[2696] | 1 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~! |
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| 2 | ! Rosenbrock - Implementation of several Rosenbrock methods: ! |
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| 3 | ! * Ros2 ! |
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| 4 | ! * Ros3 ! |
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| 5 | ! * Ros4 ! |
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| 6 | ! * Rodas3 ! |
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| 7 | ! * Rodas4 ! |
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| 8 | ! By default the code employs the KPP sparse linear algebra routines ! |
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| 9 | ! Compile with -DFULL_ALGEBRA to use full linear algebra (LAPACK) ! |
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| 10 | ! ! |
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| 11 | ! (C) Adrian Sandu, August 2004 ! |
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| 12 | ! Virginia Polytechnic Institute and State University ! |
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| 13 | ! Contact: sandu@cs.vt.edu ! |
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| 14 | ! Revised by Philipp Miehe and Adrian Sandu, May 2006 ! ! |
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| 15 | ! This implementation is part of KPP - the Kinetic PreProcessor ! |
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| 16 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~! |
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| 17 | |
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| 18 | MODULE KPP_ROOT_Integrator |
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| 19 | |
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| 20 | USE KPP_ROOT_Parameters, ONLY: NVAR, NFIX, NSPEC, LU_NONZERO |
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| 21 | USE KPP_ROOT_Global |
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| 22 | IMPLICIT NONE |
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| 23 | PUBLIC |
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| 24 | SAVE |
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| 25 | |
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| 26 | !~~~> Statistics on the work performed by the Rosenbrock method |
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| 27 | INTEGER, PARAMETER :: Nfun=1, Njac=2, Nstp=3, Nacc=4, & |
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| 28 | Nrej=5, Ndec=6, Nsol=7, Nsng=8, & |
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| 29 | Ntexit=1, Nhexit=2, Nhnew = 3 |
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| 30 | |
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| 31 | CONTAINS |
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| 32 | |
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| 33 | SUBROUTINE INTEGRATE( TIN, TOUT, & |
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| 34 | ICNTRL_U, RCNTRL_U, ISTATUS_U, RSTATUS_U, IERR_U ) |
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| 35 | |
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| 36 | IMPLICIT NONE |
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| 37 | |
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| 38 | KPP_REAL, INTENT(IN) :: TIN ! Start Time |
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| 39 | KPP_REAL, INTENT(IN) :: TOUT ! End Time |
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| 40 | ! Optional input parameters and statistics |
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| 41 | INTEGER, INTENT(IN), OPTIONAL :: ICNTRL_U(20) |
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| 42 | KPP_REAL, INTENT(IN), OPTIONAL :: RCNTRL_U(20) |
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| 43 | INTEGER, INTENT(OUT), OPTIONAL :: ISTATUS_U(20) |
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| 44 | KPP_REAL, INTENT(OUT), OPTIONAL :: RSTATUS_U(20) |
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| 45 | INTEGER, INTENT(OUT), OPTIONAL :: IERR_U |
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| 46 | |
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| 47 | KPP_REAL :: RCNTRL(20), RSTATUS(20) |
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| 48 | INTEGER :: ICNTRL(20), ISTATUS(20), IERR |
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| 49 | |
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| 50 | INTEGER, SAVE :: Ntotal = 0 |
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| 51 | |
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| 52 | ICNTRL(:) = 0 |
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| 53 | RCNTRL(:) = 0.0_dp |
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| 54 | ISTATUS(:) = 0 |
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| 55 | RSTATUS(:) = 0.0_dp |
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| 56 | |
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| 57 | !~~~> fine-tune the integrator: |
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| 58 | ICNTRL(1) = 0 ! 0 - non-autonomous, 1 - autonomous |
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| 59 | ICNTRL(2) = 0 ! 0 - vector tolerances, 1 - scalars |
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| 60 | |
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| 61 | ! If optional parameters are given, and if they are >0, |
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| 62 | ! then they overwrite default settings. |
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| 63 | IF (PRESENT(ICNTRL_U)) THEN |
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| 64 | WHERE(ICNTRL_U(:) > 0) ICNTRL(:) = ICNTRL_U(:) |
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| 65 | END IF |
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| 66 | IF (PRESENT(RCNTRL_U)) THEN |
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| 67 | WHERE(RCNTRL_U(:) > 0) RCNTRL(:) = RCNTRL_U(:) |
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| 68 | END IF |
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| 69 | |
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| 70 | |
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| 71 | CALL Rosenbrock(NVAR,VAR,TIN,TOUT, & |
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| 72 | ATOL,RTOL, & |
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| 73 | RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR) |
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| 74 | |
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| 75 | !~~~> Debug option: show no of steps |
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| 76 | ! Ntotal = Ntotal + ISTATUS(Nstp) |
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| 77 | ! PRINT*,'NSTEPS=',ISTATUS(Nstp),' (',Ntotal,')',' O3=', VAR(ind_O3) |
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| 78 | |
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| 79 | STEPMIN = RSTATUS(Nhexit) |
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| 80 | ! if optional parameters are given for output they |
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| 81 | ! are updated with the return information |
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| 82 | IF (PRESENT(ISTATUS_U)) ISTATUS_U(:) = ISTATUS(:) |
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| 83 | IF (PRESENT(RSTATUS_U)) RSTATUS_U(:) = RSTATUS(:) |
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| 84 | IF (PRESENT(IERR_U)) IERR_U = IERR |
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| 85 | |
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| 86 | END SUBROUTINE INTEGRATE |
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| 87 | |
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| 88 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 89 | SUBROUTINE Rosenbrock(N,Y,Tstart,Tend, & |
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| 90 | AbsTol,RelTol, & |
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| 91 | RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR) |
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| 92 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 93 | ! |
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| 94 | ! Solves the system y'=F(t,y) using a Rosenbrock method defined by: |
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| 95 | ! |
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| 96 | ! G = 1/(H*gamma(1)) - Jac(t0,Y0) |
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| 97 | ! T_i = t0 + Alpha(i)*H |
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| 98 | ! Y_i = Y0 + \sum_{j=1}^{i-1} A(i,j)*K_j |
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| 99 | ! G * K_i = Fun( T_i, Y_i ) + \sum_{j=1}^S C(i,j)/H * K_j + |
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| 100 | ! gamma(i)*dF/dT(t0, Y0) |
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| 101 | ! Y1 = Y0 + \sum_{j=1}^S M(j)*K_j |
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| 102 | ! |
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| 103 | ! For details on Rosenbrock methods and their implementation consult: |
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| 104 | ! E. Hairer and G. Wanner |
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| 105 | ! "Solving ODEs II. Stiff and differential-algebraic problems". |
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| 106 | ! Springer series in computational mathematics, Springer-Verlag, 1996. |
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| 107 | ! The codes contained in the book inspired this implementation. |
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| 108 | ! |
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| 109 | ! (C) Adrian Sandu, August 2004 |
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| 110 | ! Virginia Polytechnic Institute and State University |
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| 111 | ! Contact: sandu@cs.vt.edu |
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| 112 | ! Revised by Philipp Miehe and Adrian Sandu, May 2006 |
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| 113 | ! This implementation is part of KPP - the Kinetic PreProcessor |
