max_no_steps <= 0) THEN |
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| 344 | WRITE(6,*) 'ICNTRL(4)=',ICNTRL(4) |
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| 345 | CALL RK_ErrorMsg(-1,T,ZERO,IERR) |
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| 346 | END IF |
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| 347 | END IF |
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| 348 | !~~~> NewtonMaxit maximal number of Newton iterations |
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| 349 | IF (ICNTRL(5) == 0) THEN |
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| 350 | NewtonMaxit = 8 |
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| 351 | ELSE |
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| 352 | NewtonMaxit=ICNTRL(5) |
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| 353 | IF (NewtonMaxit <= 0) THEN |
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| 354 | WRITE(6,*) 'ICNTRL(5)=',ICNTRL(5) |
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| 355 | CALL RK_ErrorMsg(-2,T,ZERO,IERR) |
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| 356 | END IF |
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| 357 | END IF |
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| 358 | !~~~> StartNewton: Use extrapolation for starting values of Newton iterations |
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| 359 | IF (ICNTRL(6) == 0) THEN |
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| 360 | StartNewton = .TRUE. |
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| 361 | ELSE |
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| 362 | StartNewton = .FALSE. |
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| 363 | END IF |
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| 364 | !~~~> Solve TLM equations directly or by Newton iterations |
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| 365 | IF (ICNTRL(7) == 0) THEN |
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| 366 | TLMDirect = .FALSE. |
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| 367 | ELSE |
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| 368 | TLMDirect = .TRUE. |
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| 369 | END IF |
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| 370 | !~~~> Newton iteration error control selection |
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| 371 | IF (ICNTRL(9) == 0) THEN |
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| 372 | TLMNewtonEst = .FALSE. |
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| 373 | ELSE |
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| 374 | TLMNewtonEst = .TRUE. |
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| 375 | END IF |
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| 376 | !~~~> Gustafsson: step size controller |
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| 377 | IF(ICNTRL(11) == 0)THEN |
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| 378 | Gustafsson = .TRUE. |
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| 379 | ELSE |
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| 380 | Gustafsson = .FALSE. |
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| 381 | END IF |
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| 382 | !~~~> TLM truncation error control selection |
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| 383 | IF (ICNTRL(12) == 0) THEN |
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| 384 | TLMtruncErr = .FALSE. |
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| 385 | ELSE |
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| 386 | TLMtruncErr = .TRUE. |
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| 387 | END IF |
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| 388 | |
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| 389 | !~~~> Roundoff: smallest number s.t. 1.0 + Roundoff > 1.0 |
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| 390 | Roundoff=WLAMCH('E'); |
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| 391 | |
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| 392 | !~~~> Hmin = minimal step size |
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| 393 | IF (RCNTRL(1) == ZERO) THEN |
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| 394 | Hmin = ZERO |
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| 395 | ELSE |
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| 396 | Hmin = MIN(ABS(RCNTRL(1)),ABS(Tend-T)) |
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| 397 | END IF |
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| 398 | !~~~> Hmax = maximal step size |
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| 399 | IF (RCNTRL(2) == ZERO) THEN |
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| 400 | Hmax = ABS(Tend-T) |
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| 401 | ELSE |
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| 402 | Hmax = MIN(ABS(RCNTRL(2)),ABS(Tend-T)) |
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| 403 | END IF |
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| 404 | !~~~> Hstart = starting step size |
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| 405 | IF (RCNTRL(3) == ZERO) THEN |
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| 406 | Hstart = ZERO |
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| 407 | ELSE |
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| 408 | Hstart = MIN(ABS(RCNTRL(3)),ABS(Tend-T)) |
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| 409 | END IF |
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| 410 | !~~~> FacMin: lower bound on step decrease factor |
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| 411 | IF(RCNTRL(4) == ZERO)THEN |
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| 412 | FacMin = 0.2d0 |
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| 413 | ELSE |
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| 414 | FacMin = RCNTRL(4) |
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| 415 | END IF |
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| 416 | !~~~> FacMax: upper bound on step increase factor |
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| 417 | IF(RCNTRL(5) == ZERO)THEN |
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| 418 | FacMax = 8.D0 |
---|
| 419 | ELSE |
---|
| 420 | FacMax = RCNTRL(5) |
---|
| 421 | END IF |
---|
| 422 | !~~~> FacRej: step decrease factor after 2 consecutive rejections |
---|
| 423 | IF(RCNTRL(6) == ZERO)THEN |
---|
| 424 | FacRej = 0.1d0 |
---|
| 425 | ELSE |
---|
| 426 | FacRej = RCNTRL(6) |
---|
| 427 | END IF |
---|
| 428 | !~~~> FacSafe: by which the new step is slightly smaller |
---|
| 429 | ! than the predicted value |
---|
| 430 | IF (RCNTRL(7) == ZERO) THEN |
---|
| 431 | FacSafe=0.9d0 |
---|
| 432 | ELSE |
---|
| 433 | FacSafe=RCNTRL(7) |
---|
| 434 | END IF |
---|
| 435 | IF ( (FacMax < ONE) .OR. (FacMin > ONE) .OR. & |
---|
| 436 | (FacSafe <= 1.0d-3) .OR. (FacSafe >= ONE) ) THEN |
---|
| 437 | WRITE(6,*)'RCNTRL(4:7)=',RCNTRL(4:7) |
---|
| 438 | CALL RK_ErrorMsg(-4,T,ZERO,IERR) |
---|
| 439 | END IF |
---|
| 440 | |
---|
| 441 | !~~~> ThetaMin: decides whether the Jacobian should be recomputed |
---|
| 442 | IF (RCNTRL(8) == ZERO) THEN |
---|
| 443 | ThetaMin = 1.0d-3 |
---|
| 444 | ELSE |
---|
| 445 | ThetaMin=RCNTRL(8) |
---|
| 446 | IF (ThetaMin <= 0.0d0 .OR. ThetaMin >= 1.0d0) THEN |
---|
| 447 | WRITE(6,*) 'RCNTRL(8)=', RCNTRL(8) |
---|
| 448 | CALL RK_ErrorMsg(-5,T,ZERO,IERR) |
---|
| 449 | END IF |
---|
| 450 | END IF |
---|
| 451 | !~~~> NewtonTol: stopping crierion for Newton's method |
---|
| 452 | IF (RCNTRL(9) == ZERO) THEN |
---|
| 453 | NewtonTol = 3.0d-2 |
---|
| 454 | ELSE |
---|
| 455 | NewtonTol = RCNTRL(9) |
---|
| 456 | IF (NewtonTol <= Roundoff) THEN |
---|
| 457 | WRITE(6,*) 'RCNTRL(9)=',RCNTRL(9) |
---|
| 458 | CALL RK_ErrorMsg(-6,T,ZERO,IERR) |
---|
| 459 | END IF |
---|
| 460 | END IF |
---|
| 461 | !~~~> Qmin AND Qmax: IF Qmin < Hnew/Hold < Qmax then step size = const. |
---|
| 462 | IF (RCNTRL(10) == ZERO) THEN |
---|
| 463 | Qmin=1.D0 |
---|
| 464 | ELSE |
---|
| 465 | Qmin=RCNTRL(10) |
---|
| 466 | END IF |
---|
| 467 | IF (RCNTRL(11) == ZERO) THEN |
---|
| 468 | Qmax=1.2D0 |
---|
| 469 | ELSE |
---|
| 470 | Qmax=RCNTRL(11) |
---|
| 471 | END IF |
---|
| 472 | IF (Qmin > ONE .OR. Qmax < ONE) THEN |
---|
| 473 | WRITE(6,*) 'RCNTRL(10:11)=',Qmin,Qmax |
---|
| 474 | CALL RK_ErrorMsg(-7,T,ZERO,IERR) |
---|
| 475 | END IF |
---|
| 476 | !~~~> Check if tolerances are reasonable |
---|
| 477 | IF (ITOL == 0) THEN |
---|
| 478 | IF (AbsTol(1) <= ZERO.OR.RelTol(1) <= 10.d0*Roundoff) THEN |
---|
| 479 | WRITE (6,*) 'AbsTol/RelTol=',AbsTol,RelTol |
---|
| 480 | CALL RK_ErrorMsg(-8,T,ZERO,IERR) |
---|
| 481 | END IF |
---|
| 482 | ELSE |
---|
| 483 | DO i=1,N |
---|
| 484 | IF (AbsTol(i) <= ZERO.OR.RelTol(i) <= 10.d0*Roundoff) THEN |
---|
| 485 | WRITE (6,*) 'AbsTol/RelTol(',i,')=',AbsTol(i),RelTol(i) |
---|
| 486 | CALL RK_ErrorMsg(-8,T,ZERO,IERR) |
---|
| 487 | END IF |
---|
| 488 | END DO |
---|
| 489 | END IF |
---|
| 490 | |
---|
| 491 | !~~~> Parameters are wrong |
---|
| 492 | IF (IERR < 0) RETURN |
---|
| 493 | |
---|
| 494 | !~~~> Call the core method |
---|
| 495 | CALL RK_IntegratorTLM( N,NTLM,T,Tend,Y,Y_tlm,IERR ) |
---|
| 496 | |
---|
| 497 | CONTAINS ! Internal procedures to RungeKutta |
---|
| 498 | |
---|
| 499 | |
---|
| 500 | |
---|
| 501 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 502 | SUBROUTINE RK_IntegratorTLM( N,NTLM,T,Tend,Y,Y_tlm,IERR ) |
---|
| 503 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 504 | |
---|
| 505 | IMPLICIT NONE |
---|
| 506 | !~~~> Arguments |
---|
| 507 | INTEGER, INTENT(IN) :: N, NTLM |
---|
| 508 | KPP_REAL, INTENT(IN) :: Tend |
---|
| 509 | KPP_REAL, INTENT(INOUT) :: T, Y(N), Y_tlm(NVAR,NTLM) |
---|
| 510 | INTEGER, INTENT(OUT) :: IERR |
---|
| 511 | |
---|
| 512 | !~~~> Local variables |
---|
| 513 | #ifdef FULL_ALGEBRA |
---|
| 514 | KPP_REAL, DIMENSION(NVAR,NVAR) :: & |
---|
| 515 | FJAC, E1, Jac1, Jac2, Jac3 |
---|
| 516 | COMPLEX(kind=dp), DIMENSION(NVAR,NVAR) :: E2 |
---|
| 517 | KPP_REAL, DIMENSION(3*NVAR,3*NVAR) :: Jbig, Ebig |
---|
| 518 | KPP_REAL, DIMENSION(3*NVAR) :: Zbig |
---|
| 519 | INTEGER, DIMENSION(3*NVAR) :: IPbig |
---|
| 520 | #else |
---|
| 521 | KPP_REAL, DIMENSION(LU_NONZERO) :: & |
---|
| 522 | FJAC, E1, Jac1, Jac2, Jac3 |
---|
| 523 | COMPLEX(kind=dp), DIMENSION(LU_NONZERO) :: E2 |
---|
| 524 | ! Next 3 commented lines for sparse big linear algebra: |
---|
| 525 | ! KPP_REAL, DIMENSION(3,3,LU_NONZERO) :: Jbig |
---|
| 526 | ! KPP_REAL, DIMENSION(3,NVAR) :: Zbig |
---|
| 527 | ! INTEGER, DIMENSION(3,NVAR) :: IPbig |
---|
| 528 | KPP_REAL, DIMENSION(3*NVAR,3*NVAR) :: Jbig, Ebig |
---|
| 529 | KPP_REAL, DIMENSION(3*NVAR) :: Zbig |
---|
| 530 | INTEGER, DIMENSION(3*NVAR) :: IPbig |
