[2696] | 1 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~! |
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| 2 | ! RungeKutta - Fully Implicit 3-stage Runge-Kutta methods based on: ! |
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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, ONLY: LU_DIAG |
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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 | SUBROUTINE INTEGRATE( TIN, TOUT, & |
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| 38 | ICNTRL_U, RCNTRL_U, ISTATUS_U, RSTATUS_U, IERR_U ) |
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| 39 | |
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| 40 | USE KPP_ROOT_Parameters, ONLY: NVAR |
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| 41 | USE KPP_ROOT_Global, ONLY: ATOL,RTOL,VAR |
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| 42 | |
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| 43 | IMPLICIT NONE |
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| 44 | |
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| 45 | KPP_REAL :: TIN ! TIN - Start Time |
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| 46 | KPP_REAL :: TOUT ! TOUT - End Time |
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| 47 | INTEGER, INTENT(IN), OPTIONAL :: ICNTRL_U(20) |
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| 48 | KPP_REAL, INTENT(IN), OPTIONAL :: RCNTRL_U(20) |
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| 49 | INTEGER, INTENT(OUT), OPTIONAL :: ISTATUS_U(20) |
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| 50 | KPP_REAL, INTENT(OUT), OPTIONAL :: RSTATUS_U(20) |
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| 51 | INTEGER, INTENT(OUT), OPTIONAL :: IERR_U |
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| 52 | |
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| 53 | INTEGER :: IERR |
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| 54 | |
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| 55 | KPP_REAL :: RCNTRL(20), RSTATUS(20), T1, T2 |
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| 56 | INTEGER :: ICNTRL(20), ISTATUS(20) |
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| 57 | INTEGER, SAVE :: Ntotal = 0 |
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| 58 | |
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| 59 | RCNTRL(1:20) = 0.0_dp |
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| 60 | ICNTRL(1:20) = 0 |
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| 61 | |
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| 62 | !~~~> fine-tune the integrator: |
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| 63 | ICNTRL(2) = 0 ! 0=vector tolerances, 1=scalar tolerances |
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| 64 | ICNTRL(5) = 8 ! Max no. of Newton iterations |
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| 65 | ICNTRL(6) = 0 ! Starting values for Newton are interpolated (0) or zero (1) |
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| 66 | ICNTRL(10) = 1 ! 0 - classic or 1 - SDIRK error estimation |
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| 67 | ICNTRL(11) = 0 ! Gustaffson (0) or classic(1) controller |
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| 68 | |
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| 69 | !~~~> if optional parameters are given, and if they are >0, |
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| 70 | ! then use them to overwrite default settings |
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| 71 | IF (PRESENT(ICNTRL_U)) THEN |
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| 72 | WHERE(ICNTRL_U(:) > 0) ICNTRL(:) = ICNTRL_U(:) |
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| 73 | END IF |
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| 74 | IF (PRESENT(RCNTRL_U)) THEN |
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| 75 | WHERE(RCNTRL_U(:) > 0) RCNTRL(:) = RCNTRL_U(:) |
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| 76 | END IF |
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| 77 | |
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| 78 | |
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| 79 | T1 = TIN; T2 = TOUT |
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| 80 | CALL RungeKutta( NVAR, T1, T2, VAR, RTOL, ATOL, & |
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| 81 | RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR ) |
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| 82 | |
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| 83 | Ntotal = Ntotal + ISTATUS(Nstp) |
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| 84 | PRINT*,'NSTEPS=',ISTATUS(Nstp),' (',Ntotal,')',' O3=', VAR(ind_O3) |
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| 85 | |
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| 86 | ! if optional parameters are given for output |
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| 87 | ! use them to store information in them |
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| 88 | IF (PRESENT(ISTATUS_U)) ISTATUS_U(:) = ISTATUS(:) |
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| 89 | IF (PRESENT(RSTATUS_U)) RSTATUS_U(:) = RSTATUS(:) |
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| 90 | IF (PRESENT(IERR_U)) IERR_U = IERR |
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| 91 | |
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| 92 | IF (IERR < 0) THEN |
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| 93 | PRINT *,'Runge-Kutta: Unsuccessful exit at T=', TIN,' (IERR=',IERR,')' |
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| 94 | ENDIF |
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| 95 | |
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| 96 | END SUBROUTINE INTEGRATE |
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| 97 | |
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| 98 | |
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| 99 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 100 | SUBROUTINE RungeKutta( N,T,Tend,Y,RelTol,AbsTol, & |
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| 101 | RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR ) |
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| 102 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 103 | ! |
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| 104 | ! This implementation is based on the book and the code Radau5: |
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| 105 | ! |
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| 106 | ! E. HAIRER AND G. WANNER |
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| 107 | ! "SOLVING ORDINARY DIFFERENTIAL EQUATIONS II. |
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| 108 | ! STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS." |
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| 109 | ! SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS 14, |
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| 110 | ! SPRINGER-VERLAG (1991) |
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| 111 | ! |
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| 112 | ! UNIVERSITE DE GENEVE, DEPT. DE MATHEMATIQUES |
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| 113 | ! CH-1211 GENEVE 24, SWITZERLAND |
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| 114 | ! E-MAIL: HAIRER@DIVSUN.UNIGE.CH, WANNER@DIVSUN.UNIGE.CH |
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| 115 | ! |
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| 116 | ! Methods: |
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| 117 | ! * Radau-2A quadrature (order 5) |
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| 118 | ! * Radau-1A quadrature (order 5) |
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| 119 | ! * Lobatto-3C quadrature (order 4) |
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| 120 | ! * Gauss quadrature (order 6) |
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| 121 | ! |
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| 122 | ! (C) Adrian Sandu, August 2005 |
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| 123 | ! Virginia Polytechnic Institute and State University |
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| 124 | ! Contact: sandu@cs.vt.edu |
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| 125 | ! Revised by Philipp Miehe and Adrian Sandu, May 2006 |
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| 126 | ! This implementation is part of KPP - the Kinetic PreProcessor |
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| 127 | ! |
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| 128 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 129 | ! |
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| 130 | !~~~> INPUT ARGUMENTS: |
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| 131 | ! ---------------- |
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| 132 | ! |
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| 133 | ! Note: For input parameters equal to zero the default values of the |
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| 134 | ! corresponding variables are used. |
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| 135 | ! |
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| 136 | ! N Dimension of the system |
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| 137 | ! T Initial time value |
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| 138 | ! |
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| 139 | ! Tend Final T value (Tend-T may be positive or negative) |
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| 140 | ! |
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| 141 | ! Y(N) Initial values for Y |
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| 142 | ! |
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| 143 | ! RelTol,AbsTol Relative and absolute error tolerances. |
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| 144 | ! for ICNTRL(2) = 0: AbsTol, RelTol are N-dimensional vectors |
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| 145 | ! = 1: AbsTol, RelTol are scalars |
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| 146 | ! |
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| 147 | !~~~> Integer input parameters: |
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| 148 | ! |
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| 149 | ! ICNTRL(1) = not used |
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| 150 | ! |
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| 151 | ! ICNTRL(2) = 0: AbsTol, RelTol are NVAR-dimensional vectors |
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| 152 | ! = 1: AbsTol, RelTol are scalars |
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| 153 | ! |
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| 154 | ! ICNTRL(3) = RK method selection |
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| 155 | ! = 1: Radau-2A (the default) |
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| 156 | ! = 2: Lobatto-3C |
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| 157 | ! = 3: Gauss |
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| 158 | ! = 4: Radau-1A |
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| 159 | ! = 5: Lobatto-3A (not yet implemented) |
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| 160 | ! |
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| 161 | ! ICNTRL(4) -> maximum number of integration steps |
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| 162 | ! For ICNTRL(4)=0 the default value of 10000 is used |
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| 163 | ! |
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| 164 | ! ICNTRL(5) -> maximum number of Newton iterations |
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| 165 | ! For ICNTRL(5)=0 the default value of 8 is used |
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| 166 | ! |
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| 167 | ! ICNTRL(6) -> starting values of Newton iterations: |
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| 168 | ! ICNTRL(6)=0 : starting values are obtained from |
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| 169 | ! the extrapolated collocation solution |
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| 170 | ! (the default) |
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| 171 | ! ICNTRL(6)=1 : starting values are zero |
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| 172 | ! |
