[2696] | 1 | SUBROUTINE Rosenbrock(Y,Tstart,Tend, & |
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| 2 | AbsTol,RelTol, & |
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| 3 | RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR) |
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| 4 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 5 | ! |
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| 6 | ! Solves the system y'=F(t,y) using a Rosenbrock method defined by: |
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| 7 | ! |
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| 8 | ! G = 1/(H*gamma(1)) - Jac(t0,Y0) |
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| 9 | ! T_i = t0 + Alpha(i)*H |
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| 10 | ! Y_i = Y0 + \sum_{j=1}^{i-1} A(i,j)*K_j |
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| 11 | ! G * K_i = Fun( T_i, Y_i ) + \sum_{j=1}^S C(i,j)/H * K_j + |
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| 12 | ! gamma(i)*dF/dT(t0, Y0) |
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| 13 | ! Y1 = Y0 + \sum_{j=1}^S M(j)*K_j |
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| 14 | ! |
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| 15 | ! For details on Rosenbrock methods and their implementation consult: |
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| 16 | ! E. Hairer and G. Wanner |
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| 17 | ! "Solving ODEs II. Stiff and differential-algebraic problems". |
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| 18 | ! Springer series in computational mathematics, Springer-Verlag, 1996. |
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| 19 | ! The codes contained in the book inspired this implementation. |
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| 20 | ! |
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| 21 | ! (C) Adrian Sandu, August 2004 |
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| 22 | ! Virginia Polytechnic Institute and State University |
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| 23 | ! Contact: sandu@cs.vt.edu |
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| 24 | ! This implementation is part of KPP - the Kinetic PreProcessor |
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| 25 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 26 | ! |
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| 27 | !~~~> INPUT ARGUMENTS: |
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| 28 | ! |
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| 29 | !- Y(NVAR) = vector of initial conditions (at T=Tstart) |
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| 30 | !- [Tstart,Tend] = time range of integration |
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| 31 | ! (if Tstart>Tend the integration is performed backwards in time) |
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| 32 | !- RelTol, AbsTol = user precribed accuracy |
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| 33 | !- SUBROUTINE Fun( T, Y, Ydot ) = ODE function, |
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| 34 | ! returns Ydot = Y' = F(T,Y) |
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| 35 | !- SUBROUTINE Jac( T, Y, Jcb ) = Jacobian of the ODE function, |
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| 36 | ! returns Jcb = dFun/dY |
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| 37 | !- ICNTRL(1:20) = integer inputs parameters |
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| 38 | !- RCNTRL(1:20) = real inputs parameters |
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| 39 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 40 | ! |
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| 41 | !~~~> OUTPUT ARGUMENTS: |
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| 42 | ! |
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| 43 | !- Y(NVAR) -> vector of final states (at T->Tend) |
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| 44 | !- ISTATUS(1:20) -> integer output parameters |
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| 45 | !- RSTATUS(1:20) -> real output parameters |
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| 46 | !- IERR -> job status upon return |
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| 47 | ! success (positive value) or |
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| 48 | ! failure (negative value) |
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| 49 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 50 | ! |
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| 51 | !~~~> INPUT PARAMETERS: |
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| 52 | ! |
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| 53 | ! Note: For input parameters equal to zero the default values of the |
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| 54 | ! corresponding variables are used. |
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| 55 | ! |
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| 56 | ! ICNTRL(1) = 1: F = F(y) Independent of T (AUTONOMOUS) |
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| 57 | ! = 0: F = F(t,y) Depends on T (NON-AUTONOMOUS) |
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| 58 | ! |
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| 59 | ! ICNTRL(2) = 0: AbsTol, RelTol are NVAR-dimensional vectors |
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| 60 | ! = 1: AbsTol, RelTol are scalars |
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| 61 | ! |
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| 62 | ! ICNTRL(3) -> selection of a particular Rosenbrock method |
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| 63 | ! = 0 : default method is Rodas3 |
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| 64 | ! = 1 : method is Ros2 |
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| 65 | ! = 2 : method is Ros3 |
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| 66 | ! = 3 : method is Ros4 |
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| 67 | ! = 4 : method is Rodas3 |
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| 68 | ! = 5: method is Rodas4 |
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| 69 | ! |
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| 70 | ! ICNTRL(4) -> maximum number of integration steps |
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| 71 | ! For ICNTRL(4)=0) the default value of 100000 is used |
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| 72 | ! |
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| 73 | ! RCNTRL(1) -> Hmin, lower bound for the integration step size |
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| 74 | ! It is strongly recommended to keep Hmin = ZERO |
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| 75 | ! RCNTRL(2) -> Hmax, upper bound for the integration step size |
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| 76 | ! RCNTRL(3) -> Hstart, starting value for the integration step size |
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| 77 | ! |
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| 78 | ! RCNTRL(4) -> FacMin, lower bound on step decrease factor (default=0.2) |
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| 79 | ! RCNTRL(5) -> FacMax, upper bound on step increase factor (default=6) |
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| 80 | ! RCNTRL(6) -> FacRej, step decrease factor after multiple rejections |
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| 81 | ! (default=0.1) |
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| 82 | ! RCNTRL(7) -> FacSafe, by which the new step is slightly smaller |
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| 83 | ! than the predicted value (default=0.9) |
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| 84 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 85 | ! |
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| 86 | !~~~> OUTPUT PARAMETERS: |
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| 87 | ! |
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| 88 | ! Note: each call to Rosenbrock adds the current no. of fcn calls |
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| 89 | ! to previous value of ISTATUS(1), and similar for the other params. |
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| 90 | ! Set ISTATUS(1:20) = 0 before call to avoid this accumulation. |
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| 91 | ! |
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| 92 | ! ISTATUS(1) = No. of function calls |
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| 93 | ! ISTATUS(2) = No. of jacobian calls |
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| 94 | ! ISTATUS(3) = No. of steps |
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| 95 | ! ISTATUS(4) = No. of accepted steps |
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| 96 | ! ISTATUS(5) = No. of rejected steps (except at the beginning) |
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| 97 | ! ISTATUS(6) = No. of LU decompositions |
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| 98 | ! ISTATUS(7) = No. of forward/backward substitutions |
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| 99 | ! ISTATUS(8) = No. of singular matrix decompositions |
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| 100 | ! |
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| 101 | ! RSTATUS(1) -> Texit, the time corresponding to the |
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| 102 | ! computed Y upon return |
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| 103 | ! RSTATUS(2) -> Hexit, last accepted step before exit |