---|
| 531 | #endif |
---|
| 532 | KPP_REAL, DIMENSION(NVAR) :: Z1,Z2,Z3,Z4,SCAL,DZ1,DZ2,DZ3,DZ4, & |
---|
| 533 | G,TMP,SCAL_tlm |
---|
| 534 | KPP_REAL, DIMENSION(NVAR,NTLM) :: Z1_tlm,Z2_tlm,Z3_tlm |
---|
| 535 | KPP_REAL :: CONT(NVAR,3), Tdirection, H, Hacc, Hnew, Hold, Fac, & |
---|
| 536 | FacGus, Theta, Err, ErrOld, NewtonRate, NewtonIncrement, & |
---|
| 537 | Hratio, Qnewton, NewtonPredictedErr,NewtonIncrementOld, & |
---|
| 538 | ThetaTLM, ThetaSD |
---|
| 539 | INTEGER :: i,j,IP1(NVAR),IP2(NVAR),NewtonIter, ISING, Nconsecutive, itlm |
---|
| 540 | INTEGER :: saveNiter, NewtonIterTLM, info |
---|
| 541 | LOGICAL :: Reject, FirstStep, SkipJac, NewtonDone, SkipLU |
---|
| 542 | |
---|
| 543 | |
---|
| 544 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 545 | !~~~> Initial setting |
---|
| 546 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 547 | Tdirection = SIGN(ONE,Tend-T) |
---|
| 548 | H = MIN( MAX(ABS(Hmin),ABS(Hstart)) , Hmax ) |
---|
| 549 | IF (ABS(H) <= 10.d0*Roundoff) H = 1.0d-6 |
---|
| 550 | H = SIGN(H,Tdirection) |
---|
| 551 | Hold = H |
---|
| 552 | Reject = .FALSE. |
---|
| 553 | FirstStep = .TRUE. |
---|
| 554 | SkipJac = .FALSE. |
---|
| 555 | SkipLU = .FALSE. |
---|
| 556 | IF ((T+H*1.0001D0-Tend)*Tdirection >= ZERO) THEN |
---|
| 557 | H = Tend-T |
---|
| 558 | END IF |
---|
| 559 | Nconsecutive = 0 |
---|
| 560 | CALL RK_ErrorScale(N,ITOL,AbsTol,RelTol,Y,SCAL) |
---|
| 561 | |
---|
| 562 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 563 | !~~~> Time loop begins |
---|
| 564 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 565 | Tloop: DO WHILE ( (Tend-T)*Tdirection - Roundoff > ZERO ) |
---|
| 566 | |
---|
| 567 | IF ( .NOT.SkipLU ) THEN ! This time around skip the Jac update and LU |
---|
| 568 | !~~~> Compute the Jacobian matrix |
---|
| 569 | IF ( .NOT.SkipJac ) THEN |
---|
| 570 | CALL JAC_CHEM(T,Y,FJAC) |
---|
| 571 | ISTATUS(Njac) = ISTATUS(Njac) + 1 |
---|
| 572 | END IF |
---|
| 573 | !~~~> Compute the matrices E1 and E2 and their decompositions |
---|
| 574 | CALL RK_Decomp(N,H,FJAC,E1,IP1,E2,IP2,ISING) |
---|
| 575 | IF (ISING /= 0) THEN |
---|
| 576 | ISTATUS(Nsng) = ISTATUS(Nsng) + 1; Nconsecutive = Nconsecutive + 1 |
---|
| 577 | IF (Nconsecutive >= 5) THEN |
---|
| 578 | CALL RK_ErrorMsg(-12,T,H,IERR); RETURN |
---|
| 579 | END IF |
---|
| 580 | H=H*0.5d0; Reject=.TRUE.; SkipJac = .TRUE.; SkipLU = .FALSE. |
---|
| 581 | CYCLE Tloop |
---|
| 582 | ELSE |
---|
| 583 | Nconsecutive = 0 |
---|
| 584 | END IF |
---|
| 585 | END IF ! SkipLU |
---|
| 586 | |
---|
| 587 | ISTATUS(Nstp) = ISTATUS(Nstp) + 1 |
---|
| 588 | IF (ISTATUS(Nstp) > Max_no_steps) THEN |
---|
| 589 | PRINT*,'Max number of time steps is ',Max_no_steps |
---|
| 590 | CALL RK_ErrorMsg(-9,T,H,IERR); RETURN |
---|
| 591 | END IF |
---|
| 592 | IF (0.1D0*ABS(H) <= ABS(T)*Roundoff) THEN |
---|
| 593 | CALL RK_ErrorMsg(-10,T,H,IERR); RETURN |
---|
| 594 | END IF |
---|
| 595 | |
---|
| 596 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 597 | !~~~> Loop for the simplified Newton iterations |
---|
| 598 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 599 | |
---|
| 600 | !~~~> Starting values for Newton iteration |
---|
| 601 | IF ( FirstStep .OR. (.NOT.StartNewton) ) THEN |
---|
| 602 | CALL Set2zero(N,Z1) |
---|
| 603 | CALL Set2zero(N,Z2) |
---|
| 604 | CALL Set2zero(N,Z3) |
---|
| 605 | ELSE |
---|
| 606 | ! Evaluate quadratic polynomial |
---|
| 607 | CALL RK_Interpolate('eval',N,H,Hold,Z1,Z2,Z3,CONT) |
---|
| 608 | END IF |
---|
| 609 | |
---|
| 610 | !~~~> Initializations for Newton iteration |
---|
| 611 | NewtonDone = .FALSE. |
---|
| 612 | Fac = 0.5d0 ! Step reduction if too many iterations |
---|
| 613 | |
---|
| 614 | NewtonLoop:DO NewtonIter = 1, NewtonMaxit |
---|
| 615 | |
---|
| 616 | !~~~> Prepare the right-hand side |
---|
| 617 | CALL RK_PrepareRHS(N,T,H,Y,Z1,Z2,Z3,DZ1,DZ2,DZ3) |
---|
| 618 | |
---|
| 619 | !~~~> Solve the linear systems |
---|
| 620 | CALL RK_Solve( N,H,E1,IP1,E2,IP2,DZ1,DZ2,DZ3,ISING ) |
---|
| 621 | |
---|
| 622 | NewtonIncrement = SQRT( ( RK_ErrorNorm(N,SCAL,DZ1)**2 + & |
---|
| 623 | RK_ErrorNorm(N,SCAL,DZ2)**2 + & |
---|
| 624 | RK_ErrorNorm(N,SCAL,DZ3)**2 )/3.0d0 ) |
---|
| 625 | |
---|
| 626 | IF ( NewtonIter == 1 ) THEN |
---|
| 627 | Theta = ABS(ThetaMin) |
---|
| 628 | NewtonRate = 2.0d0 |
---|
| 629 | ELSE |
---|
| 630 | Theta = NewtonIncrement/NewtonIncrementOld |
---|
| 631 | IF (Theta < 0.99d0) THEN |
---|
| 632 | NewtonRate = Theta/(ONE-Theta) |
---|
| 633 | ELSE ! Non-convergence of Newton: Theta too large |
---|
| 634 | EXIT NewtonLoop |
---|
| 635 | END IF |
---|
| 636 | IF ( NewtonIter < NewtonMaxit ) THEN |
---|
| 637 | ! Predict error at the end of Newton process |
---|
| 638 | NewtonPredictedErr = NewtonIncrement & |
---|
| 639 | *Theta**(NewtonMaxit-NewtonIter)/(ONE-Theta) |
---|
| 640 | IF (NewtonPredictedErr >= NewtonTol) THEN |
---|
| 641 | ! Non-convergence of Newton: predicted error too large |
---|
| 642 | Qnewton = MIN(10.0d0,NewtonPredictedErr/NewtonTol) |
---|
| 643 | Fac=0.8d0*Qnewton**(-ONE/(1+NewtonMaxit-NewtonIter)) |
---|
| 644 | EXIT NewtonLoop |
---|
| 645 | END IF |
---|
| 646 | END IF |
---|
| 647 | END IF |
---|
| 648 | |
---|
| 649 | NewtonIncrementOld = MAX(NewtonIncrement,Roundoff) |
---|
| 650 | ! Update solution |
---|
| 651 | CALL WAXPY(N,-ONE,DZ1,1,Z1,1) ! Z1 <- Z1 - DZ1 |
---|
| 652 | CALL WAXPY(N,-ONE,DZ2,1,Z2,1) ! Z2 <- Z2 - DZ2 |
---|
| 653 | CALL WAXPY(N,-ONE,DZ3,1,Z3,1) ! Z3 <- Z3 - DZ3 |
---|
| 654 | |
---|
| 655 | ! Check error in Newton iterations |
---|
| 656 | NewtonDone = (NewtonRate*NewtonIncrement <= NewtonTol) |
---|
| 657 | IF (NewtonDone) THEN |
---|
| 658 | ! Tune error in TLM variables by defining the minimal number of Newton iterations. |
---|
| 659 | saveNiter = NewtonIter+1 |
---|
| 660 | EXIT NewtonLoop |
---|
| 661 | END IF |
---|
| 662 | IF (NewtonIter == NewtonMaxit) THEN |
---|
| 663 | PRINT*, 'Slow or no convergence in Newton Iteration: Max no. of', & |
---|
| 664 | 'Newton iterations reached' |
---|
| 665 | END IF |
---|
| 666 | |
---|
| 667 | END DO NewtonLoop |
---|
| 668 | |
---|
| 669 | IF (.NOT.NewtonDone) THEN |
---|
| 670 | !CALL RK_ErrorMsg(-12,T,H,IERR); |
---|
| 671 | H = Fac*H; Reject=.TRUE.; SkipJac = .TRUE.; SkipLU = .FALSE. |
---|
| 672 | CYCLE Tloop |
---|
| 673 | END IF |
---|
| 674 | |
---|
| 675 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 676 | !~~~> SDIRK Stage |
---|
| 677 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 678 | IF (SdirkError) THEN |
---|
| 679 | |
---|
| 680 | !~~~> Starting values for Newton iterations |
---|
| 681 | Z4(1:N) = Z3(1:N) |
---|
| 682 | |
---|
| 683 | !~~~> Prepare the loop-independent part of the right-hand side |
---|
| 684 | CALL FUN_CHEM(T,Y,DZ4) |
---|
| 685 | ISTATUS(Nfun) = ISTATUS(Nfun) + 1 |
---|
| 686 | |
---|
| 687 | ! G = H*rkBgam(0)*DZ4 + rkTheta(1)*Z1 + rkTheta(2)*Z2 + rkTheta(3)*Z3 |
---|
| 688 | CALL Set2Zero(N, G) |
---|
| 689 | CALL WAXPY(N,rkBgam(0)*H, DZ4,1,G,1) |
---|
| 690 | CALL WAXPY(N,rkTheta(1),Z1,1,G,1) |
---|
| 691 | CALL WAXPY(N,rkTheta(2),Z2,1,G,1) |
---|
| 692 | CALL WAXPY(N,rkTheta(3),Z3,1,G,1) |
---|
| 693 | |
---|
| 694 | !~~~> Initializations for Newton iteration |
---|
| 695 | NewtonDone = .FALSE. |
---|
| 696 | Fac = 0.5d0 ! Step reduction factor if too many iterations |
---|
| 697 | |
---|
| 698 | SDNewtonLoop:DO NewtonIter = 1, NewtonMaxit |
---|
| 699 | |
---|
| 700 | !~~~> Prepare the loop-dependent part of the right-hand side |
---|
| 701 | CALL WADD(N,Y,Z4,TMP) ! TMP <- Y + Z4 |
---|
| 702 | CALL FUN_CHEM(T+H,TMP,DZ4) ! DZ4 <- Fun(Y+Z4) |
---|
| 703 | ISTATUS(Nfun) = ISTATUS(Nfun) + 1 |
---|
| 704 | ! DZ4(1:N) = (G(1:N)-Z4(1:N))*(rkGamma/H) + DZ4(1:N) |
---|
| 705 | CALL WAXPY (N, -ONE*rkGamma/H, Z4, 1, DZ4, 1) |
---|
| 706 | CALL WAXPY (N, rkGamma/H, G,1, DZ4,1) |
---|
| 707 | |
---|
| 708 | !~~~> Solve the linear system |
---|
| 709 | #ifdef FULL_ALGEBRA |
---|
| 710 | CALL DGETRS( 'N', N, 1, E1, N, IP1, DZ4, N, ISING ) |
---|
| 711 | #else |
---|
| 712 | CALL KppSolve(E1, DZ4) |
---|
| 713 | #endif |
---|
| 714 | |
---|
| 715 | !~~~> Check convergence of Newton iterations |
---|
| 716 | NewtonIncrement = RK_ErrorNorm(N,SCAL,DZ4) |
---|
| 717 | IF ( NewtonIter == 1 ) THEN |
---|
| 718 | ThetaSD = ABS(ThetaMin) |
---|
| 719 | NewtonRate = 2.0d0 |
---|
| 720 | ELSE |
---|
| 721 | ThetaSD = NewtonIncrement/NewtonIncrementOld |
---|
| 722 | IF (ThetaSD < 0.99d0) THEN |
---|
| 723 | NewtonRate = ThetaSD/(ONE-ThetaSD) |
---|
| 724 | ! Predict error at the end of Newton process |
---|
| 725 | NewtonPredictedErr = NewtonIncrement & |
---|
| 726 | *ThetaSD**(NewtonMaxit-NewtonIter)/(ONE-ThetaSD) |
---|
| 727 | IF (NewtonPredictedErr >= NewtonTol) THEN |
---|
| 728 | ! Non-convergence of Newton: predicted error too large |
---|
| 729 | Qnewton = MIN(10.0d0,NewtonPredictedErr/NewtonTol) |
---|
| 730 | Fac = 0.8d0*Qnewton**(-ONE/(1+NewtonMaxit-NewtonIter)) |
---|
| 731 | EXIT SDNewtonLoop |
---|
| 732 | END IF |
---|
| 733 | ELSE ! Non-convergence of Newton: Theta too large |
---|
| 734 | print*,'Theta too large: ',ThetaSD |
---|
| 735 | EXIT SDNewtonLoop |
---|
| 736 | END IF |
---|
| 737 | END IF |
---|
| 738 | NewtonIncrementOld = NewtonIncrement |
---|
| 739 | ! Update solution: Z4 <-- Z4 + DZ4 |
---|
| 740 | CALL WAXPY(N,ONE,DZ4,1,Z4,1) |
---|
| 741 | |
---|
| 742 | ! Check error in Newton iterations |
---|
| 743 | NewtonDone = (NewtonRate*NewtonIncrement <= NewtonTol) |
---|
| 744 | IF (NewtonDone) EXIT SDNewtonLoop |
---|
| 745 | |
---|
| 746 | END DO SDNewtonLoop |
---|
| 747 | |
---|
| 748 | IF (.NOT.NewtonDone) THEN |
---|
| 749 | H = Fac*H; Reject=.TRUE.; SkipJac = .TRUE.; SkipLU = .FALSE. |
---|
| 750 | CYCLE Tloop |
---|
| 751 | END IF |
---|
| 752 | |
---|
| 753 | !~~~> End of implified SDIRK Newton iterations |
---|
| 754 | END IF |
---|
| 755 | |
---|
| 756 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 757 | !~~~> Error estimation, forward solution |
---|
| 758 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 759 | IF (SdirkError) THEN |
---|
| 760 | ! DZ4(1:N) = rkD(1)*Z1 + rkD(2)*Z2 + rkD(3)*Z3 - Z4 |
---|
| 761 | CALL Set2Zero(N, DZ4) |
---|
| 762 | IF (rkD(1) /= ZERO) CALL WAXPY(N, rkD(1), Z1, 1, DZ4, 1) |
---|
| 763 | IF (rkD(2) /= ZERO) CALL WAXPY(N, rkD(2), Z2, 1, DZ4, 1) |
---|
| 764 | IF (rkD(3) /= ZERO) CALL WAXPY(N, rkD(3), Z3, 1, DZ4, 1) |
---|
| 765 | CALL WAXPY(N, -ONE, Z4, 1, DZ4, 1) |
---|
| 766 | Err = RK_ErrorNorm(N,SCAL,DZ4) |
---|
| 767 | ELSE |
---|
| 768 | CALL RK_ErrorEstimate(N,H,Y,T, & |
---|
| 769 | E1,IP1,Z1,Z2,Z3,SCAL,Err,FirstStep,Reject) |
---|
| 770 | END IF |
---|
| 771 | |
---|
| 772 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 773 | !~~~> If error small enough, compute TLM solution |
---|
| 774 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 775 | IF (Err < ONE) THEN |
---|
| 776 | |
---|
| 777 | !~~~> Jacobian values |