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| 173 | ! ICNTRL(10) -> switch for error estimation strategy |
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| 174 | ! ICNTRL(10) = 0: one additional stage at c=0, |
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| 175 | ! see Hairer (default) |
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| 176 | ! ICNTRL(10) = 1: two additional stages at c=0 |
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| 177 | ! and SDIRK at c=1, stiffly accurate |
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| 178 | ! |
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| 179 | ! ICNTRL(11) -> switch for step size strategy |
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| 180 | ! ICNTRL(11)=0: mod. predictive controller (Gustafsson, default) |
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| 181 | ! ICNTRL(11)=1: classical step size control |
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| 182 | ! the choice 1 seems to produce safer results; |
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| 183 | ! for simple problems, the choice 2 produces |
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| 184 | ! often slightly faster runs |
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| 185 | ! |
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| 186 | !~~~> Real input parameters: |
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| 187 | ! |
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| 188 | ! RCNTRL(1) -> Hmin, lower bound for the integration step size |
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| 189 | ! (highly recommended to keep Hmin = ZERO, the default) |
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| 190 | ! |
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| 191 | ! RCNTRL(2) -> Hmax, upper bound for the integration step size |
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| 192 | ! |
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| 193 | ! RCNTRL(3) -> Hstart, the starting step size |
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| 194 | ! |
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| 195 | ! RCNTRL(4) -> FacMin, lower bound on step decrease factor (default=0.2) |
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| 196 | ! |
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| 197 | ! RCNTRL(5) -> FacMax, upper bound on step increase factor (default=6) |
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| 198 | ! |
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| 199 | ! RCNTRL(6) -> FacRej, step decrease factor after multiple rejections |
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| 200 | ! (default=0.1) |
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| 201 | ! |
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| 202 | ! RCNTRL(7) -> FacSafe, by which the new step is slightly smaller |
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| 203 | ! than the predicted value (default=0.9) |
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| 204 | ! |
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| 205 | ! RCNTRL(8) -> ThetaMin. If Newton convergence rate smaller |
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| 206 | ! than ThetaMin the Jacobian is not recomputed; |
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| 207 | ! (default=0.001) |
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| 208 | ! |
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| 209 | ! RCNTRL(9) -> NewtonTol, stopping criterion for Newton's method |
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| 210 | ! (default=0.03) |
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| 211 | ! |
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| 212 | ! RCNTRL(10) -> Qmin |
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| 213 | ! |
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| 214 | ! RCNTRL(11) -> Qmax. If Qmin < Hnew/Hold < Qmax, then the |
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| 215 | ! step size is kept constant and the LU factorization |
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| 216 | ! reused (default Qmin=1, Qmax=1.2) |
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| 217 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 218 | ! |
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| 219 | ! |
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| 220 | ! OUTPUT ARGUMENTS: |
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| 221 | ! ----------------- |
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| 222 | ! |
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| 223 | ! T -> T value for which the solution has been computed |
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| 224 | ! (after successful return T=Tend). |
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| 225 | ! |
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| 226 | ! Y(N) -> Numerical solution at T |
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| 227 | ! |
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| 228 | ! IERR -> Reports on successfulness upon return: |
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| 229 | ! = 1 for success |
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| 230 | ! < 0 for error (value equals error code) |
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| 231 | ! |
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| 232 | ! ISTATUS(1) -> No. of function calls |
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| 233 | ! ISTATUS(2) -> No. of Jacobian calls |
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| 234 | ! ISTATUS(3) -> No. of steps |
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| 235 | ! ISTATUS(4) -> No. of accepted steps |
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| 236 | ! ISTATUS(5) -> No. of rejected steps (except at very beginning) |
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| 237 | ! ISTATUS(6) -> No. of LU decompositions |
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| 238 | ! ISTATUS(7) -> No. of forward/backward substitutions |
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| 239 | ! ISTATUS(8) -> No. of singular matrix decompositions |
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| 240 | ! |
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| 241 | ! RSTATUS(1) -> Texit, the time corresponding to the |
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| 242 | ! computed Y upon return |
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| 243 | ! RSTATUS(2) -> Hexit, last accepted step before exit |
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| 244 | ! RSTATUS(3) -> Hnew, last predicted step (not yet taken) |
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| 245 | ! For multiple restarts, use Hnew as Hstart |
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| 246 | ! in the subsequent run |
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| 247 | ! |
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| 248 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 249 | |
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| 250 | IMPLICIT NONE |
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| 251 | |
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| 252 | INTEGER :: N |
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| 253 | KPP_REAL :: Y(N),AbsTol(N),RelTol(N),RCNTRL(20),RSTATUS(20) |
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| 254 | INTEGER :: ICNTRL(20), ISTATUS(20) |
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| 255 | LOGICAL :: StartNewton, Gustafsson, SdirkError |
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| 256 | INTEGER :: IERR, ITOL |
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| 257 | KPP_REAL :: T,Tend |
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| 258 | |
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| 259 | !~~~> Control arguments |
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| 260 | INTEGER :: Max_no_steps, NewtonMaxit, rkMethod |
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| 261 | KPP_REAL :: Hmin,Hmax,Hstart,Qmin,Qmax |
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| 262 | KPP_REAL :: Roundoff, ThetaMin, NewtonTol |
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| 263 | KPP_REAL :: FacSafe,FacMin,FacMax,FacRej |
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| 264 | ! Runge-Kutta method parameters |
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| 265 | INTEGER, PARAMETER :: R2A=1, R1A=2, L3C=3, GAU=4, L3A=5 |
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| 266 | KPP_REAL :: rkT(3,3), rkTinv(3,3), rkTinvAinv(3,3), rkAinvT(3,3), & |
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| 267 | rkA(0:3,0:3), rkB(0:3), rkC(0:3), rkD(0:3), rkE(0:3), & |
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| 268 | rkBgam(0:4), rkBhat(0:4), rkTheta(0:3), rkF(0:4), & |
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| 269 | rkGamma, rkAlpha, rkBeta, rkELO |
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| 270 | !~~~> Local variables |
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| 271 | INTEGER :: i |
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| 272 | KPP_REAL, PARAMETER :: ZERO = 0.0d0, ONE = 1.0d0 |
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| 273 | |
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| 274 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 275 | ! SETTING THE PARAMETERS |
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| 276 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 277 | IERR = 0 |
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| 278 | ISTATUS(1:20) = 0 |
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| 279 | RSTATUS(1:20) = ZERO |
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| 280 | |
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| 281 | !~~~> ICNTRL(1) - autonomous system - not used |
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| 282 | !~~~> ITOL: 1 for vector and 0 for scalar AbsTol/RelTol |
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| 283 | IF (ICNTRL(2) == 0) THEN |
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| 284 | ITOL = 1 |
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| 285 | ELSE |
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| 286 | ITOL = 0 |
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| 287 | END IF |
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| 288 | !~~~> Error control selection |
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| 289 | IF (ICNTRL(10) == 0) THEN |
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| 290 | SdirkError = .FALSE. |
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| 291 | ELSE |
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| 292 | SdirkError = .TRUE. |
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| 293 | END IF |
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| 294 | !~~~> Method selection |
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| 295 | SELECT CASE (ICNTRL(3)) |
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| 296 | CASE (0,1) |
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| 297 | CALL Radau2A_Coefficients |
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| 298 | CASE (2) |
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| 299 | CALL Lobatto3C_Coefficients |
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| 300 | CASE (3) |
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| 301 | CALL Gauss_Coefficients |
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| 302 | CASE (4) |
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| 303 | CALL Radau1A_Coefficients |
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| 304 | CASE (5) |
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| 305 | CALL Lobatto3A_Coefficients |
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| 306 | CASE DEFAULT |
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| 307 | WRITE(6,*) 'ICNTRL(3)=',ICNTRL(3) |
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| 308 | CALL RK_ErrorMsg(-13,T,ZERO,IERR) |
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| 309 | END SELECT |
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| 310 | !~~~> Max_no_steps: the maximal number of time steps |
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| 311 | IF (ICNTRL(4) == 0) THEN |
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| 312 | Max_no_steps = 200000 |