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| 114 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 115 | ! |
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| 116 | !~~~> INPUT ARGUMENTS: |
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| 117 | ! |
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| 118 | !- Y(N) = vector of initial conditions (at T=Tstart) |
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| 119 | !- [Tstart,Tend] = time range of integration |
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| 120 | ! (if Tstart>Tend the integration is performed backwards in time) |
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| 121 | !- RelTol, AbsTol = user precribed accuracy |
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| 122 | !- SUBROUTINE Fun( T, Y, Ydot ) = ODE function, |
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| 123 | ! returns Ydot = Y' = F(T,Y) |
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| 124 | !- SUBROUTINE Jac( T, Y, Jcb ) = Jacobian of the ODE function, |
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| 125 | ! returns Jcb = dFun/dY |
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| 126 | !- ICNTRL(1:20) = integer inputs parameters |
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| 127 | !- RCNTRL(1:20) = real inputs parameters |
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| 128 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 129 | ! |
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| 130 | !~~~> OUTPUT ARGUMENTS: |
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| 131 | ! |
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| 132 | !- Y(N) -> vector of final states (at T->Tend) |
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| 133 | !- ISTATUS(1:20) -> integer output parameters |
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| 134 | !- RSTATUS(1:20) -> real output parameters |
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| 135 | !- IERR -> job status upon return |
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| 136 | ! success (positive value) or |
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| 137 | ! failure (negative value) |
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| 138 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 139 | ! |
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| 140 | !~~~> INPUT PARAMETERS: |
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| 141 | ! |
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| 142 | ! Note: For input parameters equal to zero the default values of the |
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| 143 | ! corresponding variables are used. |
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| 144 | ! |
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| 145 | ! ICNTRL(1) = 1: F = F(y) Independent of T (AUTONOMOUS) |
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| 146 | ! = 0: F = F(t,y) Depends on T (NON-AUTONOMOUS) |
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| 147 | ! |
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| 148 | ! ICNTRL(2) = 0: AbsTol, RelTol are N-dimensional vectors |
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| 149 | ! = 1: AbsTol, RelTol are scalars |
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| 150 | ! |
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| 151 | ! ICNTRL(3) -> selection of a particular Rosenbrock method |
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| 152 | ! = 0 : Rodas3 (default) |
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| 153 | ! = 1 : Ros2 |
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| 154 | ! = 2 : Ros3 |
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| 155 | ! = 3 : Ros4 |
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| 156 | ! = 4 : Rodas3 |
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| 157 | ! = 5 : Rodas4 |
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| 158 | ! |
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| 159 | ! ICNTRL(4) -> maximum number of integration steps |
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| 160 | ! For ICNTRL(4)=0) the default value of 100000 is used |
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| 161 | ! |
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| 162 | ! RCNTRL(1) -> Hmin, lower bound for the integration step size |
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| 163 | ! It is strongly recommended to keep Hmin = ZERO |
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| 164 | ! RCNTRL(2) -> Hmax, upper bound for the integration step size |
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| 165 | ! RCNTRL(3) -> Hstart, starting value for the integration step size |
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| 166 | ! |
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| 167 | ! RCNTRL(4) -> FacMin, lower bound on step decrease factor (default=0.2) |
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| 168 | ! RCNTRL(5) -> FacMax, upper bound on step increase factor (default=6) |
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| 169 | ! RCNTRL(6) -> FacRej, step decrease factor after multiple rejections |
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| 170 | ! (default=0.1) |
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| 171 | ! RCNTRL(7) -> FacSafe, by which the new step is slightly smaller |
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| 172 | ! than the predicted value (default=0.9) |
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| 173 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 174 | ! |
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| 175 | ! |
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| 176 | ! OUTPUT ARGUMENTS: |
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| 177 | ! ----------------- |
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| 178 | ! |
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| 179 | ! T -> T value for which the solution has been computed |
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| 180 | ! (after successful return T=Tend). |
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| 181 | ! |
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| 182 | ! Y(N) -> Numerical solution at T |
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| 183 | ! |
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| 184 | ! IDID -> Reports on successfulness upon return: |
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| 185 | ! = 1 for success |
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| 186 | ! < 0 for error (value equals error code) |
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| 187 | ! |
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| 188 | ! ISTATUS(1) -> No. of function calls |
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| 189 | ! ISTATUS(2) -> No. of jacobian calls |
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| 190 | ! ISTATUS(3) -> No. of steps |
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| 191 | ! ISTATUS(4) -> No. of accepted steps |
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| 192 | ! ISTATUS(5) -> No. of rejected steps (except at very beginning) |
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| 193 | ! ISTATUS(6) -> No. of LU decompositions |
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| 194 | ! ISTATUS(7) -> No. of forward/backward substitutions |
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| 195 | ! ISTATUS(8) -> No. of singular matrix decompositions |
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| 196 | ! |
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| 197 | ! RSTATUS(1) -> Texit, the time corresponding to the |
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| 198 | ! computed Y upon return |
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| 199 | ! RSTATUS(2) -> Hexit, last accepted step before exit |
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| 200 | ! RSTATUS(3) -> Hnew, last predicted step (not yet taken) |
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| 201 | ! For multiple restarts, use Hnew as Hstart |
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| 202 | ! in the subsequent run |
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| 203 | ! |
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| 204 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 205 | |
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| 206 | USE KPP_ROOT_Parameters |
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| 207 | USE KPP_ROOT_LinearAlgebra |
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| 208 | IMPLICIT NONE |
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| 209 | |
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| 210 | !~~~> Arguments |
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| 211 | INTEGER, INTENT(IN) :: N |
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| 212 | KPP_REAL, INTENT(INOUT) :: Y(N) |
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| 213 | KPP_REAL, INTENT(IN) :: Tstart,Tend |
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| 214 | KPP_REAL, INTENT(IN) :: AbsTol(N),RelTol(N) |
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| 215 | INTEGER, INTENT(IN) :: ICNTRL(20) |
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| 216 | KPP_REAL, INTENT(IN) :: RCNTRL(20) |
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| 217 | INTEGER, INTENT(INOUT) :: ISTATUS(20) |
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| 218 | KPP_REAL, INTENT(INOUT) :: RSTATUS(20) |
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| 219 | INTEGER, INTENT(OUT) :: IERR |
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| 220 | !~~~> Parameters of the Rosenbrock method, up to 6 stages |
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| 221 | INTEGER :: ros_S, rosMethod |
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| 222 | INTEGER, PARAMETER :: RS2=1, RS3=2, RS4=3, RD3=4, RD4=5, RG3=6 |
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| 223 | KPP_REAL :: ros_A(15), ros_C(15), ros_M(6), ros_E(6), & |