---|
| 778 | CALL WADD(N,Y,Z1,TMP) ! TMP <- Y + Z1 |
---|
| 779 | CALL JAC_CHEM(T+rkC(1)*H,TMP,Jac1) ! Jac1 <- Jac(Y+Z1) |
---|
| 780 | CALL WADD(N,Y,Z2,TMP) ! TMP <- Y + Z2 |
---|
| 781 | CALL JAC_CHEM(T+rkC(2)*H,TMP,Jac2) ! Jac2 <- Jac(Y+Z2) |
---|
| 782 | CALL WADD(N,Y,Z3,TMP) ! TMP <- Y + Z3 |
---|
| 783 | CALL JAC_CHEM(T+rkC(3)*H,TMP,Jac3) ! Jac3 <- Jac(Y+Z3) |
---|
| 784 | |
---|
| 785 | TLMDIR: IF (TLMDirect) THEN |
---|
| 786 | |
---|
| 787 | #ifdef FULL_ALGEBRA |
---|
| 788 | !~~~> Construct the full Jacobian |
---|
| 789 | DO j=1,N |
---|
| 790 | DO i=1,N |
---|
| 791 | Jbig(i,j) = -H*rkA(1,1)*Jac1(i,j) |
---|
| 792 | Jbig(N+i,j) = -H*rkA(2,1)*Jac1(i,j) |
---|
| 793 | Jbig(2*N+i,j) = -H*rkA(3,1)*Jac1(i,j) |
---|
| 794 | Jbig(i,N+j) = -H*rkA(1,2)*Jac2(i,j) |
---|
| 795 | Jbig(N+i,N+j) = -H*rkA(2,2)*Jac2(i,j) |
---|
| 796 | Jbig(2*N+i,N+j) = -H*rkA(3,2)*Jac2(i,j) |
---|
| 797 | Jbig(i,2*N+j) = -H*rkA(1,3)*Jac3(i,j) |
---|
| 798 | Jbig(N+i,2*N+j) = -H*rkA(2,3)*Jac3(i,j) |
---|
| 799 | Jbig(2*N+i,2*N+j) = -H*rkA(3,3)*Jac3(i,j) |
---|
| 800 | END DO |
---|
| 801 | END DO |
---|
| 802 | Ebig = -Jbig |
---|
| 803 | DO i=1,3*NVAR |
---|
| 804 | Jbig(i,i) = ONE + Jbig(i,i) |
---|
| 805 | END DO |
---|
| 806 | !~~~> Solve the big system |
---|
| 807 | ! CALL DGETRF(3*NVAR,3*NVAR,Jbig,3*NVAR,IPbig,j) |
---|
| 808 | CALL WGEFA(3*N,Jbig,IPbig,info) |
---|
| 809 | IF (info /= 0) THEN |
---|
| 810 | PRINT*,'Big big guy is singular'; STOP |
---|
| 811 | END IF |
---|
| 812 | DO itlm = 1, NTLM |
---|
| 813 | DO j=1,NVAR |
---|
| 814 | Zbig(j) = Y_tlm(j,itlm) |
---|
| 815 | Zbig(NVAR+j) = Y_tlm(j,itlm) |
---|
| 816 | Zbig(2*NVAR+j) = Y_tlm(j,itlm) |
---|
| 817 | END DO |
---|
| 818 | Zbig = MATMUL(Ebig,Zbig) |
---|
| 819 | !CALL DGETRS ('N',3*NVAR,1,Jbig,3*NVAR,IPbig,Zbig,3*NVAR,0) |
---|
| 820 | CALL WGESL('N',3*N,Jbig,IPbig,Zbig) |
---|
| 821 | DO j=1,NVAR |
---|
| 822 | Z1_tlm(j,itlm) = Zbig(j) |
---|
| 823 | Z2_tlm(j,itlm) = Zbig(NVAR+j) |
---|
| 824 | Z3_tlm(j,itlm) = Zbig(2*NVAR+j) |
---|
| 825 | END DO |
---|
| 826 | END DO |
---|
| 827 | #else |
---|
| 828 | ! Commented code for sparse big linear algebra: |
---|
| 829 | ! !~~~> Construct the big Jacobian |
---|
| 830 | ! DO j=1,LU_NONZERO |
---|
| 831 | ! DO i=1,3 |
---|
| 832 | ! Jbig(i,1,j) = -H*rkA(i,1)*Jac1(j) |
---|
| 833 | ! Jbig(i,2,j) = -H*rkA(i,2)*Jac2(j) |
---|
| 834 | ! Jbig(i,3,j) = -H*rkA(i,3)*Jac3(j) |
---|
| 835 | ! END DO |
---|
| 836 | ! END DO |
---|
| 837 | ! DO j=1,NVAR |
---|
| 838 | ! DO i=1,3 |
---|
| 839 | ! Jbig(i,i,LU_DIAG(j)) = ONE + Jbig(i,i,LU_DIAG(j)) |
---|
| 840 | ! END DO |
---|
| 841 | ! END DO |
---|
| 842 | ! !~~~> Solve the big system |
---|
| 843 | ! CALL KppDecompBig( Jbig, IPbig, j ) |
---|
| 844 | ! Use full big linear algebra: |
---|
| 845 | Jbig(1:3*N,1:3*N) = 0.0d0 |
---|
| 846 | DO i=1,LU_NONZERO |
---|
| 847 | Jbig(LU_irow(i),LU_icol(i)) = -H*rkA(1,1)*Jac1(i) |
---|
| 848 | Jbig(LU_irow(i),N+LU_icol(i)) = -H*rkA(1,2)*Jac2(i) |
---|
| 849 | Jbig(LU_irow(i),2*N+LU_icol(i)) = -H*rkA(1,3)*Jac3(i) |
---|
| 850 | Jbig(N+LU_irow(i),LU_icol(i)) = -H*rkA(2,1)*Jac1(i) |
---|
| 851 | Jbig(N+LU_irow(i),N+LU_icol(i)) = -H*rkA(2,2)*Jac2(i) |
---|
| 852 | Jbig(N+LU_irow(i),2*N+LU_icol(i)) = -H*rkA(2,3)*Jac3(i) |
---|
| 853 | Jbig(2*N+LU_irow(i),LU_icol(i)) = -H*rkA(3,1)*Jac1(i) |
---|
| 854 | Jbig(2*N+LU_irow(i),N+LU_icol(i)) = -H*rkA(3,2)*Jac2(i) |
---|
| 855 | Jbig(2*N+LU_irow(i),2*N+LU_icol(i)) = -H*rkA(3,3)*Jac3(i) |
---|
| 856 | END DO |
---|
| 857 | Ebig = -Jbig |
---|
| 858 | DO i=1, 3*N |
---|
| 859 | Jbig(i,i) = ONE + Jbig(i,i) |
---|
| 860 | END DO |
---|
| 861 | ! CALL DGETRF(3*N,3*N,Jbig,3*N,IPbig,info) |
---|
| 862 | CALL WGEFA(3*N,Jbig,IPbig,info) |
---|
| 863 | IF (info /= 0) THEN |
---|
| 864 | PRINT*,'Big guy is singular'; STOP |
---|
| 865 | END IF |
---|
| 866 | |
---|
| 867 | DO itlm = 1, NTLM |
---|
| 868 | ! Commented code for sparse big linear algebra: |
---|
| 869 | ! ! Compute RHS |
---|
| 870 | ! CALL RK_PrepareRHS_TLMdirect(N,H,Jac1,Jac2,Jac3,Y_tlm(1,itlm),Zbig) |
---|
| 871 | ! ! Solve the system |
---|
| 872 | ! CALL KppSolveBig( Jbig, IPbig, Zbig ) |
---|
| 873 | ! DO j=1,NVAR |
---|
| 874 | ! Z1_tlm(j,itlm) = Zbig(1,j) |
---|
| 875 | ! Z2_tlm(j,itlm) = Zbig(2,j) |
---|
| 876 | ! Z3_tlm(j,itlm) = Zbig(3,j) |
---|
| 877 | ! END DO |
---|
| 878 | ! Compute RHS |
---|
| 879 | CALL RK_PrepareRHS_TLMdirect(N,H,Jac1,Jac2,Jac3,Y_tlm(1,itlm),Zbig) |
---|
| 880 | ! Solve the system |
---|
| 881 | ! CALL DGETRS('N',3*N,1,Jbig,3*N,IPbig,Zbig,3*N,0) |
---|
| 882 | CALL WGESL('N',3*N,Jbig,IPbig,Zbig) |
---|
| 883 | Z1_tlm(1:NVAR,itlm) = Zbig(1:NVAR) |
---|
| 884 | Z2_tlm(1:NVAR,itlm) = Zbig(NVAR+1:2*NVAR) |
---|
| 885 | Z3_tlm(1:NVAR,itlm) = Zbig(2*NVAR+1:3*NVAR) |
---|
| 886 | END DO |
---|
| 887 | #endif |
---|
| 888 | |
---|
| 889 | ELSE TLMDIR |
---|
| 890 | |
---|
| 891 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 892 | !~~~> Loop for Newton iterations, TLM variables |
---|
| 893 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 894 | |
---|
| 895 | Tlm:DO itlm = 1, NTLM |
---|
| 896 | |
---|
| 897 | !~~~> Starting values for Newton iteration |
---|
| 898 | CALL Set2zero(N,Z1_tlm(1,itlm)) |
---|
| 899 | CALL Set2zero(N,Z2_tlm(1,itlm)) |
---|
| 900 | CALL Set2zero(N,Z3_tlm(1,itlm)) |
---|
| 901 | |
---|
| 902 | !~~~> Initializations for Newton iteration |
---|
| 903 | IF (TLMNewtonEst) THEN |
---|
| 904 | NewtonDone = .FALSE. |
---|
| 905 | Fac = 0.5d0 ! Step reduction if too many iterations |
---|
| 906 | |
---|
| 907 | CALL RK_ErrorScale(N,ITOL,AbsTol_tlm(1,itlm),RelTol_tlm(1,itlm), & |
---|
| 908 | Y_tlm(1,itlm),SCAL_tlm) |
---|
| 909 | END IF |
---|
| 910 | |
---|
| 911 | NewtonLoopTLM:DO NewtonIterTLM = 1, NewtonMaxit |
---|
| 912 | |
---|
| 913 | !~~~> Prepare the right-hand side |
---|
| 914 | CALL RK_PrepareRHS_TLM(N,H,Jac1,Jac2,Jac3,Y_tlm(1,itlm), & |
---|
| 915 | Z1_tlm(1,itlm),Z2_tlm(1,itlm),Z3_tlm(1,itlm), & |
---|
| 916 | DZ1,DZ2,DZ3) |
---|
| 917 | |
---|
| 918 | !~~~> Solve the linear systems |
---|
| 919 | CALL RK_Solve( N,H,E1,IP1,E2,IP2,DZ1,DZ2,DZ3,ISING ) |
---|
| 920 | |
---|
| 921 | IF (TLMNewtonEst) THEN |
---|
| 922 | !~~~> Check convergence of Newton iterations |
---|
| 923 | NewtonIncrement = SQRT( ( RK_ErrorNorm(N,SCAL_tlm,DZ1)**2 + & |
---|
| 924 | RK_ErrorNorm(N,SCAL_tlm,DZ2)**2 + & |
---|
| 925 | RK_ErrorNorm(N,SCAL_tlm,DZ3)**2 )/3.0d0 ) |
---|
| 926 | IF ( NewtonIterTLM == 1 ) THEN |
---|
| 927 | ThetaTLM = ABS(ThetaMin) |
---|
| 928 | NewtonRate = 2.0d0 |
---|
| 929 | ELSE |
---|
| 930 | ThetaTLM = NewtonIncrement/NewtonIncrementOld |
---|
| 931 | IF (ThetaTLM < 0.99d0) THEN |
---|
| 932 | NewtonRate = ThetaTLM/(ONE-ThetaTLM) |
---|
| 933 | ELSE ! Non-convergence of Newton: ThetaTLM too large |
---|
| 934 | EXIT NewtonLoopTLM |
---|
| 935 | END IF |
---|
| 936 | IF ( NewtonIterTLM < NewtonMaxit ) THEN |
---|
| 937 | ! Predict error at the end of Newton process |
---|
| 938 | NewtonPredictedErr = NewtonIncrement & |
---|
| 939 | *ThetaTLM**(NewtonMaxit-NewtonIterTLM)/(ONE-ThetaTLM) |
---|
| 940 | IF (NewtonPredictedErr >= NewtonTol) THEN |
---|
| 941 | ! Non-convergence of Newton: predicted error too large |
---|
| 942 | Qnewton = MIN(10.0d0,NewtonPredictedErr/NewtonTol) |
---|
| 943 | Fac=0.8d0*Qnewton**(-ONE/(1+NewtonMaxit-NewtonIterTLM)) |
---|
| 944 | EXIT NewtonLoopTLM |
---|
| 945 | END IF |
---|
| 946 | END IF |
---|
| 947 | END IF |
---|
| 948 | |
---|
| 949 | NewtonIncrementOld = MAX(NewtonIncrement,Roundoff) |
---|
| 950 | END IF !(TLMNewtonEst) |
---|
| 951 | ! Update solution |
---|
| 952 | CALL WAXPY(N,-ONE,DZ1,1,Z1_tlm(1,itlm),1) ! Z1 <- Z1 - DZ1 |
---|
| 953 | CALL WAXPY(N,-ONE,DZ2,1,Z2_tlm(1,itlm),1) ! Z2 <- Z2 - DZ2 |
---|
| 954 | CALL WAXPY(N,-ONE,DZ3,1,Z3_tlm(1,itlm),1) ! Z3 <- Z3 - DZ3 |
---|
| 955 | |
---|
| 956 | ! Check error in Newton iterations |
---|
| 957 | IF (TLMNewtonEst) THEN |
---|
| 958 | NewtonDone = (NewtonRate*NewtonIncrement <= NewtonTol) |
---|
| 959 | IF (NewtonDone) EXIT NewtonLoopTLM |
---|
| 960 | ELSE |
---|
| 961 | ! Minimum number of iterations same as FWD iterations |
---|
| 962 | IF (NewtonIterTLM>=saveNiter) EXIT NewtonLoopTLM |
---|
| 963 | END IF |
---|
| 964 | |
---|
| 965 | END DO NewtonLoopTLM |
---|
| 966 | |
---|
| 967 | IF ((TLMNewtonEst) .AND. (.NOT.NewtonDone)) THEN |
---|
| 968 | !CALL RK_ErrorMsg(-12,T,H,IERR); |
---|
| 969 | H = Fac*H; Reject=.TRUE.; SkipJac = .TRUE.; SkipLU = .FALSE. |
---|
| 970 | CYCLE Tloop |
---|
| 971 | END IF |
---|
| 972 | |
---|
| 973 | END DO Tlm |
---|
| 974 | |
---|
| 975 | END IF TLMDIR |
---|
| 976 | |
---|
| 977 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 978 | !~~~> Error estimation, TLM solution |
---|
| 979 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 980 | IF (TLMtruncErr) THEN |
---|
| 981 | CALL RK_ErrorEstimate_tlm(N,NTLM,T,H,FJAC,Y, Y_tlm, & |
---|
| 982 | E1,IP1,Z1_tlm,Z2_tlm,Z3_tlm,Err,FirstStep,Reject) |
---|
| 983 | END IF |
---|
| 984 | |
---|
| 985 | END IF ! (Err<ONE) |
---|
| 986 | |
---|
| 987 | |
---|
| 988 | !~~~> Computation of new step size Hnew |
---|
| 989 | Fac = Err**(-ONE/rkELO)* & |
---|
| 990 | MIN(FacSafe,(ONE+2*NewtonMaxit)/(NewtonIter+2*NewtonMaxit)) |
---|
| 991 | Fac = MIN(FacMax,MAX(FacMin,Fac)) |
---|
| 992 | Hnew = Fac*H |
---|
| 993 | |
---|
| 994 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 995 | !~~~> Accept/reject step |
---|
| 996 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 997 | accept:IF (Err < ONE) THEN !~~~> STEP IS ACCEPTED |
---|
| 998 | |
---|
| 999 | FirstStep=.FALSE. |
---|
| 1000 | ISTATUS(Nacc) = ISTATUS(Nacc) + 1 |
---|
| 1001 | IF (Gustafsson) THEN |
---|
| 1002 | !~~~> Predictive controller of Gustafsson |
---|
| 1003 | IF (ISTATUS(Nacc) > 1) THEN |
---|
| 1004 | FacGus=FacSafe*(H/Hacc)*(Err**2/ErrOld)**(-0.25d0) |
---|
| 1005 | FacGus=MIN(FacMax,MAX(FacMin,FacGus)) |
---|
| 1006 | Fac=MIN(Fac,FacGus) |
---|
| 1007 | Hnew = Fac*H |
---|
| 1008 | END IF |
---|
| 1009 | Hacc=H |
---|
| 1010 | ErrOld=MAX(1.0d-2,Err) |
---|
| 1011 | END IF |
---|
| 1012 | Hold = H |
---|
| 1013 | T = T+H |
---|
| 1014 | ! Update solution: Y <- Y + sum(d_i Z_i) |
---|
| 1015 | IF (rkD(1)/=ZERO) CALL WAXPY(N,rkD(1),Z1,1,Y,1) |
---|
| 1016 | IF (rkD(2)/=ZERO) CALL WAXPY(N,rkD(2),Z2,1,Y,1) |
---|
| 1017 | IF (rkD(3)/=ZERO) CALL WAXPY(N,rkD(3),Z3,1,Y,1) |
---|
| 1018 | ! Update TLM solution: Y <- Y + sum(d_i*Z_i_tlm) |
---|
| 1019 | DO itlm = 1,NTLM |
---|
| 1020 | IF (rkD(1)/=ZERO) CALL WAXPY(N,rkD(1),Z1_tlm(1,itlm),1,Y_tlm(1,itlm),1) |
---|
| 1021 | IF (rkD(2)/=ZERO) CALL WAXPY(N,rkD(2),Z2_tlm(1,itlm),1,Y_tlm(1,itlm),1) |
---|
| 1022 | IF (rkD(3)/=ZERO) CALL WAXPY(N,rkD(3),Z3_tlm(1,itlm),1,Y_tlm(1,itlm),1) |
---|
| 1023 | END DO |
---|
| 1024 | ! Construct the solution quadratic interpolant Q(c_i) = Z_i, i=1:3 |
---|
| 1025 | IF (StartNewton) CALL RK_Interpolate('make',N,H,Hold,Z1,Z2,Z3,CONT) |
---|
| 1026 | CALL RK_ErrorScale(N,ITOL,AbsTol,RelTol,Y,SCAL) |