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| 104 | ! For multiple restarts, use Hexit as Hstart in the following run |
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| 105 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 106 | |
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| 107 | IMPLICIT NONE |
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| 108 | |
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| 109 | !~~~> Arguments |
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| 110 | REAL(kind=dp), INTENT(INOUT) :: Y(:,:) |
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| 111 | REAL(kind=dp), INTENT(IN) :: Tstart,Tend |
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| 112 | REAL(kind=dp), INTENT(IN) :: AbsTol(NVAR),RelTol(NVAR) |
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| 113 | INTEGER, INTENT(IN) :: ICNTRL(20) |
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| 114 | REAL(kind=dp), INTENT(IN) :: RCNTRL(20) |
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| 115 | INTEGER, INTENT(INOUT) :: ISTATUS(20) |
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| 116 | REAL(kind=dp), INTENT(INOUT) :: RSTATUS(20) |
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| 117 | INTEGER, INTENT(OUT) :: IERR |
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| 118 | !~~~> The method parameters |
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| 119 | INTEGER, PARAMETER :: Smax = 6 |
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| 120 | INTEGER :: Method, ros_S |
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| 121 | REAL(kind=dp), DIMENSION(Smax) :: ros_M, ros_E, ros_Alpha, ros_Gamma |
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| 122 | REAL(kind=dp), DIMENSION(Smax*(Smax-1)/2) :: ros_A, ros_C |
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| 123 | REAL(kind=dp) :: ros_ELO |
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| 124 | LOGICAL, DIMENSION(Smax) :: ros_NewF |
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| 125 | CHARACTER(LEN=12) :: ros_Name |
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| 126 | !~~~> Local variables |
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| 127 | REAL(kind=dp) :: Roundoff, FacMin, FacMax, FacRej, FacSafe |
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| 128 | REAL(kind=dp) :: Hmin, Hmax, Hstart, Hexit |
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| 129 | REAL(kind=dp) :: Texit |
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| 130 | INTEGER :: i, UplimTol, Max_no_steps |
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| 131 | LOGICAL :: Autonomous, VectorTol |
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| 132 | !~~~> Parameters |
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| 133 | REAL(kind=dp), PARAMETER :: ZERO = 0.0_dp, ONE = 1.0_dp |
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| 134 | REAL(kind=dp), PARAMETER :: DeltaMin = 1.0E-5_dp |
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| 135 | |
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| 136 | !~~~> Initialize statistics |
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| 137 | Nfun = ISTATUS(ifun) |
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| 138 | Njac = ISTATUS(ijac) |
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| 139 | Nstp = ISTATUS(istp) |
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| 140 | Nacc = ISTATUS(iacc) |
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| 141 | Nrej = ISTATUS(irej) |
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| 142 | Ndec = ISTATUS(idec) |
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| 143 | Nsol = ISTATUS(isol) |
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| 144 | Nsng = ISTATUS(isng) |
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| 145 | |
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| 146 | !~~~> Autonomous or time dependent ODE. Default is time dependent. |
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| 147 | Autonomous = .NOT.(ICNTRL(1) == 0) |
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| 148 | |
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| 149 | !~~~> For Scalar tolerances (ICNTRL(2).NE.0) the code uses AbsTol(1) and RelTol(1) |
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| 150 | ! For Vector tolerances (ICNTRL(2) == 0) the code uses AbsTol(1:NVAR) and RelTol(1:NVAR) |
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| 151 | IF (ICNTRL(2) == 0) THEN |
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| 152 | VectorTol = .TRUE. |
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| 153 | UplimTol = NVAR |
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| 154 | ELSE |
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| 155 | VectorTol = .FALSE. |
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| 156 | UplimTol = 1 |
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| 157 | END IF |
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| 158 | |
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| 159 | !~~~> The particular Rosenbrock method chosen |
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| 160 | IF (ICNTRL(3) == 0) THEN |
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| 161 | Method = 4 |
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| 162 | ELSEIF ( (ICNTRL(3) >= 1).AND.(ICNTRL(3) <= 5) ) THEN |
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| 163 | Method = ICNTRL(3) |
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| 164 | ELSE |
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| 165 | PRINT * , 'User-selected Rosenbrock method: ICNTRL(3)=', Method |
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| 166 | CALL ros_ErrorMsg(-2,Tstart,ZERO,IERR) |
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| 167 | RETURN |
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| 168 | END IF |
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| 169 | |
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| 170 | !~~~> The maximum number of steps admitted |
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| 171 | IF (ICNTRL(4) == 0) THEN |
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| 172 | Max_no_steps = 100000 ! 200 |
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| 173 | ELSEIF (ICNTRL(4) > 0) THEN |
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| 174 | Max_no_steps=ICNTRL(4) |
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| 175 | ELSE |
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| 176 | PRINT * ,'User-selected max no. of steps: ICNTRL(4)=',ICNTRL(4) |
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| 177 | CALL ros_ErrorMsg(-1,Tstart,ZERO,IERR) |
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| 178 | RETURN |
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| 179 | END IF |
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| 180 | |
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| 181 | !~~~> Unit roundoff (1+Roundoff>1) |
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| 182 | Roundoff = WLAMCH('E') |
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| 183 | |
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| 184 | !~~~> Lower bound on the step size: (positive value) |
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| 185 | IF (RCNTRL(1) == ZERO) THEN |
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| 186 | Hmin = ZERO |
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| 187 | ELSEIF (RCNTRL(1) > ZERO) THEN |
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| 188 | Hmin = RCNTRL(1) |
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| 189 | ELSE |
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| 190 | PRINT * , 'User-selected Hmin: RCNTRL(1)=', RCNTRL(1) |
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| 191 | CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR) |
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| 192 | RETURN |
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| 193 | END IF |
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| 194 | !~~~> Upper bound on the step size: (positive value) |
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| 195 | IF (RCNTRL(2) == ZERO) THEN |
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| 196 | Hmax = ABS(Tend-Tstart) |
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| 197 | ELSEIF (RCNTRL(2) > ZERO) THEN |
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| 198 | Hmax = MIN(ABS(RCNTRL(2)),ABS(Tend-Tstart)) |
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| 199 | ELSE |
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| 200 | PRINT * , 'User-selected Hmax: RCNTRL(2)=', RCNTRL(2) |
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| 201 | CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR) |
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| 202 | RETURN |
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| 203 | END IF |
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| 204 | !~~~> Starting step size: (positive value) |
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| 205 | IF (RCNTRL(3) == ZERO) THEN |
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| 206 | Hstart = MAX(Hmin,DeltaMin) |
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| 207 | ELSEIF (RCNTRL(3) > ZERO) THEN |
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| 208 | Hstart = MIN(ABS(RCNTRL(3)),ABS(Tend-Tstart)) |
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| 209 | ELSE |
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| 210 | PRINT * , 'User-selected Hstart: RCNTRL(3)=', RCNTRL(3) |