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| 313 | ELSE |
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| 314 | Max_no_steps=ICNTRL(4) |
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| 315 | IF (Max_no_steps <= 0) THEN |
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| 316 | WRITE(6,*) 'ICNTRL(4)=',ICNTRL(4) |
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| 317 | CALL RK_ErrorMsg(-1,T,ZERO,IERR) |
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| 318 | END IF |
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| 319 | END IF |
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| 320 | !~~~> NewtonMaxit maximal number of Newton iterations |
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| 321 | IF (ICNTRL(5) == 0) THEN |
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| 322 | NewtonMaxit = 8 |
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| 323 | ELSE |
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| 324 | NewtonMaxit=ICNTRL(5) |
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| 325 | IF (NewtonMaxit <= 0) THEN |
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| 326 | WRITE(6,*) 'ICNTRL(5)=',ICNTRL(5) |
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| 327 | CALL RK_ErrorMsg(-2,T,ZERO,IERR) |
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| 328 | END IF |
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| 329 | END IF |
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| 330 | !~~~> StartNewton: Use extrapolation for starting values of Newton iterations |
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| 331 | IF (ICNTRL(6) == 0) THEN |
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| 332 | StartNewton = .TRUE. |
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| 333 | ELSE |
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| 334 | StartNewton = .FALSE. |
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| 335 | END IF |
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| 336 | !~~~> Gustafsson: step size controller |
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| 337 | IF(ICNTRL(11) == 0)THEN |
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| 338 | Gustafsson = .TRUE. |
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| 339 | ELSE |
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| 340 | Gustafsson = .FALSE. |
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| 341 | END IF |
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| 342 | |
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| 343 | !~~~> Roundoff: smallest number s.t. 1.0 + Roundoff > 1.0 |
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| 344 | Roundoff=WLAMCH('E'); |
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| 345 | |
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| 346 | !~~~> Hmin = minimal step size |
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| 347 | IF (RCNTRL(1) == ZERO) THEN |
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| 348 | Hmin = ZERO |
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| 349 | ELSE |
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| 350 | Hmin = MIN(ABS(RCNTRL(1)),ABS(Tend-T)) |
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| 351 | END IF |
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| 352 | !~~~> Hmax = maximal step size |
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| 353 | IF (RCNTRL(2) == ZERO) THEN |
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| 354 | Hmax = ABS(Tend-T) |
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| 355 | ELSE |
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| 356 | Hmax = MIN(ABS(RCNTRL(2)),ABS(Tend-T)) |
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| 357 | END IF |
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| 358 | !~~~> Hstart = starting step size |
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| 359 | IF (RCNTRL(3) == ZERO) THEN |
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| 360 | Hstart = ZERO |
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| 361 | ELSE |
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| 362 | Hstart = MIN(ABS(RCNTRL(3)),ABS(Tend-T)) |
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| 363 | END IF |
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| 364 | !~~~> FacMin: lower bound on step decrease factor |
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| 365 | IF(RCNTRL(4) == ZERO)THEN |
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| 366 | FacMin = 0.2d0 |
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| 367 | ELSE |
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| 368 | FacMin = RCNTRL(4) |
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| 369 | END IF |
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| 370 | !~~~> FacMax: upper bound on step increase factor |
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| 371 | IF(RCNTRL(5) == ZERO)THEN |
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| 372 | FacMax = 8.D0 |
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| 373 | ELSE |
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| 374 | FacMax = RCNTRL(5) |
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| 375 | END IF |
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| 376 | !~~~> FacRej: step decrease factor after 2 consecutive rejections |
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| 377 | IF(RCNTRL(6) == ZERO)THEN |
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| 378 | FacRej = 0.1d0 |
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| 379 | ELSE |
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| 380 | FacRej = RCNTRL(6) |
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| 381 | END IF |
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| 382 | !~~~> FacSafe: by which the new step is slightly smaller |
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| 383 | ! than the predicted value |
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| 384 | IF (RCNTRL(7) == ZERO) THEN |
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| 385 | FacSafe=0.9d0 |
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| 386 | ELSE |
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| 387 | FacSafe=RCNTRL(7) |
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| 388 | END IF |
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| 389 | IF ( (FacMax < ONE) .OR. (FacMin > ONE) .OR. & |
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| 390 | (FacSafe <= 1.0d-3) .OR. (FacSafe >= ONE) ) THEN |
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| 391 | WRITE(6,*)'RCNTRL(4:7)=',RCNTRL(4:7) |
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| 392 | CALL RK_ErrorMsg(-4,T,ZERO,IERR) |
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| 393 | END IF |
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| 394 | |
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| 395 | !~~~> ThetaMin: decides whether the Jacobian should be recomputed |
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| 396 | IF (RCNTRL(8) == ZERO) THEN |
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| 397 | ThetaMin = 1.0d-3 |
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| 398 | ELSE |
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| 399 | ThetaMin=RCNTRL(8) |
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| 400 | IF (ThetaMin <= 0.0d0 .OR. ThetaMin >= 1.0d0) THEN |
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| 401 | WRITE(6,*) 'RCNTRL(8)=', RCNTRL(8) |
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| 402 | CALL RK_ErrorMsg(-5,T,ZERO,IERR) |
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| 403 | END IF |
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| 404 | END IF |
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| 405 | !~~~> NewtonTol: stopping crierion for Newton's method |
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| 406 | IF (RCNTRL(9) == ZERO) THEN |
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| 407 | NewtonTol = 3.0d-2 |
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| 408 | ELSE |
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| 409 | NewtonTol = RCNTRL(9) |
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| 410 | IF (NewtonTol <= Roundoff) THEN |
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| 411 | WRITE(6,*) 'RCNTRL(9)=',RCNTRL(9) |
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| 412 | CALL RK_ErrorMsg(-6,T,ZERO,IERR) |
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| 413 | END IF |
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| 414 | END IF |
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| 415 | !~~~> Qmin AND Qmax: IF Qmin < Hnew/Hold < Qmax then step size = const. |
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| 416 | IF (RCNTRL(10) == ZERO) THEN |
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| 417 | Qmin=1.D0 |
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| 418 | ELSE |
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| 419 | Qmin=RCNTRL(10) |
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| 420 | END IF |
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| 421 | IF (RCNTRL(11) == ZERO) THEN |
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| 422 | Qmax=1.2D0 |
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| 423 | ELSE |
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| 424 | Qmax=RCNTRL(11) |
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| 425 | END IF |
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| 426 | IF (Qmin > ONE .OR. Qmax < ONE) THEN |
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| 427 | WRITE(6,*) 'RCNTRL(10:11)=',Qmin,Qmax |
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| 428 | CALL RK_ErrorMsg(-7,T,ZERO,IERR) |
---|
| 429 | END IF |
---|
| 430 | !~~~> Check if tolerances are reasonable |
---|
| 431 | IF (ITOL == 0) THEN |
---|
| 432 | IF (AbsTol(1) <= ZERO.OR.RelTol(1) <= 10.d0*Roundoff) THEN |
---|
| 433 | WRITE (6,*) 'AbsTol/RelTol=',AbsTol,RelTol |
---|
| 434 | CALL RK_ErrorMsg(-8,T,ZERO,IERR) |
---|
| 435 | END IF |
---|
| 436 | ELSE |
---|
| 437 | DO i=1,N |
---|
| 438 | IF (AbsTol(i) <= ZERO.OR.RelTol(i) <= 10.d0*Roundoff) THEN |
---|
| 439 | WRITE (6,*) 'AbsTol/RelTol(',i,')=',AbsTol(i),RelTol(i) |
---|
| 440 | CALL RK_ErrorMsg(-8,T,ZERO,IERR) |
---|
| 441 | END IF |
---|
| 442 | END DO |
---|
| 443 | END IF |
---|
| 444 | |
---|
| 445 | !~~~> Parameters are wrong |
---|
| 446 | IF (IERR < 0) RETURN |
---|
| 447 | |
---|
| 448 | !~~~> Call the core method |
---|
| 449 | CALL RK_Integrator( N,T,Tend,Y,IERR ) |
---|
| 450 | |
---|
| 451 | CONTAINS ! Internal procedures to RungeKutta |
---|
| 452 | |
---|
| 453 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 454 | SUBROUTINE RK_Integrator( N,T,Tend,Y,IERR ) |
---|
| 455 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 456 | |
---|
| 457 | IMPLICIT NONE |
---|
| 458 | !~~~> Arguments |
---|
| 459 | INTEGER, INTENT(IN) :: N |
---|
| 460 | KPP_REAL, INTENT(IN) :: Tend |
---|
| 461 | KPP_REAL, INTENT(INOUT) :: T, Y(NVAR) |
---|
| 462 | INTEGER, INTENT(OUT) :: IERR |
---|
| 463 | |
---|
| 464 | !~~~> Local variables |
---|
| 465 | #ifdef FULL_ALGEBRA |
---|
| 466 | KPP_REAL :: FJAC(NVAR,NVAR), E1(NVAR,NVAR) |
---|
| 467 | COMPLEX(kind=dp) :: E2(NVAR,NVAR) |
---|
| 468 | #else |
---|
| 469 | KPP_REAL :: FJAC(LU_NONZERO), E1(LU_NONZERO) |
---|
| 470 | COMPLEX(kind=dp) :: E2(LU_NONZERO) |
---|
| 471 | #endif |
---|
| 472 | KPP_REAL, DIMENSION(NVAR) :: Z1,Z2,Z3,Z4,SCAL,DZ1,DZ2,DZ3,DZ4, & |
---|
| 473 | G,TMP,F0 |
---|
| 474 | KPP_REAL :: CONT(NVAR,3), Tdirection, H, Hacc, Hnew, Hold, Fac, & |
---|
| 475 | FacGus, Theta, Err, ErrOld, NewtonRate, NewtonIncrement, & |
---|
| 476 | Hratio, Qnewton, NewtonPredictedErr,NewtonIncrementOld, ThetaSD |
---|
| 477 | INTEGER :: IP1(NVAR),IP2(NVAR),NewtonIter, ISING, Nconsecutive |
---|
| 478 | LOGICAL :: Reject, FirstStep, SkipJac, NewtonDone, SkipLU |
---|
| 479 | |
---|
| 480 | |
---|
| 481 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 482 | !~~~> Initial setting |
---|
| 483 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 484 | Tdirection = SIGN(ONE,Tend-T) |
---|
| 485 | H = MIN( MAX(ABS(Hmin),ABS(Hstart)) , Hmax ) |
---|
| 486 | IF (ABS(H) <= 10.d0*Roundoff) H = 1.0d-6 |
---|
| 487 | H = SIGN(H,Tdirection) |
---|
| 488 | Hold = H |
---|
| 489 | Reject = .FALSE. |
---|
| 490 | FirstStep = .TRUE. |
---|
| 491 | SkipJac = .FALSE. |
---|
| 492 | SkipLU = .FALSE. |
---|
| 493 | IF ((T+H*1.0001D0-Tend)*Tdirection >= ZERO) THEN |
---|
| 494 | H = Tend-T |
---|