---|
| 1027 | RSTATUS(Ntexit) = T |
---|
| 1028 | RSTATUS(Nhnew) = Hnew |
---|
| 1029 | RSTATUS(Nhacc) = H |
---|
| 1030 | Hnew = Tdirection*MIN( MAX(ABS(Hnew),Hmin) , Hmax ) |
---|
| 1031 | IF (Reject) Hnew = Tdirection*MIN(ABS(Hnew),ABS(H)) |
---|
| 1032 | Reject = .FALSE. |
---|
| 1033 | IF ((T+Hnew/Qmin-Tend)*Tdirection >= ZERO) THEN |
---|
| 1034 | H = Tend-T |
---|
| 1035 | ELSE |
---|
| 1036 | Hratio=Hnew/H |
---|
| 1037 | ! Reuse the LU decomposition |
---|
| 1038 | SkipLU = (Theta<=ThetaMin) .AND. (Hratio>=Qmin) .AND. (Hratio<=Qmax) |
---|
| 1039 | ! For TLM: do not skip LU (decrease TLM error) |
---|
| 1040 | SkipLU = .FALSE. |
---|
| 1041 | IF (.NOT.SkipLU) H=Hnew |
---|
| 1042 | END IF |
---|
| 1043 | ! If convergence is fast enough, do not update Jacobian |
---|
| 1044 | ! SkipJac = (Theta <= ThetaMin) |
---|
| 1045 | SkipJac = .FALSE. ! For TLM: do not skip jac |
---|
| 1046 | |
---|
| 1047 | ELSE accept !~~~> Step is rejected |
---|
| 1048 | |
---|
| 1049 | IF (FirstStep .OR. Reject) THEN |
---|
| 1050 | H = FacRej*H |
---|
| 1051 | ELSE |
---|
| 1052 | H = Hnew |
---|
| 1053 | END IF |
---|
| 1054 | Reject = .TRUE. |
---|
| 1055 | SkipJac = .TRUE. |
---|
| 1056 | SkipLU = .FALSE. |
---|
| 1057 | IF (ISTATUS(Nacc) >= 1) ISTATUS(Nrej) = ISTATUS(Nrej) + 1 |
---|
| 1058 | |
---|
| 1059 | END IF accept |
---|
| 1060 | |
---|
| 1061 | END DO Tloop |
---|
| 1062 | |
---|
| 1063 | ! Successful exit |
---|
| 1064 | IERR = 1 |
---|
| 1065 | |
---|
| 1066 | END SUBROUTINE RK_IntegratorTLM |
---|
| 1067 | |
---|
| 1068 | |
---|
| 1069 | |
---|
| 1070 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1071 | SUBROUTINE RK_ErrorMsg(Code,T,H,IERR) |
---|
| 1072 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1073 | ! Handles all error messages |
---|
| 1074 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1075 | |
---|
| 1076 | IMPLICIT NONE |
---|
| 1077 | KPP_REAL, INTENT(IN) :: T, H |
---|
| 1078 | INTEGER, INTENT(IN) :: Code |
---|
| 1079 | INTEGER, INTENT(OUT) :: IERR |
---|
| 1080 | |
---|
| 1081 | IERR = Code |
---|
| 1082 | PRINT * , & |
---|
| 1083 | 'Forced exit from RungeKutta due to the following error:' |
---|
| 1084 | |
---|
| 1085 | |
---|
| 1086 | SELECT CASE (Code) |
---|
| 1087 | CASE (-1) |
---|
| 1088 | PRINT * , '--> Improper value for maximal no of steps' |
---|
| 1089 | CASE (-2) |
---|
| 1090 | PRINT * , '--> Improper value for maximal no of Newton iterations' |
---|
| 1091 | CASE (-3) |
---|
| 1092 | PRINT * , '--> Hmin/Hmax/Hstart must be positive' |
---|
| 1093 | CASE (-4) |
---|
| 1094 | PRINT * , '--> Improper values for FacMin/FacMax/FacSafe/FacRej' |
---|
| 1095 | CASE (-5) |
---|
| 1096 | PRINT * , '--> Improper value for ThetaMin' |
---|
| 1097 | CASE (-6) |
---|
| 1098 | PRINT * , '--> Newton stopping tolerance too small' |
---|
| 1099 | CASE (-7) |
---|
| 1100 | PRINT * , '--> Improper values for Qmin, Qmax' |
---|
| 1101 | CASE (-8) |
---|
| 1102 | PRINT * , '--> Tolerances are too small' |
---|
| 1103 | CASE (-9) |
---|
| 1104 | PRINT * , '--> No of steps exceeds maximum bound' |
---|
| 1105 | CASE (-10) |
---|
| 1106 | PRINT * , '--> Step size too small: T + 10*H = T', & |
---|
| 1107 | ' or H < Roundoff' |
---|
| 1108 | CASE (-11) |
---|
| 1109 | PRINT * , '--> Matrix is repeatedly singular' |
---|
| 1110 | CASE (-12) |
---|
| 1111 | PRINT * , '--> Non-convergence of Newton iterations' |
---|
| 1112 | CASE (-13) |
---|
| 1113 | PRINT * , '--> Requested RK method not implemented' |
---|
| 1114 | CASE DEFAULT |
---|
| 1115 | PRINT *, 'Unknown Error code: ', Code |
---|
| 1116 | END SELECT |
---|
| 1117 | |
---|
| 1118 | WRITE(6,FMT="(5X,'T=',E12.5,' H=',E12.5)") T, H |
---|
| 1119 | |
---|
| 1120 | END SUBROUTINE RK_ErrorMsg |
---|
| 1121 | |
---|
| 1122 | |
---|
| 1123 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1124 | SUBROUTINE RK_ErrorScale(N,ITOL,AbsTol,RelTol,Y,SCAL) |
---|
| 1125 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1126 | ! Handles all error messages |
---|
| 1127 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1128 | IMPLICIT NONE |
---|
| 1129 | INTEGER, INTENT(IN) :: N, ITOL |
---|
| 1130 | KPP_REAL, INTENT(IN) :: AbsTol(*), RelTol(*), Y(N) |
---|
| 1131 | KPP_REAL, INTENT(OUT) :: SCAL(N) |
---|
| 1132 | INTEGER :: i |
---|
| 1133 | |
---|
| 1134 | IF (ITOL==0) THEN |
---|
| 1135 | DO i=1,N |
---|
| 1136 | SCAL(i)= ONE/(AbsTol(1)+RelTol(1)*ABS(Y(i))) |
---|
| 1137 | END DO |
---|
| 1138 | ELSE |
---|
| 1139 | DO i=1,N |
---|
| 1140 | SCAL(i)=ONE/(AbsTol(i)+RelTol(i)*ABS(Y(i))) |
---|
| 1141 | END DO |
---|
| 1142 | END IF |
---|
| 1143 | |
---|
| 1144 | END SUBROUTINE RK_ErrorScale |
---|
| 1145 | |
---|
| 1146 | |
---|
| 1147 | !!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1148 | ! SUBROUTINE RK_Transform(N,Tr,Z1,Z2,Z3,W1,W2,W3) |
---|
| 1149 | !!~~~> W <-- Tr x Z |
---|
| 1150 | !!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1151 | ! IMPLICIT NONE |
---|
| 1152 | ! INTEGER :: N, i |
---|
| 1153 | ! KPP_REAL :: Tr(3,3),Z1(N),Z2(N),Z3(N),W1(N),W2(N),W3(N) |
---|
| 1154 | ! KPP_REAL :: x1, x2, x3 |
---|
| 1155 | ! DO i=1,N |
---|
| 1156 | ! x1 = Z1(i); x2 = Z2(i); x3 = Z3(i) |
---|
| 1157 | ! W1(i) = Tr(1,1)*x1 + Tr(1,2)*x2 + Tr(1,3)*x3 |
---|
| 1158 | ! W2(i) = Tr(2,1)*x1 + Tr(2,2)*x2 + Tr(2,3)*x3 |
---|
| 1159 | ! W3(i) = Tr(3,1)*x1 + Tr(3,2)*x2 + Tr(3,3)*x3 |
---|
| 1160 | ! END DO |
---|
| 1161 | ! END SUBROUTINE RK_Transform |
---|
| 1162 | |
---|
| 1163 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1164 | SUBROUTINE RK_Interpolate(action,N,H,Hold,Z1,Z2,Z3,CONT) |
---|
| 1165 | !~~~> Constructs or evaluates a quadratic polynomial |
---|
| 1166 | ! that interpolates the Z solution at current step |
---|
| 1167 | ! and provides starting values for the next step |
---|
| 1168 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1169 | INTEGER, INTENT(IN) :: N |
---|
| 1170 | KPP_REAL, INTENT(IN) :: H,Hold |
---|
| 1171 | KPP_REAL, INTENT(INOUT) :: Z1(N),Z2(N),Z3(N),CONT(N,3) |
---|
| 1172 | CHARACTER(LEN=4), INTENT(IN) :: action |
---|
| 1173 | KPP_REAL :: r, x1, x2, x3, den |
---|
| 1174 | INTEGER :: i |
---|
| 1175 | |
---|
| 1176 | SELECT CASE (action) |
---|
| 1177 | CASE ('make') |
---|
| 1178 | ! Construct the solution quadratic interpolant Q(c_i) = Z_i, i=1:3 |
---|
| 1179 | den = (rkC(3)-rkC(2))*(rkC(2)-rkC(1))*(rkC(1)-rkC(3)) |
---|
| 1180 | DO i=1,N |
---|
| 1181 | CONT(i,1)=(-rkC(3)**2*rkC(2)*Z1(i)+Z3(i)*rkC(2)*rkC(1)**2 & |
---|
| 1182 | +rkC(2)**2*rkC(3)*Z1(i)-rkC(2)**2*rkC(1)*Z3(i) & |
---|
| 1183 | +rkC(3)**2*rkC(1)*Z2(i)-Z2(i)*rkC(3)*rkC(1)**2)& |
---|
| 1184 | /den-Z3(i) |
---|
| 1185 | CONT(i,2)= -( rkC(1)**2*(Z3(i)-Z2(i)) + rkC(2)**2*(Z1(i) & |
---|
| 1186 | -Z3(i)) +rkC(3)**2*(Z2(i)-Z1(i)) )/den |
---|
| 1187 | CONT(i,3)= ( rkC(1)*(Z3(i)-Z2(i)) + rkC(2)*(Z1(i)-Z3(i)) & |
---|
| 1188 | +rkC(3)*(Z2(i)-Z1(i)) )/den |
---|
| 1189 | END DO |
---|
| 1190 | CASE ('eval') |
---|
| 1191 | ! Evaluate quadratic polynomial |
---|
| 1192 | r = H/Hold |
---|
| 1193 | x1 = ONE + rkC(1)*r |
---|
| 1194 | x2 = ONE + rkC(2)*r |
---|
| 1195 | x3 = ONE + rkC(3)*r |
---|
| 1196 | DO i=1,N |
---|
| 1197 | Z1(i) = CONT(i,1)+x1*(CONT(i,2)+x1*CONT(i,3)) |
---|
| 1198 | Z2(i) = CONT(i,1)+x2*(CONT(i,2)+x2*CONT(i,3)) |
---|
| 1199 | Z3(i) = CONT(i,1)+x3*(CONT(i,2)+x3*CONT(i,3)) |
---|
| 1200 | END DO |
---|
| 1201 | END SELECT |
---|
| 1202 | END SUBROUTINE RK_Interpolate |
---|
| 1203 | |
---|
| 1204 | |
---|
| 1205 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1206 | SUBROUTINE RK_PrepareRHS(N,T,H,Y,Z1,Z2,Z3,R1,R2,R3) |
---|
| 1207 | !~~~> Prepare the right-hand side for Newton iterations |
---|
| 1208 | ! R = Z - hA x F |
---|
| 1209 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1210 | IMPLICIT NONE |
---|
| 1211 | |
---|
| 1212 | INTEGER, INTENT(IN) :: N |
---|
| 1213 | KPP_REAL, INTENT(IN) :: T, H |
---|
| 1214 | KPP_REAL, INTENT(IN), DIMENSION(N) :: Y,Z1,Z2,Z3 |
---|
| 1215 | KPP_REAL, INTENT(INOUT), DIMENSION(N) :: R1,R2,R3 |
---|
| 1216 | KPP_REAL, DIMENSION(N) :: F, TMP |
---|
| 1217 | |
---|
| 1218 | CALL WCOPY(N,Z1,1,R1,1) ! R1 <- Z1 |
---|
| 1219 | CALL WCOPY(N,Z2,1,R2,1) ! R2 <- Z2 |
---|
| 1220 | CALL WCOPY(N,Z3,1,R3,1) ! R3 <- Z3 |
---|
| 1221 | |
---|
| 1222 | CALL WADD(N,Y,Z1,TMP) ! TMP <- Y + Z1 |
---|
| 1223 | CALL FUN_CHEM(T+rkC(1)*H,TMP,F) ! F1 <- Fun(Y+Z1) |
---|
| 1224 | CALL WAXPY(N,-H*rkA(1,1),F,1,R1,1) ! R1 <- R1 - h*A_11*F1 |
---|
| 1225 | CALL WAXPY(N,-H*rkA(2,1),F,1,R2,1) ! R2 <- R2 - h*A_21*F1 |
---|
| 1226 | CALL WAXPY(N,-H*rkA(3,1),F,1,R3,1) ! R3 <- R3 - h*A_31*F1 |
---|
| 1227 | |
---|
| 1228 | CALL WADD(N,Y,Z2,TMP) ! TMP <- Y + Z2 |
---|
| 1229 | CALL FUN_CHEM(T+rkC(2)*H,TMP,F) ! F2 <- Fun(Y+Z2) |
---|
| 1230 | CALL WAXPY(N,-H*rkA(1,2),F,1,R1,1) ! R1 <- R1 - h*A_12*F2 |
---|
| 1231 | CALL WAXPY(N,-H*rkA(2,2),F,1,R2,1) ! R2 <- R2 - h*A_22*F2 |
---|
| 1232 | CALL WAXPY(N,-H*rkA(3,2),F,1,R3,1) ! R3 <- R3 - h*A_32*F2 |
---|
| 1233 | |
---|
| 1234 | CALL WADD(N,Y,Z3,TMP) ! TMP <- Y + Z3 |
---|
| 1235 | CALL FUN_CHEM(T+rkC(3)*H,TMP,F) ! F3 <- Fun(Y+Z3) |
---|
| 1236 | CALL WAXPY(N,-H*rkA(1,3),F,1,R1,1) ! R1 <- R1 - h*A_13*F3 |
---|
| 1237 | CALL WAXPY(N,-H*rkA(2,3),F,1,R2,1) ! R2 <- R2 - h*A_23*F3 |
---|
| 1238 | CALL WAXPY(N,-H*rkA(3,3),F,1,R3,1) ! R3 <- R3 - h*A_33*F3 |
---|
| 1239 | |
---|
| 1240 | END SUBROUTINE RK_PrepareRHS |
---|
| 1241 | |
---|
| 1242 | |
---|
| 1243 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1244 | SUBROUTINE RK_PrepareRHS_TLM(N,H,Jac1,Jac2,Jac3,Y_tlm, & |
---|
| 1245 | Z1_tlm,Z2_tlm,Z3_tlm,R1,R2,R3) |
---|
| 1246 | !~~~> Prepare the right-hand side for Newton iterations |
---|
| 1247 | ! R = Z_tlm - hA x Jac*Z_tlm |
---|
| 1248 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1249 | IMPLICIT NONE |
---|
| 1250 | |
---|
| 1251 | INTEGER, INTENT(IN) :: N |
---|
| 1252 | KPP_REAL, INTENT(IN) :: H |
---|
| 1253 | KPP_REAL, INTENT(IN), DIMENSION(N) :: Y_tlm,Z1_tlm,Z2_tlm,Z3_tlm |
---|
| 1254 | KPP_REAL, INTENT(INOUT), DIMENSION(N) :: R1,R2,R3 |
---|
| 1255 | #ifdef FULL_ALGEBRA |
---|
| 1256 | KPP_REAL, INTENT(IN), DIMENSION(NVAR,NVAR) :: Jac1, Jac2, Jac3 |
---|
| 1257 | #else |
---|
| 1258 | KPP_REAL, INTENT(IN), DIMENSION(LU_NONZERO) :: Jac1, Jac2, Jac3 |
---|
| 1259 | #endif |
---|
| 1260 | KPP_REAL, DIMENSION(N) :: F, TMP |
---|
| 1261 | |
---|
| 1262 | CALL WCOPY(N,Z1_tlm,1,R1,1) ! R1 <- Z1_tlm |
---|
| 1263 | CALL WCOPY(N,Z2_tlm,1,R2,1) ! R2 <- Z2_tlm |
---|
| 1264 | CALL WCOPY(N,Z3_tlm,1,R3,1) ! R3 <- Z3_tlm |
---|
| 1265 | |
---|
| 1266 | CALL WADD(N,Y_tlm,Z1_tlm,TMP) ! TMP <- Y + Z1 |
---|
| 1267 | #ifdef FULL_ALGEBRA |
---|
| 1268 | F = MATMUL(Jac1,TMP) |
---|
| 1269 | #else |
---|
| 1270 | CALL Jac_SP_Vec ( Jac1, TMP, F ) ! F1 <- Jac(Y+Z1)*(Y_tlm+Z1_tlm) |
---|
| 1271 | #endif |
---|
| 1272 | CALL WAXPY(N,-H*rkA(1,1),F,1,R1,1) ! R1 <- R1 - h*A_11*F1 |
---|
| 1273 | CALL WAXPY(N,-H*rkA(2,1),F,1,R2,1) ! R2 <- R2 - h*A_21*F1 |
---|