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| 211 | CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR) |
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| 212 | RETURN |
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| 213 | END IF |
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| 214 | !~~~> Step size can be changed s.t. FacMin < Hnew/Hexit < FacMax |
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| 215 | IF (RCNTRL(4) == ZERO) THEN |
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| 216 | FacMin = 0.2_dp |
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| 217 | ELSEIF (RCNTRL(4) > ZERO) THEN |
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| 218 | FacMin = RCNTRL(4) |
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| 219 | ELSE |
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| 220 | PRINT * , 'User-selected FacMin: RCNTRL(4)=', RCNTRL(4) |
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| 221 | CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR) |
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| 222 | RETURN |
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| 223 | END IF |
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| 224 | IF (RCNTRL(5) == ZERO) THEN |
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| 225 | FacMax = 6.0_dp |
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| 226 | ELSEIF (RCNTRL(5) > ZERO) THEN |
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| 227 | FacMax = RCNTRL(5) |
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| 228 | ELSE |
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| 229 | PRINT * , 'User-selected FacMax: RCNTRL(5)=', RCNTRL(5) |
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| 230 | CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR) |
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| 231 | RETURN |
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| 232 | END IF |
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| 233 | !~~~> FacRej: Factor to decrease step after 2 succesive rejections |
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| 234 | IF (RCNTRL(6) == ZERO) THEN |
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| 235 | FacRej = 0.1_dp |
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| 236 | ELSEIF (RCNTRL(6) > ZERO) THEN |
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| 237 | FacRej = RCNTRL(6) |
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| 238 | ELSE |
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| 239 | PRINT * , 'User-selected FacRej: RCNTRL(6)=', RCNTRL(6) |
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| 240 | CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR) |
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| 241 | RETURN |
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| 242 | END IF |
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| 243 | !~~~> FacSafe: Safety Factor in the computation of new step size |
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| 244 | IF (RCNTRL(7) == ZERO) THEN |
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| 245 | FacSafe = 0.9_dp |
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| 246 | ELSEIF (RCNTRL(7) > ZERO) THEN |
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| 247 | FacSafe = RCNTRL(7) |
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| 248 | ELSE |
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| 249 | PRINT * , 'User-selected FacSafe: RCNTRL(7)=', RCNTRL(7) |
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| 250 | CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR) |
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| 251 | RETURN |
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| 252 | END IF |
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| 253 | !~~~> Check if tolerances are reasonable |
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| 254 | DO i=1,UplimTol |
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| 255 | IF ( (AbsTol(i) <= ZERO) .OR. (RelTol(i) <= 10.0_dp*Roundoff) & |
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| 256 | .OR. (RelTol(i) >= 1.0_dp) ) THEN |
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| 257 | PRINT * , ' AbsTol(',i,') = ',AbsTol(i) |
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| 258 | PRINT * , ' RelTol(',i,') = ',RelTol(i) |
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| 259 | CALL ros_ErrorMsg(-5,Tstart,ZERO,IERR) |
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| 260 | RETURN |
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| 261 | END IF |
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| 262 | END DO |
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| 263 | |
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| 264 | |
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| 265 | !~~~> Initialize the particular Rosenbrock method |
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| 266 | SELECT CASE (Method) |
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| 267 | CASE (1) |
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| 268 | CALL Ros2(ros_S, ros_A, ros_C, ros_M, ros_E, & |
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| 269 | ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name) |
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| 270 | CASE (2) |
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| 271 | CALL Ros3(ros_S, ros_A, ros_C, ros_M, ros_E, & |
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| 272 | ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name) |
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| 273 | CASE (3) |
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| 274 | CALL Ros4(ros_S, ros_A, ros_C, ros_M, ros_E, & |
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| 275 | ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name) |
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| 276 | CASE (4) |
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| 277 | CALL Rodas3(ros_S, ros_A, ros_C, ros_M, ros_E, & |
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| 278 | ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name) |
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| 279 | CASE (5) |
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| 280 | CALL Rodas4(ros_S, ros_A, ros_C, ros_M, ros_E, & |
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| 281 | ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name) |
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| 282 | CASE DEFAULT |
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| 283 | PRINT * , 'Unknown Rosenbrock method: ICNTRL(3)=', Method |
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| 284 | CALL ros_ErrorMsg(-2,Tstart,ZERO,IERR) |
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| 285 | RETURN |
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| 286 | END SELECT |
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| 287 | |
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| 288 | !~~~> CALL Rosenbrock method |
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| 289 | CALL ros_Integrator(Y,Tstart,Tend,Texit, & |
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| 290 | AbsTol, RelTol, & |
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| 291 | ! Rosenbrock method coefficients |
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| 292 | ros_S, ros_M, ros_E, ros_A, ros_C, & |
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| 293 | ros_Alpha, ros_Gamma, ros_ELO, ros_NewF, & |
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| 294 | ! Integration parameters |
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| 295 | Autonomous, VectorTol, Max_no_steps, & |
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| 296 | Roundoff, Hmin, Hmax, Hstart, Hexit, & |
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| 297 | FacMin, FacMax, FacRej, FacSafe, & |
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| 298 | ! Error indicator |
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| 299 | IERR) |
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| 300 | |
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| 301 | |
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| 302 | !~~~> Collect run statistics |
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| 303 | ISTATUS(ifun) = Nfun |
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| 304 | ISTATUS(ijac) = Njac |
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| 305 | ISTATUS(istp) = Nstp |
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| 306 | ISTATUS(iacc) = Nacc |
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| 307 | ISTATUS(irej) = Nrej |
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| 308 | ISTATUS(idec) = Ndec |
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| 309 | ISTATUS(isol) = Nsol |
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| 310 | ISTATUS(isng) = Nsng |
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| 311 | !~~~> Last T and H |
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| 312 | RSTATUS(itexit) = Texit |
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| 313 | RSTATUS(ihexit) = Hexit |
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| 314 | |
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| 315 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 316 | CONTAINS ! SUBROUTINES internal to Rosenbrock |
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| 317 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 318 | |
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| 319 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 320 | SUBROUTINE ros_ErrorMsg(Code,T,H,IERR) |