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| 224 | ros_Alpha(6), ros_Gamma(6), ros_ELO |
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| 225 | LOGICAL :: ros_NewF(6) |
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| 226 | CHARACTER(LEN=12) :: ros_Name |
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| 227 | !~~~> Local variables |
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| 228 | KPP_REAL :: Roundoff, FacMin, FacMax, FacRej, FacSafe |
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| 229 | KPP_REAL :: Hmin, Hmax, Hstart |
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| 230 | KPP_REAL :: Texit |
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| 231 | INTEGER :: i, UplimTol, Max_no_steps |
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| 232 | LOGICAL :: Autonomous, VectorTol |
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| 233 | !~~~> Parameters |
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| 234 | KPP_REAL, PARAMETER :: ZERO = 0.0_dp, ONE = 1.0_dp |
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| 235 | KPP_REAL, PARAMETER :: DeltaMin = 1.0E-5_dp |
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| 236 | |
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| 237 | !~~~> Initialize statistics |
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| 238 | ISTATUS(1:8) = 0 |
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| 239 | RSTATUS(1:3) = ZERO |
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| 240 | |
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| 241 | !~~~> Autonomous or time dependent ODE. Default is time dependent. |
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| 242 | Autonomous = .NOT.(ICNTRL(1) == 0) |
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| 243 | |
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| 244 | !~~~> For Scalar tolerances (ICNTRL(2).NE.0) the code uses AbsTol(1) and RelTol(1) |
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| 245 | ! For Vector tolerances (ICNTRL(2) == 0) the code uses AbsTol(1:N) and RelTol(1:N) |
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| 246 | IF (ICNTRL(2) == 0) THEN |
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| 247 | VectorTol = .TRUE. |
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| 248 | UplimTol = N |
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| 249 | ELSE |
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| 250 | VectorTol = .FALSE. |
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| 251 | UplimTol = 1 |
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| 252 | END IF |
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| 253 | |
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| 254 | !~~~> Initialize the particular Rosenbrock method selected |
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| 255 | SELECT CASE (ICNTRL(3)) |
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| 256 | CASE (1) |
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| 257 | CALL Ros2 |
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| 258 | CASE (2) |
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| 259 | CALL Ros3 |
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| 260 | CASE (3) |
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| 261 | CALL Ros4 |
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| 262 | CASE (0,4) |
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| 263 | CALL Rodas3 |
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| 264 | CASE (5) |
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| 265 | CALL Rodas4 |
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| 266 | CASE (6) |
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| 267 | CALL Rang3 |
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| 268 | CASE DEFAULT |
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| 269 | PRINT * , 'Unknown Rosenbrock method: ICNTRL(3)=',ICNTRL(3) |
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| 270 | CALL ros_ErrorMsg(-2,Tstart,ZERO,IERR) |
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| 271 | RETURN |
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| 272 | END SELECT |
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| 273 | |
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| 274 | !~~~> The maximum number of steps admitted |
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| 275 | IF (ICNTRL(4) == 0) THEN |
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| 276 | Max_no_steps = 200000 |
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| 277 | ELSEIF (ICNTRL(4) > 0) THEN |
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| 278 | Max_no_steps=ICNTRL(4) |
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| 279 | ELSE |
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| 280 | PRINT * ,'User-selected max no. of steps: ICNTRL(4)=',ICNTRL(4) |
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| 281 | CALL ros_ErrorMsg(-1,Tstart,ZERO,IERR) |
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| 282 | RETURN |
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| 283 | END IF |
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| 284 | |
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| 285 | !~~~> Unit roundoff (1+Roundoff>1) |
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| 286 | Roundoff = WLAMCH('E') |
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| 287 | |
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| 288 | !~~~> Lower bound on the step size: (positive value) |
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| 289 | IF (RCNTRL(1) == ZERO) THEN |
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| 290 | Hmin = ZERO |
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| 291 | ELSEIF (RCNTRL(1) > ZERO) THEN |
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| 292 | Hmin = RCNTRL(1) |
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| 293 | ELSE |
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| 294 | PRINT * , 'User-selected Hmin: RCNTRL(1)=', RCNTRL(1) |
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| 295 | CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR) |
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| 296 | RETURN |
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| 297 | END IF |
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| 298 | !~~~> Upper bound on the step size: (positive value) |
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| 299 | IF (RCNTRL(2) == ZERO) THEN |
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| 300 | Hmax = ABS(Tend-Tstart) |
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| 301 | ELSEIF (RCNTRL(2) > ZERO) THEN |
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| 302 | Hmax = MIN(ABS(RCNTRL(2)),ABS(Tend-Tstart)) |
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| 303 | ELSE |
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| 304 | PRINT * , 'User-selected Hmax: RCNTRL(2)=', RCNTRL(2) |
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| 305 | CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR) |
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| 306 | RETURN |
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| 307 | END IF |
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| 308 | !~~~> Starting step size: (positive value) |
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| 309 | IF (RCNTRL(3) == ZERO) THEN |
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| 310 | Hstart = MAX(Hmin,DeltaMin) |
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| 311 | ELSEIF (RCNTRL(3) > ZERO) THEN |
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| 312 | Hstart = MIN(ABS(RCNTRL(3)),ABS(Tend-Tstart)) |
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| 313 | ELSE |
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| 314 | PRINT * , 'User-selected Hstart: RCNTRL(3)=', RCNTRL(3) |
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| 315 | CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR) |
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| 316 | RETURN |
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| 317 | END IF |
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| 318 | !~~~> Step size can be changed s.t. FacMin < Hnew/Hold < FacMax |
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| 319 | IF (RCNTRL(4) == ZERO) THEN |
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| 320 | FacMin = 0.2_dp |
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| 321 | ELSEIF (RCNTRL(4) > ZERO) THEN |
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| 322 | FacMin = RCNTRL(4) |
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| 323 | ELSE |
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| 324 | PRINT * , 'User-selected FacMin: RCNTRL(4)=', RCNTRL(4) |
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| 325 | CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR) |
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| 326 | RETURN |
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| 327 | END IF |
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| 328 | IF (RCNTRL(5) == ZERO) THEN |
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| 329 | FacMax = 6.0_dp |
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| 330 | ELSEIF (RCNTRL(5) > ZERO) THEN |
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| 331 | FacMax = RCNTRL(5) |
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| 332 | ELSE |
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| 333 | PRINT * , 'User-selected FacMax: RCNTRL(5)=', RCNTRL(5) |
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| 334 | CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR) |
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| 335 | RETURN |
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| 336 | END IF |
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| 337 | !~~~> FacRej: Factor to decrease step after 2 succesive rejections |
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| 338 | IF (RCNTRL(6) == ZERO) THEN |
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| 339 | FacRej = 0.1_dp |