| 1274 | CALL WAXPY(N,-H*rkA(3,1),F,1,R3,1) ! R3 <- R3 - h*A_31*F1 |
---|
| 1275 | |
---|
| 1276 | CALL WADD(N,Y_tlm,Z2_tlm,TMP) ! TMP <- Y + Z2 |
---|
| 1277 | #ifdef FULL_ALGEBRA |
---|
| 1278 | F = MATMUL(Jac2,TMP) |
---|
| 1279 | #else |
---|
| 1280 | CALL Jac_SP_Vec ( Jac2, TMP, F ) ! F2 <- Jac(Y+Z2)*(Y_tlm+Z2_tlm) |
---|
| 1281 | #endif |
---|
| 1282 | CALL WAXPY(N,-H*rkA(1,2),F,1,R1,1) ! R1 <- R1 - h*A_12*F2 |
---|
| 1283 | CALL WAXPY(N,-H*rkA(2,2),F,1,R2,1) ! R2 <- R2 - h*A_22*F2 |
---|
| 1284 | CALL WAXPY(N,-H*rkA(3,2),F,1,R3,1) ! R3 <- R3 - h*A_32*F2 |
---|
| 1285 | |
---|
| 1286 | CALL WADD(N,Y_tlm,Z3_tlm,TMP) ! TMP <- Y + Z3 |
---|
| 1287 | #ifdef FULL_ALGEBRA |
---|
| 1288 | F = MATMUL(Jac3,TMP) |
---|
| 1289 | #else |
---|
| 1290 | CALL Jac_SP_Vec ( Jac3, TMP, F ) ! F3 <- Jac(Y+Z3)*(Y_tlm+Z3_tlm) |
---|
| 1291 | #endif |
---|
| 1292 | CALL WAXPY(N,-H*rkA(1,3),F,1,R1,1) ! R1 <- R1 - h*A_13*F3 |
---|
| 1293 | CALL WAXPY(N,-H*rkA(2,3),F,1,R2,1) ! R2 <- R2 - h*A_23*F3 |
---|
| 1294 | CALL WAXPY(N,-H*rkA(3,3),F,1,R3,1) ! R3 <- R3 - h*A_33*F3 |
---|
| 1295 | |
---|
| 1296 | END SUBROUTINE RK_PrepareRHS_TLM |
---|
| 1297 | |
---|
| 1298 | |
---|
| 1299 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1300 | SUBROUTINE RK_PrepareRHS_TLMdirect(N,H,Jac1,Jac2,Jac3,Y_tlm,Zbig) |
---|
| 1301 | !~~~> Prepare the right-hand side for direct solution |
---|
| 1302 | ! Z = hA x Jac*Y_tlm |
---|
| 1303 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1304 | IMPLICIT NONE |
---|
| 1305 | |
---|
| 1306 | INTEGER, INTENT(IN) :: N |
---|
| 1307 | KPP_REAL, INTENT(IN) :: H |
---|
| 1308 | KPP_REAL, INTENT(IN), DIMENSION(N) :: Y_tlm |
---|
| 1309 | ! Commented code for sparse big linear algebra: |
---|
| 1310 | ! KPP_REAL, INTENT(OUT), DIMENSION(3,N) :: Zbig |
---|
| 1311 | KPP_REAL, INTENT(OUT), DIMENSION(3*N) :: Zbig |
---|
| 1312 | #ifdef FULL_ALGEBRA |
---|
| 1313 | KPP_REAL, INTENT(IN), DIMENSION(NVAR,NVAR) :: Jac1, Jac2, Jac3 |
---|
| 1314 | #else |
---|
| 1315 | KPP_REAL, INTENT(IN), DIMENSION(LU_NONZERO) :: Jac1, Jac2, Jac3 |
---|
| 1316 | #endif |
---|
| 1317 | KPP_REAL, DIMENSION(N) :: F, TMP |
---|
| 1318 | |
---|
| 1319 | #ifdef FULL_ALGEBRA |
---|
| 1320 | F = MATMUL(Jac1,Y_tlm) |
---|
| 1321 | #else |
---|
| 1322 | CALL Jac_SP_Vec ( Jac1, Y_tlm, F ) ! F1 <- Jac(Y+Z1)*(Y_tlm) |
---|
| 1323 | #endif |
---|
| 1324 | ! Commented code for sparse big linear algebra: |
---|
| 1325 | ! Zbig(1,1:N) = H*rkA(1,1)*F(1:N) |
---|
| 1326 | ! Zbig(2,1:N) = H*rkA(2,1)*F(1:N) |
---|
| 1327 | ! Zbig(3,1:N) = H*rkA(3,1)*F(1:N) |
---|
| 1328 | Zbig(1:N) = H*rkA(1,1)*F(1:N) |
---|
| 1329 | Zbig(N+1:2*N) = H*rkA(2,1)*F(1:N) |
---|
| 1330 | Zbig(2*N+1:3*N) = H*rkA(3,1)*F(1:N) |
---|
| 1331 | |
---|
| 1332 | #ifdef FULL_ALGEBRA |
---|
| 1333 | F = MATMUL(Jac2,Y_tlm) |
---|
| 1334 | #else |
---|
| 1335 | CALL Jac_SP_Vec ( Jac2, Y_tlm, F ) ! F2 <- Jac(Y+Z2)*(Y_tlm) |
---|
| 1336 | #endif |
---|
| 1337 | ! Commented code for sparse big linear algebra: |
---|
| 1338 | ! Zbig(1,1:N) = Zbig(1,1:N) + H*rkA(1,2)*F(1:N) |
---|
| 1339 | ! Zbig(2,1:N) = Zbig(2,1:N) + H*rkA(2,2)*F(1:N) |
---|
| 1340 | ! Zbig(3,1:N) = Zbig(3,1:N) + H*rkA(3,2)*F(1:N) |
---|
| 1341 | Zbig(1:N) = Zbig(1:N) + H*rkA(1,2)*F(1:N) |
---|
| 1342 | Zbig(N+1:2*N) = Zbig(N+1:2*N) + H*rkA(2,2)*F(1:N) |
---|
| 1343 | Zbig(2*N+1:3*N) = Zbig(2*N+1:3*N) + H*rkA(3,2)*F(1:N) |
---|
| 1344 | |
---|
| 1345 | #ifdef FULL_ALGEBRA |
---|
| 1346 | F = MATMUL(Jac3,Y_tlm) |
---|
| 1347 | #else |
---|
| 1348 | CALL Jac_SP_Vec ( Jac3, Y_tlm, F ) ! F3 <- Jac(Y+Z3)*(Y_tlm) |
---|
| 1349 | #endif |
---|
| 1350 | ! Commented code for sparse big linear algebra: |
---|
| 1351 | ! Zbig(1,1:N) = Zbig(1,1:N) + H*rkA(1,3)*F(1:N) |
---|
| 1352 | ! Zbig(2,1:N) = Zbig(2,1:N) + H*rkA(2,3)*F(1:N) |
---|
| 1353 | ! Zbig(3,1:N) = Zbig(3,1:N) + H*rkA(3,3)*F(1:N) |
---|
| 1354 | Zbig(1:N) = Zbig(1:N) + H*rkA(1,3)*F(1:N) |
---|
| 1355 | Zbig(N+1:2*N) = Zbig(N+1:2*N) + H*rkA(2,3)*F(1:N) |
---|
| 1356 | Zbig(2*N+1:3*N) = Zbig(2*N+1:3*N) + H*rkA(3,3)*F(1:N) |
---|
| 1357 | |
---|
| 1358 | END SUBROUTINE RK_PrepareRHS_TLMdirect |
---|
| 1359 | |
---|
| 1360 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1361 | SUBROUTINE RK_Decomp(N,H,FJAC,E1,IP1,E2,IP2,ISING) |
---|
| 1362 | !~~~> Compute the matrices E1 and E2 and their decompositions |
---|
| 1363 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1364 | IMPLICIT NONE |
---|
| 1365 | |
---|
| 1366 | INTEGER, INTENT(IN) :: N |
---|
| 1367 | KPP_REAL, INTENT(IN) :: H |
---|
| 1368 | #ifdef FULL_ALGEBRA |
---|
| 1369 | KPP_REAL, INTENT(IN) :: FJAC(NVAR,NVAR) |
---|
| 1370 | KPP_REAL, INTENT(OUT) ::E1(NVAR,NVAR) |
---|
| 1371 | COMPLEX(kind=dp), INTENT(OUT) :: E2(N,N) |
---|
| 1372 | #else |
---|
| 1373 | KPP_REAL, INTENT(IN) :: FJAC(LU_NONZERO) |
---|
| 1374 | KPP_REAL, INTENT(OUT) :: E1(LU_NONZERO) |
---|
| 1375 | COMPLEX(kind=dp),INTENT(OUT) :: E2(LU_NONZERO) |
---|
| 1376 | #endif |
---|
| 1377 | INTEGER, INTENT(OUT) :: IP1(N), IP2(N), ISING |
---|
| 1378 | |
---|
| 1379 | INTEGER :: i, j |
---|
| 1380 | KPP_REAL :: Alpha, Beta, Gamma |
---|
| 1381 | |
---|
| 1382 | Gamma = rkGamma/H |
---|
| 1383 | Alpha = rkAlpha/H |
---|
| 1384 | Beta = rkBeta /H |
---|
| 1385 | |
---|
| 1386 | #ifdef FULL_ALGEBRA |
---|
| 1387 | DO j=1,N |
---|
| 1388 | DO i=1,N |
---|
| 1389 | E1(i,j)=-FJAC(i,j) |
---|
| 1390 | END DO |
---|
| 1391 | E1(j,j)=E1(j,j)+Gamma |
---|
| 1392 | END DO |
---|
| 1393 | CALL DGETRF(N,N,E1,N,IP1,ISING) |
---|
| 1394 | #else |
---|
| 1395 | DO i=1,LU_NONZERO |
---|
| 1396 | E1(i)=-FJAC(i) |
---|
| 1397 | END DO |
---|
| 1398 | DO i=1,NVAR |
---|
| 1399 | j=LU_DIAG(i); E1(j)=E1(j)+Gamma |
---|
| 1400 | END DO |
---|
| 1401 | CALL KppDecomp(E1,ISING) |
---|
| 1402 | #endif |
---|
| 1403 | |
---|
| 1404 | IF (ISING /= 0) THEN |
---|
| 1405 | ISTATUS(Ndec) = ISTATUS(Ndec) + 1 |
---|
| 1406 | RETURN |
---|
| 1407 | END IF |
---|
| 1408 | |
---|
| 1409 | #ifdef FULL_ALGEBRA |
---|
| 1410 | DO j=1,N |
---|
| 1411 | DO i=1,N |
---|
| 1412 | E2(i,j) = DCMPLX( -FJAC(i,j), ZERO ) |
---|
| 1413 | END DO |
---|
| 1414 | E2(j,j) = E2(j,j) + CMPLX( Alpha, Beta ) |
---|
| 1415 | END DO |
---|
| 1416 | CALL ZGETRF(N,N,E2,N,IP2,ISING) |
---|
| 1417 | #else |
---|
| 1418 | DO i=1,LU_NONZERO |
---|
| 1419 | E2(i) = DCMPLX( -FJAC(i), ZERO ) |
---|
| 1420 | END DO |
---|
| 1421 | DO i=1,NVAR |
---|
| 1422 | j = LU_DIAG(i); E2(j)=E2(j) + CMPLX( Alpha, Beta ) |
---|
| 1423 | END DO |
---|
| 1424 | CALL KppDecompCmplx(E2,ISING) |
---|
| 1425 | #endif |
---|
| 1426 | ISTATUS(Ndec) = ISTATUS(Ndec) + 1 |
---|
| 1427 | |
---|
| 1428 | END SUBROUTINE RK_Decomp |
---|
| 1429 | |
---|
| 1430 | |
---|
| 1431 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1432 | SUBROUTINE RK_Solve(N,H,E1,IP1,E2,IP2,R1,R2,R3,ISING) |
---|
| 1433 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1434 | IMPLICIT NONE |
---|
| 1435 | INTEGER, INTENT(IN) :: N,IP1(NVAR),IP2(NVAR) |
---|
| 1436 | INTEGER, INTENT(OUT) :: ISING |
---|
| 1437 | #ifdef FULL_ALGEBRA |
---|
| 1438 | KPP_REAL, INTENT(IN) :: E1(NVAR,NVAR) |
---|
| 1439 | COMPLEX(kind=dp), INTENT(IN) :: E2(NVAR,NVAR) |
---|
| 1440 | #else |
---|
| 1441 | KPP_REAL, INTENT(IN) :: E1(LU_NONZERO) |
---|
| 1442 | COMPLEX(kind=dp), INTENT(IN) :: E2(LU_NONZERO) |
---|
| 1443 | #endif |
---|
| 1444 | KPP_REAL, INTENT(IN) :: H |
---|
| 1445 | KPP_REAL, INTENT(INOUT) :: R1(N),R2(N),R3(N) |
---|
| 1446 | |
---|
| 1447 | KPP_REAL :: x1, x2, x3 |
---|
| 1448 | COMPLEX(kind=dp) :: BC(N) |
---|
| 1449 | INTEGER :: i |
---|
| 1450 | ! |
---|
| 1451 | ! Z <- h^{-1) T^{-1) A^{-1) x Z |
---|
| 1452 | DO i=1,N |
---|
| 1453 | x1 = R1(i)/H; x2 = R2(i)/H; x3 = R3(i)/H |
---|
| 1454 | R1(i) = rkTinvAinv(1,1)*x1 + rkTinvAinv(1,2)*x2 + rkTinvAinv(1,3)*x3 |
---|
| 1455 | R2(i) = rkTinvAinv(2,1)*x1 + rkTinvAinv(2,2)*x2 + rkTinvAinv(2,3)*x3 |
---|
| 1456 | R3(i) = rkTinvAinv(3,1)*x1 + rkTinvAinv(3,2)*x2 + rkTinvAinv(3,3)*x3 |
---|
| 1457 | END DO |
---|
| 1458 | |
---|
| 1459 | #ifdef FULL_ALGEBRA |
---|
| 1460 | CALL DGETRS ('N',N,1,E1,N,IP1,R1,N,0) |
---|
| 1461 | #else |
---|
| 1462 | CALL KppSolve (E1,R1) |
---|
| 1463 | #endif |
---|
| 1464 | ! |
---|
| 1465 | DO i=1,N |
---|
| 1466 | BC(i) = DCMPLX(R2(i),R3(i)) |
---|
| 1467 | END DO |
---|
| 1468 | #ifdef FULL_ALGEBRA |
---|
| 1469 | CALL ZGETRS ('N',N,1,E2,N,IP2,BC,N,0) |
---|
| 1470 | #else |
---|
| 1471 | CALL KppSolveCmplx (E2,BC) |
---|
| 1472 | #endif |
---|
| 1473 | DO i=1,N |
---|
| 1474 | R2(i) = DBLE( BC(i) ) |
---|
| 1475 | R3(i) = AIMAG( BC(i) ) |
---|
| 1476 | END DO |
---|
| 1477 | |
---|
| 1478 | ! Z <- T x Z |
---|
| 1479 | DO i=1,N |
---|
| 1480 | x1 = R1(i); x2 = R2(i); x3 = R3(i) |
---|
| 1481 | R1(i) = rkT(1,1)*x1 + rkT(1,2)*x2 + rkT(1,3)*x3 |
---|
| 1482 | R2(i) = rkT(2,1)*x1 + rkT(2,2)*x2 + rkT(2,3)*x3 |
---|
| 1483 | R3(i) = rkT(3,1)*x1 + rkT(3,2)*x2 + rkT(3,3)*x3 |
---|
| 1484 | END DO |
---|
| 1485 | |
---|
| 1486 | ISTATUS(Nsol) = ISTATUS(Nsol) + 1 |
---|
| 1487 | |
---|
| 1488 | END SUBROUTINE RK_Solve |
---|
| 1489 | |
---|
| 1490 | |
---|
| 1491 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1492 | SUBROUTINE RK_ErrorEstimate(N,H,Y,T, & |
---|
| 1493 | E1,IP1,Z1,Z2,Z3,SCAL,Err, & |
---|
| 1494 | FirstStep,Reject) |
---|
| 1495 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1496 | IMPLICIT NONE |
---|
| 1497 | |
---|
| 1498 | INTEGER, INTENT(IN) :: N, IP1(N) |
---|
| 1499 | #ifdef FULL_ALGEBRA |
---|
| 1500 | KPP_REAL, INTENT(IN) :: E1(NVAR,NVAR) |
---|
| 1501 | #else |
---|
| 1502 | KPP_REAL, INTENT(IN) :: E1(LU_NONZERO) |
---|
| 1503 | #endif |
---|
| 1504 | KPP_REAL, INTENT(IN) :: T,H,SCAL(N),Z1(N),Z2(N),Z3(N),Y(N) |
---|
| 1505 | LOGICAL,INTENT(IN) :: FirstStep,Reject |
---|
| 1506 | KPP_REAL, INTENT(INOUT) :: Err |
---|
| 1507 | |
---|
| 1508 | KPP_REAL :: F1(N),F2(N),F0(N),TMP(N) |
---|
| 1509 | INTEGER :: i |
---|
| 1510 | KPP_REAL :: HEE1,HEE2,HEE3 |
---|
| 1511 | |
---|
| 1512 | HEE1 = rkE(1)/H |
---|
| 1513 | HEE2 = rkE(2)/H |
---|
| 1514 | HEE3 = rkE(3)/H |
---|
| 1515 | |
---|
| 1516 | CALL FUN_CHEM(T,Y,F0) |
---|
| 1517 | ISTATUS(Nfun) = ISTATUS(Nfun) + 1 |
---|
| 1518 | |
---|
| 1519 | DO i=1,N |
---|
| 1520 | F2(i) = HEE1*Z1(i)+HEE2*Z2(i)+HEE3*Z3(i) |
---|
| 1521 | TMP(i) = rkE(0)*F0(i) + F2(i) |
---|
| 1522 | END DO |
---|
| 1523 | |
---|
| 1524 | #ifdef FULL_ALGEBRA |
---|
| 1525 | CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) |
---|
| 1526 | IF ((rkMethod==R1A).OR.(rkMethod==GAU).OR.(rkMethod==L3A)) CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) |
---|
| 1527 | IF (rkMethod==GAU) CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) |
---|
| 1528 | #else |
---|
| 1529 | CALL KppSolve (E1, TMP) |
---|
| 1530 | IF ((rkMethod==R1A).OR.(rkMethod==GAU).OR.(rkMethod==L3A)) CALL KppSolve (E1,TMP) |
---|
| 1531 | IF (rkMethod==GAU) CALL KppSolve (E1,TMP) |
---|
| 1532 | #endif |
---|
| 1533 | |
---|
| 1534 | Err = RK_ErrorNorm(N,SCAL,TMP) |
---|
| 1535 | ! |
---|
| 1536 | IF (Err < ONE) RETURN |