| 495 | END IF |
---|
| 496 | Nconsecutive = 0 |
---|
| 497 | CALL RK_ErrorScale(N,ITOL,AbsTol,RelTol,Y,SCAL) |
---|
| 498 | |
---|
| 499 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 500 | !~~~> Time loop begins |
---|
| 501 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 502 | Tloop: DO WHILE ( (Tend-T)*Tdirection - Roundoff > ZERO ) |
---|
| 503 | |
---|
| 504 | !IF ( .NOT.Reject ) THEN |
---|
| 505 | CALL FUN_CHEM(T,Y,F0) |
---|
| 506 | ISTATUS(Nfun) = ISTATUS(Nfun) + 1 |
---|
| 507 | !END IF |
---|
| 508 | |
---|
| 509 | IF ( .NOT.SkipLU ) THEN ! This time around skip the Jac update and LU |
---|
| 510 | !~~~> Compute the Jacobian matrix |
---|
| 511 | IF ( .NOT.SkipJac ) THEN |
---|
| 512 | CALL JAC_CHEM(T,Y,FJAC) |
---|
| 513 | ISTATUS(Njac) = ISTATUS(Njac) + 1 |
---|
| 514 | END IF |
---|
| 515 | !~~~> Compute the matrices E1 and E2 and their decompositions |
---|
| 516 | CALL RK_Decomp(N,H,FJAC,E1,IP1,E2,IP2,ISING) |
---|
| 517 | IF (ISING /= 0) THEN |
---|
| 518 | ISTATUS(Nsng) = ISTATUS(Nsng) + 1; Nconsecutive = Nconsecutive + 1 |
---|
| 519 | IF (Nconsecutive >= 5) THEN |
---|
| 520 | CALL RK_ErrorMsg(-12,T,H,IERR); RETURN |
---|
| 521 | END IF |
---|
| 522 | H=H*0.5d0; Reject=.TRUE.; SkipJac = .TRUE.; SkipLU = .FALSE. |
---|
| 523 | CYCLE Tloop |
---|
| 524 | ELSE |
---|
| 525 | Nconsecutive = 0 |
---|
| 526 | END IF |
---|
| 527 | END IF ! SkipLU |
---|
| 528 | |
---|
| 529 | ISTATUS(Nstp) = ISTATUS(Nstp) + 1 |
---|
| 530 | IF (ISTATUS(Nstp) > Max_no_steps) THEN |
---|
| 531 | PRINT*,'Max number of time steps is ',Max_no_steps |
---|
| 532 | CALL RK_ErrorMsg(-9,T,H,IERR); RETURN |
---|
| 533 | END IF |
---|
| 534 | IF (0.1D0*ABS(H) <= ABS(T)*Roundoff) THEN |
---|
| 535 | CALL RK_ErrorMsg(-10,T,H,IERR); RETURN |
---|
| 536 | END IF |
---|
| 537 | |
---|
| 538 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 539 | !~~~> Loop for the simplified Newton iterations |
---|
| 540 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 541 | |
---|
| 542 | !~~~> Starting values for Newton iteration |
---|
| 543 | IF ( FirstStep .OR. (.NOT.StartNewton) ) THEN |
---|
| 544 | CALL Set2zero(N,Z1) |
---|
| 545 | CALL Set2zero(N,Z2) |
---|
| 546 | CALL Set2zero(N,Z3) |
---|
| 547 | ELSE |
---|
| 548 | ! Evaluate quadratic polynomial |
---|
| 549 | CALL RK_Interpolate('eval',N,H,Hold,Z1,Z2,Z3,CONT) |
---|
| 550 | END IF |
---|
| 551 | |
---|
| 552 | !~~~> Initializations for Newton iteration |
---|
| 553 | NewtonDone = .FALSE. |
---|
| 554 | Fac = 0.5d0 ! Step reduction if too many iterations |
---|
| 555 | |
---|
| 556 | NewtonLoop:DO NewtonIter = 1, NewtonMaxit |
---|
| 557 | |
---|
| 558 | !~~~> Prepare the right-hand side |
---|
| 559 | CALL RK_PrepareRHS(N,T,H,Y,F0,Z1,Z2,Z3,DZ1,DZ2,DZ3) |
---|
| 560 | |
---|
| 561 | !~~~> Solve the linear systems |
---|
| 562 | CALL RK_Solve( N,H,E1,IP1,E2,IP2,DZ1,DZ2,DZ3,ISING ) |
---|
| 563 | |
---|
| 564 | NewtonIncrement = SQRT( ( RK_ErrorNorm(N,SCAL,DZ1)**2 + & |
---|
| 565 | RK_ErrorNorm(N,SCAL,DZ2)**2 + & |
---|
| 566 | RK_ErrorNorm(N,SCAL,DZ3)**2 )/3.0d0 ) |
---|
| 567 | |
---|
| 568 | IF ( NewtonIter == 1 ) THEN |
---|
| 569 | Theta = ABS(ThetaMin) |
---|
| 570 | NewtonRate = 2.0d0 |
---|
| 571 | ELSE |
---|
| 572 | Theta = NewtonIncrement/NewtonIncrementOld |
---|
| 573 | IF (Theta < 0.99d0) THEN |
---|
| 574 | NewtonRate = Theta/(ONE-Theta) |
---|
| 575 | ELSE ! Non-convergence of Newton: Theta too large |
---|
| 576 | EXIT NewtonLoop |
---|
| 577 | END IF |
---|
| 578 | IF ( NewtonIter < NewtonMaxit ) THEN |
---|
| 579 | ! Predict error at the end of Newton process |
---|
| 580 | NewtonPredictedErr = NewtonIncrement & |
---|
| 581 | *Theta**(NewtonMaxit-NewtonIter)/(ONE-Theta) |
---|
| 582 | IF (NewtonPredictedErr >= NewtonTol) THEN |
---|
| 583 | ! Non-convergence of Newton: predicted error too large |
---|
| 584 | Qnewton = MIN(10.0d0,NewtonPredictedErr/NewtonTol) |
---|
| 585 | Fac=0.8d0*Qnewton**(-ONE/(1+NewtonMaxit-NewtonIter)) |
---|
| 586 | EXIT NewtonLoop |
---|
| 587 | END IF |
---|
| 588 | END IF |
---|
| 589 | END IF |
---|
| 590 | |
---|
| 591 | NewtonIncrementOld = MAX(NewtonIncrement,Roundoff) |
---|
| 592 | ! Update solution |
---|
| 593 | CALL WAXPY(N,-ONE,DZ1,1,Z1,1) ! Z1 <- Z1 - DZ1 |
---|
| 594 | CALL WAXPY(N,-ONE,DZ2,1,Z2,1) ! Z2 <- Z2 - DZ2 |
---|
| 595 | CALL WAXPY(N,-ONE,DZ3,1,Z3,1) ! Z3 <- Z3 - DZ3 |
---|
| 596 | |
---|
| 597 | ! Check error in Newton iterations |
---|
| 598 | NewtonDone = (NewtonRate*NewtonIncrement <= NewtonTol) |
---|
| 599 | IF (NewtonDone) EXIT NewtonLoop |
---|
| 600 | IF (NewtonIter == NewtonMaxit) THEN |
---|
| 601 | PRINT*, 'Slow or no convergence in Newton Iteration: Max no. of', & |
---|
| 602 | 'Newton iterations reached' |
---|
| 603 | END IF |
---|
| 604 | |
---|
| 605 | END DO NewtonLoop |
---|
| 606 | |
---|
| 607 | IF (.NOT.NewtonDone) THEN |
---|
| 608 | !CALL RK_ErrorMsg(-12,T,H,IERR); |
---|
| 609 | H = Fac*H; Reject=.TRUE.; SkipJac = .TRUE.; SkipLU = .FALSE. |
---|
| 610 | CYCLE Tloop |
---|
| 611 | END IF |
---|
| 612 | |
---|
| 613 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 614 | !~~~> SDIRK Stage |
---|
| 615 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 616 | IF (SdirkError) THEN |
---|
| 617 | |
---|
| 618 | !~~~> Starting values for Newton iterations |
---|
| 619 | Z4(1:N) = Z3(1:N) |
---|
| 620 | |
---|
| 621 | !~~~> Prepare the loop-independent part of the right-hand side |
---|
| 622 | ! G = H*rkBgam(0)*F0 + rkTheta(1)*Z1 + rkTheta(2)*Z2 + rkTheta(3)*Z3 |
---|
| 623 | CALL Set2Zero(N, G) |
---|
| 624 | IF (rkMethod/=L3A) CALL WAXPY(N,rkBgam(0)*H, F0,1,G,1) |
---|
| 625 | CALL WAXPY(N,rkTheta(1),Z1,1,G,1) |
---|
| 626 | CALL WAXPY(N,rkTheta(2),Z2,1,G,1) |
---|
| 627 | CALL WAXPY(N,rkTheta(3),Z3,1,G,1) |
---|
| 628 | |
---|
| 629 | !~~~> Initializations for Newton iteration |
---|
| 630 | NewtonDone = .FALSE. |
---|
| 631 | Fac = 0.5d0 ! Step reduction factor if too many iterations |
---|
| 632 | |
---|
| 633 | SDNewtonLoop:DO NewtonIter = 1, NewtonMaxit |
---|
| 634 | |
---|
| 635 | !~~~> Prepare the loop-dependent part of the right-hand side |
---|
| 636 | CALL WADD(N,Y,Z4,TMP) ! TMP <- Y + Z4 |
---|
| 637 | CALL FUN_CHEM(T+H,TMP,DZ4) ! DZ4 <- Fun(Y+Z4) |
---|
| 638 | ISTATUS(Nfun) = ISTATUS(Nfun) + 1 |
---|
| 639 | ! DZ4(1:N) = (G(1:N)-Z4(1:N))*(rkGamma/H) + DZ4(1:N) |
---|
| 640 | CALL WAXPY (N, -ONE*rkGamma/H, Z4, 1, DZ4, 1) |
---|
| 641 | CALL WAXPY (N, rkGamma/H, G,1, DZ4,1) |
---|
| 642 | |
---|
| 643 | !~~~> Solve the linear system |
---|
| 644 | #ifdef FULL_ALGEBRA |
---|
| 645 | CALL DGETRS( 'N', N, 1, E1, N, IP1, DZ4, N, ISING ) |
---|
| 646 | #else |
---|
| 647 | CALL KppSolve(E1, DZ4) |
---|
| 648 | #endif |
---|
| 649 | |
---|
| 650 | !~~~> Check convergence of Newton iterations |
---|
| 651 | NewtonIncrement = RK_ErrorNorm(N,SCAL,DZ4) |
---|
| 652 | IF ( NewtonIter == 1 ) THEN |
---|
| 653 | ThetaSD = ABS(ThetaMin) |
---|
| 654 | NewtonRate = 2.0d0 |
---|
| 655 | ELSE |
---|
| 656 | ThetaSD = NewtonIncrement/NewtonIncrementOld |
---|
| 657 | IF (ThetaSD < 0.99d0) THEN |
---|
| 658 | NewtonRate = ThetaSD/(ONE-ThetaSD) |
---|
| 659 | ! Predict error at the end of Newton process |
---|
| 660 | NewtonPredictedErr = NewtonIncrement & |
---|
| 661 | *ThetaSD**(NewtonMaxit-NewtonIter)/(ONE-ThetaSD) |
---|
| 662 | IF (NewtonPredictedErr >= NewtonTol) THEN |
---|
| 663 | ! Non-convergence of Newton: predicted error too large |
---|
| 664 | !PRINT*,'Error too large: ', NewtonPredictedErr |
---|
| 665 | Qnewton = MIN(10.0d0,NewtonPredictedErr/NewtonTol) |
---|
| 666 | Fac = 0.8d0*Qnewton**(-ONE/(1+NewtonMaxit-NewtonIter)) |
---|
| 667 | EXIT SDNewtonLoop |
---|
| 668 | END IF |
---|
| 669 | ELSE ! Non-convergence of Newton: Theta too large |
---|
| 670 | !PRINT*,'Theta too large: ',ThetaSD |
---|
| 671 | EXIT SDNewtonLoop |
---|
| 672 | END IF |
---|
| 673 | END IF |
---|
| 674 | NewtonIncrementOld = NewtonIncrement |
---|
| 675 | ! Update solution: Z4 <-- Z4 + DZ4 |
---|
| 676 | CALL WAXPY(N,ONE,DZ4,1,Z4,1) |
---|
| 677 | |
---|
| 678 | ! Check error in Newton iterations |
---|
| 679 | NewtonDone = (NewtonRate*NewtonIncrement <= NewtonTol) |
---|
| 680 | IF (NewtonDone) EXIT SDNewtonLoop |
---|
| 681 | |
---|
| 682 | END DO SDNewtonLoop |
---|
| 683 | |
---|
| 684 | IF (.NOT.NewtonDone) THEN |
---|
| 685 | H = Fac*H; Reject=.TRUE.; SkipJac = .TRUE.; SkipLU = .FALSE. |
---|
| 686 | CYCLE Tloop |
---|
| 687 | END IF |
---|
| 688 | END IF |
---|
| 689 | !~~~> End of implified SDIRK Newton iterations |
---|
| 690 | |
---|
| 691 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 692 | !~~~> Error estimation |
---|
| 693 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 694 | IF (SdirkError) THEN |
---|
| 695 | CALL Set2Zero(N, DZ4) |
---|
| 696 | IF (rkMethod==L3A) THEN |
---|
| 697 | DZ4(1:N) = H*rkF(0)*F0(1:N) |
---|
| 698 | IF (rkF(1) /= ZERO) CALL WAXPY(N, rkF(1), Z1, 1, DZ4, 1) |
---|
| 699 | IF (rkF(2) /= ZERO) CALL WAXPY(N, rkF(2), Z2, 1, DZ4, 1) |
---|
| 700 | IF (rkF(3) /= ZERO) CALL WAXPY(N, rkF(3), Z3, 1, DZ4, 1) |
---|
| 701 | TMP = Y + Z4 |
---|
| 702 | CALL FUN_CHEM(T+H,TMP,DZ1) |
---|
| 703 | CALL WAXPY(N, H*rkBgam(4), DZ1, 1, DZ4, 1) |
---|
| 704 | ELSE |
---|
| 705 | ! DZ4(1:N) = rkD(1)*Z1 + rkD(2)*Z2 + rkD(3)*Z3 - Z4 |
---|
| 706 | IF (rkD(1) /= ZERO) CALL WAXPY(N, rkD(1), Z1, 1, DZ4, 1) |
---|
| 707 | IF (rkD(2) /= ZERO) CALL WAXPY(N, rkD(2), Z2, 1, DZ4, 1) |
---|
| 708 | IF (rkD(3) /= ZERO) CALL WAXPY(N, rkD(3), Z3, 1, DZ4, 1) |
---|
| 709 | CALL WAXPY(N, -ONE, Z4, 1, DZ4, 1) |
---|
| 710 | END IF |
---|
| 711 | Err = RK_ErrorNorm(N,SCAL,DZ4) |
---|
| 712 | ELSE |
---|
| 713 | CALL RK_ErrorEstimate(N,H,T,Y,F0, & |
---|
| 714 | E1,IP1,Z1,Z2,Z3,SCAL,Err,FirstStep,Reject) |
---|
| 715 | END IF |
---|
| 716 | |
---|
| 717 | !~~~> Computation of new step size Hnew |
---|
| 718 | Fac = Err**(-ONE/rkELO)* & |
---|
| 719 | MIN(FacSafe,(ONE+2*NewtonMaxit)/(NewtonIter+2*NewtonMaxit)) |
---|
| 720 | Fac = MIN(FacMax,MAX(FacMin,Fac)) |
---|
| 721 | Hnew = Fac*H |
---|
| 722 | |
---|
| 723 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 724 | !~~~> Accept/reject step |
---|
| 725 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 726 | accept:IF (Err < ONE) THEN !~~~> STEP IS ACCEPTED |
---|
| 727 | FirstStep=.FALSE. |
---|
| 728 | ISTATUS(Nacc) = ISTATUS(Nacc) + 1 |
---|
| 729 | IF (Gustafsson) THEN |
---|
| 730 | !~~~> Predictive controller of Gustafsson |
---|
| 731 | IF (ISTATUS(Nacc) > 1) THEN |
---|
| 732 | FacGus=FacSafe*(H/Hacc)*(Err**2/ErrOld)**(-0.25d0) |
---|
| 733 | FacGus=MIN(FacMax,MAX(FacMin,FacGus)) |
---|
| 734 | Fac=MIN(Fac,FacGus) |
---|
| 735 | Hnew = Fac*H |
---|
| 736 | END IF |
---|
| 737 | Hacc=H |
---|
| 738 | ErrOld=MAX(1.0d-2,Err) |
---|
| 739 | END IF |
---|
| 740 | Hold = H |
---|
| 741 | T = T+H |
---|
| 742 | ! Update solution: Y <- Y + sum(d_i Z_i) |
---|
| 743 | IF (rkD(1) /= ZERO) CALL WAXPY(N,rkD(1),Z1,1,Y,1) |
---|
| 744 | IF (rkD(2) /= ZERO) CALL WAXPY(N,rkD(2),Z2,1,Y,1) |
---|
| 745 | IF (rkD(3) /= ZERO) CALL WAXPY(N,rkD(3),Z3,1,Y,1) |
---|
| 746 | ! Construct the solution quadratic interpolant Q(c_i) = Z_i, i=1:3 |
---|
| 747 | IF (StartNewton) CALL RK_Interpolate('make',N,H,Hold,Z1,Z2,Z3,CONT) |
---|
| 748 | CALL RK_ErrorScale(N,ITOL,AbsTol,RelTol,Y,SCAL) |
---|
| 749 | RSTATUS(Ntexit) = T |
---|
| 750 | RSTATUS(Nhnew) = Hnew |
---|
| 751 | RSTATUS(Nhacc) = H |
---|
| 752 | Hnew = Tdirection*MIN( MAX(ABS(Hnew),Hmin) , Hmax ) |
---|
| 753 | IF (Reject) Hnew = Tdirection*MIN(ABS(Hnew),ABS(H)) |
---|
| 754 | Reject = .FALSE. |
---|
| 755 | IF ((T+Hnew/Qmin-Tend)*Tdirection >= ZERO) THEN |
---|
| 756 | H = Tend-T |
---|
| 757 | ELSE |
---|
| 758 | Hratio=Hnew/H |
---|
| 759 | ! Reuse the LU decomposition |
---|
| 760 | SkipLU = (Theta<=ThetaMin) .AND. (Hratio>=Qmin) .AND. (Hratio<=Qmax) |
---|
| 761 | IF (.NOT.SkipLU) H=Hnew |
---|
| 762 | END IF |
---|
| 763 | ! If convergence is fast enough, do not update Jacobian |
---|
| 764 | ! SkipJac = (Theta <= ThetaMin) |
---|
| 765 | SkipJac = .FALSE. |
---|
| 766 | |
---|
| 767 | ELSE accept !~~~> Step is rejected |
---|
| 768 | IF (FirstStep .OR. Reject) THEN |
---|
| 769 | H = FacRej*H |
---|
| 770 | ELSE |
---|
| 771 | H = Hnew |
---|
| 772 | END IF |
---|
| 773 | Reject = .TRUE. |
---|
| 774 | SkipJac = .TRUE. ! Skip if rejected - Jac is independent of H |
---|
| 775 | SkipLU = .FALSE. |
---|
| 776 | IF (ISTATUS(Nacc) >= 1) ISTATUS(Nrej) = ISTATUS(Nrej) + 1 |
---|
| 777 | END IF accept |
---|
| 778 | |
---|
| 779 | END DO Tloop |
---|
| 780 | |
---|
| 781 | ! Successful exit |
---|
| 782 | IERR = 1 |
---|
| 783 | |
---|
| 784 | END SUBROUTINE RK_Integrator |
---|
| 785 | |
---|
| 786 | |
---|