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| 340 | ELSEIF (RCNTRL(6) > ZERO) THEN |
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| 341 | FacRej = RCNTRL(6) |
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| 342 | ELSE |
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| 343 | PRINT * , 'User-selected FacRej: RCNTRL(6)=', RCNTRL(6) |
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| 344 | CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR) |
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| 345 | RETURN |
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| 346 | END IF |
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| 347 | !~~~> FacSafe: Safety Factor in the computation of new step size |
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| 348 | IF (RCNTRL(7) == ZERO) THEN |
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| 349 | FacSafe = 0.9_dp |
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| 350 | ELSEIF (RCNTRL(7) > ZERO) THEN |
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| 351 | FacSafe = RCNTRL(7) |
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| 352 | ELSE |
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| 353 | PRINT * , 'User-selected FacSafe: RCNTRL(7)=', RCNTRL(7) |
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| 354 | CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR) |
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| 355 | RETURN |
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| 356 | END IF |
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| 357 | !~~~> Check if tolerances are reasonable |
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| 358 | DO i=1,UplimTol |
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| 359 | IF ( (AbsTol(i) <= ZERO) .OR. (RelTol(i) <= 10.0_dp*Roundoff) & |
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| 360 | .OR. (RelTol(i) >= 1.0_dp) ) THEN |
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| 361 | PRINT * , ' AbsTol(',i,') = ',AbsTol(i) |
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| 362 | PRINT * , ' RelTol(',i,') = ',RelTol(i) |
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| 363 | CALL ros_ErrorMsg(-5,Tstart,ZERO,IERR) |
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| 364 | RETURN |
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| 365 | END IF |
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| 366 | END DO |
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| 367 | |
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| 368 | |
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| 369 | !~~~> CALL Rosenbrock method |
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| 370 | CALL ros_Integrator(Y, Tstart, Tend, Texit, & |
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| 371 | AbsTol, RelTol, & |
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| 372 | ! Integration parameters |
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| 373 | Autonomous, VectorTol, Max_no_steps, & |
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| 374 | Roundoff, Hmin, Hmax, Hstart, & |
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| 375 | FacMin, FacMax, FacRej, FacSafe, & |
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| 376 | ! Error indicator |
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| 377 | IERR) |
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| 378 | |
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| 379 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 380 | CONTAINS ! SUBROUTINES internal to Rosenbrock |
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| 381 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 382 | |
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| 383 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 384 | SUBROUTINE ros_ErrorMsg(Code,T,H,IERR) |
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| 385 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 386 | ! Handles all error messages |
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| 387 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 388 | |
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| 389 | KPP_REAL, INTENT(IN) :: T, H |
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| 390 | INTEGER, INTENT(IN) :: Code |
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| 391 | INTEGER, INTENT(OUT) :: IERR |
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| 392 | |
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| 393 | IERR = Code |
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| 394 | PRINT * , & |
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| 395 | 'Forced exit from Rosenbrock due to the following error:' |
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| 396 | |
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| 397 | SELECT CASE (Code) |
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| 398 | CASE (-1) |
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| 399 | PRINT * , '--> Improper value for maximal no of steps' |
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| 400 | CASE (-2) |
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| 401 | PRINT * , '--> Selected Rosenbrock method not implemented' |
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| 402 | CASE (-3) |
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| 403 | PRINT * , '--> Hmin/Hmax/Hstart must be positive' |
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| 404 | CASE (-4) |
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| 405 | PRINT * , '--> FacMin/FacMax/FacRej must be positive' |
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| 406 | CASE (-5) |
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| 407 | PRINT * , '--> Improper tolerance values' |
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| 408 | CASE (-6) |
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| 409 | PRINT * , '--> No of steps exceeds maximum bound' |
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| 410 | CASE (-7) |
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| 411 | PRINT * , '--> Step size too small: T + 10*H = T', & |
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| 412 | ' or H < Roundoff' |
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| 413 | CASE (-8) |
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| 414 | PRINT * , '--> Matrix is repeatedly singular' |
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| 415 | CASE DEFAULT |
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| 416 | PRINT *, 'Unknown Error code: ', Code |
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| 417 | END SELECT |
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| 418 | |
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| 419 | PRINT *, "T=", T, "and H=", H |
---|
| 420 | |
---|
| 421 | END SUBROUTINE ros_ErrorMsg |
---|
| 422 | |
---|
| 423 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 424 | SUBROUTINE ros_Integrator (Y, Tstart, Tend, T, & |
---|
| 425 | AbsTol, RelTol, & |
---|
| 426 | !~~~> Integration parameters |
---|
| 427 | Autonomous, VectorTol, Max_no_steps, & |
---|
| 428 | Roundoff, Hmin, Hmax, Hstart, & |
---|
| 429 | FacMin, FacMax, FacRej, FacSafe, & |
---|
| 430 | !~~~> Error indicator |
---|
| 431 | IERR ) |
---|
| 432 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 433 | ! Template for the implementation of a generic Rosenbrock method |
---|
| 434 | ! defined by ros_S (no of stages) |
---|
| 435 | ! and its coefficients ros_{A,C,M,E,Alpha,Gamma} |
---|
| 436 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 437 | |
---|
| 438 | IMPLICIT NONE |
---|
| 439 | |
---|
| 440 | !~~~> Input: the initial condition at Tstart; Output: the solution at T |
---|
| 441 | KPP_REAL, INTENT(INOUT) :: Y(N) |
---|
| 442 | !~~~> Input: integration interval |
---|
| 443 | KPP_REAL, INTENT(IN) :: Tstart,Tend |
---|
| 444 | !~~~> Output: time at which the solution is returned (T=Tend if success) |
---|
| 445 | KPP_REAL, INTENT(OUT) :: T |
---|
| 446 | !~~~> Input: tolerances |
---|
| 447 | KPP_REAL, INTENT(IN) :: AbsTol(N), RelTol(N) |
---|
| 448 | !~~~> Input: integration parameters |
---|
| 449 | LOGICAL, INTENT(IN) :: Autonomous, VectorTol |
---|
| 450 | KPP_REAL, INTENT(IN) :: Hstart, Hmin, Hmax |
---|
| 451 | INTEGER, INTENT(IN) :: Max_no_steps |
---|
| 452 | KPP_REAL, INTENT(IN) :: Roundoff, FacMin, FacMax, FacRej, FacSafe |
---|
| 453 | !~~~> Output: Error indicator |
---|
| 454 | INTEGER, INTENT(OUT) :: IERR |
---|
| 455 | ! ~~~~ Local variables |
---|
| 456 | KPP_REAL :: Ynew(N), Fcn0(N), Fcn(N) |
---|
| 457 | KPP_REAL :: K(N*ros_S), dFdT(N) |
---|
| 458 | #ifdef FULL_ALGEBRA |
---|
| 459 | KPP_REAL :: Jac0(N,N), Ghimj(N,N) |
---|
| 460 | #else |
---|
| 461 | KPP_REAL :: Jac0(LU_NONZERO), Ghimj(LU_NONZERO) |
---|
| 462 | #endif |
---|
| 463 | KPP_REAL :: H, Hnew, HC, HG, Fac, Tau |
---|
| 464 | KPP_REAL :: Err, Yerr(N) |
---|
| 465 | INTEGER :: Pivot(N), Direction, ioffset, j, istage |
---|
| 466 | LOGICAL :: RejectLastH, RejectMoreH, Singular |
---|
| 467 | !~~~> Local parameters |
---|
| 468 | KPP_REAL, PARAMETER :: ZERO = 0.0_dp, ONE = 1.0_dp |
---|
| 469 | KPP_REAL, PARAMETER :: DeltaMin = 1.0E-5_dp |
---|
| 470 | !~~~> Locally called functions |
---|
| 471 | ! KPP_REAL WLAMCH |
---|
| 472 | ! EXTERNAL WLAMCH |
---|
| 473 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 474 | |
---|
| 475 | |
---|
| 476 | !~~~> Initial preparations |
---|
| 477 | T = Tstart |
---|
| 478 | RSTATUS(Nhexit) = ZERO |
---|
| 479 | H = MIN( MAX(ABS(Hmin),ABS(Hstart)) , ABS(Hmax) ) |
---|
| 480 | IF (ABS(H) <= 10.0_dp*Roundoff) H = DeltaMin |
---|
| 481 | |
---|
| 482 | IF (Tend >= Tstart) THEN |
---|
| 483 | Direction = +1 |
---|
| 484 | ELSE |
---|
| 485 | Direction = -1 |
---|
| 486 | END IF |
---|
| 487 | H = Direction*H |
---|
| 488 | |
---|
| 489 | RejectLastH=.FALSE. |
---|
| 490 | RejectMoreH=.FALSE. |
---|
| 491 | |
---|
| 492 | !~~~> Time loop begins below |
---|
| 493 | |
---|
| 494 | TimeLoop: DO WHILE ( (Direction > 0).AND.((T-Tend)+Roundoff <= ZERO) & |
---|
| 495 | .OR. (Direction < 0).AND.((Tend-T)+Roundoff <= ZERO) ) |
---|
| 496 | |
---|