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| 321 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 322 | ! Handles all error messages |
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| 323 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 324 | |
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| 325 | REAL(kind=dp), INTENT(IN) :: T, H |
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| 326 | INTEGER, INTENT(IN) :: Code |
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| 327 | INTEGER, INTENT(OUT) :: IERR |
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| 328 | |
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| 329 | IERR = Code |
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| 330 | PRINT * , & |
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| 331 | 'Forced exit from Rosenbrock due to the following error:' |
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| 332 | IF ((Code>=-8).AND.(Code<=-1)) THEN |
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| 333 | PRINT *, IERR_NAMES(Code) |
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| 334 | ELSE |
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| 335 | PRINT *, 'Unknown Error code: ', Code |
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| 336 | ENDIF |
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| 337 | |
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| 338 | PRINT *, "T=", T, "and H=", H |
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| 339 | |
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| 340 | END SUBROUTINE ros_ErrorMsg |
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| 341 | |
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| 342 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 343 | SUBROUTINE ros_Integrator (Y, Tstart, Tend, Tout, & |
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| 344 | AbsTol, RelTol, & |
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| 345 | !~~~> Rosenbrock method coefficients |
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| 346 | ros_S, ros_M, ros_E, ros_A, ros_C, & |
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| 347 | ros_Alpha, ros_Gamma, ros_ELO, ros_NewF, & |
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| 348 | !~~~> Integration parameters |
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| 349 | Autonomous, VectorTol, Max_no_steps, & |
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| 350 | Roundoff, Hmin, Hmax, Hstart, Hexit, & |
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| 351 | FacMin, FacMax, FacRej, FacSafe, & |
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| 352 | !~~~> Error indicator |
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| 353 | IERR ) |
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| 354 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 355 | ! Template for the implementation of a generic Rosenbrock method |
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| 356 | ! defined by ros_S (no of stages) |
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| 357 | ! and its coefficients ros_{A,C,M,E,Alpha,Gamma} |
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| 358 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 359 | |
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| 360 | USE messy_kpp_compress, ONLY: kco_initialize, kco_compress, & |
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| 361 | kco_finalize, cell_done |
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| 362 | IMPLICIT NONE |
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| 363 | |
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| 364 | !~~~> Input: the initial condition at Tstart; Output: the solution at T |
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| 365 | REAL(kind=dp), INTENT(INOUT) :: Y(:,:) |
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| 366 | !~~~> Input: integration interval |
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| 367 | REAL(kind=dp), INTENT(IN) :: Tstart,Tend |
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| 368 | !~~~> Output: time at which the solution is returned (T=Tend if success) |
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| 369 | REAL(kind=dp), INTENT(OUT) :: Tout |
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| 370 | !~~~> Input: tolerances |
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| 371 | REAL(kind=dp), INTENT(IN) :: AbsTol(NVAR), RelTol(NVAR) |
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| 372 | !~~~> Input: The Rosenbrock method parameters |
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| 373 | INTEGER, INTENT(IN) :: ros_S |
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| 374 | REAL(kind=dp), INTENT(IN) :: ros_M(ros_S), ros_E(ros_S), & |
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| 375 | ros_Alpha(ros_S), ros_A(ros_S*(ros_S-1)/2), & |
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| 376 | ros_Gamma(ros_S), ros_C(ros_S*(ros_S-1)/2), ros_ELO |
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| 377 | LOGICAL, INTENT(IN) :: ros_NewF(ros_S) |
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| 378 | !~~~> Input: integration parameters |
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| 379 | LOGICAL, INTENT(IN) :: Autonomous, VectorTol |
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| 380 | REAL(kind=dp), INTENT(IN) :: Hstart, Hmin, Hmax |
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| 381 | INTEGER, INTENT(IN) :: Max_no_steps |
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| 382 | REAL(kind=dp), INTENT(IN) :: Roundoff, FacMin, FacMax, FacRej, FacSafe |
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| 383 | !~~~> Output: last accepted step |
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| 384 | REAL(kind=dp), INTENT(OUT) :: Hexit |
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| 385 | !~~~> Output: Error indicator |
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| 386 | INTEGER, INTENT(OUT) :: IERR |
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| 387 | ! ~~~~ Local variables |
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| 388 | REAL(kind=dp) :: Ynew(VL_DIM,NVAR), Fcn0(VL_DIM,NVAR), Fcn(VL_DIM,NVAR) |
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| 389 | REAL(kind=dp) :: K(VL_DIM,NVAR*ros_S) |
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| 390 | REAL(kind=dp) :: Jac0(VL_DIM,LU_NONZERO), Ghimj(VL_DIM,LU_NONZERO) |
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| 391 | REAL(kind=dp) :: Yerr(VL_DIM,NVAR) |
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| 392 | |
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| 393 | !~~~> Local parameters |
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| 394 | INTEGER :: ioffset, i, j, istage |
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| 395 | REAL(kind=dp), PARAMETER :: ZERO = 0.0_dp, ONE = 1.0_dp |
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| 396 | REAL(kind=dp), PARAMETER :: DeltaMin = 1.0E-5_dp |
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| 397 | !~~~> Locally called functions |
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| 398 | ! REAL(kind=dp) WLAMCH |
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| 399 | ! EXTERNAL WLAMCH |
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| 400 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 401 | real(kind=dp) :: deltat |
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| 402 | !kk logical,parameter :: fixed_step=.true. |
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| 403 | |
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| 404 | integer,save :: icount=0 |
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| 405 | |
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| 406 | ! vector compression algorithm |
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| 407 | |
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| 408 | REAL(kind=dp), dimension(VL_dim) :: T |
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| 409 | REAL(kind=dp), dimension(VL_dim) :: H, Hnew, HC, HC2, HG, Fac, Tau, Err |
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| 410 | LOGICAL, dimension(VL_dim) :: RejectLastH, RejectMoreH |
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| 411 | LOGICAL, dimension(VL_dim) :: Accept_step, error_flag |
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| 412 | integer :: vl_save |
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| 413 | |
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| 414 | REAL(kind=dp), dimension(size(FIX,1),size(FIX,2)) :: FIX_lo |
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| 415 | |
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| 416 | !~~~> Initial preparations |
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| 417 | T(1:VL) = Tstart |
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| 418 | Hexit = 0.0_dp |
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| 419 | H(1:VL) = MIN(Hstart,Hmax) |
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| 420 | Tout = Tend |
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| 421 | icount = icount+1 |
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| 422 | where (ABS(H(1:VL)) <= 10.0_dp*Roundoff) H(1:VL) = DeltaMin |
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| 423 | |
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| 424 | deltat=Tend-Tstart |
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| 425 | |
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| 426 | RejectLastH(1:VL)=.FALSE. |
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| 427 | RejectMoreH(1:VL)=.FALSE. |