| 787 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 788 | SUBROUTINE RK_ErrorMsg(Code,T,H,IERR) |
---|
| 789 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 790 | ! Handles all error messages |
---|
| 791 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 792 | |
---|
| 793 | IMPLICIT NONE |
---|
| 794 | KPP_REAL, INTENT(IN) :: T, H |
---|
| 795 | INTEGER, INTENT(IN) :: Code |
---|
| 796 | INTEGER, INTENT(OUT) :: IERR |
---|
| 797 | |
---|
| 798 | IERR = Code |
---|
| 799 | PRINT * , & |
---|
| 800 | 'Forced exit from RungeKutta due to the following error:' |
---|
| 801 | |
---|
| 802 | |
---|
| 803 | SELECT CASE (Code) |
---|
| 804 | CASE (-1) |
---|
| 805 | PRINT * , '--> Improper value for maximal no of steps' |
---|
| 806 | CASE (-2) |
---|
| 807 | PRINT * , '--> Improper value for maximal no of Newton iterations' |
---|
| 808 | CASE (-3) |
---|
| 809 | PRINT * , '--> Hmin/Hmax/Hstart must be positive' |
---|
| 810 | CASE (-4) |
---|
| 811 | PRINT * , '--> Improper values for FacMin/FacMax/FacSafe/FacRej' |
---|
| 812 | CASE (-5) |
---|
| 813 | PRINT * , '--> Improper value for ThetaMin' |
---|
| 814 | CASE (-6) |
---|
| 815 | PRINT * , '--> Newton stopping tolerance too small' |
---|
| 816 | CASE (-7) |
---|
| 817 | PRINT * , '--> Improper values for Qmin, Qmax' |
---|
| 818 | CASE (-8) |
---|
| 819 | PRINT * , '--> Tolerances are too small' |
---|
| 820 | CASE (-9) |
---|
| 821 | PRINT * , '--> No of steps exceeds maximum bound' |
---|
| 822 | CASE (-10) |
---|
| 823 | PRINT * , '--> Step size too small: T + 10*H = T', & |
---|
| 824 | ' or H < Roundoff' |
---|
| 825 | CASE (-11) |
---|
| 826 | PRINT * , '--> Matrix is repeatedly singular' |
---|
| 827 | CASE (-12) |
---|
| 828 | PRINT * , '--> Non-convergence of Newton iterations' |
---|
| 829 | CASE (-13) |
---|
| 830 | PRINT * , '--> Requested RK method not implemented' |
---|
| 831 | CASE DEFAULT |
---|
| 832 | PRINT *, 'Unknown Error code: ', Code |
---|
| 833 | END SELECT |
---|
| 834 | |
---|
| 835 | WRITE(6,FMT="(5X,'T=',E12.5,' H=',E12.5)") T, H |
---|
| 836 | |
---|
| 837 | END SUBROUTINE RK_ErrorMsg |
---|
| 838 | |
---|
| 839 | |
---|
| 840 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 841 | SUBROUTINE RK_ErrorScale(N,ITOL,AbsTol,RelTol,Y,SCAL) |
---|
| 842 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 843 | ! Handles all error messages |
---|
| 844 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 845 | IMPLICIT NONE |
---|
| 846 | INTEGER, INTENT(IN) :: N, ITOL |
---|
| 847 | KPP_REAL, INTENT(IN) :: AbsTol(*), RelTol(*), Y(N) |
---|
| 848 | KPP_REAL, INTENT(OUT) :: SCAL(N) |
---|
| 849 | INTEGER :: i |
---|
| 850 | |
---|
| 851 | IF (ITOL==0) THEN |
---|
| 852 | DO i=1,N |
---|
| 853 | SCAL(i)= ONE/(AbsTol(1)+RelTol(1)*ABS(Y(i))) |
---|
| 854 | END DO |
---|
| 855 | ELSE |
---|
| 856 | DO i=1,N |
---|
| 857 | SCAL(i)=ONE/(AbsTol(i)+RelTol(i)*ABS(Y(i))) |
---|
| 858 | END DO |
---|
| 859 | END IF |
---|
| 860 | |
---|
| 861 | END SUBROUTINE RK_ErrorScale |
---|
| 862 | |
---|
| 863 | |
---|
| 864 | !!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 865 | ! SUBROUTINE RK_Transform(N,Tr,Z1,Z2,Z3,W1,W2,W3) |
---|
| 866 | !!~~~> W <-- Tr x Z |
---|
| 867 | !!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 868 | ! IMPLICIT NONE |
---|
| 869 | ! INTEGER :: N, i |
---|
| 870 | ! KPP_REAL :: Tr(3,3),Z1(N),Z2(N),Z3(N),W1(N),W2(N),W3(N) |
---|
| 871 | ! KPP_REAL :: x1, x2, x3 |
---|
| 872 | ! DO i=1,N |
---|
| 873 | ! x1 = Z1(i); x2 = Z2(i); x3 = Z3(i) |
---|
| 874 | ! W1(i) = Tr(1,1)*x1 + Tr(1,2)*x2 + Tr(1,3)*x3 |
---|
| 875 | ! W2(i) = Tr(2,1)*x1 + Tr(2,2)*x2 + Tr(2,3)*x3 |
---|
| 876 | ! W3(i) = Tr(3,1)*x1 + Tr(3,2)*x2 + Tr(3,3)*x3 |
---|
| 877 | ! END DO |
---|
| 878 | ! END SUBROUTINE RK_Transform |
---|
| 879 | |
---|
| 880 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 881 | SUBROUTINE RK_Interpolate(action,N,H,Hold,Z1,Z2,Z3,CONT) |
---|
| 882 | !~~~> Constructs or evaluates a quadratic polynomial |
---|
| 883 | ! that interpolates the Z solution at current step |
---|
| 884 | ! and provides starting values for the next step |
---|
| 885 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 886 | INTEGER :: N, i |
---|
| 887 | KPP_REAL :: H,Hold,Z1(N),Z2(N),Z3(N),CONT(N,3) |
---|
| 888 | KPP_REAL :: r, x1, x2, x3, den |
---|
| 889 | CHARACTER(LEN=4) :: action |
---|
| 890 | |
---|
| 891 | SELECT CASE (action) |
---|
| 892 | CASE ('make') |
---|
| 893 | ! Construct the solution quadratic interpolant Q(c_i) = Z_i, i=1:3 |
---|
| 894 | den = (rkC(3)-rkC(2))*(rkC(2)-rkC(1))*(rkC(1)-rkC(3)) |
---|
| 895 | DO i=1,N |
---|
| 896 | CONT(i,1)=(-rkC(3)**2*rkC(2)*Z1(i)+Z3(i)*rkC(2)*rkC(1)**2 & |
---|
| 897 | +rkC(2)**2*rkC(3)*Z1(i)-rkC(2)**2*rkC(1)*Z3(i) & |
---|
| 898 | +rkC(3)**2*rkC(1)*Z2(i)-Z2(i)*rkC(3)*rkC(1)**2)& |
---|
| 899 | /den-Z3(i) |
---|
| 900 | CONT(i,2)= -( rkC(1)**2*(Z3(i)-Z2(i)) + rkC(2)**2*(Z1(i) & |
---|
| 901 | -Z3(i)) +rkC(3)**2*(Z2(i)-Z1(i)) )/den |
---|
| 902 | CONT(i,3)= ( rkC(1)*(Z3(i)-Z2(i)) + rkC(2)*(Z1(i)-Z3(i)) & |
---|
| 903 | +rkC(3)*(Z2(i)-Z1(i)) )/den |
---|
| 904 | END DO |
---|
| 905 | CASE ('eval') |
---|
| 906 | ! Evaluate quadratic polynomial |
---|
| 907 | r = H/Hold |
---|
| 908 | x1 = ONE + rkC(1)*r |
---|
| 909 | x2 = ONE + rkC(2)*r |
---|
| 910 | x3 = ONE + rkC(3)*r |
---|
| 911 | DO i=1,N |
---|
| 912 | Z1(i) = CONT(i,1)+x1*(CONT(i,2)+x1*CONT(i,3)) |
---|
| 913 | Z2(i) = CONT(i,1)+x2*(CONT(i,2)+x2*CONT(i,3)) |
---|
| 914 | Z3(i) = CONT(i,1)+x3*(CONT(i,2)+x3*CONT(i,3)) |
---|
| 915 | END DO |
---|
| 916 | END SELECT |
---|
| 917 | END SUBROUTINE RK_Interpolate |
---|
| 918 | |
---|
| 919 | |
---|
| 920 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 921 | SUBROUTINE RK_PrepareRHS(N,T,H,Y,F0,Z1,Z2,Z3,R1,R2,R3) |
---|
| 922 | !~~~> Prepare the right-hand side for Newton iterations |
---|
| 923 | ! R = Z - hA x F |
---|
| 924 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 925 | IMPLICIT NONE |
---|
| 926 | |
---|
| 927 | INTEGER :: N |
---|
| 928 | KPP_REAL :: T, H |
---|
| 929 | KPP_REAL, DIMENSION(N) :: Y,Z1,Z2,Z3,F0,F,R1,R2,R3,TMP |
---|
| 930 | |
---|
| 931 | CALL WCOPY(N,Z1,1,R1,1) ! R1 <- Z1 |
---|
| 932 | CALL WCOPY(N,Z2,1,R2,1) ! R2 <- Z2 |
---|
| 933 | CALL WCOPY(N,Z3,1,R3,1) ! R3 <- Z3 |
---|
| 934 | |
---|
| 935 | IF (rkMethod==L3A) THEN |
---|
| 936 | CALL WAXPY(N,-H*rkA(1,0),F0,1,R1,1) ! R1 <- R1 - h*A_10*F0 |
---|
| 937 | CALL WAXPY(N,-H*rkA(2,0),F0,1,R2,1) ! R2 <- R2 - h*A_20*F0 |
---|
| 938 | CALL WAXPY(N,-H*rkA(3,0),F0,1,R3,1) ! R3 <- R3 - h*A_30*F0 |
---|
| 939 | END IF |
---|
| 940 | |
---|
| 941 | CALL WADD(N,Y,Z1,TMP) ! TMP <- Y + Z1 |
---|
| 942 | CALL FUN_CHEM(T+rkC(1)*H,TMP,F) ! F1 <- Fun(Y+Z1) |
---|
| 943 | CALL WAXPY(N,-H*rkA(1,1),F,1,R1,1) ! R1 <- R1 - h*A_11*F1 |
---|
| 944 | CALL WAXPY(N,-H*rkA(2,1),F,1,R2,1) ! R2 <- R2 - h*A_21*F1 |
---|
| 945 | CALL WAXPY(N,-H*rkA(3,1),F,1,R3,1) ! R3 <- R3 - h*A_31*F1 |
---|
| 946 | |
---|
| 947 | CALL WADD(N,Y,Z2,TMP) ! TMP <- Y + Z2 |
---|
| 948 | CALL FUN_CHEM(T+rkC(2)*H,TMP,F) ! F2 <- Fun(Y+Z2) |
---|
| 949 | CALL WAXPY(N,-H*rkA(1,2),F,1,R1,1) ! R1 <- R1 - h*A_12*F2 |
---|
| 950 | CALL WAXPY(N,-H*rkA(2,2),F,1,R2,1) ! R2 <- R2 - h*A_22*F2 |
---|
| 951 | CALL WAXPY(N,-H*rkA(3,2),F,1,R3,1) ! R3 <- R3 - h*A_32*F2 |
---|
| 952 | |
---|
| 953 | CALL WADD(N,Y,Z3,TMP) ! TMP <- Y + Z3 |
---|
| 954 | CALL FUN_CHEM(T+rkC(3)*H,TMP,F) ! F3 <- Fun(Y+Z3) |
---|
| 955 | CALL WAXPY(N,-H*rkA(1,3),F,1,R1,1) ! R1 <- R1 - h*A_13*F3 |
---|
| 956 | CALL WAXPY(N,-H*rkA(2,3),F,1,R2,1) ! R2 <- R2 - h*A_23*F3 |
---|
| 957 | CALL WAXPY(N,-H*rkA(3,3),F,1,R3,1) ! R3 <- R3 - h*A_33*F3 |
---|
| 958 | |
---|
| 959 | END SUBROUTINE RK_PrepareRHS |
---|
| 960 | |
---|
| 961 | |
---|
| 962 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 963 | SUBROUTINE RK_Decomp(N,H,FJAC,E1,IP1,E2,IP2,ISING) |
---|
| 964 | !~~~> Compute the matrices E1 and E2 and their decompositions |
---|
| 965 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 966 | IMPLICIT NONE |
---|
| 967 | |
---|
| 968 | INTEGER :: N, ISING |
---|
| 969 | KPP_REAL :: H, Alpha, Beta, Gamma |
---|
| 970 | #ifdef FULL_ALGEBRA |
---|
| 971 | KPP_REAL :: FJAC(NVAR,NVAR),E1(NVAR,NVAR) |
---|
| 972 | COMPLEX(kind=dp) :: E2(N,N) |
---|
| 973 | #else |
---|
| 974 | KPP_REAL :: FJAC(LU_NONZERO),E1(LU_NONZERO) |
---|
| 975 | COMPLEX(kind=dp) :: E2(LU_NONZERO) |
---|
| 976 | #endif |
---|
| 977 | INTEGER :: IP1(N), IP2(N), i, j |
---|
| 978 | |
---|
| 979 | Gamma = rkGamma/H |
---|
| 980 | Alpha = rkAlpha/H |
---|
| 981 | Beta = rkBeta /H |
---|
| 982 | |
---|
| 983 | #ifdef FULL_ALGEBRA |
---|
| 984 | DO j=1,N |
---|
| 985 | DO i=1,N |
---|
| 986 | E1(i,j)=-FJAC(i,j) |
---|
| 987 | END DO |
---|
| 988 | E1(j,j)=E1(j,j)+Gamma |
---|
| 989 | END DO |
---|
| 990 | CALL DGETRF(N,N,E1,N,IP1,ISING) |
---|
| 991 | #else |
---|
| 992 | DO i=1,LU_NONZERO |
---|
| 993 | E1(i)=-FJAC(i) |
---|
| 994 | END DO |
---|
| 995 | DO i=1,NVAR |
---|
| 996 | j=LU_DIAG(i); E1(j)=E1(j)+Gamma |
---|
| 997 | END DO |
---|
| 998 | CALL KppDecomp(E1,ISING) |
---|
| 999 | #endif |
---|
| 1000 | |
---|
| 1001 | IF (ISING /= 0) THEN |
---|
| 1002 | ISTATUS(Ndec) = ISTATUS(Ndec) + 1 |
---|
| 1003 | RETURN |
---|
| 1004 | END IF |
---|
| 1005 | |
---|
| 1006 | #ifdef FULL_ALGEBRA |
---|
| 1007 | DO j=1,N |
---|
| 1008 | DO i=1,N |
---|
| 1009 | E2(i,j) = DCMPLX( -FJAC(i,j), ZERO ) |
---|
| 1010 | END DO |
---|
| 1011 | E2(j,j) = E2(j,j) + CMPLX( Alpha, Beta ) |
---|
| 1012 | END DO |
---|
| 1013 | CALL ZGETRF(N,N,E2,N,IP2,ISING) |
---|
| 1014 | #else |
---|
| 1015 | DO i=1,LU_NONZERO |
---|
| 1016 | E2(i) = DCMPLX( -FJAC(i), ZERO ) |
---|
| 1017 | END DO |
---|
| 1018 | DO i=1,NVAR |
---|
| 1019 | j=LU_DIAG(i); E2(j)=E2(j) + CMPLX( Alpha, Beta ) |
---|
| 1020 | END DO |
---|
| 1021 | CALL KppDecompCmplx(E2,ISING) |
---|
| 1022 | #endif |
---|
| 1023 | ISTATUS(Ndec) = ISTATUS(Ndec) + 1 |
---|
| 1024 | |
---|
| 1025 | END SUBROUTINE RK_Decomp |
---|
| 1026 | |
---|
| 1027 | |
---|
| 1028 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1029 | SUBROUTINE RK_Solve(N,H,E1,IP1,E2,IP2,R1,R2,R3,ISING) |
---|
| 1030 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1031 | IMPLICIT NONE |
---|
| 1032 | INTEGER :: N,IP1(NVAR),IP2(NVAR),ISING |
---|
| 1033 | #ifdef FULL_ALGEBRA |
---|
| 1034 | KPP_REAL :: E1(NVAR,NVAR) |
---|
| 1035 | COMPLEX(kind=dp) :: E2(NVAR,NVAR) |
---|
| 1036 | #else |
---|
| 1037 | KPP_REAL :: E1(LU_NONZERO) |
---|
| 1038 | COMPLEX(kind=dp) :: E2(LU_NONZERO) |
---|
| 1039 | #endif |
---|
| 1040 | KPP_REAL :: R1(N),R2(N),R3(N) |
---|
| 1041 | KPP_REAL :: H, x1, x2, x3 |
---|
| 1042 | COMPLEX(kind=dp) :: BC(N) |
---|
| 1043 | INTEGER :: i |
---|
| 1044 | ! |
---|
| 1045 | ! Z <- h^{-1) T^{-1) A^{-1) x Z |
---|
| 1046 | DO i=1,N |
---|
| 1047 | x1 = R1(i)/H; x2 = R2(i)/H; x3 = R3(i)/H |
---|
| 1048 | R1(i) = rkTinvAinv(1,1)*x1 + rkTinvAinv(1,2)*x2 + rkTinvAinv(1,3)*x3 |
---|
| 1049 | R2(i) = rkTinvAinv(2,1)*x1 + rkTinvAinv(2,2)*x2 + rkTinvAinv(2,3)*x3 |
---|
| 1050 | R3(i) = rkTinvAinv(3,1)*x1 + rkTinvAinv(3,2)*x2 + rkTinvAinv(3,3)*x3 |
---|
| 1051 | END DO |
---|
| 1052 | |
---|
| 1053 | #ifdef FULL_ALGEBRA |
---|
| 1054 | CALL DGETRS ('N',N,1,E1,N,IP1,R1,N,0) |
---|
| 1055 | #else |
---|
| 1056 | CALL KppSolve (E1,R1) |
---|
| 1057 | #endif |
---|
| 1058 | ! |
---|
| 1059 | DO i=1,N |
---|
| 1060 | BC(i) = DCMPLX(R2(i),R3(i)) |
---|
| 1061 | END DO |
---|
| 1062 | #ifdef FULL_ALGEBRA |
---|
| 1063 | CALL ZGETRS ('N',N,1,E2,N,IP2,BC,N,0) |
---|
| 1064 | #else |
---|
| 1065 | CALL KppSolveCmplx (E2,BC) |
---|
| 1066 | #endif |
---|
| 1067 | DO i=1,N |
---|
| 1068 | R2(i) = DBLE( BC(i) ) |
---|
| 1069 | R3(i) = AIMAG( BC(i) ) |
---|
| 1070 | END DO |
---|
| 1071 | |
---|
| 1072 | ! Z <- T x Z |
---|
| 1073 | DO i=1,N |
---|
| 1074 | x1 = R1(i); x2 = R2(i); x3 = R3(i) |
---|
| 1075 | R1(i) = rkT(1,1)*x1 + rkT(1,2)*x2 + rkT(1,3)*x3 |
---|
| 1076 | R2(i) = rkT(2,1)*x1 + rkT(2,2)*x2 + rkT(2,3)*x3 |
---|
| 1077 | R3(i) = rkT(3,1)*x1 + rkT(3,2)*x2 + rkT(3,3)*x3 |
---|
| 1078 | END DO |
---|
| 1079 | |
---|
| 1080 | ISTATUS(Nsol) = ISTATUS(Nsol) + 1 |
---|
| 1081 | |
---|
| 1082 | END SUBROUTINE RK_Solve |
---|
| 1083 | |
---|
| 1084 | |
---|
| 1085 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1086 | SUBROUTINE RK_ErrorEstimate(N,H,T,Y,F0, & |
---|
| 1087 | E1,IP1,Z1,Z2,Z3,SCAL,Err, & |
---|
| 1088 | FirstStep,Reject) |
---|
| 1089 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1090 | IMPLICIT NONE |
---|
| 1091 | |
---|
| 1092 | INTEGER :: N |
---|
| 1093 | #ifdef FULL_ALGEBRA |
---|
| 1094 | KPP_REAL :: E1(NVAR,NVAR) |
---|
| 1095 | #else |
---|
| 1096 | KPP_REAL :: E1(LU_NONZERO) |
---|
| 1097 | #endif |
---|
| 1098 | KPP_REAL :: SCAL(N),Z1(N),Z2(N),Z3(N),F1(N),F2(N), & |
---|
| 1099 | F0(N),Y(N),TMP(N),T,H |