| 497 | IF ( ISTATUS(Nstp) > Max_no_steps ) THEN ! Too many steps |
---|
| 498 | CALL ros_ErrorMsg(-6,T,H,IERR) |
---|
| 499 | RETURN |
---|
| 500 | END IF |
---|
| 501 | IF ( ((T+0.1_dp*H) == T).OR.(H <= Roundoff) ) THEN ! Step size too small |
---|
| 502 | CALL ros_ErrorMsg(-7,T,H,IERR) |
---|
| 503 | RETURN |
---|
| 504 | END IF |
---|
| 505 | |
---|
| 506 | !~~~> Limit H if necessary to avoid going beyond Tend |
---|
| 507 | H = MIN(H,ABS(Tend-T)) |
---|
| 508 | |
---|
| 509 | !~~~> Compute the function at current time |
---|
| 510 | CALL FunTemplate(T,Y,Fcn0) |
---|
| 511 | ISTATUS(Nfun) = ISTATUS(Nfun) + 1 |
---|
| 512 | |
---|
| 513 | !~~~> Compute the function derivative with respect to T |
---|
| 514 | IF (.NOT.Autonomous) THEN |
---|
| 515 | CALL ros_FunTimeDerivative ( T, Roundoff, Y, & |
---|
| 516 | Fcn0, dFdT ) |
---|
| 517 | END IF |
---|
| 518 | |
---|
| 519 | !~~~> Compute the Jacobian at current time |
---|
| 520 | CALL JacTemplate(T,Y,Jac0) |
---|
| 521 | ISTATUS(Njac) = ISTATUS(Njac) + 1 |
---|
| 522 | |
---|
| 523 | !~~~> Repeat step calculation until current step accepted |
---|
| 524 | UntilAccepted: DO |
---|
| 525 | |
---|
| 526 | CALL ros_PrepareMatrix(H,Direction,ros_Gamma(1), & |
---|
| 527 | Jac0,Ghimj,Pivot,Singular) |
---|
| 528 | IF (Singular) THEN ! More than 5 consecutive failed decompositions |
---|
| 529 | CALL ros_ErrorMsg(-8,T,H,IERR) |
---|
| 530 | RETURN |
---|
| 531 | END IF |
---|
| 532 | |
---|
| 533 | !~~~> Compute the stages |
---|
| 534 | Stage: DO istage = 1, ros_S |
---|
| 535 | |
---|
| 536 | ! Current istage offset. Current istage vector is K(ioffset+1:ioffset+N) |
---|
| 537 | ioffset = N*(istage-1) |
---|
| 538 | |
---|
| 539 | ! For the 1st istage the function has been computed previously |
---|
| 540 | IF ( istage == 1 ) THEN |
---|
| 541 | !slim: CALL WCOPY(N,Fcn0,1,Fcn,1) |
---|
| 542 | Fcn(1:N) = Fcn0(1:N) |
---|
| 543 | ! istage>1 and a new function evaluation is needed at the current istage |
---|
| 544 | ELSEIF ( ros_NewF(istage) ) THEN |
---|
| 545 | !slim: CALL WCOPY(N,Y,1,Ynew,1) |
---|
| 546 | Ynew(1:N) = Y(1:N) |
---|
| 547 | DO j = 1, istage-1 |
---|
| 548 | CALL WAXPY(N,ros_A((istage-1)*(istage-2)/2+j), & |
---|
| 549 | K(N*(j-1)+1),1,Ynew,1) |
---|
| 550 | END DO |
---|
| 551 | Tau = T + ros_Alpha(istage)*Direction*H |
---|
| 552 | CALL FunTemplate(Tau,Ynew,Fcn) |
---|
| 553 | ISTATUS(Nfun) = ISTATUS(Nfun) + 1 |
---|
| 554 | END IF ! if istage == 1 elseif ros_NewF(istage) |
---|
| 555 | !slim: CALL WCOPY(N,Fcn,1,K(ioffset+1),1) |
---|
| 556 | K(ioffset+1:ioffset+N) = Fcn(1:N) |
---|
| 557 | DO j = 1, istage-1 |
---|
| 558 | HC = ros_C((istage-1)*(istage-2)/2+j)/(Direction*H) |
---|
| 559 | CALL WAXPY(N,HC,K(N*(j-1)+1),1,K(ioffset+1),1) |
---|
| 560 | END DO |
---|
| 561 | IF ((.NOT. Autonomous).AND.(ros_Gamma(istage).NE.ZERO)) THEN |
---|
| 562 | HG = Direction*H*ros_Gamma(istage) |
---|
| 563 | CALL WAXPY(N,HG,dFdT,1,K(ioffset+1),1) |
---|
| 564 | END IF |
---|
| 565 | CALL ros_Solve(Ghimj, Pivot, K(ioffset+1)) |
---|
| 566 | |
---|
| 567 | END DO Stage |
---|
| 568 | |
---|
| 569 | |
---|
| 570 | !~~~> Compute the new solution |
---|
| 571 | !slim: CALL WCOPY(N,Y,1,Ynew,1) |
---|
| 572 | Ynew(1:N) = Y(1:N) |
---|
| 573 | DO j=1,ros_S |
---|
| 574 | CALL WAXPY(N,ros_M(j),K(N*(j-1)+1),1,Ynew,1) |
---|
| 575 | END DO |
---|
| 576 | |
---|
| 577 | !~~~> Compute the error estimation |
---|
| 578 | !slim: CALL WSCAL(N,ZERO,Yerr,1) |
---|
| 579 | Yerr(1:N) = ZERO |
---|
| 580 | DO j=1,ros_S |
---|
| 581 | CALL WAXPY(N,ros_E(j),K(N*(j-1)+1),1,Yerr,1) |
---|
| 582 | END DO |
---|
| 583 | Err = ros_ErrorNorm ( Y, Ynew, Yerr, AbsTol, RelTol, VectorTol ) |
---|
| 584 | |
---|
| 585 | !~~~> New step size is bounded by FacMin <= Hnew/H <= FacMax |
---|
| 586 | Fac = MIN(FacMax,MAX(FacMin,FacSafe/Err**(ONE/ros_ELO))) |
---|
| 587 | Hnew = H*Fac |
---|
| 588 | |
---|
| 589 | !~~~> Check the error magnitude and adjust step size |
---|
| 590 | ISTATUS(Nstp) = ISTATUS(Nstp) + 1 |
---|
| 591 | IF ( (Err <= ONE).OR.(H <= Hmin) ) THEN !~~~> Accept step |
---|
| 592 | ISTATUS(Nacc) = ISTATUS(Nacc) + 1 |
---|
| 593 | !slim: CALL WCOPY(N,Ynew,1,Y,1) |
---|
| 594 | Y(1:N) = Ynew(1:N) |
---|
| 595 | T = T + Direction*H |
---|
| 596 | Hnew = MAX(Hmin,MIN(Hnew,Hmax)) |
---|
| 597 | IF (RejectLastH) THEN ! No step size increase after a rejected step |
---|
| 598 | Hnew = MIN(Hnew,H) |
---|
| 599 | END IF |
---|
| 600 | RSTATUS(Nhexit) = H |
---|
| 601 | RSTATUS(Nhnew) = Hnew |
---|
| 602 | RSTATUS(Ntexit) = T |
---|
| 603 | RejectLastH = .FALSE. |
---|
| 604 | RejectMoreH = .FALSE. |
---|
| 605 | H = Hnew |
---|
| 606 | EXIT UntilAccepted ! EXIT THE LOOP: WHILE STEP NOT ACCEPTED |
---|
| 607 | ELSE !~~~> Reject step |
---|
| 608 | IF (RejectMoreH) THEN |
---|
| 609 | Hnew = H*FacRej |
---|
| 610 | END IF |
---|
| 611 | RejectMoreH = RejectLastH |
---|
| 612 | RejectLastH = .TRUE. |
---|
| 613 | H = Hnew |
---|
| 614 | IF (ISTATUS(Nacc) >= 1) ISTATUS(Nrej) = ISTATUS(Nrej) + 1 |
---|
| 615 | END IF ! Err <= 1 |
---|
| 616 | |
---|
| 617 | END DO UntilAccepted |
---|
| 618 | |
---|
| 619 | END DO TimeLoop |
---|
| 620 | |
---|
| 621 | !~~~> Succesful exit |
---|
| 622 | IERR = 1 !~~~> The integration was successful |
---|
| 623 | |
---|
| 624 | END SUBROUTINE ros_Integrator |
---|
| 625 | |
---|
| 626 | |
---|
| 627 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 628 | KPP_REAL FUNCTION ros_ErrorNorm ( Y, Ynew, Yerr, & |
---|
| 629 | AbsTol, RelTol, VectorTol ) |
---|
| 630 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 631 | !~~~> Computes the "scaled norm" of the error vector Yerr |
---|
| 632 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 633 | IMPLICIT NONE |
---|
| 634 | |
---|
| 635 | ! Input arguments |
---|
| 636 | KPP_REAL, INTENT(IN) :: Y(N), Ynew(N), & |
---|
| 637 | Yerr(N), AbsTol(N), RelTol(N) |
---|
| 638 | LOGICAL, INTENT(IN) :: VectorTol |
---|
| 639 | ! Local variables |
---|
| 640 | KPP_REAL :: Err, Scale, Ymax |
---|
| 641 | INTEGER :: i |
---|
| 642 | KPP_REAL, PARAMETER :: ZERO = 0.0_dp |
---|
| 643 | |
---|
| 644 | Err = ZERO |
---|
| 645 | DO i=1,N |
---|
| 646 | Ymax = MAX(ABS(Y(i)),ABS(Ynew(i))) |
---|
| 647 | IF (VectorTol) THEN |
---|
| 648 | Scale = AbsTol(i)+RelTol(i)*Ymax |
---|
| 649 | ELSE |
---|
| 650 | Scale = AbsTol(1)+RelTol(1)*Ymax |
---|
| 651 | END IF |
---|
| 652 | Err = Err+(Yerr(i)/Scale)**2 |
---|
| 653 | END DO |
---|
| 654 | Err = SQRT(Err/N) |
---|
| 655 | |
---|
| 656 | ros_ErrorNorm = MAX(Err,1.0d-10) |
---|
| 657 | |
---|
| 658 | END FUNCTION ros_ErrorNorm |
---|
| 659 | |
---|
| 660 | |
---|
| 661 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 662 | SUBROUTINE ros_FunTimeDerivative ( T, Roundoff, Y, & |
---|
| 663 | Fcn0, dFdT ) |
---|
| 664 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 665 | !~~~> The time partial derivative of the function by finite differences |
---|
| 666 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 667 | IMPLICIT NONE |
---|
| 668 | |
---|
| 669 | !~~~> Input arguments |
---|
| 670 | KPP_REAL, INTENT(IN) :: T, Roundoff, Y(N), Fcn0(N) |
---|
| 671 | !~~~> Output arguments |
---|
| 672 | KPP_REAL, INTENT(OUT) :: dFdT(N) |
---|
| 673 | !~~~> Local variables |
---|
| 674 | KPP_REAL :: Delta |
---|
| 675 | KPP_REAL, PARAMETER :: ONE = 1.0_dp, DeltaMin = 1.0E-6_dp |
---|
| 676 | |
---|
| 677 | Delta = SQRT(Roundoff)*MAX(DeltaMin,ABS(T)) |
---|
| 678 | CALL FunTemplate(T+Delta,Y,dFdT) |
---|
| 679 | ISTATUS(Nfun) = ISTATUS(Nfun) + 1 |
---|
| 680 | CALL WAXPY(N,(-ONE),Fcn0,1,dFdT,1) |
---|
| 681 | CALL WSCAL(N,(ONE/Delta),dFdT,1) |
---|
| 682 | |
---|
| 683 | END SUBROUTINE ros_FunTimeDerivative |
---|
| 684 | |
---|
| 685 | |
---|
| 686 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 687 | SUBROUTINE ros_PrepareMatrix ( H, Direction, gam, & |
---|
| 688 | Jac0, Ghimj, Pivot, Singular ) |
---|
| 689 | ! --- --- --- --- --- --- --- --- --- --- --- --- --- |
---|
| 690 | ! Prepares the LHS matrix for stage calculations |
---|
| 691 | ! 1. Construct Ghimj = 1/(H*ham) - Jac0 |
---|
| 692 | ! "(Gamma H) Inverse Minus Jacobian" |
---|
| 693 | ! 2. Repeat LU decomposition of Ghimj until successful. |
---|
| 694 | ! -half the step size if LU decomposition fails and retry |
---|
| 695 | ! -exit after 5 consecutive fails |
---|
| 696 | ! --- --- --- --- --- --- --- --- --- --- --- --- --- |
---|
| 697 | IMPLICIT NONE |
---|
| 698 | |
---|
| 699 | !~~~> Input arguments |
---|
| 700 | #ifdef FULL_ALGEBRA |
---|
| 701 | KPP_REAL, INTENT(IN) :: Jac0(N,N) |
---|
| 702 | #else |
---|
| 703 | KPP_REAL, INTENT(IN) :: Jac0(LU_NONZERO) |
---|
| 704 | #endif |
---|
| 705 | KPP_REAL, INTENT(IN) :: gam |
---|
| 706 | INTEGER, INTENT(IN) :: Direction |
---|
| 707 | !~~~> Output arguments |
---|
| 708 | #ifdef FULL_ALGEBRA |
---|
| 709 | KPP_REAL, INTENT(OUT) :: Ghimj(N,N) |
---|
| 710 | #else |
---|
| 711 | KPP_REAL, INTENT(OUT) :: Ghimj(LU_NONZERO) |
---|
| 712 | #endif |
---|
| 713 | LOGICAL, INTENT(OUT) :: Singular |
---|
| 714 | INTEGER, INTENT(OUT) :: Pivot(N) |
---|
| 715 | !~~~> Inout arguments |
---|
| 716 | KPP_REAL, INTENT(INOUT) :: H ! step size is decreased when LU fails |
---|
| 717 | !~~~> Local variables |
---|
| 718 | INTEGER :: i, ISING, Nconsecutive |
---|
| 719 | KPP_REAL :: ghinv |
---|
| 720 | KPP_REAL, PARAMETER :: ONE = 1.0_dp, HALF = 0.5_dp |
---|
| 721 | |
---|
| 722 | Nconsecutive = 0 |
---|
| 723 | Singular = .TRUE. |
---|
| 724 | |
---|
| 725 | DO WHILE (Singular) |
---|
| 726 | |
---|
| 727 | !~~~> Construct Ghimj = 1/(H*gam) - Jac0 |
---|
| 728 | #ifdef FULL_ALGEBRA |
---|
| 729 | !slim: CALL WCOPY(N*N,Jac0,1,Ghimj,1) |
---|
| 730 | !slim: CALL WSCAL(N*N,(-ONE),Ghimj,1) |
---|
| 731 | Ghimj = -Jac0 |
---|
| 732 | ghinv = ONE/(Direction*H*gam) |
---|
| 733 | DO i=1,N |
---|
| 734 | Ghimj(i,i) = Ghimj(i,i)+ghinv |
---|
| 735 | END DO |
---|
| 736 | #else |
---|
| 737 | !slim: CALL WCOPY(LU_NONZERO,Jac0,1,Ghimj,1) |
---|
| 738 | !slim: CALL WSCAL(LU_NONZERO,(-ONE),Ghimj,1) |
---|
| 739 | Ghimj(1:LU_NONZERO) = -Jac0(1:LU_NONZERO) |
---|
| 740 | ghinv = ONE/(Direction*H*gam) |
---|
| 741 | DO i=1,N |
---|
| 742 | Ghimj(LU_DIAG(i)) = Ghimj(LU_DIAG(i))+ghinv |
---|
| 743 | END DO |
---|
| 744 | #endif |
---|
| 745 | !~~~> Compute LU decomposition |
---|
| 746 | CALL ros_Decomp( Ghimj, Pivot, ISING ) |
---|