---|
| 1537 | firej:IF (FirstStep.OR.Reject) THEN |
---|
| 1538 | DO i=1,N |
---|
| 1539 | TMP(i)=Y(i)+TMP(i) |
---|
| 1540 | END DO |
---|
| 1541 | CALL FUN_CHEM(T,TMP,F1) |
---|
| 1542 | ISTATUS(Nfun) = ISTATUS(Nfun) + 1 |
---|
| 1543 | DO i=1,N |
---|
| 1544 | TMP(i)=F1(i)+F2(i) |
---|
| 1545 | END DO |
---|
| 1546 | |
---|
| 1547 | #ifdef FULL_ALGEBRA |
---|
| 1548 | CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) |
---|
| 1549 | #else |
---|
| 1550 | CALL KppSolve (E1, TMP) |
---|
| 1551 | #endif |
---|
| 1552 | Err = RK_ErrorNorm(N,SCAL,TMP) |
---|
| 1553 | END IF firej |
---|
| 1554 | |
---|
| 1555 | END SUBROUTINE RK_ErrorEstimate |
---|
| 1556 | |
---|
| 1557 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1558 | SUBROUTINE RK_ErrorEstimate_tlm(N,NTLM,T,H,FJAC,Y,Y_tlm, & |
---|
| 1559 | E1,IP1,Z1_tlm,Z2_tlm,Z3_tlm,FWD_Err, & |
---|
| 1560 | FirstStep,Reject) |
---|
| 1561 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1562 | IMPLICIT NONE |
---|
| 1563 | |
---|
| 1564 | INTEGER, INTENT(IN) :: N,NTLM,IP1(N) |
---|
| 1565 | #ifdef FULL_ALGEBRA |
---|
| 1566 | KPP_REAL, INTENT(IN) :: FJAC(N,N), E1(N,N) |
---|
| 1567 | KPP_REAL :: J0(N,N) |
---|
| 1568 | #else |
---|
| 1569 | KPP_REAL, INTENT(IN) :: FJAC(LU_NONZERO), E1(LU_NONZERO) |
---|
| 1570 | KPP_REAL :: J0(LU_NONZERO) |
---|
| 1571 | #endif |
---|
| 1572 | KPP_REAL, INTENT(IN) :: T,H, Z1_tlm(N,NTLM),Z2_tlm(N,NTLM),Z3_tlm(N,NTLM), & |
---|
| 1573 | Y_tlm(N,NTLM), Y(N) |
---|
| 1574 | LOGICAL, INTENT(IN) :: FirstStep, Reject |
---|
| 1575 | KPP_REAL, INTENT(INOUT) :: FWD_Err |
---|
| 1576 | |
---|
| 1577 | INTEGER :: itlm |
---|
| 1578 | KPP_REAL :: HEE1,HEE2,HEE3, SCAL_tlm(N), Err, TMP(N), TMP2(N), JY_tlm(N) |
---|
| 1579 | |
---|
| 1580 | HEE1 = rkE(1)/H |
---|
| 1581 | HEE2 = rkE(2)/H |
---|
| 1582 | HEE3 = rkE(3)/H |
---|
| 1583 | |
---|
| 1584 | DO itlm=1,NTLM |
---|
| 1585 | CALL RK_ErrorScale(N,ITOL,AbsTol_tlm(1,itlm),RelTol_tlm(1,itlm), & |
---|
| 1586 | Y_tlm(1,itlm),SCAL_tlm) |
---|
| 1587 | |
---|
| 1588 | CALL JAC_CHEM(T,Y,J0) |
---|
| 1589 | ISTATUS(Njac) = ISTATUS(Njac) + 1 |
---|
| 1590 | CALL JAC_SP_Vec(J0,Y_tlm(1,itlm),JY_tlm) |
---|
| 1591 | |
---|
| 1592 | DO i=1,N |
---|
| 1593 | TMP2(i) = HEE1*Z1_tlm(i,itlm)+HEE2*Z2_tlm(i,itlm)+HEE3*Z3_tlm(i,itlm) |
---|
| 1594 | TMP(i) = rkE(0)*JY_tlm(i) + TMP2(i) |
---|
| 1595 | END DO |
---|
| 1596 | |
---|
| 1597 | #ifdef FULL_ALGEBRA |
---|
| 1598 | CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) |
---|
| 1599 | ! IF ((ICNTRL(3)==3).OR.(ICNTRL(3)==4)) CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) |
---|
| 1600 | ! IF (ICNTRL(3)==3) CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) |
---|
| 1601 | #else |
---|
| 1602 | CALL KppSolve (E1, TMP) |
---|
| 1603 | ! IF ((ICNTRL(3)==3).OR.(ICNTRL(3)==4)) THEN |
---|
| 1604 | ! CALL KppSolve (E1,TMP) |
---|
| 1605 | ! END IF |
---|
| 1606 | ! IF (ICNTRL(3)==3) THEN |
---|
| 1607 | ! CALL KppSolve (E1,TMP) |
---|
| 1608 | ! END IF |
---|
| 1609 | #endif |
---|
| 1610 | |
---|
| 1611 | Err = RK_ErrorNorm(N,SCAL_tlm,TMP) |
---|
| 1612 | ! |
---|
| 1613 | FWD_Err = MAX(FWD_Err, Err) |
---|
| 1614 | END DO |
---|
| 1615 | |
---|
| 1616 | END SUBROUTINE RK_ErrorEstimate_tlm |
---|
| 1617 | |
---|
| 1618 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1619 | KPP_REAL FUNCTION RK_ErrorNorm(N,SCAL,DY) |
---|
| 1620 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1621 | IMPLICIT NONE |
---|
| 1622 | |
---|
| 1623 | INTEGER, INTENT(IN) :: N |
---|
| 1624 | KPP_REAL, INTENT(IN) :: SCAL(N),DY(N) |
---|
| 1625 | INTEGER :: i |
---|
| 1626 | |
---|
| 1627 | RK_ErrorNorm = ZERO |
---|
| 1628 | DO i=1,N |
---|
| 1629 | RK_ErrorNorm = RK_ErrorNorm + (DY(i)*SCAL(i))**2 |
---|
| 1630 | END DO |
---|
| 1631 | RK_ErrorNorm = MAX( SQRT(RK_ErrorNorm/N), 1.0d-10 ) |
---|
| 1632 | |
---|
| 1633 | END FUNCTION RK_ErrorNorm |
---|
| 1634 | |
---|
| 1635 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1636 | SUBROUTINE Radau2A_Coefficients |
---|
| 1637 | ! The coefficients of the 3-stage Radau-2A method |
---|
| 1638 | ! (given to ~30 accurate digits) |
---|
| 1639 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1640 | IMPLICIT NONE |
---|
| 1641 | ! The coefficients of the Radau2A method |
---|
| 1642 | KPP_REAL :: b0 |
---|
| 1643 | |
---|
| 1644 | ! b0 = 1.0d0 |
---|
| 1645 | IF (SdirkError) THEN |
---|
| 1646 | b0 = 0.2d-1 |
---|
| 1647 | ELSE |
---|
| 1648 | b0 = 0.5d-1 |
---|
| 1649 | END IF |
---|
| 1650 | |
---|
| 1651 | ! The coefficients of the Radau2A method |
---|
| 1652 | rkMethod = R2A |
---|
| 1653 | |
---|
| 1654 | rkA(1,1) = 1.968154772236604258683861429918299d-1 |
---|
| 1655 | rkA(1,2) = -6.55354258501983881085227825696087d-2 |
---|
| 1656 | rkA(1,3) = 2.377097434822015242040823210718965d-2 |
---|
| 1657 | rkA(2,1) = 3.944243147390872769974116714584975d-1 |
---|
| 1658 | rkA(2,2) = 2.920734116652284630205027458970589d-1 |
---|
| 1659 | rkA(2,3) = -4.154875212599793019818600988496743d-2 |
---|
| 1660 | rkA(3,1) = 3.764030627004672750500754423692808d-1 |
---|
| 1661 | rkA(3,2) = 5.124858261884216138388134465196080d-1 |
---|
| 1662 | rkA(3,3) = 1.111111111111111111111111111111111d-1 |
---|
| 1663 | |
---|
| 1664 | rkB(1) = 3.764030627004672750500754423692808d-1 |
---|
| 1665 | rkB(2) = 5.124858261884216138388134465196080d-1 |
---|
| 1666 | rkB(3) = 1.111111111111111111111111111111111d-1 |
---|
| 1667 | |
---|
| 1668 | rkC(1) = 1.550510257216821901802715925294109d-1 |
---|
| 1669 | rkC(2) = 6.449489742783178098197284074705891d-1 |
---|
| 1670 | rkC(3) = 1.0d0 |
---|
| 1671 | |
---|
| 1672 | ! New solution: H* Sum B_j*f(Z_j) = Sum D_j*Z_j |
---|
| 1673 | rkD(1) = 0.0d0 |
---|
| 1674 | rkD(2) = 0.0d0 |
---|
| 1675 | rkD(3) = 1.0d0 |
---|
| 1676 | |
---|
| 1677 | ! Classical error estimator: |
---|
| 1678 | ! H* Sum (B_j-Bhat_j)*f(Z_j) = H*E(0)*f(0) + Sum E_j*Z_j |
---|
| 1679 | rkE(0) = 1.0d0*b0 |
---|
| 1680 | rkE(1) = -10.04880939982741556246032950764708d0*b0 |
---|
| 1681 | rkE(2) = 1.382142733160748895793662840980412d0*b0 |
---|
| 1682 | rkE(3) = -.3333333333333333333333333333333333d0*b0 |
---|
| 1683 | |
---|
| 1684 | ! Sdirk error estimator |
---|
| 1685 | rkBgam(0) = b0 |
---|
| 1686 | rkBgam(1) = .3764030627004672750500754423692807d0-1.558078204724922382431975370686279d0*b0 |
---|
| 1687 | rkBgam(2) = .8914115380582557157653087040196118d0*b0+.5124858261884216138388134465196077d0 |
---|
| 1688 | rkBgam(3) = -.1637777184845662566367174924883037d0-.3333333333333333333333333333333333d0*b0 |
---|
| 1689 | rkBgam(4) = .2748888295956773677478286035994148d0 |
---|
| 1690 | |
---|
| 1691 | ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j |
---|
| 1692 | rkTheta(1) = -1.520677486405081647234271944611547d0-10.04880939982741556246032950764708d0*b0 |
---|
| 1693 | rkTheta(2) = 2.070455145596436382729929151810376d0+1.382142733160748895793662840980413d0*b0 |
---|
| 1694 | rkTheta(3) = -.3333333333333333333333333333333333d0*b0-.3744441479783868387391430179970741d0 |
---|
| 1695 | |
---|
| 1696 | ! Local order of error estimator |
---|
| 1697 | IF (b0==0.0d0) THEN |
---|
| 1698 | rkELO = 6.0d0 |
---|
| 1699 | ELSE |
---|
| 1700 | rkELO = 4.0d0 |
---|
| 1701 | END IF |
---|
| 1702 | |
---|
| 1703 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1704 | !~~~> Diagonalize the RK matrix: |
---|
| 1705 | ! rkTinv * inv(rkA) * rkT = |
---|
| 1706 | ! | rkGamma 0 0 | |
---|
| 1707 | ! | 0 rkAlpha -rkBeta | |
---|
| 1708 | ! | 0 rkBeta rkAlpha | |
---|
| 1709 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1710 | |
---|
| 1711 | rkGamma = 3.637834252744495732208418513577775d0 |
---|
| 1712 | rkAlpha = 2.681082873627752133895790743211112d0 |
---|
| 1713 | rkBeta = 3.050430199247410569426377624787569d0 |
---|
| 1714 | |
---|
| 1715 | rkT(1,1) = 9.443876248897524148749007950641664d-2 |
---|
| 1716 | rkT(1,2) = -1.412552950209542084279903838077973d-1 |
---|
| 1717 | rkT(1,3) = -3.00291941051474244918611170890539d-2 |
---|
| 1718 | rkT(2,1) = 2.502131229653333113765090675125018d-1 |
---|
| 1719 | rkT(2,2) = 2.041293522937999319959908102983381d-1 |
---|
| 1720 | rkT(2,3) = 3.829421127572619377954382335998733d-1 |
---|
| 1721 | rkT(3,1) = 1.0d0 |
---|
| 1722 | rkT(3,2) = 1.0d0 |
---|
| 1723 | rkT(3,3) = 0.0d0 |
---|
| 1724 | |
---|
| 1725 | rkTinv(1,1) = 4.178718591551904727346462658512057d0 |
---|
| 1726 | rkTinv(1,2) = 3.27682820761062387082533272429617d-1 |
---|
| 1727 | rkTinv(1,3) = 5.233764454994495480399309159089876d-1 |
---|
| 1728 | rkTinv(2,1) = -4.178718591551904727346462658512057d0 |
---|
| 1729 | rkTinv(2,2) = -3.27682820761062387082533272429617d-1 |
---|
| 1730 | rkTinv(2,3) = 4.766235545005504519600690840910124d-1 |
---|
| 1731 | rkTinv(3,1) = -5.02872634945786875951247343139544d-1 |
---|
| 1732 | rkTinv(3,2) = 2.571926949855605429186785353601676d0 |
---|
| 1733 | rkTinv(3,3) = -5.960392048282249249688219110993024d-1 |
---|
| 1734 | |
---|
| 1735 | rkTinvAinv(1,1) = 1.520148562492775501049204957366528d+1 |
---|
| 1736 | rkTinvAinv(1,2) = 1.192055789400527921212348994770778d0 |
---|
| 1737 | rkTinvAinv(1,3) = 1.903956760517560343018332287285119d0 |
---|
| 1738 | rkTinvAinv(2,1) = -9.669512977505946748632625374449567d0 |
---|
| 1739 | rkTinvAinv(2,2) = -8.724028436822336183071773193986487d0 |
---|
| 1740 | rkTinvAinv(2,3) = 3.096043239482439656981667712714881d0 |
---|
| 1741 | rkTinvAinv(3,1) = -1.409513259499574544876303981551774d+1 |
---|
| 1742 | rkTinvAinv(3,2) = 5.895975725255405108079130152868952d0 |
---|
| 1743 | rkTinvAinv(3,3) = -1.441236197545344702389881889085515d-1 |
---|
| 1744 | |
---|
| 1745 | rkAinvT(1,1) = .3435525649691961614912493915818282d0 |
---|
| 1746 | rkAinvT(1,2) = -.4703191128473198422370558694426832d0 |
---|
| 1747 | rkAinvT(1,3) = .3503786597113668965366406634269080d0 |
---|
| 1748 | rkAinvT(2,1) = .9102338692094599309122768354288852d0 |
---|
| 1749 | rkAinvT(2,2) = 1.715425895757991796035292755937326d0 |
---|
| 1750 | rkAinvT(2,3) = .4040171993145015239277111187301784d0 |
---|
| 1751 | rkAinvT(3,1) = 3.637834252744495732208418513577775d0 |
---|
| 1752 | rkAinvT(3,2) = 2.681082873627752133895790743211112d0 |
---|
| 1753 | rkAinvT(3,3) = -3.050430199247410569426377624787569d0 |
---|
| 1754 | |
---|
| 1755 | END SUBROUTINE Radau2A_Coefficients |
---|
| 1756 | |
---|
| 1757 | |
---|
| 1758 | |
---|
| 1759 | |
---|
| 1760 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1761 | SUBROUTINE Lobatto3C_Coefficients |
---|
| 1762 | ! The coefficients of the 3-stage Lobatto-3C method |
---|
| 1763 | ! (given to ~30 accurate digits) |
---|
| 1764 | ! The parameter b0 can be chosen to tune the error estimator |
---|
| 1765 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1766 | IMPLICIT NONE |
---|
| 1767 | KPP_REAL :: b0 |
---|
| 1768 | |
---|
| 1769 | rkMethod = L3C |
---|
| 1770 | |
---|
| 1771 | ! b0 = 1.0d0 |
---|
| 1772 | IF (SdirkError) THEN |
---|
| 1773 | b0 = 0.2d0 |
---|
| 1774 | ELSE |
---|