---|
| 1100 | INTEGER :: IP1(N), i |
---|
| 1101 | LOGICAL FirstStep,Reject |
---|
| 1102 | KPP_REAL :: HrkE1,HrkE2,HrkE3,Err |
---|
| 1103 | |
---|
| 1104 | HrkE1 = rkE(1)/H |
---|
| 1105 | HrkE2 = rkE(2)/H |
---|
| 1106 | HrkE3 = rkE(3)/H |
---|
| 1107 | |
---|
| 1108 | DO i=1,N |
---|
| 1109 | F2(i) = HrkE1*Z1(i)+HrkE2*Z2(i)+HrkE3*Z3(i) |
---|
| 1110 | TMP(i) = rkE(0)*F0(i) + F2(i) |
---|
| 1111 | END DO |
---|
| 1112 | |
---|
| 1113 | |
---|
| 1114 | #ifdef FULL_ALGEBRA |
---|
| 1115 | CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) |
---|
| 1116 | IF ((rkMethod==R1A).OR.(rkMethod==GAU).OR.(rkMethod==L3A)) CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) |
---|
| 1117 | IF (rkMethod==GAU) CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) |
---|
| 1118 | #else |
---|
| 1119 | CALL KppSolve (E1, TMP) |
---|
| 1120 | IF ((rkMethod==R1A).OR.(rkMethod==GAU).OR.(rkMethod==L3A)) CALL KppSolve (E1,TMP) |
---|
| 1121 | IF (rkMethod==GAU) CALL KppSolve (E1,TMP) |
---|
| 1122 | #endif |
---|
| 1123 | |
---|
| 1124 | Err = RK_ErrorNorm(N,SCAL,TMP) |
---|
| 1125 | ! |
---|
| 1126 | IF (Err < ONE) RETURN |
---|
| 1127 | firej:IF (FirstStep.OR.Reject) THEN |
---|
| 1128 | DO i=1,N |
---|
| 1129 | TMP(i)=Y(i)+TMP(i) |
---|
| 1130 | END DO |
---|
| 1131 | CALL FUN_CHEM(T,TMP,F1) |
---|
| 1132 | ISTATUS(Nfun) = ISTATUS(Nfun) + 1 |
---|
| 1133 | DO i=1,N |
---|
| 1134 | TMP(i)=F1(i)+F2(i) |
---|
| 1135 | END DO |
---|
| 1136 | |
---|
| 1137 | #ifdef FULL_ALGEBRA |
---|
| 1138 | CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) |
---|
| 1139 | #else |
---|
| 1140 | CALL KppSolve (E1, TMP) |
---|
| 1141 | #endif |
---|
| 1142 | Err = RK_ErrorNorm(N,SCAL,TMP) |
---|
| 1143 | END IF firej |
---|
| 1144 | |
---|
| 1145 | END SUBROUTINE RK_ErrorEstimate |
---|
| 1146 | |
---|
| 1147 | |
---|
| 1148 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1149 | KPP_REAL FUNCTION RK_ErrorNorm(N,SCAL,DY) |
---|
| 1150 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1151 | IMPLICIT NONE |
---|
| 1152 | |
---|
| 1153 | INTEGER :: N |
---|
| 1154 | KPP_REAL :: SCAL(N),DY(N) |
---|
| 1155 | INTEGER :: i |
---|
| 1156 | |
---|
| 1157 | RK_ErrorNorm = ZERO |
---|
| 1158 | DO i=1,N |
---|
| 1159 | RK_ErrorNorm = RK_ErrorNorm + (DY(i)*SCAL(i))**2 |
---|
| 1160 | END DO |
---|
| 1161 | RK_ErrorNorm = MAX( SQRT(RK_ErrorNorm/N), 1.0d-10 ) |
---|
| 1162 | |
---|
| 1163 | END FUNCTION RK_ErrorNorm |
---|
| 1164 | |
---|
| 1165 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1166 | SUBROUTINE Radau2A_Coefficients |
---|
| 1167 | ! The coefficients of the 3-stage Radau-2A method |
---|
| 1168 | ! (given to ~30 accurate digits) |
---|
| 1169 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1170 | IMPLICIT NONE |
---|
| 1171 | ! The coefficients of the Radau2A method |
---|
| 1172 | KPP_REAL :: b0 |
---|
| 1173 | |
---|
| 1174 | ! b0 = 1.0d0 |
---|
| 1175 | IF (SdirkError) THEN |
---|
| 1176 | b0 = 0.2d-1 |
---|
| 1177 | ELSE |
---|
| 1178 | b0 = 0.5d-1 |
---|
| 1179 | END IF |
---|
| 1180 | |
---|
| 1181 | ! The coefficients of the Radau2A method |
---|
| 1182 | rkMethod = R2A |
---|
| 1183 | |
---|
| 1184 | rkA(1,1) = 1.968154772236604258683861429918299d-1 |
---|
| 1185 | rkA(1,2) = -6.55354258501983881085227825696087d-2 |
---|
| 1186 | rkA(1,3) = 2.377097434822015242040823210718965d-2 |
---|
| 1187 | rkA(2,1) = 3.944243147390872769974116714584975d-1 |
---|
| 1188 | rkA(2,2) = 2.920734116652284630205027458970589d-1 |
---|
| 1189 | rkA(2,3) = -4.154875212599793019818600988496743d-2 |
---|
| 1190 | rkA(3,1) = 3.764030627004672750500754423692808d-1 |
---|
| 1191 | rkA(3,2) = 5.124858261884216138388134465196080d-1 |
---|
| 1192 | rkA(3,3) = 1.111111111111111111111111111111111d-1 |
---|
| 1193 | |
---|
| 1194 | rkB(1) = 3.764030627004672750500754423692808d-1 |
---|
| 1195 | rkB(2) = 5.124858261884216138388134465196080d-1 |
---|
| 1196 | rkB(3) = 1.111111111111111111111111111111111d-1 |
---|
| 1197 | |
---|
| 1198 | rkC(1) = 1.550510257216821901802715925294109d-1 |
---|
| 1199 | rkC(2) = 6.449489742783178098197284074705891d-1 |
---|
| 1200 | rkC(3) = 1.0d0 |
---|
| 1201 | |
---|
| 1202 | ! New solution: H* Sum B_j*f(Z_j) = Sum D_j*Z_j |
---|
| 1203 | rkD(1) = 0.0d0 |
---|
| 1204 | rkD(2) = 0.0d0 |
---|
| 1205 | rkD(3) = 1.0d0 |
---|
| 1206 | |
---|
| 1207 | ! Classical error estimator: |
---|
| 1208 | ! H* Sum (B_j-Bhat_j)*f(Z_j) = H*E(0)*f(0) + Sum E_j*Z_j |
---|
| 1209 | rkE(0) = 1.0d0*b0 |
---|
| 1210 | rkE(1) = -10.04880939982741556246032950764708d0*b0 |
---|
| 1211 | rkE(2) = 1.382142733160748895793662840980412d0*b0 |
---|
| 1212 | rkE(3) = -.3333333333333333333333333333333333d0*b0 |
---|
| 1213 | |
---|
| 1214 | ! Sdirk error estimator |
---|
| 1215 | rkBgam(0) = b0 |
---|
| 1216 | rkBgam(1) = .3764030627004672750500754423692807d0-1.558078204724922382431975370686279d0*b0 |
---|
| 1217 | rkBgam(2) = .8914115380582557157653087040196118d0*b0+.5124858261884216138388134465196077d0 |
---|
| 1218 | rkBgam(3) = -.1637777184845662566367174924883037d0-.3333333333333333333333333333333333d0*b0 |
---|
| 1219 | rkBgam(4) = .2748888295956773677478286035994148d0 |
---|
| 1220 | |
---|
| 1221 | ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j |
---|
| 1222 | rkTheta(1) = -1.520677486405081647234271944611547d0-10.04880939982741556246032950764708d0*b0 |
---|
| 1223 | rkTheta(2) = 2.070455145596436382729929151810376d0+1.382142733160748895793662840980413d0*b0 |
---|
| 1224 | rkTheta(3) = -.3333333333333333333333333333333333d0*b0-.3744441479783868387391430179970741d0 |
---|
| 1225 | |
---|
| 1226 | ! Local order of error estimator |
---|
| 1227 | IF (b0==0.0d0) THEN |
---|
| 1228 | rkELO = 6.0d0 |
---|
| 1229 | ELSE |
---|
| 1230 | rkELO = 4.0d0 |
---|
| 1231 | END IF |
---|
| 1232 | |
---|
| 1233 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1234 | !~~~> Diagonalize the RK matrix: |
---|
| 1235 | ! rkTinv * inv(rkA) * rkT = |
---|
| 1236 | ! | rkGamma 0 0 | |
---|
| 1237 | ! | 0 rkAlpha -rkBeta | |
---|
| 1238 | ! | 0 rkBeta rkAlpha | |
---|
| 1239 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1240 | |
---|
| 1241 | rkGamma = 3.637834252744495732208418513577775d0 |
---|
| 1242 | rkAlpha = 2.681082873627752133895790743211112d0 |
---|
| 1243 | rkBeta = 3.050430199247410569426377624787569d0 |
---|
| 1244 | |
---|
| 1245 | rkT(1,1) = 9.443876248897524148749007950641664d-2 |
---|
| 1246 | rkT(1,2) = -1.412552950209542084279903838077973d-1 |
---|
| 1247 | rkT(1,3) = -3.00291941051474244918611170890539d-2 |
---|
| 1248 | rkT(2,1) = 2.502131229653333113765090675125018d-1 |
---|
| 1249 | rkT(2,2) = 2.041293522937999319959908102983381d-1 |
---|
| 1250 | rkT(2,3) = 3.829421127572619377954382335998733d-1 |
---|
| 1251 | rkT(3,1) = 1.0d0 |
---|
| 1252 | rkT(3,2) = 1.0d0 |
---|
| 1253 | rkT(3,3) = 0.0d0 |
---|
| 1254 | |
---|
| 1255 | rkTinv(1,1) = 4.178718591551904727346462658512057d0 |
---|
| 1256 | rkTinv(1,2) = 3.27682820761062387082533272429617d-1 |
---|
| 1257 | rkTinv(1,3) = 5.233764454994495480399309159089876d-1 |
---|
| 1258 | rkTinv(2,1) = -4.178718591551904727346462658512057d0 |
---|
| 1259 | rkTinv(2,2) = -3.27682820761062387082533272429617d-1 |
---|
| 1260 | rkTinv(2,3) = 4.766235545005504519600690840910124d-1 |
---|
| 1261 | rkTinv(3,1) = -5.02872634945786875951247343139544d-1 |
---|
| 1262 | rkTinv(3,2) = 2.571926949855605429186785353601676d0 |
---|
| 1263 | rkTinv(3,3) = -5.960392048282249249688219110993024d-1 |
---|
| 1264 | |
---|
| 1265 | rkTinvAinv(1,1) = 1.520148562492775501049204957366528d+1 |
---|
| 1266 | rkTinvAinv(1,2) = 1.192055789400527921212348994770778d0 |
---|
| 1267 | rkTinvAinv(1,3) = 1.903956760517560343018332287285119d0 |
---|
| 1268 | rkTinvAinv(2,1) = -9.669512977505946748632625374449567d0 |
---|
| 1269 | rkTinvAinv(2,2) = -8.724028436822336183071773193986487d0 |
---|
| 1270 | rkTinvAinv(2,3) = 3.096043239482439656981667712714881d0 |
---|
| 1271 | rkTinvAinv(3,1) = -1.409513259499574544876303981551774d+1 |
---|
| 1272 | rkTinvAinv(3,2) = 5.895975725255405108079130152868952d0 |
---|
| 1273 | rkTinvAinv(3,3) = -1.441236197545344702389881889085515d-1 |
---|
| 1274 | |
---|
| 1275 | rkAinvT(1,1) = .3435525649691961614912493915818282d0 |
---|
| 1276 | rkAinvT(1,2) = -.4703191128473198422370558694426832d0 |
---|
| 1277 | rkAinvT(1,3) = .3503786597113668965366406634269080d0 |
---|
| 1278 | rkAinvT(2,1) = .9102338692094599309122768354288852d0 |
---|
| 1279 | rkAinvT(2,2) = 1.715425895757991796035292755937326d0 |
---|
| 1280 | rkAinvT(2,3) = .4040171993145015239277111187301784d0 |
---|
| 1281 | rkAinvT(3,1) = 3.637834252744495732208418513577775d0 |
---|
| 1282 | rkAinvT(3,2) = 2.681082873627752133895790743211112d0 |
---|
| 1283 | rkAinvT(3,3) = -3.050430199247410569426377624787569d0 |
---|
| 1284 | |
---|
| 1285 | END SUBROUTINE Radau2A_Coefficients |
---|
| 1286 | |
---|
| 1287 | |
---|
| 1288 | |
---|
| 1289 | |
---|
| 1290 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1291 | SUBROUTINE Lobatto3C_Coefficients |
---|
| 1292 | ! The coefficients of the 3-stage Lobatto-3C method |
---|
| 1293 | ! (given to ~30 accurate digits) |
---|
| 1294 | ! The parameter b0 can be chosen to tune the error estimator |
---|
| 1295 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1296 | IMPLICIT NONE |
---|
| 1297 | KPP_REAL :: b0 |
---|
| 1298 | |
---|
| 1299 | rkMethod = L3C |
---|
| 1300 | |
---|
| 1301 | ! b0 = 1.0d0 |
---|
| 1302 | IF (SdirkError) THEN |
---|
| 1303 | b0 = 0.2d0 |
---|
| 1304 | ELSE |
---|
| 1305 | b0 = 0.5d0 |
---|
| 1306 | END IF |
---|
| 1307 | ! The coefficients of the Lobatto3C method |
---|
| 1308 | |
---|
| 1309 | rkA(1,1) = .1666666666666666666666666666666667d0 |
---|
| 1310 | rkA(1,2) = -.3333333333333333333333333333333333d0 |
---|
| 1311 | rkA(1,3) = .1666666666666666666666666666666667d0 |
---|
| 1312 | rkA(2,1) = .1666666666666666666666666666666667d0 |
---|
| 1313 | rkA(2,2) = .4166666666666666666666666666666667d0 |
---|
| 1314 | rkA(2,3) = -.8333333333333333333333333333333333d-1 |
---|
| 1315 | rkA(3,1) = .1666666666666666666666666666666667d0 |
---|
| 1316 | rkA(3,2) = .6666666666666666666666666666666667d0 |
---|
| 1317 | rkA(3,3) = .1666666666666666666666666666666667d0 |
---|
| 1318 | |
---|
| 1319 | rkB(1) = .1666666666666666666666666666666667d0 |
---|
| 1320 | rkB(2) = .6666666666666666666666666666666667d0 |
---|
| 1321 | rkB(3) = .1666666666666666666666666666666667d0 |
---|
| 1322 | |
---|
| 1323 | rkC(1) = 0.0d0 |
---|
| 1324 | rkC(2) = 0.5d0 |
---|
| 1325 | rkC(3) = 1.0d0 |
---|
| 1326 | |
---|
| 1327 | ! Classical error estimator, embedded solution: |
---|
| 1328 | rkBhat(0) = b0 |
---|
| 1329 | rkBhat(1) = .16666666666666666666666666666666667d0-b0 |
---|
| 1330 | rkBhat(2) = .66666666666666666666666666666666667d0 |
---|
| 1331 | rkBhat(3) = .16666666666666666666666666666666667d0 |
---|
| 1332 | |
---|
| 1333 | ! New solution: h Sum_j b_j f(Z_j) = sum d_j Z_j |
---|
| 1334 | rkD(1) = 0.0d0 |
---|
| 1335 | rkD(2) = 0.0d0 |
---|
| 1336 | rkD(3) = 1.0d0 |
---|
| 1337 | |
---|
| 1338 | ! Classical error estimator: |
---|
| 1339 | ! H* Sum (B_j-Bhat_j)*f(Z_j) = H*E(0)*f(0) + Sum E_j*Z_j |
---|
| 1340 | rkE(0) = .3808338772072650364017425226487022*b0 |
---|
| 1341 | rkE(1) = -1.142501631621795109205227567946107*b0 |
---|
| 1342 | rkE(2) = -1.523335508829060145606970090594809*b0 |
---|
| 1343 | rkE(3) = .3808338772072650364017425226487022*b0 |
---|
| 1344 | |
---|
| 1345 | ! Sdirk error estimator |
---|
| 1346 | rkBgam(0) = b0 |
---|
| 1347 | rkBgam(1) = .1666666666666666666666666666666667d0-1.d0*b0 |
---|
| 1348 | rkBgam(2) = .6666666666666666666666666666666667d0 |
---|
| 1349 | rkBgam(3) = -.2141672105405983697350758559820354d0 |
---|
| 1350 | rkBgam(4) = .3808338772072650364017425226487021d0 |
---|
| 1351 | |
---|
| 1352 | ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j |
---|
| 1353 | rkTheta(1) = -3.d0*b0-.3808338772072650364017425226487021d0 |
---|
| 1354 | rkTheta(2) = -4.d0*b0+1.523335508829060145606970090594808d0 |
---|
| 1355 | rkTheta(3) = -.142501631621795109205227567946106d0+b0 |
---|
| 1356 | |
---|
| 1357 | ! Local order of error estimator |
---|
| 1358 | IF (b0==0.0d0) THEN |
---|
| 1359 | rkELO = 5.0d0 |
---|
| 1360 | ELSE |
---|
| 1361 | rkELO = 4.0d0 |
---|
| 1362 | END IF |
---|
| 1363 | |
---|
| 1364 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1365 | !~~~> Diagonalize the RK matrix: |
---|
| 1366 | ! rkTinv * inv(rkA) * rkT = |
---|
| 1367 | ! | rkGamma 0 0 | |
---|
| 1368 | ! | 0 rkAlpha -rkBeta | |
---|
| 1369 | ! | 0 rkBeta rkAlpha | |
---|
| 1370 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1371 | |
---|
| 1372 | rkGamma = 2.625816818958466716011888933765284d0 |
---|
| 1373 | rkAlpha = 1.687091590520766641994055533117359d0 |