| 747 | IF (ISING == 0) THEN |
---|
| 748 | !~~~> If successful done |
---|
| 749 | Singular = .FALSE. |
---|
| 750 | ELSE ! ISING .ne. 0 |
---|
| 751 | !~~~> If unsuccessful half the step size; if 5 consecutive fails then return |
---|
| 752 | ISTATUS(Nsng) = ISTATUS(Nsng) + 1 |
---|
| 753 | Nconsecutive = Nconsecutive+1 |
---|
| 754 | Singular = .TRUE. |
---|
| 755 | PRINT*,'Warning: LU Decomposition returned ISING = ',ISING |
---|
| 756 | IF (Nconsecutive <= 5) THEN ! Less than 5 consecutive failed decompositions |
---|
| 757 | H = H*HALF |
---|
| 758 | ELSE ! More than 5 consecutive failed decompositions |
---|
| 759 | RETURN |
---|
| 760 | END IF ! Nconsecutive |
---|
| 761 | END IF ! ISING |
---|
| 762 | |
---|
| 763 | END DO ! WHILE Singular |
---|
| 764 | |
---|
| 765 | END SUBROUTINE ros_PrepareMatrix |
---|
| 766 | |
---|
| 767 | |
---|
| 768 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 769 | SUBROUTINE ros_Decomp( A, Pivot, ISING ) |
---|
| 770 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 771 | ! Template for the LU decomposition |
---|
| 772 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 773 | IMPLICIT NONE |
---|
| 774 | !~~~> Inout variables |
---|
| 775 | #ifdef FULL_ALGEBRA |
---|
| 776 | KPP_REAL, INTENT(INOUT) :: A(N,N) |
---|
| 777 | #else |
---|
| 778 | KPP_REAL, INTENT(INOUT) :: A(LU_NONZERO) |
---|
| 779 | #endif |
---|
| 780 | !~~~> Output variables |
---|
| 781 | INTEGER, INTENT(OUT) :: Pivot(N), ISING |
---|
| 782 | |
---|
| 783 | #ifdef FULL_ALGEBRA |
---|
| 784 | CALL DGETRF( N, N, A, N, Pivot, ISING ) |
---|
| 785 | #else |
---|
| 786 | CALL KppDecomp ( A, ISING ) |
---|
| 787 | Pivot(1) = 1 |
---|
| 788 | #endif |
---|
| 789 | ISTATUS(Ndec) = ISTATUS(Ndec) + 1 |
---|
| 790 | |
---|
| 791 | END SUBROUTINE ros_Decomp |
---|
| 792 | |
---|
| 793 | |
---|
| 794 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 795 | SUBROUTINE ros_Solve( A, Pivot, b ) |
---|
| 796 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 797 | ! Template for the forward/backward substitution (using pre-computed LU decomposition) |
---|
| 798 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 799 | IMPLICIT NONE |
---|
| 800 | !~~~> Input variables |
---|
| 801 | #ifdef FULL_ALGEBRA |
---|
| 802 | KPP_REAL, INTENT(IN) :: A(N,N) |
---|
| 803 | INTEGER :: ISING |
---|
| 804 | #else |
---|
| 805 | KPP_REAL, INTENT(IN) :: A(LU_NONZERO) |
---|
| 806 | #endif |
---|
| 807 | INTEGER, INTENT(IN) :: Pivot(N) |
---|
| 808 | !~~~> InOut variables |
---|
| 809 | KPP_REAL, INTENT(INOUT) :: b(N) |
---|
| 810 | |
---|
| 811 | #ifdef FULL_ALGEBRA |
---|
| 812 | CALL DGETRS( 'N', N , 1, A, N, Pivot, b, N, ISING ) |
---|
| 813 | IF ( Info < 0 ) THEN |
---|
| 814 | PRINT*,"Error in DGETRS. ISING=",ISING |
---|
| 815 | END IF |
---|
| 816 | #else |
---|
| 817 | CALL KppSolve( A, b ) |
---|
| 818 | #endif |
---|
| 819 | |
---|
| 820 | ISTATUS(Nsol) = ISTATUS(Nsol) + 1 |
---|
| 821 | |
---|
| 822 | END SUBROUTINE ros_Solve |
---|
| 823 | |
---|
| 824 | |
---|
| 825 | |
---|
| 826 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 827 | SUBROUTINE Ros2 |
---|
| 828 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 829 | ! --- AN L-STABLE METHOD, 2 stages, order 2 |
---|
| 830 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 831 | |
---|
| 832 | IMPLICIT NONE |
---|
| 833 | DOUBLE PRECISION g |
---|
| 834 | |
---|
| 835 | g = 1.0_dp + 1.0_dp/SQRT(2.0_dp) |
---|
| 836 | rosMethod = RS2 |
---|
| 837 | !~~~> Name of the method |
---|
| 838 | ros_Name = 'ROS-2' |
---|
| 839 | !~~~> Number of stages |
---|
| 840 | ros_S = 2 |
---|
| 841 | |
---|
| 842 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 843 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 844 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 845 | ! The general mapping formula is: |
---|
| 846 | ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 847 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 848 | |
---|
| 849 | ros_A(1) = (1.0_dp)/g |
---|
| 850 | ros_C(1) = (-2.0_dp)/g |
---|
| 851 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 852 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 853 | ros_NewF(1) = .TRUE. |
---|
| 854 | ros_NewF(2) = .TRUE. |
---|
| 855 | !~~~> M_i = Coefficients for new step solution |
---|
| 856 | ros_M(1)= (3.0_dp)/(2.0_dp*g) |
---|
| 857 | ros_M(2)= (1.0_dp)/(2.0_dp*g) |
---|
| 858 | ! E_i = Coefficients for error estimator |
---|
| 859 | ros_E(1) = 1.0_dp/(2.0_dp*g) |
---|
| 860 | ros_E(2) = 1.0_dp/(2.0_dp*g) |
---|
| 861 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 862 | ! main and the embedded scheme orders plus one |
---|
| 863 | ros_ELO = 2.0_dp |
---|
| 864 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 865 | ros_Alpha(1) = 0.0_dp |
---|
| 866 | ros_Alpha(2) = 1.0_dp |
---|
| 867 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 868 | ros_Gamma(1) = g |
---|
| 869 | ros_Gamma(2) =-g |
---|
| 870 | |
---|
| 871 | END SUBROUTINE Ros2 |
---|
| 872 | |
---|
| 873 | |
---|
| 874 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 875 | SUBROUTINE Ros3 |
---|
| 876 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 877 | ! --- AN L-STABLE METHOD, 3 stages, order 3, 2 function evaluations |
---|
| 878 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 879 | |
---|
| 880 | IMPLICIT NONE |
---|
| 881 | rosMethod = RS3 |
---|
| 882 | !~~~> Name of the method |
---|
| 883 | ros_Name = 'ROS-3' |
---|
| 884 | !~~~> Number of stages |
---|
| 885 | ros_S = 3 |
---|
| 886 | |
---|
| 887 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 888 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 889 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 890 | ! The general mapping formula is: |
---|
| 891 | ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 892 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 893 | |
---|
| 894 | ros_A(1)= 1.0_dp |
---|
| 895 | ros_A(2)= 1.0_dp |
---|
| 896 | ros_A(3)= 0.0_dp |
---|
| 897 | |
---|
| 898 | ros_C(1) = -0.10156171083877702091975600115545E+01_dp |
---|
| 899 | ros_C(2) = 0.40759956452537699824805835358067E+01_dp |
---|
| 900 | ros_C(3) = 0.92076794298330791242156818474003E+01_dp |
---|
| 901 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 902 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 903 | ros_NewF(1) = .TRUE. |
---|
| 904 | ros_NewF(2) = .TRUE. |
---|
| 905 | ros_NewF(3) = .FALSE. |
---|
| 906 | !~~~> M_i = Coefficients for new step solution |
---|
| 907 | ros_M(1) = 0.1E+01_dp |
---|
| 908 | ros_M(2) = 0.61697947043828245592553615689730E+01_dp |
---|
| 909 | ros_M(3) = -0.42772256543218573326238373806514_dp |
---|
| 910 | ! E_i = Coefficients for error estimator |
---|
| 911 | ros_E(1) = 0.5_dp |
---|
| 912 | ros_E(2) = -0.29079558716805469821718236208017E+01_dp |
---|
| 913 | ros_E(3) = 0.22354069897811569627360909276199_dp |
---|
| 914 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 915 | ! main and the embedded scheme orders plus 1 |
---|
| 916 | ros_ELO = 3.0_dp |
---|
| 917 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 918 | ros_Alpha(1)= 0.0_dp |
---|
| 919 | ros_Alpha(2)= 0.43586652150845899941601945119356_dp |
---|
| 920 | ros_Alpha(3)= 0.43586652150845899941601945119356_dp |
---|
| 921 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 922 | ros_Gamma(1)= 0.43586652150845899941601945119356_dp |
---|
| 923 | ros_Gamma(2)= 0.24291996454816804366592249683314_dp |
---|
| 924 | ros_Gamma(3)= 0.21851380027664058511513169485832E+01_dp |
---|
| 925 | |
---|
| 926 | END SUBROUTINE Ros3 |
---|
| 927 | |
---|
| 928 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 929 | |
---|
| 930 | |
---|
| 931 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 932 | SUBROUTINE Ros4 |
---|
| 933 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 934 | ! L-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 4 STAGES |
---|
| 935 | ! L-STABLE EMBEDDED ROSENBROCK METHOD OF ORDER 3 |
---|
| 936 | ! |
---|
| 937 | ! E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL |
---|
| 938 | ! EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS. |
---|
| 939 | ! SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS, |
---|
| 940 | ! SPRINGER-VERLAG (1990) |
---|
| 941 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 942 | |
---|
| 943 | IMPLICIT NONE |
---|
| 944 | |
---|
| 945 | rosMethod = RS4 |
---|
| 946 | !~~~> Name of the method |
---|
| 947 | ros_Name = 'ROS-4' |
---|
| 948 | !~~~> Number of stages |
---|
| 949 | ros_S = 4 |
---|
| 950 | |
---|
| 951 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 952 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 953 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 954 | ! The general mapping formula is: |
---|
| 955 | ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 956 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 957 | |
---|
| 958 | ros_A(1) = 0.2000000000000000E+01_dp |
---|
| 959 | ros_A(2) = 0.1867943637803922E+01_dp |
---|
| 960 | ros_A(3) = 0.2344449711399156_dp |
---|
| 961 | ros_A(4) = ros_A(2) |
---|
| 962 | ros_A(5) = ros_A(3) |
---|
| 963 | ros_A(6) = 0.0_dp |
---|
| 964 | |
---|
| 965 | ros_C(1) =-0.7137615036412310E+01_dp |
---|
| 966 | ros_C(2) = 0.2580708087951457E+01_dp |
---|
| 967 | ros_C(3) = 0.6515950076447975_dp |
---|
| 968 | ros_C(4) =-0.2137148994382534E+01_dp |
---|
| 969 | ros_C(5) =-0.3214669691237626_dp |
---|
| 970 | ros_C(6) =-0.6949742501781779_dp |
---|
| 971 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 972 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 973 | ros_NewF(1) = .TRUE. |
---|
| 974 | ros_NewF(2) = .TRUE. |
---|
| 975 | ros_NewF(3) = .TRUE. |
---|
| 976 | ros_NewF(4) = .FALSE. |
---|
| 977 | !~~~> M_i = Coefficients for new step solution |
---|
| 978 | ros_M(1) = 0.2255570073418735E+01_dp |
---|
| 979 | ros_M(2) = 0.2870493262186792_dp |
---|
| 980 | ros_M(3) = 0.4353179431840180_dp |
---|
| 981 | ros_M(4) = 0.1093502252409163E+01_dp |
---|
| 982 | !~~~> E_i = Coefficients for error estimator |
---|
| 983 | ros_E(1) =-0.2815431932141155_dp |
---|
| 984 | ros_E(2) =-0.7276199124938920E-01_dp |
---|
| 985 | ros_E(3) =-0.1082196201495311_dp |
---|
| 986 | ros_E(4) =-0.1093502252409163E+01_dp |
---|
| 987 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 988 | ! main and the embedded scheme orders plus 1 |