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| 428 | |
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| 429 | !~~~> Time loop begins below |
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| 430 | |
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| 431 | Nstp = 0 |
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| 432 | error_flag(1:VL) = .false. |
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| 433 | vl_save = VL |
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| 434 | FIX_lo(1:VL,:) = FIX(1:VL,:) |
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| 435 | Kacc(1:VL) = 0 |
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| 436 | Krej(1:VL) = 0 |
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| 437 | |
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| 438 | ! intialize compression algorithm |
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| 439 | |
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| 440 | IF (.not. l_fixed_step) then |
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| 441 | if( return_A ) then |
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| 442 | call kco_initialize (VL, y, Kacc, Krej, a=a) |
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| 443 | else |
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| 444 | call kco_initialize (VL, y, Kacc, Krej) |
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| 445 | end if |
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| 446 | end if |
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| 447 | |
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| 448 | TimeLoop: DO |
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| 449 | |
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| 450 | IF ( Nstp > Max_no_steps ) THEN ! Too many steps |
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| 451 | CALL ros_ErrorMsg(-6,T(1),H(1),IERR) |
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| 452 | ! mz_pj_20070514: replace 'call abort()' by |
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| 453 | ! error_flag(1:VL) = .TRUE. |
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| 454 | ! NOTE: Nstp is the maximum number of steps of the remaining current vector |
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| 455 | call abort() !Vorlaeufig |
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| 456 | RETURN |
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| 457 | END IF |
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| 458 | |
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| 459 | Where ( (T(1:VL)+0.1_dp*H(1:VL)) == T(1:VL)) ! Step size too small |
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| 460 | error_flag(1:VL) = .TRUE. |
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| 461 | endwhere |
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| 462 | |
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| 463 | if( any(error_flag)) then |
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| 464 | CALL ros_ErrorMsg(-7,T(1),H(1),IERR) |
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| 465 | call abort() !Vorlaeufig |
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| 466 | RETURN |
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| 467 | end if |
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| 468 | |
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| 469 | !~~~> Limit H if necessary to avoid going beyond Tend |
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| 470 | !kk Hexit = H |
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| 471 | H(1:VL) = MIN(H(1:VL),ABS(Tend-T(1:VL))) |
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| 472 | |
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| 473 | !~~~> Compute the function at current time |
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| 474 | CALL Fun( Y, FIX_lo, RCONST, Fcn0) |
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| 475 | |
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| 476 | |
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| 477 | !~~~> Compute the Jacobian at current time |
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| 478 | CALL Jac_SP( Y, FIX_lo, RCONST, Jac0) |
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| 479 | |
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| 480 | |
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| 481 | !~~~> Repeat step calculation until current step accepted |
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| 482 | |
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| 483 | CALL ros_PrepareMatrix(H,ros_Gamma(1), Jac0,Ghimj) |
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| 484 | |
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| 485 | !~~~> Compute the stages |
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| 486 | Stage: DO istage = 1, ros_S |
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| 487 | |
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| 488 | ! Current istage offset. Current istage vector is K(ioffset+1:ioffset+NVAR) |
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| 489 | ioffset = NVAR*(istage-1) |
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| 490 | |
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| 491 | ! For the 1st istage the function has been computed previously |
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| 492 | IF ( istage == 1 ) THEN |
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| 493 | Fcn(1:VL,1:NVAR) = Fcn0(1:VL,1:NVAR) |
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| 494 | ! istage>1 and a new function evaluation is needed at the current istage |
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| 495 | ELSEIF ( ros_NewF(istage) ) THEN |
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| 496 | select case (istage-1) |
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| 497 | case (1) |
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| 498 | Ynew(1:VL,1:NVAR) = Y(1:VL,1:NVAR) |
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| 499 | Ynew(1:VL,1:NVAR) = Ynew(1:VL,1:NVAR) + K(1:VL,NVAR*(1- 1)+ 1:NVAR*1) & |
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| 500 | *ros_A((istage-1)*(istage-2)/2+ 1) |
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| 501 | case (2) |
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| 502 | Ynew(1:VL,1:NVAR) = Y(1:VL,1:NVAR) |
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| 503 | Ynew(1:VL,1:NVAR) = Ynew(1:VL,1:NVAR) + K(1:VL,NVAR*(2- 1)+ 1:NVAR*2) & |
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| 504 | *ros_A((istage-1)*(istage-2)/2+ 2) |
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| 505 | Ynew(1:VL,1:NVAR) = Ynew(1:VL,1:NVAR) + K(1:VL,NVAR*(2- 1)+ 1:NVAR*2) & |
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| 506 | *ros_A((istage-1)*(istage-2)/2+ 2) |
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| 507 | case DEFAULT |
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| 508 | Ynew(1:VL,1:NVAR) = Y(1:VL,1:NVAR) |
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| 509 | DO j = 1, istage-1 |
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| 510 | Ynew(1:VL,1:NVAR) = Ynew(1:VL,1:NVAR) + K(1:VL,NVAR*(j- 1)+ 1:NVAR*j) & |
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| 511 | *ros_A((istage-1)*(istage-2)/2+ j) |
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| 512 | END DO |
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| 513 | end select |
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| 514 | |
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| 515 | CALL Fun( Ynew, FIX_lo, RCONST, Fcn) |
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| 516 | END IF ! if istage == 1 elseif ros_NewF(istage) |
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| 517 | |
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| 518 | select case (istage-1) |
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| 519 | case (1) |
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| 520 | K(1:VL,ioffset+ 1:ioffset+ NVAR) = Fcn(1:VL,1:NVAR) |
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| 521 | j = 1 |
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| 522 | HC(1:VL) = ros_C((istage-1)*(istage-2)/2+ j)/(H(1:VL)) |
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| 523 | do i=1,NVAR |
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| 524 | K(1:VL,ioffset+ i) = K(1:VL,ioffset+ i) + K(1:VL,NVAR*(j- 1)+ i) *HC(1:VL) |
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| 525 | end do |
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| 526 | case (2) |
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| 527 | K(1:VL,ioffset+ 1:ioffset+ NVAR) = Fcn(1:VL,1:NVAR) |
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| 528 | HC(1:VL) = ros_C((istage-1)*(istage-2)/2+ 1)/(H(1:VL)) |
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| 529 | HC2(1:VL) = ros_C((istage-1)*(istage-2)/2+ 2)/(H(1:VL)) |
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| 530 | do i=1,NVAR |
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| 531 | K(1:VL,ioffset+ i) = K(1:VL,ioffset+ i) + K(1:VL,NVAR*(1- 1)+ i) *HC(1:VL) |
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| 532 | K(1:VL,ioffset+ i) = K(1:VL,ioffset+ i) + K(1:VL,NVAR*(2- 1)+ i) *HC2(1:VL) |
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| 533 | end do |
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| 534 | case DEFAULT |
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| 535 | K(1:VL,ioffset+ 1:ioffset+ NVAR) = Fcn(1:VL,1:NVAR) |
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| 536 | DO j = 1, istage-1 |