---|
| 1374 | rkBeta = 2.508731754924880510838743672432351d0 |
---|
| 1375 | |
---|
| 1376 | rkT(1,1) = 1.d0 |
---|
| 1377 | rkT(1,2) = 1.d0 |
---|
| 1378 | rkT(1,3) = 0.d0 |
---|
| 1379 | rkT(2,1) = .4554100411010284672111720348287483d0 |
---|
| 1380 | rkT(2,2) = -.6027050205505142336055860174143743d0 |
---|
| 1381 | rkT(2,3) = -.4309321229203225731070721341350346d0 |
---|
| 1382 | rkT(3,1) = 2.195823345445647152832799205549709d0 |
---|
| 1383 | rkT(3,2) = -1.097911672722823576416399602774855d0 |
---|
| 1384 | rkT(3,3) = .7850032632435902184104551358922130d0 |
---|
| 1385 | |
---|
| 1386 | rkTinv(1,1) = .4205559181381766909344950150991349d0 |
---|
| 1387 | rkTinv(1,2) = .3488903392193734304046467270632057d0 |
---|
| 1388 | rkTinv(1,3) = .1915253879645878102698098373933487d0 |
---|
| 1389 | rkTinv(2,1) = .5794440818618233090655049849008650d0 |
---|
| 1390 | rkTinv(2,2) = -.3488903392193734304046467270632057d0 |
---|
| 1391 | rkTinv(2,3) = -.1915253879645878102698098373933487d0 |
---|
| 1392 | rkTinv(3,1) = -.3659705575742745254721332009249516d0 |
---|
| 1393 | rkTinv(3,2) = -1.463882230297098101888532803699806d0 |
---|
| 1394 | rkTinv(3,3) = .4702733607340189781407813565524989d0 |
---|
| 1395 | |
---|
| 1396 | rkTinvAinv(1,1) = 1.104302803159744452668648155627548d0 |
---|
| 1397 | rkTinvAinv(1,2) = .916122120694355522658740710823143d0 |
---|
| 1398 | rkTinvAinv(1,3) = .5029105849749601702795812241441172d0 |
---|
| 1399 | rkTinvAinv(2,1) = 1.895697196840255547331351844372453d0 |
---|
| 1400 | rkTinvAinv(2,2) = 3.083877879305644477341259289176857d0 |
---|
| 1401 | rkTinvAinv(2,3) = -1.502910584974960170279581224144117d0 |
---|
| 1402 | rkTinvAinv(3,1) = .8362439183082935036129145574774502d0 |
---|
| 1403 | rkTinvAinv(3,2) = -3.344975673233174014451658229909802d0 |
---|
| 1404 | rkTinvAinv(3,3) = .312908409479233358005944466882642d0 |
---|
| 1405 | |
---|
| 1406 | rkAinvT(1,1) = 2.625816818958466716011888933765282d0 |
---|
| 1407 | rkAinvT(1,2) = 1.687091590520766641994055533117358d0 |
---|
| 1408 | rkAinvT(1,3) = -2.508731754924880510838743672432351d0 |
---|
| 1409 | rkAinvT(2,1) = 1.195823345445647152832799205549710d0 |
---|
| 1410 | rkAinvT(2,2) = -2.097911672722823576416399602774855d0 |
---|
| 1411 | rkAinvT(2,3) = .7850032632435902184104551358922130d0 |
---|
| 1412 | rkAinvT(3,1) = 5.765829871932827589653709477334136d0 |
---|
| 1413 | rkAinvT(3,2) = .1170850640335862051731452613329320d0 |
---|
| 1414 | rkAinvT(3,3) = 4.078738281412060947659653944216779d0 |
---|
| 1415 | |
---|
| 1416 | END SUBROUTINE Lobatto3C_Coefficients |
---|
| 1417 | |
---|
| 1418 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1419 | SUBROUTINE Gauss_Coefficients |
---|
| 1420 | ! The coefficients of the 3-stage Gauss method |
---|
| 1421 | ! (given to ~30 accurate digits) |
---|
| 1422 | ! The parameter b3 can be chosen by the user |
---|
| 1423 | ! to tune the error estimator |
---|
| 1424 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1425 | IMPLICIT NONE |
---|
| 1426 | KPP_REAL :: b0 |
---|
| 1427 | ! The coefficients of the Gauss method |
---|
| 1428 | |
---|
| 1429 | |
---|
| 1430 | rkMethod = GAU |
---|
| 1431 | |
---|
| 1432 | ! b0 = 4.0d0 |
---|
| 1433 | b0 = 0.1d0 |
---|
| 1434 | |
---|
| 1435 | ! The coefficients of the Gauss method |
---|
| 1436 | |
---|
| 1437 | rkA(1,1) = .1388888888888888888888888888888889d0 |
---|
| 1438 | rkA(1,2) = -.359766675249389034563954710966045d-1 |
---|
| 1439 | rkA(1,3) = .97894440153083260495800422294756d-2 |
---|
| 1440 | rkA(2,1) = .3002631949808645924380249472131556d0 |
---|
| 1441 | rkA(2,2) = .2222222222222222222222222222222222d0 |
---|
| 1442 | rkA(2,3) = -.224854172030868146602471694353778d-1 |
---|
| 1443 | rkA(3,1) = .2679883337624694517281977355483022d0 |
---|
| 1444 | rkA(3,2) = .4804211119693833479008399155410489d0 |
---|
| 1445 | rkA(3,3) = .1388888888888888888888888888888889d0 |
---|
| 1446 | |
---|
| 1447 | rkB(1) = .2777777777777777777777777777777778d0 |
---|
| 1448 | rkB(2) = .4444444444444444444444444444444444d0 |
---|
| 1449 | rkB(3) = .2777777777777777777777777777777778d0 |
---|
| 1450 | |
---|
| 1451 | rkC(1) = .1127016653792583114820734600217600d0 |
---|
| 1452 | rkC(2) = .5000000000000000000000000000000000d0 |
---|
| 1453 | rkC(3) = .8872983346207416885179265399782400d0 |
---|
| 1454 | |
---|
| 1455 | ! Classical error estimator, embedded solution: |
---|
| 1456 | rkBhat(0) = b0 |
---|
| 1457 | rkBhat(1) =-1.4788305577012361475298775666303999d0*b0 & |
---|
| 1458 | +.27777777777777777777777777777777778d0 |
---|
| 1459 | rkBhat(2) = .44444444444444444444444444444444444d0 & |
---|
| 1460 | +.66666666666666666666666666666666667d0*b0 |
---|
| 1461 | rkBhat(3) = -.18783610896543051913678910003626672d0*b0 & |
---|
| 1462 | +.27777777777777777777777777777777778d0 |
---|
| 1463 | |
---|
| 1464 | ! New solution: h Sum_j b_j f(Z_j) = sum d_j Z_j |
---|
| 1465 | rkD(1) = .1666666666666666666666666666666667d1 |
---|
| 1466 | rkD(2) = -.1333333333333333333333333333333333d1 |
---|
| 1467 | rkD(3) = .1666666666666666666666666666666667d1 |
---|
| 1468 | |
---|
| 1469 | ! Classical error estimator: |
---|
| 1470 | ! H* Sum (B_j-Bhat_j)*f(Z_j) = H*E(0)*f(0) + Sum E_j*Z_j |
---|
| 1471 | rkE(0) = .2153144231161121782447335303806954d0*b0 |
---|
| 1472 | rkE(1) = -2.825278112319014084275808340593191d0*b0 |
---|
| 1473 | rkE(2) = .2870858974881495709929780405075939d0*b0 |
---|
| 1474 | rkE(3) = -.4558086256248162565397206448274867d-1*b0 |
---|
| 1475 | |
---|
| 1476 | ! Sdirk error estimator |
---|
| 1477 | rkBgam(0) = 0.d0 |
---|
| 1478 | rkBgam(1) = .2373339543355109188382583162660537d0 |
---|
| 1479 | rkBgam(2) = .5879873931885192299409334646982414d0 |
---|
| 1480 | rkBgam(3) = -.4063577064014232702392531134499046d-1 |
---|
| 1481 | rkBgam(4) = .2153144231161121782447335303806955d0 |
---|
| 1482 | |
---|
| 1483 | ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j |
---|
| 1484 | rkTheta(1) = -2.594040933093095272574031876464493d0 |
---|
| 1485 | rkTheta(2) = 1.824611539036311947589425112250199d0 |
---|
| 1486 | rkTheta(3) = .1856563166634371860478043996459493d0 |
---|
| 1487 | |
---|
| 1488 | ! ELO = local order of classical error estimator |
---|
| 1489 | rkELO = 4.0d0 |
---|
| 1490 | |
---|
| 1491 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1492 | !~~~> Diagonalize the RK matrix: |
---|
| 1493 | ! rkTinv * inv(rkA) * rkT = |
---|
| 1494 | ! | rkGamma 0 0 | |
---|
| 1495 | ! | 0 rkAlpha -rkBeta | |
---|
| 1496 | ! | 0 rkBeta rkAlpha | |
---|
| 1497 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1498 | |
---|
| 1499 | rkGamma = 4.644370709252171185822941421408064d0 |
---|
| 1500 | rkAlpha = 3.677814645373914407088529289295970d0 |
---|
| 1501 | rkBeta = 3.508761919567443321903661209182446d0 |
---|
| 1502 | |
---|
| 1503 | rkT(1,1) = .7215185205520017032081769924397664d-1 |
---|
| 1504 | rkT(1,2) = -.8224123057363067064866206597516454d-1 |
---|
| 1505 | rkT(1,3) = -.6012073861930850173085948921439054d-1 |
---|
| 1506 | rkT(2,1) = .1188325787412778070708888193730294d0 |
---|
| 1507 | rkT(2,2) = .5306509074206139504614411373957448d-1 |
---|
| 1508 | rkT(2,3) = .3162050511322915732224862926182701d0 |
---|
| 1509 | rkT(3,1) = 1.d0 |
---|
| 1510 | rkT(3,2) = 1.d0 |
---|
| 1511 | rkT(3,3) = 0.d0 |
---|
| 1512 | |
---|
| 1513 | rkTinv(1,1) = 5.991698084937800775649580743981285d0 |
---|
| 1514 | rkTinv(1,2) = 1.139214295155735444567002236934009d0 |
---|
| 1515 | rkTinv(1,3) = .4323121137838583855696375901180497d0 |
---|
| 1516 | rkTinv(2,1) = -5.991698084937800775649580743981285d0 |
---|
| 1517 | rkTinv(2,2) = -1.139214295155735444567002236934009d0 |
---|
| 1518 | rkTinv(2,3) = .5676878862161416144303624098819503d0 |
---|
| 1519 | rkTinv(3,1) = -1.246213273586231410815571640493082d0 |
---|
| 1520 | rkTinv(3,2) = 2.925559646192313662599230367054972d0 |
---|
| 1521 | rkTinv(3,3) = -.2577352012734324923468722836888244d0 |
---|
| 1522 | |
---|
| 1523 | rkTinvAinv(1,1) = 27.82766708436744962047620566703329d0 |
---|
| 1524 | rkTinvAinv(1,2) = 5.290933503982655311815946575100597d0 |
---|
| 1525 | rkTinvAinv(1,3) = 2.007817718512643701322151051660114d0 |
---|
| 1526 | rkTinvAinv(2,1) = -17.66368928942422710690385180065675d0 |
---|
| 1527 | rkTinvAinv(2,2) = -14.45491129892587782538830044147713d0 |
---|
| 1528 | rkTinvAinv(2,3) = 2.992182281487356298677848948339886d0 |
---|
| 1529 | rkTinvAinv(3,1) = -25.60678350282974256072419392007303d0 |
---|
| 1530 | rkTinvAinv(3,2) = 6.762434375611708328910623303779923d0 |
---|
| 1531 | rkTinvAinv(3,3) = 1.043979339483109825041215970036771d0 |
---|
| 1532 | |
---|
| 1533 | rkAinvT(1,1) = .3350999483034677402618981153470483d0 |
---|
| 1534 | rkAinvT(1,2) = -.5134173605009692329246186488441294d0 |
---|
| 1535 | rkAinvT(1,3) = .6745196507033116204327635673208923d-1 |
---|
| 1536 | rkAinvT(2,1) = .5519025480108928886873752035738885d0 |
---|
| 1537 | rkAinvT(2,2) = 1.304651810077110066076640761092008d0 |
---|
| 1538 | rkAinvT(2,3) = .9767507983414134987545585703726984d0 |
---|
| 1539 | rkAinvT(3,1) = 4.644370709252171185822941421408064d0 |
---|
| 1540 | rkAinvT(3,2) = 3.677814645373914407088529289295970d0 |
---|
| 1541 | rkAinvT(3,3) = -3.508761919567443321903661209182446d0 |
---|
| 1542 | |
---|
| 1543 | END SUBROUTINE Gauss_Coefficients |
---|
| 1544 | |
---|
| 1545 | |
---|
| 1546 | |
---|
| 1547 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1548 | SUBROUTINE Radau1A_Coefficients |
---|
| 1549 | ! The coefficients of the 3-stage Gauss method |
---|
| 1550 | ! (given to ~30 accurate digits) |
---|
| 1551 | ! The parameter b3 can be chosen by the user |
---|
| 1552 | ! to tune the error estimator |
---|
| 1553 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1554 | IMPLICIT NONE |
---|
| 1555 | ! KPP_REAL :: b0 = 0.3d0 |
---|
| 1556 | KPP_REAL :: b0 = 0.1d0 |
---|
| 1557 | |
---|
| 1558 | ! The coefficients of the Radau1A method |
---|
| 1559 | |
---|
| 1560 | rkMethod = R1A |
---|
| 1561 | |
---|
| 1562 | rkA(1,1) = .1111111111111111111111111111111111d0 |
---|
| 1563 | rkA(1,2) = -.1916383190435098943442935597058829d0 |
---|
| 1564 | rkA(1,3) = .8052720793239878323318244859477174d-1 |
---|
| 1565 | rkA(2,1) = .1111111111111111111111111111111111d0 |
---|
| 1566 | rkA(2,2) = .2920734116652284630205027458970589d0 |
---|
| 1567 | rkA(2,3) = -.481334970546573839513422644787591d-1 |
---|
| 1568 | rkA(3,1) = .1111111111111111111111111111111111d0 |
---|
| 1569 | rkA(3,2) = .5370223859435462728402311533676479d0 |
---|
| 1570 | rkA(3,3) = .1968154772236604258683861429918299d0 |
---|
| 1571 | |
---|
| 1572 | rkB(1) = .1111111111111111111111111111111111d0 |
---|
| 1573 | rkB(2) = .5124858261884216138388134465196080d0 |
---|
| 1574 | rkB(3) = .3764030627004672750500754423692808d0 |
---|
| 1575 | |
---|
| 1576 | rkC(1) = 0.d0 |
---|
| 1577 | rkC(2) = .3550510257216821901802715925294109d0 |
---|
| 1578 | rkC(3) = .8449489742783178098197284074705891d0 |
---|
| 1579 | |
---|
| 1580 | ! Classical error estimator, embedded solution: |
---|
| 1581 | rkBhat(0) = b0 |
---|
| 1582 | rkBhat(1) = .11111111111111111111111111111111111d0-b0 |
---|
| 1583 | rkBhat(2) = .51248582618842161383881344651960810d0 |
---|
| 1584 | rkBhat(3) = .37640306270046727505007544236928079d0 |
---|
| 1585 | |
---|
| 1586 | ! New solution: H* Sum B_j*f(Z_j) = Sum D_j*Z_j |
---|
| 1587 | rkD(1) = .3333333333333333333333333333333333d0 |
---|
| 1588 | rkD(2) = -.8914115380582557157653087040196127d0 |
---|
| 1589 | rkD(3) = .1558078204724922382431975370686279d1 |
---|
| 1590 | |
---|
| 1591 | ! Classical error estimator: |
---|
| 1592 | ! H* Sum (b_j-bhat_j) f(Z_j) = H*E(0)*F(0) + Sum E_j Z_j |
---|
| 1593 | rkE(0) = .2748888295956773677478286035994148d0*b0 |
---|
| 1594 | rkE(1) = -1.374444147978386838739143017997074d0*b0 |
---|
| 1595 | rkE(2) = -1.335337922441686804550326197041126d0*b0 |
---|
| 1596 | rkE(3) = .235782604058977333559011782643466d0*b0 |
---|
| 1597 | |
---|
| 1598 | ! Sdirk error estimator |
---|
| 1599 | rkBgam(0) = 0.0d0 |
---|
| 1600 | rkBgam(1) = .1948150124588532186183490991130616d-1 |
---|
| 1601 | rkBgam(2) = .7575249005733381398986810981093584d0 |
---|
| 1602 | rkBgam(3) = -.518952314149008295083446116200793d-1 |
---|
| 1603 | rkBgam(4) = .2748888295956773677478286035994148d0 |
---|
| 1604 | |
---|
| 1605 | ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j |
---|
| 1606 | rkTheta(1) = -1.224370034375505083904362087063351d0 |
---|
| 1607 | rkTheta(2) = .9340045331532641409047527962010133d0 |
---|
| 1608 | rkTheta(3) = .4656990124352088397561234800640929d0 |
---|
| 1609 | |
---|
| 1610 | ! ELO = local order of classical error estimator |
---|
| 1611 | rkELO = 4.0d0 |
---|
| 1612 | |
---|