---|
| 989 | ros_ELO = 4.0_dp |
---|
| 990 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 991 | ros_Alpha(1) = 0.0_dp |
---|
| 992 | ros_Alpha(2) = 0.1145640000000000E+01_dp |
---|
| 993 | ros_Alpha(3) = 0.6552168638155900_dp |
---|
| 994 | ros_Alpha(4) = ros_Alpha(3) |
---|
| 995 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 996 | ros_Gamma(1) = 0.5728200000000000_dp |
---|
| 997 | ros_Gamma(2) =-0.1769193891319233E+01_dp |
---|
| 998 | ros_Gamma(3) = 0.7592633437920482_dp |
---|
| 999 | ros_Gamma(4) =-0.1049021087100450_dp |
---|
| 1000 | |
---|
| 1001 | END SUBROUTINE Ros4 |
---|
| 1002 | |
---|
| 1003 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1004 | SUBROUTINE Rodas3 |
---|
| 1005 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1006 | ! --- A STIFFLY-STABLE METHOD, 4 stages, order 3 |
---|
| 1007 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1008 | |
---|
| 1009 | IMPLICIT NONE |
---|
| 1010 | |
---|
| 1011 | rosMethod = RD3 |
---|
| 1012 | !~~~> Name of the method |
---|
| 1013 | ros_Name = 'RODAS-3' |
---|
| 1014 | !~~~> Number of stages |
---|
| 1015 | ros_S = 4 |
---|
| 1016 | |
---|
| 1017 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 1018 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 1019 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 1020 | ! The general mapping formula is: |
---|
| 1021 | ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 1022 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 1023 | |
---|
| 1024 | ros_A(1) = 0.0_dp |
---|
| 1025 | ros_A(2) = 2.0_dp |
---|
| 1026 | ros_A(3) = 0.0_dp |
---|
| 1027 | ros_A(4) = 2.0_dp |
---|
| 1028 | ros_A(5) = 0.0_dp |
---|
| 1029 | ros_A(6) = 1.0_dp |
---|
| 1030 | |
---|
| 1031 | ros_C(1) = 4.0_dp |
---|
| 1032 | ros_C(2) = 1.0_dp |
---|
| 1033 | ros_C(3) =-1.0_dp |
---|
| 1034 | ros_C(4) = 1.0_dp |
---|
| 1035 | ros_C(5) =-1.0_dp |
---|
| 1036 | ros_C(6) =-(8.0_dp/3.0_dp) |
---|
| 1037 | |
---|
| 1038 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 1039 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 1040 | ros_NewF(1) = .TRUE. |
---|
| 1041 | ros_NewF(2) = .FALSE. |
---|
| 1042 | ros_NewF(3) = .TRUE. |
---|
| 1043 | ros_NewF(4) = .TRUE. |
---|
| 1044 | !~~~> M_i = Coefficients for new step solution |
---|
| 1045 | ros_M(1) = 2.0_dp |
---|
| 1046 | ros_M(2) = 0.0_dp |
---|
| 1047 | ros_M(3) = 1.0_dp |
---|
| 1048 | ros_M(4) = 1.0_dp |
---|
| 1049 | !~~~> E_i = Coefficients for error estimator |
---|
| 1050 | ros_E(1) = 0.0_dp |
---|
| 1051 | ros_E(2) = 0.0_dp |
---|
| 1052 | ros_E(3) = 0.0_dp |
---|
| 1053 | ros_E(4) = 1.0_dp |
---|
| 1054 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 1055 | ! main and the embedded scheme orders plus 1 |
---|
| 1056 | ros_ELO = 3.0_dp |
---|
| 1057 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 1058 | ros_Alpha(1) = 0.0_dp |
---|
| 1059 | ros_Alpha(2) = 0.0_dp |
---|
| 1060 | ros_Alpha(3) = 1.0_dp |
---|
| 1061 | ros_Alpha(4) = 1.0_dp |
---|
| 1062 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 1063 | ros_Gamma(1) = 0.5_dp |
---|
| 1064 | ros_Gamma(2) = 1.5_dp |
---|
| 1065 | ros_Gamma(3) = 0.0_dp |
---|
| 1066 | ros_Gamma(4) = 0.0_dp |
---|
| 1067 | |
---|
| 1068 | END SUBROUTINE Rodas3 |
---|
| 1069 | |
---|
| 1070 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1071 | SUBROUTINE Rodas4 |
---|
| 1072 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1073 | ! STIFFLY-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 6 STAGES |
---|
| 1074 | ! |
---|
| 1075 | ! E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL |
---|
| 1076 | ! EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS. |
---|
| 1077 | ! SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS, |
---|
| 1078 | ! SPRINGER-VERLAG (1996) |
---|
| 1079 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1080 | |
---|
| 1081 | IMPLICIT NONE |
---|
| 1082 | |
---|
| 1083 | rosMethod = RD4 |
---|
| 1084 | !~~~> Name of the method |
---|
| 1085 | ros_Name = 'RODAS-4' |
---|
| 1086 | !~~~> Number of stages |
---|
| 1087 | ros_S = 6 |
---|
| 1088 | |
---|
| 1089 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 1090 | ros_Alpha(1) = 0.000_dp |
---|
| 1091 | ros_Alpha(2) = 0.386_dp |
---|
| 1092 | ros_Alpha(3) = 0.210_dp |
---|
| 1093 | ros_Alpha(4) = 0.630_dp |
---|
| 1094 | ros_Alpha(5) = 1.000_dp |
---|
| 1095 | ros_Alpha(6) = 1.000_dp |
---|
| 1096 | |
---|
| 1097 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 1098 | ros_Gamma(1) = 0.2500000000000000_dp |
---|
| 1099 | ros_Gamma(2) =-0.1043000000000000_dp |
---|
| 1100 | ros_Gamma(3) = 0.1035000000000000_dp |
---|
| 1101 | ros_Gamma(4) =-0.3620000000000023E-01_dp |
---|
| 1102 | ros_Gamma(5) = 0.0_dp |
---|
| 1103 | ros_Gamma(6) = 0.0_dp |
---|
| 1104 | |
---|
| 1105 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 1106 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 1107 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 1108 | ! The general mapping formula is: A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 1109 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 1110 | |
---|
| 1111 | ros_A(1) = 0.1544000000000000E+01_dp |
---|
| 1112 | ros_A(2) = 0.9466785280815826_dp |
---|
| 1113 | ros_A(3) = 0.2557011698983284_dp |
---|
| 1114 | ros_A(4) = 0.3314825187068521E+01_dp |
---|
| 1115 | ros_A(5) = 0.2896124015972201E+01_dp |
---|
| 1116 | ros_A(6) = 0.9986419139977817_dp |
---|
| 1117 | ros_A(7) = 0.1221224509226641E+01_dp |
---|
| 1118 | ros_A(8) = 0.6019134481288629E+01_dp |
---|
| 1119 | ros_A(9) = 0.1253708332932087E+02_dp |
---|
| 1120 | ros_A(10) =-0.6878860361058950_dp |
---|
| 1121 | ros_A(11) = ros_A(7) |
---|
| 1122 | ros_A(12) = ros_A(8) |
---|
| 1123 | ros_A(13) = ros_A(9) |
---|
| 1124 | ros_A(14) = ros_A(10) |
---|
| 1125 | ros_A(15) = 1.0_dp |
---|
| 1126 | |
---|
| 1127 | ros_C(1) =-0.5668800000000000E+01_dp |
---|
| 1128 | ros_C(2) =-0.2430093356833875E+01_dp |
---|
| 1129 | ros_C(3) =-0.2063599157091915_dp |
---|
| 1130 | ros_C(4) =-0.1073529058151375_dp |
---|
| 1131 | ros_C(5) =-0.9594562251023355E+01_dp |
---|
| 1132 | ros_C(6) =-0.2047028614809616E+02_dp |
---|
| 1133 | ros_C(7) = 0.7496443313967647E+01_dp |
---|
| 1134 | ros_C(8) =-0.1024680431464352E+02_dp |
---|
| 1135 | ros_C(9) =-0.3399990352819905E+02_dp |
---|
| 1136 | ros_C(10) = 0.1170890893206160E+02_dp |
---|
| 1137 | ros_C(11) = 0.8083246795921522E+01_dp |
---|
| 1138 | ros_C(12) =-0.7981132988064893E+01_dp |
---|
| 1139 | ros_C(13) =-0.3152159432874371E+02_dp |
---|
| 1140 | ros_C(14) = 0.1631930543123136E+02_dp |
---|
| 1141 | ros_C(15) =-0.6058818238834054E+01_dp |
---|
| 1142 | |
---|
| 1143 | !~~~> M_i = Coefficients for new step solution |
---|
| 1144 | ros_M(1) = ros_A(7) |
---|
| 1145 | ros_M(2) = ros_A(8) |
---|
| 1146 | ros_M(3) = ros_A(9) |
---|
| 1147 | ros_M(4) = ros_A(10) |
---|
| 1148 | ros_M(5) = 1.0_dp |
---|
| 1149 | ros_M(6) = 1.0_dp |
---|
| 1150 | |
---|
| 1151 | !~~~> E_i = Coefficients for error estimator |
---|
| 1152 | ros_E(1) = 0.0_dp |
---|
| 1153 | ros_E(2) = 0.0_dp |
---|
| 1154 | ros_E(3) = 0.0_dp |
---|
| 1155 | ros_E(4) = 0.0_dp |
---|
| 1156 | ros_E(5) = 0.0_dp |
---|
| 1157 | ros_E(6) = 1.0_dp |
---|
| 1158 | |
---|
| 1159 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 1160 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 1161 | ros_NewF(1) = .TRUE. |
---|
| 1162 | ros_NewF(2) = .TRUE. |
---|
| 1163 | ros_NewF(3) = .TRUE. |
---|
| 1164 | ros_NewF(4) = .TRUE. |
---|
| 1165 | ros_NewF(5) = .TRUE. |
---|
| 1166 | ros_NewF(6) = .TRUE. |
---|
| 1167 | |
---|
| 1168 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 1169 | ! main and the embedded scheme orders plus 1 |
---|
| 1170 | ros_ELO = 4.0_dp |
---|
| 1171 | |
---|
| 1172 | END SUBROUTINE Rodas4 |
---|
| 1173 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1174 | SUBROUTINE Rang3 |
---|
| 1175 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1176 | ! STIFFLY-STABLE W METHOD OF ORDER 3, WITH 4 STAGES |
---|
| 1177 | ! |
---|
| 1178 | ! J. RANG and L. ANGERMANN |
---|
| 1179 | ! NEW ROSENBROCK W-METHODS OF ORDER 3 |
---|
| 1180 | ! FOR PARTIAL DIFFERENTIAL ALGEBRAIC |
---|
| 1181 | ! EQUATIONS OF INDEX 1 |
---|
| 1182 | ! BIT Numerical Mathematics (2005) 45: 761-787 |
---|
| 1183 | ! DOI: 10.1007/s10543-005-0035-y |
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| 1184 | ! Table 4.1-4.2 |
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| 1185 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 1186 | |
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| 1187 | IMPLICIT NONE |
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| 1188 | |
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| 1189 | rosMethod = RG3 |
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| 1190 | !~~~> Name of the method |
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| 1191 | ros_Name = 'RANG-3' |
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| 1192 | !~~~> Number of stages |
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| 1193 | ros_S = 4 |
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| 1194 | |
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| 1195 | ros_A(1) = 5.09052051067020d+00; |
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| 1196 | ros_A(2) = 5.09052051067020d+00; |
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| 1197 | ros_A(3) = 0.0d0; |
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| 1198 | ros_A(4) = 4.97628111010787d+00; |
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| 1199 | ros_A(5) = 2.77268164715849d-02; |
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| 1200 | ros_A(6) = 2.29428036027904d-01; |
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| 1201 | |
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| 1202 | ros_C(1) = -1.16790812312283d+01; |
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| 1203 | ros_C(2) = -1.64057326467367d+01; |
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| 1204 | ros_C(3) = -2.77268164715850d-01; |
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| 1205 | ros_C(4) = -8.38103960500476d+00; |
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| 1206 | ros_C(5) = -8.48328409199343d-01; |