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| 537 | HC(1:VL) = ros_C((istage-1)*(istage-2)/2+ j)/(H(1:VL)) |
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| 538 | do i=1,NVAR |
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| 539 | K(1:VL,ioffset+ i) = K(1:VL,ioffset+ i) + K(1:VL,NVAR*(j- 1)+ i) *HC(1:VL) |
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| 540 | end do |
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| 541 | END DO |
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| 542 | end select |
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| 543 | |
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| 544 | CALL ros_Solve(Ghimj, K(:,ioffset+1:ioffset+NVAR)) |
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| 545 | |
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| 546 | END DO Stage |
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| 547 | |
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| 548 | |
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| 549 | if(ros_S == 3) then |
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| 550 | YNEW(1:VL,1:NVAR) = Y(1:VL,1:NVAR) |
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| 551 | Yerr(1:VL,1:NVAR) = ZERO |
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| 552 | |
---|
| 553 | !~~~> Compute the new solution |
---|
| 554 | Ynew(1:VL,1:NVAR) = Ynew(1:VL,1:NVAR) + K(1:VL,NVAR*(1-1)+1:NVAR*1) * ros_m(1) |
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| 555 | Ynew(1:VL,1:NVAR) = Ynew(1:VL,1:NVAR) + K(1:VL,NVAR*(2-1)+1:NVAR*2) * ros_m(2) |
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| 556 | Ynew(1:VL,1:NVAR) = Ynew(1:VL,1:NVAR) + K(1:VL,NVAR*(3-1)+1:NVAR*3) * ros_m(3) |
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| 557 | |
---|
| 558 | !~~~> Compute the error estimation |
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| 559 | |
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| 560 | Yerr(1:VL,1:NVAR) = Yerr(1:VL,1:NVAR) + K(1:VL,NVAR*(1-1)+1:NVAR*1) * ros_E(1) |
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| 561 | Yerr(1:VL,1:NVAR) = Yerr(1:VL,1:NVAR) + K(1:VL,NVAR*(2-1)+1:NVAR*2) * ros_E(2) |
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| 562 | Yerr(1:VL,1:NVAR) = Yerr(1:VL,1:NVAR) + K(1:VL,NVAR*(3-1)+1:NVAR*3) * ros_E(3) |
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| 563 | else |
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| 564 | YNEW(1:VL,1:NVAR) = Y(1:VL,1:NVAR) |
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| 565 | Yerr(1:VL,1:NVAR) = ZERO |
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| 566 | DO j=1,ros_S |
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| 567 | !~~~> Compute the new solution |
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| 568 | Ynew(1:VL,1:NVAR) = Ynew(1:VL,1:NVAR) + & |
---|
| 569 | K(1:VL,NVAR*(j-1)+1:NVAR*j) * ros_m(j) |
---|
| 570 | |
---|
| 571 | !~~~> Compute the error estimation |
---|
| 572 | |
---|
| 573 | Yerr(1:VL,1:NVAR) = Yerr(1:VL,1:NVAR) + & |
---|
| 574 | K(1:VL,NVAR*(j-1)+1:NVAR*j) * ros_E(j) |
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| 575 | END DO |
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| 576 | end if |
---|
| 577 | |
---|
| 578 | |
---|
| 579 | !~~~> Check the error magnitude and adjust step size |
---|
| 580 | Nstp = Nstp+1 |
---|
| 581 | |
---|
| 582 | if(l_fixed_step) then |
---|
| 583 | Err = 0.0 |
---|
| 584 | H = t_steps(Nstp)*deltat |
---|
| 585 | Hnew = t_steps(Nstp)*deltat |
---|
| 586 | else |
---|
| 587 | Err(1:VL) = ros_ErrorNorm ( Y, Ynew, Yerr, AbsTol, RelTol, VectorTol ) |
---|
| 588 | !~~~> New step size is bounded by FacMin <= Hnew/H <= FacMax |
---|
| 589 | where(Err(1:VL) >= 10.0_dp*Roundoff) |
---|
| 590 | Fac(1:VL) = MIN(FacMax,MAX(FacMin,FacSafe/Err(1:VL)**(ONE/ros_ELO))) |
---|
| 591 | elsewhere |
---|
| 592 | Fac(1:VL) = 1000. !Force large time step |
---|
| 593 | endwhere |
---|
| 594 | Hnew(1:VL) = H(1:VL)*Fac(1:VL) |
---|
| 595 | end if |
---|
| 596 | |
---|
| 597 | Accept_step(1:VL) = (Err(1:VL) <= ONE).OR.(H(1:VL) <= Hmin) |
---|
| 598 | where (Accept_step(1:VL)) |
---|
| 599 | Kacc(1:VL) = Kacc(1:VL)+1 |
---|
| 600 | T(1:VL) = T(1:VL) + H(1:VL) |
---|
| 601 | Hnew(1:VL) = MAX(Hmin,MIN(Hnew(1:VL),Hmax)) |
---|
| 602 | endwhere |
---|
| 603 | do i=1,NVAR |
---|
| 604 | where (Accept_step(1:VL)) |
---|
| 605 | Y(1:VL,i) = MAX(Ynew(1:VL,i),ZERO) |
---|
| 606 | endwhere |
---|
| 607 | end do |
---|
| 608 | where (Accept_step(1:VL) .and. RejectLastH(1:VL)) |
---|
| 609 | Hnew(1:VL) = MIN(Hnew(1:VL),H(1:VL)) |
---|
| 610 | endwhere |
---|
| 611 | where (Accept_step(1:VL)) |
---|
| 612 | RejectLastH(1:VL) = .FALSE. |
---|
| 613 | RejectMoreH(1:VL) = .FALSE. |
---|
| 614 | H(1:VL) = Hnew(1:VL) |
---|
| 615 | endwhere |
---|
| 616 | !kk ELSE !~~~> Reject step |
---|
| 617 | where (.NOT. Accept_step(1:VL) .and. RejectMoreH(1:VL)) |
---|
| 618 | Hnew(1:VL) = H(1:VL)*FacRej |
---|
| 619 | endwhere |
---|
| 620 | where (.NOT. Accept_step(1:VL)) |
---|
| 621 | RejectMoreH(1:VL) = RejectLastH(1:VL) |
---|
| 622 | RejectLastH(1:VL) = .TRUE. |
---|
| 623 | H(1:VL) = Hnew(1:VL) |
---|
| 624 | endwhere |
---|
| 625 | where ( .NOT. Accept_step(1:VL) .and. Kacc(1:VL) >= 1) |
---|
| 626 | Krej(1:VL) = Krej(1:VL)+1 |
---|
| 627 | endwhere |
---|
| 628 | |
---|
| 629 | |
---|
| 630 | if(l_fixed_step) then |
---|
| 631 | if(Nstp >= Nfsteps) EXIT |
---|
| 632 | else |
---|
| 633 | cell_done(1:VL) = ( .NOT. (T(1:VL)-Tend)+Roundoff <= ZERO ) |
---|
| 634 | call kco_compress (VL, T, H, Hnew, error_flag, y, RCONST, FIX_lo, & |
---|
| 635 | RejectLastH, RejectMoreH, Kacc, Krej, a=a) |
---|
| 636 | if(VL == 0) EXIT |
---|
| 637 | end if |
---|
| 638 | |
---|
| 639 | !kk write(0,*) ' aaa ',icount,Nstp,count(cell_done),count(Accept_step),VL,minval(T) |
---|
| 640 | END DO TimeLoop |
---|
| 641 | |
---|
| 642 | !~~~> Succesful exit |
---|
| 643 | IERR = 1 !~~~> The integration was successful |
---|
| 644 | |
---|
| 645 | IF (.not. l_fixed_step) then |
---|
| 646 | call kco_finalize (y, Kacc, Krej, a=a) |
---|
| 647 | end if |
---|
| 648 | |
---|
| 649 | VL = VL_save |
---|
| 650 | |
---|
| 651 | !kk write(0,*) 'bbb ',icount,Nstp,maxval(Kacc),maxval(Krej) |
---|
| 652 | !kk if(icount >= 10) call abort() |
---|
| 653 | |
---|
| 654 | END SUBROUTINE ros_Integrator |
---|
| 655 | |
---|
| 656 | |
---|
| 657 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 658 | FUNCTION ros_ErrorNorm ( Y, Ynew, Yerr, & |
---|
| 659 | AbsTol, RelTol, VectorTol ) |
---|
| 660 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 661 | !~~~> Computes the "scaled norm" of the error vector Yerr |
---|
| 662 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 663 | IMPLICIT NONE |
---|
| 664 | |
---|
| 665 | ! Input arguments |
---|
| 666 | REAL(kind=dp), INTENT(IN) :: Y(:,:), Ynew(:,:), & |
---|
| 667 | Yerr(:,:), AbsTol(NVAR), RelTol(NVAR) |
---|
| 668 | LOGICAL, INTENT(IN) :: VectorTol |
---|
| 669 | ! Local variables |
---|
| 670 | REAL(kind=dp), dimension(VL) :: Err, Scale, Ymax |
---|
| 671 | INTEGER :: i |
---|
| 672 | REAL(kind=dp), PARAMETER :: ZERO = 0.0_dp |
---|
| 673 | REAL(kind=dp), dimension(VL) :: ros_ErrorNorm |
---|
| 674 | |
---|
| 675 | Err = ZERO |
---|
| 676 | DO i=1,NVAR |
---|
| 677 | Ymax = MAX( ABS(Y(1:VL,i)),ABS(Ynew(1:VL,i))) |
---|
| 678 | IF (VectorTol) THEN |
---|
| 679 | Scale = AbsTol(i)+ RelTol(i)*Ymax |
---|
| 680 | ELSE |
---|
| 681 | Scale = AbsTol(1)+ RelTol(1)*Ymax |
---|
| 682 | END IF |
---|
| 683 | Err = Err+ (Yerr(1:VL,i)/Scale)**2 |
---|
| 684 | END DO |
---|
| 685 | Err = SQRT(Err/NVAR) |
---|
| 686 | |
---|
| 687 | ros_ErrorNorm = Err |
---|
| 688 | |
---|
| 689 | END FUNCTION ros_ErrorNorm |
---|
| 690 | |
---|
| 691 | |
---|
| 692 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 693 | |
---|
| 694 | SUBROUTINE ros_PrepareMatrix ( H, gam, Jac0, Ghimj ) |
---|
| 695 | |
---|
| 696 | ! --- --- --- --- --- --- --- --- --- --- --- --- --- |
---|
| 697 | ! Prepares the LHS matrix for stage calculations |
---|
| 698 | ! 1. Construct Ghimj = 1/(H*ham) - Jac0 |
---|
| 699 | ! "(Gamma H) Inverse Minus Jacobian" |
---|
| 700 | ! 2. Repeat LU decomposition of Ghimj until successful. |
---|
| 701 | ! -half the step size if LU decomposition fails and retry |
---|
| 702 | ! -exit after 5 consecutive fails |
---|
| 703 | ! --- --- --- --- --- --- --- --- --- --- --- --- --- |
---|
| 704 | IMPLICIT NONE |
---|
| 705 | |
---|
| 706 | !~~~> Input arguments |
---|
| 707 | REAL(kind=dp), INTENT(IN) :: Jac0(:,:) |
---|
| 708 | REAL(kind=dp), INTENT(IN) :: gam |
---|
| 709 | !~~~> Output arguments |
---|
| 710 | REAL(kind=dp), INTENT(OUT) :: Ghimj(:,:) |
---|
| 711 | !~~~> Inout arguments |
---|
| 712 | REAL(kind=dp), INTENT(INOUT) :: H(:) ! step size is decreased when LU fails |
---|
| 713 | !~~~> Local variables |
---|
| 714 | INTEGER :: i, ising, Nconsecutive |
---|
| 715 | REAL(kind=dp), dimension(size(H)) :: ghinv |
---|
| 716 | REAL(kind=dp), PARAMETER :: ONE = 1.0_dp, HALF = 0.5_dp |
---|
| 717 | |
---|
| 718 | Nconsecutive = 0 |
---|
| 719 | |
---|
| 720 | |
---|
| 721 | !~~~> Construct Ghimj = 1/(H*gam) - Jac0 |
---|
| 722 | Ghimj(1:VL,1:LU_NONZERO) = -ONE * JAC0(1:VL,1:LU_NONZERO) |
---|
| 723 | ghinv(1:VL) = ONE/(H(1:VL)*gam) |
---|
| 724 | DO i=1,NVAR |
---|
| 725 | Ghimj(1:VL,LU_DIAG(i)) = Ghimj(1:VL,LU_DIAG(i))+ghinv(1:VL) |
---|
| 726 | END DO |
---|
| 727 | |
---|
| 728 | !~~~> Compute LU decomposition |
---|
| 729 | !CDIR noiexpand |
---|
| 730 | CALL KppDecomp( Ghimj, ising ) |
---|
| 731 | |
---|
| 732 | Ndec = Ndec + VL |
---|
| 733 | |
---|
| 734 | END SUBROUTINE ros_PrepareMatrix |
---|
| 735 | |
---|
| 736 | |
---|
| 737 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 738 | SUBROUTINE ros_Solve( A, b ) |
---|