| 1613 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1614 | !~~~> Diagonalize the RK matrix: |
---|
| 1615 | ! rkTinv * inv(rkA) * rkT = |
---|
| 1616 | ! | rkGamma 0 0 | |
---|
| 1617 | ! | 0 rkAlpha -rkBeta | |
---|
| 1618 | ! | 0 rkBeta rkAlpha | |
---|
| 1619 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1620 | |
---|
| 1621 | rkGamma = 3.637834252744495732208418513577775d0 |
---|
| 1622 | rkAlpha = 2.681082873627752133895790743211112d0 |
---|
| 1623 | rkBeta = 3.050430199247410569426377624787569d0 |
---|
| 1624 | |
---|
| 1625 | rkT(1,1) = .424293819848497965354371036408369d0 |
---|
| 1626 | rkT(1,2) = -.3235571519651980681202894497035503d0 |
---|
| 1627 | rkT(1,3) = -.522137786846287839586599927945048d0 |
---|
| 1628 | rkT(2,1) = .57594609499806128896291585429339d-1 |
---|
| 1629 | rkT(2,2) = .3148663231849760131614374283783d-2 |
---|
| 1630 | rkT(2,3) = .452429247674359778577728510381731d0 |
---|
| 1631 | rkT(3,1) = 1.d0 |
---|
| 1632 | rkT(3,2) = 1.d0 |
---|
| 1633 | rkT(3,3) = 0.d0 |
---|
| 1634 | |
---|
| 1635 | rkTinv(1,1) = 1.233523612685027760114769983066164d0 |
---|
| 1636 | rkTinv(1,2) = 1.423580134265707095505388133369554d0 |
---|
| 1637 | rkTinv(1,3) = .3946330125758354736049045150429624d0 |
---|
| 1638 | rkTinv(2,1) = -1.233523612685027760114769983066164d0 |
---|
| 1639 | rkTinv(2,2) = -1.423580134265707095505388133369554d0 |
---|
| 1640 | rkTinv(2,3) = .6053669874241645263950954849570376d0 |
---|
| 1641 | rkTinv(3,1) = -.1484438963257383124456490049673414d0 |
---|
| 1642 | rkTinv(3,2) = 2.038974794939896109682070471785315d0 |
---|
| 1643 | rkTinv(3,3) = -.544501292892686735299355831692542d-1 |
---|
| 1644 | |
---|
| 1645 | rkTinvAinv(1,1) = 4.487354449794728738538663081025420d0 |
---|
| 1646 | rkTinvAinv(1,2) = 5.178748573958397475446442544234494d0 |
---|
| 1647 | rkTinvAinv(1,3) = 1.435609490412123627047824222335563d0 |
---|
| 1648 | rkTinvAinv(2,1) = -2.854361287939276673073807031221493d0 |
---|
| 1649 | rkTinvAinv(2,2) = -1.003648660720543859000994063139137d+1 |
---|
| 1650 | rkTinvAinv(2,3) = 1.789135380979465422050817815017383d0 |
---|
| 1651 | rkTinvAinv(3,1) = -4.160768067752685525282947313530352d0 |
---|
| 1652 | rkTinvAinv(3,2) = 1.124128569859216916690209918405860d0 |
---|
| 1653 | rkTinvAinv(3,3) = 1.700644430961823796581896350418417d0 |
---|
| 1654 | |
---|
| 1655 | rkAinvT(1,1) = 1.543510591072668287198054583233180d0 |
---|
| 1656 | rkAinvT(1,2) = -2.460228411937788329157493833295004d0 |
---|
| 1657 | rkAinvT(1,3) = -.412906170450356277003910443520499d0 |
---|
| 1658 | rkAinvT(2,1) = .209519643211838264029272585946993d0 |
---|
| 1659 | rkAinvT(2,2) = 1.388545667194387164417459732995766d0 |
---|
| 1660 | rkAinvT(2,3) = 1.20339553005832004974976023130002d0 |
---|
| 1661 | rkAinvT(3,1) = 3.637834252744495732208418513577775d0 |
---|
| 1662 | rkAinvT(3,2) = 2.681082873627752133895790743211112d0 |
---|
| 1663 | rkAinvT(3,3) = -3.050430199247410569426377624787569d0 |
---|
| 1664 | |
---|
| 1665 | END SUBROUTINE Radau1A_Coefficients |
---|
| 1666 | |
---|
| 1667 | |
---|
| 1668 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1669 | SUBROUTINE Lobatto3A_Coefficients |
---|
| 1670 | ! The coefficients of the 4-stage Lobatto-3A method |
---|
| 1671 | ! (given to ~30 accurate digits) |
---|
| 1672 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1673 | IMPLICIT NONE |
---|
| 1674 | |
---|
| 1675 | ! The coefficients of the Lobatto-3A method |
---|
| 1676 | |
---|
| 1677 | rkMethod = L3A |
---|
| 1678 | |
---|
| 1679 | rkA(0,0) = 0.0d0 |
---|
| 1680 | rkA(0,1) = 0.0d0 |
---|
| 1681 | rkA(0,2) = 0.0d0 |
---|
| 1682 | rkA(0,3) = 0.0d0 |
---|
| 1683 | rkA(1,0) = .11030056647916491413674311390609397d0 |
---|
| 1684 | rkA(1,1) = .1896994335208350858632568860939060d0 |
---|
| 1685 | rkA(1,2) = -.339073642291438837776604807792215d-1 |
---|
| 1686 | rkA(1,3) = .1030056647916491413674311390609397d-1 |
---|
| 1687 | rkA(2,0) = .73032766854168419196590219427239365d-1 |
---|
| 1688 | rkA(2,1) = .4505740308958105504443271474458881d0 |
---|
| 1689 | rkA(2,2) = .2269672331458315808034097805727606d0 |
---|
| 1690 | rkA(2,3) = -.2696723314583158080340978057276063d-1 |
---|
| 1691 | rkA(3,0) = .83333333333333333333333333333333333d-1 |
---|
| 1692 | rkA(3,1) = .4166666666666666666666666666666667d0 |
---|
| 1693 | rkA(3,2) = .4166666666666666666666666666666667d0 |
---|
| 1694 | rkA(3,3) = .8333333333333333333333333333333333d-1 |
---|
| 1695 | |
---|
| 1696 | rkB(0) = .83333333333333333333333333333333333d-1 |
---|
| 1697 | rkB(1) = .4166666666666666666666666666666667d0 |
---|
| 1698 | rkB(2) = .4166666666666666666666666666666667d0 |
---|
| 1699 | rkB(3) = .8333333333333333333333333333333333d-1 |
---|
| 1700 | |
---|
| 1701 | rkC(0) = 0.0d0 |
---|
| 1702 | rkC(1) = .2763932022500210303590826331268724d0 |
---|
| 1703 | rkC(2) = .7236067977499789696409173668731276d0 |
---|
| 1704 | rkC(3) = 1.0d0 |
---|
| 1705 | |
---|
| 1706 | ! New solution: H*Sum B_j*f(Z_j) = Sum D_j*Z_j |
---|
| 1707 | rkD(0) = 0.0d0 |
---|
| 1708 | rkD(1) = 0.0d0 |
---|
| 1709 | rkD(2) = 0.0d0 |
---|
| 1710 | rkD(3) = 1.0d0 |
---|
| 1711 | |
---|
| 1712 | ! Classical error estimator, embedded solution: |
---|
| 1713 | rkBhat(0) = .90909090909090909090909090909090909d-1 |
---|
| 1714 | rkBhat(1) = .39972675774621371442114262372173276d0 |
---|
| 1715 | rkBhat(2) = .43360657558711961891219070961160058d0 |
---|
| 1716 | rkBhat(3) = .15151515151515151515151515151515152d-1 |
---|
| 1717 | |
---|
| 1718 | ! Classical error estimator: |
---|
| 1719 | ! H* Sum (B_j-Bhat_j)*f(Z_j) = H*E(0)*f(0) + Sum E_j*Z_j |
---|
| 1720 | |
---|
| 1721 | rkE(0) = .1957403846510110711315759367097231d-1 |
---|
| 1722 | rkE(1) = -.1986820345632580910316020806676438d0 |
---|
| 1723 | rkE(2) = .1660586371214229125096727578826900d0 |
---|
| 1724 | rkE(3) = -.9787019232550553556578796835486154d-1 |
---|
| 1725 | |
---|
| 1726 | ! Sdirk error estimator: |
---|
| 1727 | rkF(0) = 0.0d0 |
---|
| 1728 | rkF(1) = -.66535815876916686607437314126436349d0 |
---|
| 1729 | rkF(2) = 1.7419302743497277572980407931678409d0 |
---|
| 1730 | rkF(3) = -1.2918865386966730694684011822841728d0 |
---|
| 1731 | |
---|
| 1732 | ! ELO = local order of classical error estimator |
---|
| 1733 | rkELO = 4.0d0 |
---|
| 1734 | |
---|
| 1735 | ! Sdirk error estimator: |
---|
| 1736 | rkBgam(0) = .2950472755430528877214995073815946d-1 |
---|
| 1737 | rkBgam(1) = .5370310883226113978352873633882769d0 |
---|
| 1738 | rkBgam(2) = .2963022450107219354980459699450564d0 |
---|
| 1739 | rkBgam(3) = -.7815248400375080035021681445218837d-1 |
---|
| 1740 | rkBgam(4) = .2153144231161121782447335303806956d0 |
---|
| 1741 | |
---|
| 1742 | ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j |
---|
| 1743 | rkTheta(0) = 0.0d0 |
---|
| 1744 | rkTheta(1) = -.6653581587691668660743731412643631d0 |
---|
| 1745 | rkTheta(2) = 1.741930274349727757298040793167842d0 |
---|
| 1746 | rkTheta(3) = -.291886538696673069468401182284174d0 |
---|
| 1747 | |
---|
| 1748 | |
---|
| 1749 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1750 | !~~~> Diagonalize the RK matrix: |
---|
| 1751 | ! rkTinv * inv(rkA) * rkT = |
---|
| 1752 | ! | rkGamma 0 0 | |
---|
| 1753 | ! | 0 rkAlpha -rkBeta | |
---|
| 1754 | ! | 0 rkBeta rkAlpha | |
---|
| 1755 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1756 | |
---|
| 1757 | rkGamma = 4.644370709252171185822941421408063d0 |
---|
| 1758 | rkAlpha = 3.677814645373914407088529289295968d0 |
---|
| 1759 | rkBeta = 3.508761919567443321903661209182446d0 |
---|
| 1760 | |
---|
| 1761 | rkT(1,1) = .5303036326129938105898786144870856d-1 |
---|
| 1762 | rkT(1,2) = -.7776129960563076320631956091016914d-1 |
---|
| 1763 | rkT(1,3) = .6043307469475508514468017399717112d-2 |
---|
| 1764 | rkT(2,1) = .2637242522173698467283726114649606d0 |
---|
| 1765 | rkT(2,2) = .2193839918662961493126393244533346d0 |
---|
| 1766 | rkT(2,3) = .3198765142300936188514264752235344d0 |
---|
| 1767 | rkT(3,1) = 1.d0 |
---|
| 1768 | rkT(3,2) = 1.d0 |
---|
| 1769 | rkT(3,3) = 0.d0 |
---|
| 1770 | |
---|
| 1771 | rkTinv(1,1) = 7.695032983257654470769069079238553d0 |
---|
| 1772 | rkTinv(1,2) = -.1453793830957233720334601186354032d0 |
---|
| 1773 | rkTinv(1,3) = .6302696746849084900422461036874826d0 |
---|
| 1774 | rkTinv(2,1) = -7.695032983257654470769069079238553d0 |
---|
| 1775 | rkTinv(2,2) = .1453793830957233720334601186354032d0 |
---|
| 1776 | rkTinv(2,3) = .3697303253150915099577538963125174d0 |
---|
| 1777 | rkTinv(3,1) = -1.066660885401270392058552736086173d0 |
---|
| 1778 | rkTinv(3,2) = 3.146358406832537460764521760668932d0 |
---|
| 1779 | rkTinv(3,3) = -.7732056038202974770406168510664738d0 |
---|
| 1780 | |
---|
| 1781 | rkTinvAinv(1,1) = 35.73858579417120341641749040405149d0 |
---|
| 1782 | rkTinvAinv(1,2) = -.675195748578927863668368190236025d0 |
---|
| 1783 | rkTinvAinv(1,3) = 2.927206016036483646751158874041632d0 |
---|
| 1784 | rkTinvAinv(2,1) = -24.55824590667225493437162206039511d0 |
---|
| 1785 | rkTinvAinv(2,2) = -10.50514413892002061837750015342036 |
---|
| 1786 | rkTinvAinv(2,3) = 4.072793983963516353248841125958369d0 |
---|
| 1787 | rkTinvAinv(3,1) = -30.92301972744621647251975054630589d0 |
---|
| 1788 | rkTinvAinv(3,2) = 12.08182467154052413351908559269928d0 |
---|
| 1789 | rkTinvAinv(3,3) = -1.546411207640594954081233702132946d0 |
---|
| 1790 | |
---|
| 1791 | rkAinvT(1,1) = .2462926658317812882584158369803835d0 |
---|
| 1792 | rkAinvT(1,2) = -.2647871194157644619747121197289574d0 |
---|
| 1793 | rkAinvT(1,3) = .2950720515900466654896406799284586d0 |
---|
| 1794 | rkAinvT(2,1) = 1.224833192317784474576995878738004d0 |
---|
| 1795 | rkAinvT(2,2) = 1.929224190340981580557006261869763d0 |
---|
| 1796 | rkAinvT(2,3) = .4066803323234419988910915619080306d0 |
---|
| 1797 | rkAinvT(3,1) = 4.644370709252171185822941421408064d0 |
---|
| 1798 | rkAinvT(3,2) = 3.677814645373914407088529289295968d0 |
---|
| 1799 | rkAinvT(3,3) = -3.508761919567443321903661209182446d0 |
---|
| 1800 | |
---|
| 1801 | END SUBROUTINE Lobatto3A_Coefficients |
---|
| 1802 | |
---|
| 1803 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1804 | END SUBROUTINE RungeKutta ! and all its internal procedures |
---|
| 1805 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1806 | |
---|
| 1807 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1808 | SUBROUTINE FUN_CHEM(T, V, FCT) |
---|
| 1809 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1810 | |
---|
| 1811 | USE KPP_ROOT_Parameters |
---|
| 1812 | USE KPP_ROOT_Global |
---|
| 1813 | USE KPP_ROOT_Function, ONLY: Fun |
---|
| 1814 | USE KPP_ROOT_Rates, ONLY: Update_SUN, Update_RCONST, Update_PHOTO |
---|
| 1815 | |
---|
| 1816 | IMPLICIT NONE |
---|
| 1817 | |
---|
| 1818 | KPP_REAL :: V(NVAR), FCT(NVAR) |
---|
| 1819 | KPP_REAL :: T, Told |
---|
| 1820 | |
---|
| 1821 | Told = TIME |
---|
| 1822 | TIME = T |
---|
| 1823 | CALL Update_SUN() |
---|
| 1824 | CALL Update_RCONST() |
---|
| 1825 | CALL Update_PHOTO() |
---|
| 1826 | TIME = Told |
---|
| 1827 | |
---|
| 1828 | CALL Fun(V, FIX, RCONST, FCT) |
---|
| 1829 | |
---|
| 1830 | END SUBROUTINE FUN_CHEM |
---|
| 1831 | |
---|
| 1832 | |
---|
| 1833 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1834 | SUBROUTINE JAC_CHEM (T, V, JF) |
---|
| 1835 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1836 | |
---|
| 1837 | USE KPP_ROOT_Parameters |
---|
| 1838 | USE KPP_ROOT_Global |
---|
| 1839 | USE KPP_ROOT_JacobianSP |
---|
| 1840 | USE KPP_ROOT_Jacobian, ONLY: Jac_SP |
---|
| 1841 | USE KPP_ROOT_Rates, ONLY: Update_SUN, Update_RCONST, Update_PHOTO |
---|
| 1842 | |
---|
| 1843 | IMPLICIT NONE |
---|
| 1844 | |
---|
| 1845 | KPP_REAL :: V(NVAR), T , Told |
---|
| 1846 | #ifdef FULL_ALGEBRA |
---|
| 1847 | KPP_REAL :: JV(LU_NONZERO), JF(NVAR,NVAR) |
---|
| 1848 | INTEGER :: i, j |
---|
| 1849 | #else |
---|
| 1850 | KPP_REAL :: JF(LU_NONZERO) |
---|
| 1851 | #endif |
---|
| 1852 | |
---|
| 1853 | Told = TIME |
---|
| 1854 | TIME = T |
---|
| 1855 | CALL Update_SUN() |
---|
| 1856 | CALL Update_RCONST() |
---|
| 1857 | CALL Update_PHOTO() |
---|
| 1858 | TIME = Told |
---|
| 1859 | |
---|
| 1860 | #ifdef FULL_ALGEBRA |
---|
| 1861 | CALL Jac_SP(V, FIX, RCONST, JV) |
---|
| 1862 | DO j=1,NVAR |
---|
| 1863 | DO i=1,NVAR |
---|
| 1864 | JF(i,j) = 0.0d0 |
---|
| 1865 | END DO |
---|
| 1866 | END DO |
---|
| 1867 | DO i=1,LU_NONZERO |
---|
| 1868 | JF(LU_IROW(i),LU_ICOL(i)) = JV(i) |
---|
| 1869 | END DO |
---|
| 1870 | #else |
---|
| 1871 | CALL Jac_SP(V, FIX, RCONST, JF) |
---|
| 1872 | #endif |
---|
| 1873 | |
---|
| 1874 | END SUBROUTINE JAC_CHEM |
---|
| 1875 | |
---|
| 1876 | |
---|
| 1877 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1878 | |
---|
| 1879 | END MODULE KPP_ROOT_Integrator |
---|
| 1880 | |
---|
| 1881 | |
---|