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| 1207 | ros_C(6) = 2.87009860433106d-01; |
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| 1208 | |
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| 1209 | ros_M(1) = 5.22582761233094d+00; |
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| 1210 | ros_M(2) = -5.56971148154165d-01; |
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| 1211 | ros_M(3) = 3.57979469353645d-01; |
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| 1212 | ros_M(4) = 1.72337398521064d+00; |
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| 1213 | |
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| 1214 | ros_E(1) = -5.16845212784040d+00; |
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| 1215 | ros_E(2) = -1.26351942603842d+00; |
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| 1216 | ros_E(3) = -1.11022302462516d-16; |
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| 1217 | ros_E(4) = 2.22044604925031d-16; |
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| 1218 | |
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| 1219 | ros_Alpha(1) = 0.0d00; |
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| 1220 | ros_Alpha(2) = 2.21878746765329d+00; |
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| 1221 | ros_Alpha(3) = 2.21878746765329d+00; |
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| 1222 | ros_Alpha(4) = 1.55392337535788d+00; |
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| 1223 | |
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| 1224 | ros_Gamma(1) = 4.35866521508459d-01; |
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| 1225 | ros_Gamma(2) = -1.78292094614483d+00; |
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| 1226 | ros_Gamma(3) = -2.46541900496934d+00; |
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| 1227 | ros_Gamma(4) = -8.05529997906370d-01; |
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| 1228 | |
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| 1229 | |
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| 1230 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
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| 1231 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
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| 1232 | ros_NewF(1) = .TRUE. |
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| 1233 | ros_NewF(2) = .TRUE. |
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| 1234 | ros_NewF(3) = .TRUE. |
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| 1235 | ros_NewF(4) = .TRUE. |
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| 1236 | |
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| 1237 | !~~~> ros_ELO = estimator of local order - the minimum between the |
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| 1238 | ! main and the embedded scheme orders plus 1 |
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| 1239 | ros_ELO = 3.0_dp |
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| 1240 | |
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| 1241 | END SUBROUTINE Rang3 |
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| 1242 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 1243 | |
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| 1244 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 1245 | ! End of the set of internal Rosenbrock subroutines |
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| 1246 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 1247 | END SUBROUTINE Rosenbrock |
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| 1248 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 1249 | |
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| 1250 | |
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| 1251 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 1252 | SUBROUTINE FunTemplate( T, Y, Ydot ) |
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| 1253 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 1254 | ! Template for the ODE function call. |
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| 1255 | ! Updates the rate coefficients (and possibly the fixed species) at each call |
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| 1256 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 1257 | USE KPP_ROOT_Parameters, ONLY: NVAR, LU_NONZERO |
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| 1258 | USE KPP_ROOT_Global, ONLY: FIX, RCONST, TIME |
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| 1259 | USE KPP_ROOT_Function, ONLY: Fun |
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| 1260 | USE KPP_ROOT_Rates, ONLY: Update_SUN, Update_RCONST |
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| 1261 | !~~~> Input variables |
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| 1262 | KPP_REAL :: T, Y(NVAR) |
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| 1263 | !~~~> Output variables |
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| 1264 | KPP_REAL :: Ydot(NVAR) |
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| 1265 | !~~~> Local variables |
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| 1266 | KPP_REAL :: Told |
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| 1267 | |
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| 1268 | Told = TIME |
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| 1269 | TIME = T |
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| 1270 | CALL Update_SUN() |
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| 1271 | CALL Update_RCONST() |
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| 1272 | CALL Fun( Y, FIX, RCONST, Ydot ) |
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| 1273 | TIME = Told |
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| 1274 | |
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| 1275 | END SUBROUTINE FunTemplate |
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| 1276 | |
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| 1277 | |
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| 1278 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 1279 | SUBROUTINE JacTemplate( T, Y, Jcb ) |
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| 1280 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 1281 | ! Template for the ODE Jacobian call. |
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| 1282 | ! Updates the rate coefficients (and possibly the fixed species) at each call |
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| 1283 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 1284 | USE KPP_ROOT_Parameters, ONLY: NVAR, LU_NONZERO |
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| 1285 | USE KPP_ROOT_Global, ONLY: FIX, RCONST, TIME |
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| 1286 | USE KPP_ROOT_Jacobian, ONLY: Jac_SP, LU_IROW, LU_ICOL |
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| 1287 | USE KPP_ROOT_LinearAlgebra |
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| 1288 | USE KPP_ROOT_Rates, ONLY: Update_SUN, Update_RCONST |
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| 1289 | !~~~> Input variables |
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| 1290 | KPP_REAL :: T, Y(NVAR) |
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| 1291 | !~~~> Output variables |
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| 1292 | #ifdef FULL_ALGEBRA |
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| 1293 | KPP_REAL :: JV(LU_NONZERO), Jcb(NVAR,NVAR) |
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| 1294 | #else |
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| 1295 | KPP_REAL :: Jcb(LU_NONZERO) |
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| 1296 | #endif |
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| 1297 | !~~~> Local variables |
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| 1298 | KPP_REAL :: Told |
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| 1299 | #ifdef FULL_ALGEBRA |
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| 1300 | INTEGER :: i, j |
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| 1301 | #endif |
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| 1302 | |
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| 1303 | Told = TIME |
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| 1304 | TIME = T |
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| 1305 | CALL Update_SUN() |
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| 1306 | CALL Update_RCONST() |
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| 1307 | #ifdef FULL_ALGEBRA |
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| 1308 | CALL Jac_SP(Y, FIX, RCONST, JV) |
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| 1309 | DO j=1,NVAR |
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| 1310 | DO i=1,NVAR |
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| 1311 | Jcb(i,j) = 0.0_dp |
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| 1312 | END DO |
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| 1313 | END DO |
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| 1314 | DO i=1,LU_NONZERO |
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| 1315 | Jcb(LU_IROW(i),LU_ICOL(i)) = JV(i) |
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| 1316 | END DO |
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| 1317 | #else |
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| 1318 | CALL Jac_SP( Y, FIX, RCONST, Jcb ) |
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| 1319 | #endif |
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| 1320 | TIME = Told |
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| 1321 | |
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| 1322 | END SUBROUTINE JacTemplate |
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| 1323 | |
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| 1324 | END MODULE KPP_ROOT_Integrator |
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| 1325 | |
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| 1326 | |
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| 1327 | |
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| 1328 | |
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