| 739 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 740 | ! Template for the forward/backward substitution (using pre-computed LU decomposition) |
---|
| 741 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 742 | IMPLICIT NONE |
---|
| 743 | !~~~> Input variables |
---|
| 744 | REAL(kind=dp), INTENT(IN) :: A(:,:) |
---|
| 745 | !~~~> InOut variables |
---|
| 746 | REAL(kind=dp), INTENT(INOUT) :: b(:,:) |
---|
| 747 | |
---|
| 748 | CALL KppSolve( A, b ) |
---|
| 749 | |
---|
| 750 | Nsol = Nsol+VL |
---|
| 751 | |
---|
| 752 | END SUBROUTINE ros_Solve |
---|
| 753 | |
---|
| 754 | |
---|
| 755 | |
---|
| 756 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 757 | SUBROUTINE Ros2 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,& |
---|
| 758 | ros_Gamma,ros_NewF,ros_ELO,ros_Name) |
---|
| 759 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 760 | ! --- AN L-STABLE METHOD, 2 stages, order 2 |
---|
| 761 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 762 | |
---|
| 763 | IMPLICIT NONE |
---|
| 764 | |
---|
| 765 | INTEGER, PARAMETER :: S=2 |
---|
| 766 | INTEGER, INTENT(OUT) :: ros_S |
---|
| 767 | REAL(kind=dp), DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma |
---|
| 768 | REAL(kind=dp), DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C |
---|
| 769 | REAL(kind=dp), INTENT(OUT) :: ros_ELO |
---|
| 770 | LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF |
---|
| 771 | CHARACTER(LEN=12), INTENT(OUT) :: ros_Name |
---|
| 772 | DOUBLE PRECISION g |
---|
| 773 | |
---|
| 774 | g = 1.0_dp + 1.0_dp/SQRT(2.0_dp) |
---|
| 775 | |
---|
| 776 | !~~~> Name of the method |
---|
| 777 | ros_Name = 'ROS-2' |
---|
| 778 | !~~~> Number of stages |
---|
| 779 | ros_S = S |
---|
| 780 | |
---|
| 781 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 782 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 783 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 784 | ! The general mapping formula is: |
---|
| 785 | ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 786 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 787 | |
---|
| 788 | ros_A(1) = (1.0_dp)/g |
---|
| 789 | ros_C(1) = (-2.0_dp)/g |
---|
| 790 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 791 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 792 | ros_NewF(1) = .TRUE. |
---|
| 793 | ros_NewF(2) = .TRUE. |
---|
| 794 | !~~~> M_i = Coefficients for new step solution |
---|
| 795 | ros_M(1)= (3.0_dp)/(2.0_dp*g) |
---|
| 796 | ros_M(2)= (1.0_dp)/(2.0_dp*g) |
---|
| 797 | ! E_i = Coefficients for error estimator |
---|
| 798 | ros_E(1) = 1.0_dp/(2.0_dp*g) |
---|
| 799 | ros_E(2) = 1.0_dp/(2.0_dp*g) |
---|
| 800 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 801 | ! main and the embedded scheme orders plus one |
---|
| 802 | ros_ELO = 2.0_dp |
---|
| 803 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 804 | ros_Alpha(1) = 0.0_dp |
---|
| 805 | ros_Alpha(2) = 1.0_dp |
---|
| 806 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 807 | ros_Gamma(1) = g |
---|
| 808 | ros_Gamma(2) =-g |
---|
| 809 | |
---|
| 810 | END SUBROUTINE Ros2 |
---|
| 811 | |
---|
| 812 | |
---|
| 813 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 814 | SUBROUTINE Ros3 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,& |
---|
| 815 | ros_Gamma,ros_NewF,ros_ELO,ros_Name) |
---|
| 816 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 817 | ! --- AN L-STABLE METHOD, 3 stages, order 3, 2 function evaluations |
---|
| 818 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 819 | |
---|
| 820 | IMPLICIT NONE |
---|
| 821 | |
---|
| 822 | INTEGER, PARAMETER :: S=3 |
---|
| 823 | INTEGER, INTENT(OUT) :: ros_S |
---|
| 824 | REAL(kind=dp), DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma |
---|
| 825 | REAL(kind=dp), DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C |
---|
| 826 | REAL(kind=dp), INTENT(OUT) :: ros_ELO |
---|
| 827 | LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF |
---|
| 828 | CHARACTER(LEN=12), INTENT(OUT) :: ros_Name |
---|
| 829 | |
---|
| 830 | !~~~> Name of the method |
---|
| 831 | ros_Name = 'ROS-3' |
---|
| 832 | !~~~> Number of stages |
---|
| 833 | ros_S = S |
---|
| 834 | |
---|
| 835 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 836 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 837 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 838 | ! The general mapping formula is: |
---|
| 839 | ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 840 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 841 | |
---|
| 842 | ros_A(1)= 1.0_dp |
---|
| 843 | ros_A(2)= 1.0_dp |
---|
| 844 | ros_A(3)= 0.0_dp |
---|
| 845 | |
---|
| 846 | ros_C(1) = -0.10156171083877702091975600115545E+01_dp |
---|
| 847 | ros_C(2) = 0.40759956452537699824805835358067E+01_dp |
---|
| 848 | ros_C(3) = 0.92076794298330791242156818474003E+01_dp |
---|
| 849 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 850 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 851 | ros_NewF(1) = .TRUE. |
---|
| 852 | ros_NewF(2) = .TRUE. |
---|
| 853 | ros_NewF(3) = .FALSE. |
---|
| 854 | !~~~> M_i = Coefficients for new step solution |
---|
| 855 | ros_M(1) = 0.1E+01_dp |
---|
| 856 | ros_M(2) = 0.61697947043828245592553615689730E+01_dp |
---|
| 857 | ros_M(3) = -0.42772256543218573326238373806514E+00_dp |
---|
| 858 | ! E_i = Coefficients for error estimator |
---|
| 859 | ros_E(1) = 0.5E+00_dp |
---|
| 860 | ros_E(2) = -0.29079558716805469821718236208017E+01_dp |
---|
| 861 | ros_E(3) = 0.22354069897811569627360909276199E+00_dp |
---|
| 862 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 863 | ! main and the embedded scheme orders plus 1 |
---|
| 864 | ros_ELO = 3.0_dp |
---|
| 865 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 866 | ros_Alpha(1)= 0.0E+00_dp |
---|
| 867 | ros_Alpha(2)= 0.43586652150845899941601945119356E+00_dp |
---|
| 868 | ros_Alpha(3)= 0.43586652150845899941601945119356E+00_dp |
---|
| 869 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 870 | ros_Gamma(1)= 0.43586652150845899941601945119356E+00_dp |
---|
| 871 | ros_Gamma(2)= 0.24291996454816804366592249683314E+00_dp |
---|
| 872 | ros_Gamma(3)= 0.21851380027664058511513169485832E+01_dp |
---|
| 873 | |
---|
| 874 | END SUBROUTINE Ros3 |
---|
| 875 | |
---|
| 876 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 877 | |
---|
| 878 | |
---|
| 879 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 880 | SUBROUTINE Ros4 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,& |
---|
| 881 | ros_Gamma,ros_NewF,ros_ELO,ros_Name) |
---|
| 882 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 883 | ! L-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 4 STAGES |
---|
| 884 | ! L-STABLE EMBEDDED ROSENBROCK METHOD OF ORDER 3 |
---|
| 885 | ! |
---|
| 886 | ! E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL |
---|
| 887 | ! EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS. |
---|
| 888 | ! SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS, |
---|
| 889 | ! SPRINGER-VERLAG (1990) |
---|
| 890 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 891 | |
---|
| 892 | IMPLICIT NONE |
---|
| 893 | |
---|
| 894 | INTEGER, PARAMETER :: S=4 |
---|
| 895 | INTEGER, INTENT(OUT) :: ros_S |
---|
| 896 | REAL(kind=dp), DIMENSION(4), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma |
---|
| 897 | REAL(kind=dp), DIMENSION(6), INTENT(OUT) :: ros_A, ros_C |
---|
| 898 | REAL(kind=dp), INTENT(OUT) :: ros_ELO |
---|
| 899 | LOGICAL, DIMENSION(4), INTENT(OUT) :: ros_NewF |
---|
| 900 | CHARACTER(LEN=12), INTENT(OUT) :: ros_Name |
---|
| 901 | DOUBLE PRECISION g |
---|
| 902 | |
---|
| 903 | !~~~> Name of the method |
---|
| 904 | ros_Name = 'ROS-4' |
---|
| 905 | !~~~> Number of stages |
---|
| 906 | ros_S = S |
---|
| 907 | |
---|
| 908 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 909 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 910 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 911 | ! The general mapping formula is: |
---|
| 912 | ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 913 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 914 | |
---|
| 915 | ros_A(1) = 0.2000000000000000E+01_dp |
---|
| 916 | ros_A(2) = 0.1867943637803922E+01_dp |
---|
| 917 | ros_A(3) = 0.2344449711399156E+00_dp |
---|
| 918 | ros_A(4) = ros_A(2) |
---|
| 919 | ros_A(5) = ros_A(3) |
---|
| 920 | ros_A(6) = 0.0_dp |
---|
| 921 | |
---|
| 922 | ros_C(1) =-0.7137615036412310E+01_dp |
---|
| 923 | ros_C(2) = 0.2580708087951457E+01_dp |
---|
| 924 | ros_C(3) = 0.6515950076447975E+00_dp |
---|
| 925 | ros_C(4) =-0.2137148994382534E+01_dp |
---|
| 926 | ros_C(5) =-0.3214669691237626E+00_dp |
---|
| 927 | ros_C(6) =-0.6949742501781779E+00_dp |
---|
| 928 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 929 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 930 | ros_NewF(1) = .TRUE. |
---|
| 931 | ros_NewF(2) = .TRUE. |
---|
| 932 | ros_NewF(3) = .TRUE. |
---|
| 933 | ros_NewF(4) = .FALSE. |
---|
| 934 | !~~~> M_i = Coefficients for new step solution |
---|
| 935 | ros_M(1) = 0.2255570073418735E+01_dp |
---|
| 936 | ros_M(2) = 0.2870493262186792E+00_dp |
---|
| 937 | ros_M(3) = 0.4353179431840180E+00_dp |
---|
| 938 | ros_M(4) = 0.1093502252409163E+01_dp |
---|
| 939 | !~~~> E_i = Coefficients for error estimator |
---|
| 940 | ros_E(1) =-0.2815431932141155E+00_dp |
---|
| 941 | ros_E(2) =-0.7276199124938920E-01_dp |
---|
| 942 | ros_E(3) =-0.1082196201495311E+00_dp |
---|
| 943 | ros_E(4) =-0.1093502252409163E+01_dp |
---|
| 944 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 945 | ! main and the embedded scheme orders plus 1 |
---|
| 946 | ros_ELO = 4.0_dp |
---|
| 947 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 948 | ros_Alpha(1) = 0.0_dp |
---|
| 949 | ros_Alpha(2) = 0.1145640000000000E+01_dp |