| 1775 | b0 = 0.5d0 |
---|
| 1776 | END IF |
---|
| 1777 | ! The coefficients of the Lobatto3C method |
---|
| 1778 | |
---|
| 1779 | rkA(1,1) = .1666666666666666666666666666666667d0 |
---|
| 1780 | rkA(1,2) = -.3333333333333333333333333333333333d0 |
---|
| 1781 | rkA(1,3) = .1666666666666666666666666666666667d0 |
---|
| 1782 | rkA(2,1) = .1666666666666666666666666666666667d0 |
---|
| 1783 | rkA(2,2) = .4166666666666666666666666666666667d0 |
---|
| 1784 | rkA(2,3) = -.8333333333333333333333333333333333d-1 |
---|
| 1785 | rkA(3,1) = .1666666666666666666666666666666667d0 |
---|
| 1786 | rkA(3,2) = .6666666666666666666666666666666667d0 |
---|
| 1787 | rkA(3,3) = .1666666666666666666666666666666667d0 |
---|
| 1788 | |
---|
| 1789 | rkB(1) = .1666666666666666666666666666666667d0 |
---|
| 1790 | rkB(2) = .6666666666666666666666666666666667d0 |
---|
| 1791 | rkB(3) = .1666666666666666666666666666666667d0 |
---|
| 1792 | |
---|
| 1793 | rkC(1) = 0.0d0 |
---|
| 1794 | rkC(2) = 0.5d0 |
---|
| 1795 | rkC(3) = 1.0d0 |
---|
| 1796 | |
---|
| 1797 | ! Classical error estimator, embedded solution: |
---|
| 1798 | rkBhat(0) = b0 |
---|
| 1799 | rkBhat(1) = .16666666666666666666666666666666667d0-b0 |
---|
| 1800 | rkBhat(2) = .66666666666666666666666666666666667d0 |
---|
| 1801 | rkBhat(3) = .16666666666666666666666666666666667d0 |
---|
| 1802 | |
---|
| 1803 | ! New solution: h Sum_j b_j f(Z_j) = sum d_j Z_j |
---|
| 1804 | rkD(1) = 0.0d0 |
---|
| 1805 | rkD(2) = 0.0d0 |
---|
| 1806 | rkD(3) = 1.0d0 |
---|
| 1807 | |
---|
| 1808 | ! Classical error estimator: |
---|
| 1809 | ! H* Sum (B_j-Bhat_j)*f(Z_j) = H*E(0)*f(0) + Sum E_j*Z_j |
---|
| 1810 | rkE(0) = .3808338772072650364017425226487022*b0 |
---|
| 1811 | rkE(1) = -1.142501631621795109205227567946107*b0 |
---|
| 1812 | rkE(2) = -1.523335508829060145606970090594809*b0 |
---|
| 1813 | rkE(3) = .3808338772072650364017425226487022*b0 |
---|
| 1814 | |
---|
| 1815 | ! Sdirk error estimator |
---|
| 1816 | rkBgam(0) = b0 |
---|
| 1817 | rkBgam(1) = .1666666666666666666666666666666667d0-1.d0*b0 |
---|
| 1818 | rkBgam(2) = .6666666666666666666666666666666667d0 |
---|
| 1819 | rkBgam(3) = -.2141672105405983697350758559820354d0 |
---|
| 1820 | rkBgam(4) = .3808338772072650364017425226487021d0 |
---|
| 1821 | |
---|
| 1822 | ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j |
---|
| 1823 | rkTheta(1) = -3.d0*b0-.3808338772072650364017425226487021d0 |
---|
| 1824 | rkTheta(2) = -4.d0*b0+1.523335508829060145606970090594808d0 |
---|
| 1825 | rkTheta(3) = -.142501631621795109205227567946106d0+b0 |
---|
| 1826 | |
---|
| 1827 | ! Local order of error estimator |
---|
| 1828 | IF (b0==0.0d0) THEN |
---|
| 1829 | rkELO = 5.0d0 |
---|
| 1830 | ELSE |
---|
| 1831 | rkELO = 4.0d0 |
---|
| 1832 | END IF |
---|
| 1833 | |
---|
| 1834 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1835 | !~~~> Diagonalize the RK matrix: |
---|
| 1836 | ! rkTinv * inv(rkA) * rkT = |
---|
| 1837 | ! | rkGamma 0 0 | |
---|
| 1838 | ! | 0 rkAlpha -rkBeta | |
---|
| 1839 | ! | 0 rkBeta rkAlpha | |
---|
| 1840 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1841 | |
---|
| 1842 | rkGamma = 2.625816818958466716011888933765284d0 |
---|
| 1843 | rkAlpha = 1.687091590520766641994055533117359d0 |
---|
| 1844 | rkBeta = 2.508731754924880510838743672432351d0 |
---|
| 1845 | |
---|
| 1846 | rkT(1,1) = 1.d0 |
---|
| 1847 | rkT(1,2) = 1.d0 |
---|
| 1848 | rkT(1,3) = 0.d0 |
---|
| 1849 | rkT(2,1) = .4554100411010284672111720348287483d0 |
---|
| 1850 | rkT(2,2) = -.6027050205505142336055860174143743d0 |
---|
| 1851 | rkT(2,3) = -.4309321229203225731070721341350346d0 |
---|
| 1852 | rkT(3,1) = 2.195823345445647152832799205549709d0 |
---|
| 1853 | rkT(3,2) = -1.097911672722823576416399602774855d0 |
---|
| 1854 | rkT(3,3) = .7850032632435902184104551358922130d0 |
---|
| 1855 | |
---|
| 1856 | rkTinv(1,1) = .4205559181381766909344950150991349d0 |
---|
| 1857 | rkTinv(1,2) = .3488903392193734304046467270632057d0 |
---|
| 1858 | rkTinv(1,3) = .1915253879645878102698098373933487d0 |
---|
| 1859 | rkTinv(2,1) = .5794440818618233090655049849008650d0 |
---|
| 1860 | rkTinv(2,2) = -.3488903392193734304046467270632057d0 |
---|
| 1861 | rkTinv(2,3) = -.1915253879645878102698098373933487d0 |
---|
| 1862 | rkTinv(3,1) = -.3659705575742745254721332009249516d0 |
---|
| 1863 | rkTinv(3,2) = -1.463882230297098101888532803699806d0 |
---|
| 1864 | rkTinv(3,3) = .4702733607340189781407813565524989d0 |
---|
| 1865 | |
---|
| 1866 | rkTinvAinv(1,1) = 1.104302803159744452668648155627548d0 |
---|
| 1867 | rkTinvAinv(1,2) = .916122120694355522658740710823143d0 |
---|
| 1868 | rkTinvAinv(1,3) = .5029105849749601702795812241441172d0 |
---|
| 1869 | rkTinvAinv(2,1) = 1.895697196840255547331351844372453d0 |
---|
| 1870 | rkTinvAinv(2,2) = 3.083877879305644477341259289176857d0 |
---|
| 1871 | rkTinvAinv(2,3) = -1.502910584974960170279581224144117d0 |
---|
| 1872 | rkTinvAinv(3,1) = .8362439183082935036129145574774502d0 |
---|
| 1873 | rkTinvAinv(3,2) = -3.344975673233174014451658229909802d0 |
---|
| 1874 | rkTinvAinv(3,3) = .312908409479233358005944466882642d0 |
---|
| 1875 | |
---|
| 1876 | rkAinvT(1,1) = 2.625816818958466716011888933765282d0 |
---|
| 1877 | rkAinvT(1,2) = 1.687091590520766641994055533117358d0 |
---|
| 1878 | rkAinvT(1,3) = -2.508731754924880510838743672432351d0 |
---|
| 1879 | rkAinvT(2,1) = 1.195823345445647152832799205549710d0 |
---|
| 1880 | rkAinvT(2,2) = -2.097911672722823576416399602774855d0 |
---|
| 1881 | rkAinvT(2,3) = .7850032632435902184104551358922130d0 |
---|
| 1882 | rkAinvT(3,1) = 5.765829871932827589653709477334136d0 |
---|
| 1883 | rkAinvT(3,2) = .1170850640335862051731452613329320d0 |
---|
| 1884 | rkAinvT(3,3) = 4.078738281412060947659653944216779d0 |
---|
| 1885 | |
---|
| 1886 | END SUBROUTINE Lobatto3C_Coefficients |
---|
| 1887 | |
---|
| 1888 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1889 | SUBROUTINE Gauss_Coefficients |
---|
| 1890 | ! The coefficients of the 3-stage Gauss method |
---|
| 1891 | ! (given to ~30 accurate digits) |
---|
| 1892 | ! The parameter b3 can be chosen by the user |
---|
| 1893 | ! to tune the error estimator |
---|
| 1894 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1895 | IMPLICIT NONE |
---|
| 1896 | KPP_REAL :: b0 |
---|
| 1897 | ! The coefficients of the Gauss method |
---|
| 1898 | |
---|
| 1899 | |
---|
| 1900 | rkMethod = GAU |
---|
| 1901 | |
---|
| 1902 | ! b0 = 4.0d0 |
---|
| 1903 | b0 = 0.1d0 |
---|
| 1904 | |
---|
| 1905 | ! The coefficients of the Gauss method |
---|
| 1906 | |
---|
| 1907 | rkA(1,1) = .1388888888888888888888888888888889d0 |
---|
| 1908 | rkA(1,2) = -.359766675249389034563954710966045d-1 |
---|
| 1909 | rkA(1,3) = .97894440153083260495800422294756d-2 |
---|
| 1910 | rkA(2,1) = .3002631949808645924380249472131556d0 |
---|
| 1911 | rkA(2,2) = .2222222222222222222222222222222222d0 |
---|
| 1912 | rkA(2,3) = -.224854172030868146602471694353778d-1 |
---|
| 1913 | rkA(3,1) = .2679883337624694517281977355483022d0 |
---|
| 1914 | rkA(3,2) = .4804211119693833479008399155410489d0 |
---|
| 1915 | rkA(3,3) = .1388888888888888888888888888888889d0 |
---|
| 1916 | |
---|
| 1917 | rkB(1) = .2777777777777777777777777777777778d0 |
---|
| 1918 | rkB(2) = .4444444444444444444444444444444444d0 |
---|
| 1919 | rkB(3) = .2777777777777777777777777777777778d0 |
---|
| 1920 | |
---|
| 1921 | rkC(1) = .1127016653792583114820734600217600d0 |
---|
| 1922 | rkC(2) = .5000000000000000000000000000000000d0 |
---|
| 1923 | rkC(3) = .8872983346207416885179265399782400d0 |
---|
| 1924 | |
---|
| 1925 | ! Classical error estimator, embedded solution: |
---|
| 1926 | rkBhat(0) = b0 |
---|
| 1927 | rkBhat(1) =-1.4788305577012361475298775666303999d0*b0 & |
---|
| 1928 | +.27777777777777777777777777777777778d0 |
---|
| 1929 | rkBhat(2) = .44444444444444444444444444444444444d0 & |
---|
| 1930 | +.66666666666666666666666666666666667d0*b0 |
---|
| 1931 | rkBhat(3) = -.18783610896543051913678910003626672d0*b0 & |
---|
| 1932 | +.27777777777777777777777777777777778d0 |
---|
| 1933 | |
---|
| 1934 | ! New solution: h Sum_j b_j f(Z_j) = sum d_j Z_j |
---|
| 1935 | rkD(1) = .1666666666666666666666666666666667d1 |
---|
| 1936 | rkD(2) = -.1333333333333333333333333333333333d1 |
---|
| 1937 | rkD(3) = .1666666666666666666666666666666667d1 |
---|
| 1938 | |
---|
| 1939 | ! Classical error estimator: |
---|
| 1940 | ! H* Sum (B_j-Bhat_j)*f(Z_j) = H*E(0)*f(0) + Sum E_j*Z_j |
---|
| 1941 | rkE(0) = .2153144231161121782447335303806954d0*b0 |
---|
| 1942 | rkE(1) = -2.825278112319014084275808340593191d0*b0 |
---|
| 1943 | rkE(2) = .2870858974881495709929780405075939d0*b0 |
---|
| 1944 | rkE(3) = -.4558086256248162565397206448274867d-1*b0 |
---|
| 1945 | |
---|
| 1946 | ! Sdirk error estimator |
---|
| 1947 | rkBgam(0) = 0.d0 |
---|
| 1948 | rkBgam(1) = .2373339543355109188382583162660537d0 |
---|
| 1949 | rkBgam(2) = .5879873931885192299409334646982414d0 |
---|
| 1950 | rkBgam(3) = -.4063577064014232702392531134499046d-1 |
---|
| 1951 | rkBgam(4) = .2153144231161121782447335303806955d0 |
---|
| 1952 | |
---|
| 1953 | ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j |
---|
| 1954 | rkTheta(1) = -2.594040933093095272574031876464493d0 |
---|
| 1955 | rkTheta(2) = 1.824611539036311947589425112250199d0 |
---|
| 1956 | rkTheta(3) = .1856563166634371860478043996459493d0 |
---|
| 1957 | |
---|
| 1958 | ! ELO = local order of classical error estimator |
---|
| 1959 | rkELO = 4.0d0 |
---|
| 1960 | |
---|
| 1961 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1962 | !~~~> Diagonalize the RK matrix: |
---|
| 1963 | ! rkTinv * inv(rkA) * rkT = |
---|
| 1964 | ! | rkGamma 0 0 | |
---|
| 1965 | ! | 0 rkAlpha -rkBeta | |
---|
| 1966 | ! | 0 rkBeta rkAlpha | |
---|
| 1967 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1968 | |
---|
| 1969 | rkGamma = 4.644370709252171185822941421408064d0 |
---|
| 1970 | rkAlpha = 3.677814645373914407088529289295970d0 |
---|
| 1971 | rkBeta = 3.508761919567443321903661209182446d0 |
---|
| 1972 | |
---|
| 1973 | rkT(1,1) = .7215185205520017032081769924397664d-1 |
---|
| 1974 | rkT(1,2) = -.8224123057363067064866206597516454d-1 |
---|
| 1975 | rkT(1,3) = -.6012073861930850173085948921439054d-1 |
---|
| 1976 | rkT(2,1) = .1188325787412778070708888193730294d0 |
---|
| 1977 | rkT(2,2) = .5306509074206139504614411373957448d-1 |
---|
| 1978 | rkT(2,3) = .3162050511322915732224862926182701d0 |
---|
| 1979 | rkT(3,1) = 1.d0 |
---|
| 1980 | rkT(3,2) = 1.d0 |
---|
| 1981 | rkT(3,3) = 0.d0 |
---|
| 1982 | |
---|
| 1983 | rkTinv(1,1) = 5.991698084937800775649580743981285d0 |
---|
| 1984 | rkTinv(1,2) = 1.139214295155735444567002236934009d0 |
---|
| 1985 | rkTinv(1,3) = .4323121137838583855696375901180497d0 |
---|
| 1986 | rkTinv(2,1) = -5.991698084937800775649580743981285d0 |
---|
| 1987 | rkTinv(2,2) = -1.139214295155735444567002236934009d0 |
---|
| 1988 | rkTinv(2,3) = .5676878862161416144303624098819503d0 |