---|
| 950 | ros_Alpha(3) = 0.6552168638155900E+00_dp |
---|
| 951 | ros_Alpha(4) = ros_Alpha(3) |
---|
| 952 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 953 | ros_Gamma(1) = 0.5728200000000000E+00_dp |
---|
| 954 | ros_Gamma(2) =-0.1769193891319233E+01_dp |
---|
| 955 | ros_Gamma(3) = 0.7592633437920482E+00_dp |
---|
| 956 | ros_Gamma(4) =-0.1049021087100450E+00_dp |
---|
| 957 | |
---|
| 958 | END SUBROUTINE Ros4 |
---|
| 959 | |
---|
| 960 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 961 | SUBROUTINE Rodas3 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,& |
---|
| 962 | ros_Gamma,ros_NewF,ros_ELO,ros_Name) |
---|
| 963 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 964 | ! --- A STIFFLY-STABLE METHOD, 4 stages, order 3 |
---|
| 965 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 966 | |
---|
| 967 | IMPLICIT NONE |
---|
| 968 | |
---|
| 969 | INTEGER, PARAMETER :: S=4 |
---|
| 970 | INTEGER, INTENT(OUT) :: ros_S |
---|
| 971 | REAL(kind=dp), DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma |
---|
| 972 | REAL(kind=dp), DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C |
---|
| 973 | REAL(kind=dp), INTENT(OUT) :: ros_ELO |
---|
| 974 | LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF |
---|
| 975 | CHARACTER(LEN=12), INTENT(OUT) :: ros_Name |
---|
| 976 | DOUBLE PRECISION g |
---|
| 977 | |
---|
| 978 | !~~~> Name of the method |
---|
| 979 | ros_Name = 'RODAS-3' |
---|
| 980 | !~~~> Number of stages |
---|
| 981 | ros_S = S |
---|
| 982 | |
---|
| 983 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 984 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 985 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 986 | ! The general mapping formula is: |
---|
| 987 | ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 988 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 989 | |
---|
| 990 | ros_A(1) = 0.0E+00_dp |
---|
| 991 | ros_A(2) = 2.0E+00_dp |
---|
| 992 | ros_A(3) = 0.0E+00_dp |
---|
| 993 | ros_A(4) = 2.0E+00_dp |
---|
| 994 | ros_A(5) = 0.0E+00_dp |
---|
| 995 | ros_A(6) = 1.0E+00_dp |
---|
| 996 | |
---|
| 997 | ros_C(1) = 4.0E+00_dp |
---|
| 998 | ros_C(2) = 1.0E+00_dp |
---|
| 999 | ros_C(3) =-1.0E+00_dp |
---|
| 1000 | ros_C(4) = 1.0E+00_dp |
---|
| 1001 | ros_C(5) =-1.0E+00_dp |
---|
| 1002 | ros_C(6) =-(8.0E+00_dp/3.0E+00_dp) |
---|
| 1003 | |
---|
| 1004 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 1005 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 1006 | ros_NewF(1) = .TRUE. |
---|
| 1007 | ros_NewF(2) = .FALSE. |
---|
| 1008 | ros_NewF(3) = .TRUE. |
---|
| 1009 | ros_NewF(4) = .TRUE. |
---|
| 1010 | !~~~> M_i = Coefficients for new step solution |
---|
| 1011 | ros_M(1) = 2.0E+00_dp |
---|
| 1012 | ros_M(2) = 0.0E+00_dp |
---|
| 1013 | ros_M(3) = 1.0E+00_dp |
---|
| 1014 | ros_M(4) = 1.0E+00_dp |
---|
| 1015 | !~~~> E_i = Coefficients for error estimator |
---|
| 1016 | ros_E(1) = 0.0E+00_dp |
---|
| 1017 | ros_E(2) = 0.0E+00_dp |
---|
| 1018 | ros_E(3) = 0.0E+00_dp |
---|
| 1019 | ros_E(4) = 1.0E+00_dp |
---|
| 1020 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 1021 | ! main and the embedded scheme orders plus 1 |
---|
| 1022 | ros_ELO = 3.0E+00_dp |
---|
| 1023 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 1024 | ros_Alpha(1) = 0.0E+00_dp |
---|
| 1025 | ros_Alpha(2) = 0.0E+00_dp |
---|
| 1026 | ros_Alpha(3) = 1.0E+00_dp |
---|
| 1027 | ros_Alpha(4) = 1.0E+00_dp |
---|
| 1028 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 1029 | ros_Gamma(1) = 0.5E+00_dp |
---|
| 1030 | ros_Gamma(2) = 1.5E+00_dp |
---|
| 1031 | ros_Gamma(3) = 0.0E+00_dp |
---|
| 1032 | ros_Gamma(4) = 0.0E+00_dp |
---|
| 1033 | |
---|
| 1034 | END SUBROUTINE Rodas3 |
---|
| 1035 | |
---|
| 1036 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1037 | SUBROUTINE Rodas4 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha,& |
---|
| 1038 | ros_Gamma,ros_NewF,ros_ELO,ros_Name) |
---|
| 1039 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1040 | ! STIFFLY-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 6 STAGES |
---|
| 1041 | ! |
---|
| 1042 | ! E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL |
---|
| 1043 | ! EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS. |
---|
| 1044 | ! SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS, |
---|
| 1045 | ! SPRINGER-VERLAG (1996) |
---|
| 1046 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1047 | |
---|
| 1048 | IMPLICIT NONE |
---|
| 1049 | |
---|
| 1050 | INTEGER, PARAMETER :: S=6 |
---|
| 1051 | INTEGER, INTENT(OUT) :: ros_S |
---|
| 1052 | REAL(kind=dp), DIMENSION(S), INTENT(OUT) :: ros_M,ros_E,ros_Alpha,ros_Gamma |
---|
| 1053 | REAL(kind=dp), DIMENSION(S*(S-1)/2), INTENT(OUT) :: ros_A, ros_C |
---|
| 1054 | REAL(kind=dp), INTENT(OUT) :: ros_ELO |
---|
| 1055 | LOGICAL, DIMENSION(S), INTENT(OUT) :: ros_NewF |
---|
| 1056 | CHARACTER(LEN=12), INTENT(OUT) :: ros_Name |
---|
| 1057 | DOUBLE PRECISION g |
---|
| 1058 | |
---|
| 1059 | !~~~> Name of the method |
---|
| 1060 | ros_Name = 'RODAS-4' |
---|
| 1061 | !~~~> Number of stages |
---|
| 1062 | ros_S = 6 |
---|
| 1063 | |
---|
| 1064 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 1065 | ros_Alpha(1) = 0.000_dp |
---|
| 1066 | ros_Alpha(2) = 0.386_dp |
---|
| 1067 | ros_Alpha(3) = 0.210_dp |
---|
| 1068 | ros_Alpha(4) = 0.630_dp |
---|
| 1069 | ros_Alpha(5) = 1.000_dp |
---|
| 1070 | ros_Alpha(6) = 1.000_dp |
---|
| 1071 | |
---|
| 1072 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 1073 | ros_Gamma(1) = 0.2500000000000000E+00_dp |
---|
| 1074 | ros_Gamma(2) =-0.1043000000000000E+00_dp |
---|
| 1075 | ros_Gamma(3) = 0.1035000000000000E+00_dp |
---|
| 1076 | ros_Gamma(4) =-0.3620000000000023E-01_dp |
---|
| 1077 | ros_Gamma(5) = 0.0_dp |
---|
| 1078 | ros_Gamma(6) = 0.0_dp |
---|
| 1079 | |
---|
| 1080 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 1081 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 1082 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 1083 | ! The general mapping formula is: A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 1084 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 1085 | |
---|
| 1086 | ros_A(1) = 0.1544000000000000E+01_dp |
---|
| 1087 | ros_A(2) = 0.9466785280815826E+00_dp |
---|
| 1088 | ros_A(3) = 0.2557011698983284E+00_dp |
---|
| 1089 | ros_A(4) = 0.3314825187068521E+01_dp |
---|
| 1090 | ros_A(5) = 0.2896124015972201E+01_dp |
---|
| 1091 | ros_A(6) = 0.9986419139977817E+00_dp |
---|
| 1092 | ros_A(7) = 0.1221224509226641E+01_dp |
---|
| 1093 | ros_A(8) = 0.6019134481288629E+01_dp |
---|
| 1094 | ros_A(9) = 0.1253708332932087E+02_dp |
---|
| 1095 | ros_A(10) =-0.6878860361058950E+00_dp |
---|
| 1096 | ros_A(11) = ros_A(7) |
---|
| 1097 | ros_A(12) = ros_A(8) |
---|
| 1098 | ros_A(13) = ros_A(9) |
---|
| 1099 | ros_A(14) = ros_A(10) |
---|
| 1100 | ros_A(15) = 1.0E+00_dp |
---|
| 1101 | |
---|
| 1102 | ros_C(1) =-0.5668800000000000E+01_dp |
---|
| 1103 | ros_C(2) =-0.2430093356833875E+01_dp |
---|
| 1104 | ros_C(3) =-0.2063599157091915E+00_dp |
---|
| 1105 | ros_C(4) =-0.1073529058151375E+00_dp |
---|
| 1106 | ros_C(5) =-0.9594562251023355E+01_dp |
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| 1107 | ros_C(6) =-0.2047028614809616E+02_dp |
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| 1108 | ros_C(7) = 0.7496443313967647E+01_dp |
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| 1109 | ros_C(8) =-0.1024680431464352E+02_dp |
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| 1110 | ros_C(9) =-0.3399990352819905E+02_dp |
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| 1111 | ros_C(10) = 0.1170890893206160E+02_dp |
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| 1112 | ros_C(11) = 0.8083246795921522E+01_dp |
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| 1113 | ros_C(12) =-0.7981132988064893E+01_dp |
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| 1114 | ros_C(13) =-0.3152159432874371E+02_dp |
---|
| 1115 | ros_C(14) = 0.1631930543123136E+02_dp |
---|
| 1116 | ros_C(15) =-0.6058818238834054E+01_dp |
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| 1117 | |
---|
| 1118 | !~~~> M_i = Coefficients for new step solution |
---|
| 1119 | ros_M(1) = ros_A(7) |
---|
| 1120 | ros_M(2) = ros_A(8) |
---|
| 1121 | ros_M(3) = ros_A(9) |
---|
| 1122 | ros_M(4) = ros_A(10) |
---|
| 1123 | ros_M(5) = 1.0E+00_dp |
---|
| 1124 | ros_M(6) = 1.0E+00_dp |
---|
| 1125 | |
---|
| 1126 | !~~~> E_i = Coefficients for error estimator |
---|
| 1127 | ros_E(1) = 0.0E+00_dp |
---|
| 1128 | ros_E(2) = 0.0E+00_dp |
---|
| 1129 | ros_E(3) = 0.0E+00_dp |
---|
| 1130 | ros_E(4) = 0.0E+00_dp |
---|
| 1131 | ros_E(5) = 0.0E+00_dp |
---|
| 1132 | ros_E(6) = 1.0E+00_dp |
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| 1133 | |
---|
| 1134 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 1135 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 1136 | ros_NewF(1) = .TRUE. |
---|
| 1137 | ros_NewF(2) = .TRUE. |
---|
| 1138 | ros_NewF(3) = .TRUE. |
---|
| 1139 | ros_NewF(4) = .TRUE. |
---|
| 1140 | ros_NewF(5) = .TRUE. |
---|
| 1141 | ros_NewF(6) = .TRUE. |
---|
| 1142 | |
---|
| 1143 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 1144 | ! main and the embedded scheme orders plus 1 |
---|
| 1145 | ros_ELO = 4.0_dp |
---|
| 1146 | |
---|
| 1147 | END SUBROUTINE Rodas4 |
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| 1148 | |
---|
| 1149 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 1150 | ! End of the set of internal Rosenbrock subroutines |
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| 1151 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
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| 1152 | END SUBROUTINE Rosenbrock |
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