---|
| 1989 | rkTinv(3,1) = -1.246213273586231410815571640493082d0 |
---|
| 1990 | rkTinv(3,2) = 2.925559646192313662599230367054972d0 |
---|
| 1991 | rkTinv(3,3) = -.2577352012734324923468722836888244d0 |
---|
| 1992 | |
---|
| 1993 | rkTinvAinv(1,1) = 27.82766708436744962047620566703329d0 |
---|
| 1994 | rkTinvAinv(1,2) = 5.290933503982655311815946575100597d0 |
---|
| 1995 | rkTinvAinv(1,3) = 2.007817718512643701322151051660114d0 |
---|
| 1996 | rkTinvAinv(2,1) = -17.66368928942422710690385180065675d0 |
---|
| 1997 | rkTinvAinv(2,2) = -14.45491129892587782538830044147713d0 |
---|
| 1998 | rkTinvAinv(2,3) = 2.992182281487356298677848948339886d0 |
---|
| 1999 | rkTinvAinv(3,1) = -25.60678350282974256072419392007303d0 |
---|
| 2000 | rkTinvAinv(3,2) = 6.762434375611708328910623303779923d0 |
---|
| 2001 | rkTinvAinv(3,3) = 1.043979339483109825041215970036771d0 |
---|
| 2002 | |
---|
| 2003 | rkAinvT(1,1) = .3350999483034677402618981153470483d0 |
---|
| 2004 | rkAinvT(1,2) = -.5134173605009692329246186488441294d0 |
---|
| 2005 | rkAinvT(1,3) = .6745196507033116204327635673208923d-1 |
---|
| 2006 | rkAinvT(2,1) = .5519025480108928886873752035738885d0 |
---|
| 2007 | rkAinvT(2,2) = 1.304651810077110066076640761092008d0 |
---|
| 2008 | rkAinvT(2,3) = .9767507983414134987545585703726984d0 |
---|
| 2009 | rkAinvT(3,1) = 4.644370709252171185822941421408064d0 |
---|
| 2010 | rkAinvT(3,2) = 3.677814645373914407088529289295970d0 |
---|
| 2011 | rkAinvT(3,3) = -3.508761919567443321903661209182446d0 |
---|
| 2012 | |
---|
| 2013 | END SUBROUTINE Gauss_Coefficients |
---|
| 2014 | |
---|
| 2015 | |
---|
| 2016 | |
---|
| 2017 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 2018 | SUBROUTINE Radau1A_Coefficients |
---|
| 2019 | ! The coefficients of the 3-stage Gauss method |
---|
| 2020 | ! (given to ~30 accurate digits) |
---|
| 2021 | ! The parameter b3 can be chosen by the user |
---|
| 2022 | ! to tune the error estimator |
---|
| 2023 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 2024 | IMPLICIT NONE |
---|
| 2025 | ! KPP_REAL :: b0 = 0.3d0 |
---|
| 2026 | KPP_REAL :: b0 = 0.1d0 |
---|
| 2027 | |
---|
| 2028 | ! The coefficients of the Radau1A method |
---|
| 2029 | |
---|
| 2030 | rkMethod = R1A |
---|
| 2031 | |
---|
| 2032 | rkA(1,1) = .1111111111111111111111111111111111d0 |
---|
| 2033 | rkA(1,2) = -.1916383190435098943442935597058829d0 |
---|
| 2034 | rkA(1,3) = .8052720793239878323318244859477174d-1 |
---|
| 2035 | rkA(2,1) = .1111111111111111111111111111111111d0 |
---|
| 2036 | rkA(2,2) = .2920734116652284630205027458970589d0 |
---|
| 2037 | rkA(2,3) = -.481334970546573839513422644787591d-1 |
---|
| 2038 | rkA(3,1) = .1111111111111111111111111111111111d0 |
---|
| 2039 | rkA(3,2) = .5370223859435462728402311533676479d0 |
---|
| 2040 | rkA(3,3) = .1968154772236604258683861429918299d0 |
---|
| 2041 | |
---|
| 2042 | rkB(1) = .1111111111111111111111111111111111d0 |
---|
| 2043 | rkB(2) = .5124858261884216138388134465196080d0 |
---|
| 2044 | rkB(3) = .3764030627004672750500754423692808d0 |
---|
| 2045 | |
---|
| 2046 | rkC(1) = 0.d0 |
---|
| 2047 | rkC(2) = .3550510257216821901802715925294109d0 |
---|
| 2048 | rkC(3) = .8449489742783178098197284074705891d0 |
---|
| 2049 | |
---|
| 2050 | ! Classical error estimator, embedded solution: |
---|
| 2051 | rkBhat(0) = b0 |
---|
| 2052 | rkBhat(1) = .11111111111111111111111111111111111d0-b0 |
---|
| 2053 | rkBhat(2) = .51248582618842161383881344651960810d0 |
---|
| 2054 | rkBhat(3) = .37640306270046727505007544236928079d0 |
---|
| 2055 | |
---|
| 2056 | ! New solution: H* Sum B_j*f(Z_j) = Sum D_j*Z_j |
---|
| 2057 | rkD(1) = .3333333333333333333333333333333333d0 |
---|
| 2058 | rkD(2) = -.8914115380582557157653087040196127d0 |
---|
| 2059 | rkD(3) = .1558078204724922382431975370686279d1 |
---|
| 2060 | |
---|
| 2061 | ! Classical error estimator: |
---|
| 2062 | ! H* Sum (b_j-bhat_j) f(Z_j) = H*E(0)*F(0) + Sum E_j Z_j |
---|
| 2063 | rkE(0) = .2748888295956773677478286035994148d0*b0 |
---|
| 2064 | rkE(1) = -1.374444147978386838739143017997074d0*b0 |
---|
| 2065 | rkE(2) = -1.335337922441686804550326197041126d0*b0 |
---|
| 2066 | rkE(3) = .235782604058977333559011782643466d0*b0 |
---|
| 2067 | |
---|
| 2068 | ! Sdirk error estimator |
---|
| 2069 | rkBgam(0) = 0.0d0 |
---|
| 2070 | rkBgam(1) = .1948150124588532186183490991130616d-1 |
---|
| 2071 | rkBgam(2) = .7575249005733381398986810981093584d0 |
---|
| 2072 | rkBgam(3) = -.518952314149008295083446116200793d-1 |
---|
| 2073 | rkBgam(4) = .2748888295956773677478286035994148d0 |
---|
| 2074 | |
---|
| 2075 | ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j |
---|
| 2076 | rkTheta(1) = -1.224370034375505083904362087063351d0 |
---|
| 2077 | rkTheta(2) = .9340045331532641409047527962010133d0 |
---|
| 2078 | rkTheta(3) = .4656990124352088397561234800640929d0 |
---|
| 2079 | |
---|
| 2080 | ! ELO = local order of classical error estimator |
---|
| 2081 | rkELO = 4.0d0 |
---|
| 2082 | |
---|
| 2083 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 2084 | !~~~> Diagonalize the RK matrix: |
---|
| 2085 | ! rkTinv * inv(rkA) * rkT = |
---|
| 2086 | ! | rkGamma 0 0 | |
---|
| 2087 | ! | 0 rkAlpha -rkBeta | |
---|
| 2088 | ! | 0 rkBeta rkAlpha | |
---|
| 2089 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 2090 | |
---|
| 2091 | rkGamma = 3.637834252744495732208418513577775d0 |
---|
| 2092 | rkAlpha = 2.681082873627752133895790743211112d0 |
---|
| 2093 | rkBeta = 3.050430199247410569426377624787569d0 |
---|
| 2094 | |
---|
| 2095 | rkT(1,1) = .424293819848497965354371036408369d0 |
---|
| 2096 | rkT(1,2) = -.3235571519651980681202894497035503d0 |
---|
| 2097 | rkT(1,3) = -.522137786846287839586599927945048d0 |
---|
| 2098 | rkT(2,1) = .57594609499806128896291585429339d-1 |
---|
| 2099 | rkT(2,2) = .3148663231849760131614374283783d-2 |
---|
| 2100 | rkT(2,3) = .452429247674359778577728510381731d0 |
---|
| 2101 | rkT(3,1) = 1.d0 |
---|
| 2102 | rkT(3,2) = 1.d0 |
---|
| 2103 | rkT(3,3) = 0.d0 |
---|
| 2104 | |
---|
| 2105 | rkTinv(1,1) = 1.233523612685027760114769983066164d0 |
---|
| 2106 | rkTinv(1,2) = 1.423580134265707095505388133369554d0 |
---|
| 2107 | rkTinv(1,3) = .3946330125758354736049045150429624d0 |
---|
| 2108 | rkTinv(2,1) = -1.233523612685027760114769983066164d0 |
---|
| 2109 | rkTinv(2,2) = -1.423580134265707095505388133369554d0 |
---|
| 2110 | rkTinv(2,3) = .6053669874241645263950954849570376d0 |
---|
| 2111 | rkTinv(3,1) = -.1484438963257383124456490049673414d0 |
---|
| 2112 | rkTinv(3,2) = 2.038974794939896109682070471785315d0 |
---|
| 2113 | rkTinv(3,3) = -.544501292892686735299355831692542d-1 |
---|
| 2114 | |
---|
| 2115 | rkTinvAinv(1,1) = 4.487354449794728738538663081025420d0 |
---|
| 2116 | rkTinvAinv(1,2) = 5.178748573958397475446442544234494d0 |
---|
| 2117 | rkTinvAinv(1,3) = 1.435609490412123627047824222335563d0 |
---|
| 2118 | rkTinvAinv(2,1) = -2.854361287939276673073807031221493d0 |
---|
| 2119 | rkTinvAinv(2,2) = -1.003648660720543859000994063139137d+1 |
---|
| 2120 | rkTinvAinv(2,3) = 1.789135380979465422050817815017383d0 |
---|
| 2121 | rkTinvAinv(3,1) = -4.160768067752685525282947313530352d0 |
---|
| 2122 | rkTinvAinv(3,2) = 1.124128569859216916690209918405860d0 |
---|
| 2123 | rkTinvAinv(3,3) = 1.700644430961823796581896350418417d0 |
---|
| 2124 | |
---|
| 2125 | rkAinvT(1,1) = 1.543510591072668287198054583233180d0 |
---|
| 2126 | rkAinvT(1,2) = -2.460228411937788329157493833295004d0 |
---|
| 2127 | rkAinvT(1,3) = -.412906170450356277003910443520499d0 |
---|
| 2128 | rkAinvT(2,1) = .209519643211838264029272585946993d0 |
---|
| 2129 | rkAinvT(2,2) = 1.388545667194387164417459732995766d0 |
---|
| 2130 | rkAinvT(2,3) = 1.20339553005832004974976023130002d0 |
---|
| 2131 | rkAinvT(3,1) = 3.637834252744495732208418513577775d0 |
---|
| 2132 | rkAinvT(3,2) = 2.681082873627752133895790743211112d0 |
---|
| 2133 | rkAinvT(3,3) = -3.050430199247410569426377624787569d0 |
---|
| 2134 | |
---|
| 2135 | END SUBROUTINE Radau1A_Coefficients |
---|
| 2136 | |
---|
| 2137 | |
---|
| 2138 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 2139 | END SUBROUTINE RungeKuttaTLM ! and all its internal procedures |
---|
| 2140 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 2141 | |
---|
| 2142 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 2143 | SUBROUTINE FUN_CHEM(T, V, FCT) |
---|
| 2144 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 2145 | |
---|
| 2146 | USE KPP_ROOT_Parameters |
---|
| 2147 | USE KPP_ROOT_Global |
---|
| 2148 | USE KPP_ROOT_Function, ONLY: Fun |
---|
| 2149 | USE KPP_ROOT_Rates, ONLY: Update_SUN, Update_RCONST, Update_PHOTO |
---|
| 2150 | |
---|
| 2151 | IMPLICIT NONE |
---|
| 2152 | |
---|
| 2153 | KPP_REAL, INTENT(IN) :: V(NVAR), T |
---|
| 2154 | KPP_REAL, INTENT(INOUT) :: FCT(NVAR) |
---|
| 2155 | KPP_REAL :: Told |
---|
| 2156 | |
---|
| 2157 | Told = TIME |
---|
| 2158 | TIME = T |
---|
| 2159 | CALL Update_SUN() |
---|
| 2160 | CALL Update_RCONST() |
---|
| 2161 | CALL Update_PHOTO() |
---|
| 2162 | TIME = Told |
---|
| 2163 | |
---|
| 2164 | CALL Fun(V, FIX, RCONST, FCT) |
---|
| 2165 | |
---|
| 2166 | END SUBROUTINE FUN_CHEM |
---|
| 2167 | |
---|
| 2168 | |
---|
| 2169 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 2170 | SUBROUTINE JAC_CHEM (T, V, JF) |
---|
| 2171 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 2172 | |
---|
| 2173 | USE KPP_ROOT_Parameters |
---|
| 2174 | USE KPP_ROOT_Global |
---|
| 2175 | USE KPP_ROOT_JacobianSP |
---|
| 2176 | USE KPP_ROOT_Jacobian, ONLY: Jac_SP |
---|
| 2177 | USE KPP_ROOT_Rates, ONLY: Update_SUN, Update_RCONST, Update_PHOTO |
---|
| 2178 | |
---|
| 2179 | IMPLICIT NONE |
---|
| 2180 | |
---|
| 2181 | KPP_REAL, INTENT(IN) :: V(NVAR), T |
---|
| 2182 | #ifdef FULL_ALGEBRA |
---|
| 2183 | KPP_REAL, INTENT(INOUT) :: JF(NVAR,NVAR) |
---|
| 2184 | KPP_REAL :: JV(LU_NONZERO) |
---|
| 2185 | INTEGER :: i, j |
---|
| 2186 | #else |
---|
| 2187 | KPP_REAL, INTENT(INOUT) :: JF(LU_NONZERO) |
---|
| 2188 | #endif |
---|
| 2189 | |
---|
| 2190 | KPP_REAL :: Told |
---|
| 2191 | |
---|
| 2192 | Told = TIME |
---|
| 2193 | TIME = T |
---|
| 2194 | CALL Update_SUN() |
---|
| 2195 | CALL Update_RCONST() |
---|
| 2196 | CALL Update_PHOTO() |
---|
| 2197 | TIME = Told |
---|
| 2198 | |
---|
| 2199 | #ifdef FULL_ALGEBRA |
---|
| 2200 | CALL Jac_SP(V, FIX, RCONST, JV) |
---|
| 2201 | DO j=1,NVAR |
---|
| 2202 | DO i=1,NVAR |
---|
| 2203 | JF(i,j) = 0.0d0 |
---|
| 2204 | END DO |
---|
| 2205 | END DO |
---|
| 2206 | DO i=1,LU_NONZERO |
---|
| 2207 | JF(LU_IROW(i),LU_ICOL(i)) = JV(i) |
---|
| 2208 | END DO |
---|
| 2209 | #else |
---|
| 2210 | CALL Jac_SP(V, FIX, RCONST, JF) |
---|
| 2211 | #endif |
---|
| 2212 | |
---|
| 2213 | END SUBROUTINE JAC_CHEM |
---|
| 2214 | |
---|
| 2215 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 2216 | END MODULE KPP_ROOT_Integrator |
---|
| 2217 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 2218 | |
---|