[2696] | 1 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~! |
---|
| 2 | ! SDIRK - Singly-Diagonally-Implicit Runge-Kutta methods ! |
---|
| 3 | ! * Sdirk 2a, 2b: L-stable, 2 stages, order 2 ! |
---|
| 4 | ! * Sdirk 3a: L-stable, 3 stages, order 2, adj-invariant ! |
---|
| 5 | ! * Sdirk 4a, 4b: L-stable, 5 stages, order 4 ! |
---|
| 6 | ! By default the code employs the KPP sparse linear algebra routines ! |
---|
| 7 | ! Compile with -DFULL_ALGEBRA to use full linear algebra (LAPACK) ! |
---|
| 8 | ! ! |
---|
| 9 | ! (C) Adrian Sandu, July 2005 ! |
---|
| 10 | ! Virginia Polytechnic Institute and State University ! |
---|
| 11 | ! Contact: sandu@cs.vt.edu ! |
---|
| 12 | ! Revised by Philipp Miehe and Adrian Sandu, May 2006 ! |
---|
| 13 | ! This implementation is part of KPP - the Kinetic PreProcessor ! |
---|
| 14 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~! |
---|
| 15 | |
---|
| 16 | MODULE KPP_ROOT_Integrator |
---|
| 17 | |
---|
| 18 | USE KPP_ROOT_Precision |
---|
| 19 | USE KPP_ROOT_Global, ONLY: FIX, RCONST, TIME |
---|
| 20 | USE KPP_ROOT_Parameters, ONLY: NVAR, NSPEC, NFIX, LU_NONZERO |
---|
| 21 | USE KPP_ROOT_JacobianSP, ONLY: LU_DIAG |
---|
| 22 | USE KPP_ROOT_LinearAlgebra, ONLY: KppDecomp, & |
---|
| 23 | KppSolve, Set2zero, WLAMCH, WCOPY, WAXPY, WSCAL, WADD |
---|
| 24 | |
---|
| 25 | IMPLICIT NONE |
---|
| 26 | PUBLIC |
---|
| 27 | SAVE |
---|
| 28 | |
---|
| 29 | !~~~> Statistics on the work performed by the SDIRK method |
---|
| 30 | INTEGER, PARAMETER :: Nfun=1, Njac=2, Nstp=3, Nacc=4, & |
---|
| 31 | Nrej=5, Ndec=6, Nsol=7, Nsng=8, & |
---|
| 32 | Ntexit=1, Nhexit=2, Nhnew=3 |
---|
| 33 | |
---|
| 34 | CONTAINS |
---|
| 35 | |
---|
| 36 | SUBROUTINE INTEGRATE( TIN, TOUT, & |
---|
| 37 | ICNTRL_U, RCNTRL_U, ISTATUS_U, RSTATUS_U, Ierr_U ) |
---|
| 38 | |
---|
| 39 | USE KPP_ROOT_Parameters |
---|
| 40 | USE KPP_ROOT_Global |
---|
| 41 | IMPLICIT NONE |
---|
| 42 | |
---|
| 43 | KPP_REAL, INTENT(IN) :: TIN ! Start Time |
---|
| 44 | KPP_REAL, INTENT(IN) :: TOUT ! End Time |
---|
| 45 | ! Optional input parameters and statistics |
---|
| 46 | INTEGER, INTENT(IN), OPTIONAL :: ICNTRL_U(20) |
---|
| 47 | KPP_REAL, INTENT(IN), OPTIONAL :: RCNTRL_U(20) |
---|
| 48 | INTEGER, INTENT(OUT), OPTIONAL :: ISTATUS_U(20) |
---|
| 49 | KPP_REAL, INTENT(OUT), OPTIONAL :: RSTATUS_U(20) |
---|
| 50 | INTEGER, INTENT(OUT), OPTIONAL :: Ierr_U |
---|
| 51 | |
---|
| 52 | INTEGER, SAVE :: Ntotal = 0 |
---|
| 53 | KPP_REAL :: RCNTRL(20), RSTATUS(20), T1, T2 |
---|
| 54 | INTEGER :: ICNTRL(20), ISTATUS(20), Ierr |
---|
| 55 | |
---|
| 56 | ICNTRL(:) = 0 |
---|
| 57 | RCNTRL(:) = 0.0_dp |
---|
| 58 | ISTATUS(:) = 0 |
---|
| 59 | RSTATUS(:) = 0.0_dp |
---|
| 60 | |
---|
| 61 | !~~~> fine-tune the integrator: |
---|
| 62 | ICNTRL(2) = 0 ! 0 - vector tolerances, 1 - scalar tolerances |
---|
| 63 | ICNTRL(6) = 0 ! starting values of Newton iterations: interpolated (0), zero (1) |
---|
| 64 | ! If optional parameters are given, and if they are >0, |
---|
| 65 | ! then they overwrite default settings. |
---|
| 66 | IF (PRESENT(ICNTRL_U)) THEN |
---|
| 67 | WHERE(ICNTRL_U(:) > 0) ICNTRL(:) = ICNTRL_U(:) |
---|
| 68 | END IF |
---|
| 69 | IF (PRESENT(RCNTRL_U)) THEN |
---|
| 70 | WHERE(RCNTRL_U(:) > 0) RCNTRL(:) = RCNTRL_U(:) |
---|
| 71 | END IF |
---|
| 72 | |
---|
| 73 | |
---|
| 74 | T1 = TIN; T2 = TOUT |
---|
| 75 | CALL SDIRK( NVAR,T1,T2,VAR,RTOL,ATOL, & |
---|
| 76 | RCNTRL,ICNTRL,RSTATUS,ISTATUS,Ierr ) |
---|
| 77 | |
---|
| 78 | !~~~> Debug option: print number of steps |
---|
| 79 | ! Ntotal = Ntotal + ISTATUS(Nstp) |
---|
| 80 | ! PRINT*,'NSTEPS=',ISTATUS(Nstp), '(',Ntotal,')',' O3=',VAR(ind_O3), & |
---|
| 81 | ! ' NO2=',VAR(ind_NO2) |
---|
| 82 | |
---|
| 83 | IF (Ierr < 0) THEN |
---|
| 84 | PRINT *,'SDIRK: Unsuccessful exit at T=',TIN,' (Ierr=',Ierr,')' |
---|
| 85 | ENDIF |
---|
| 86 | |
---|
| 87 | ! if optional parameters are given for output they to return information |
---|
| 88 | IF (PRESENT(ISTATUS_U)) ISTATUS_U(:) = ISTATUS(:) |
---|
| 89 | IF (PRESENT(RSTATUS_U)) RSTATUS_U(:) = RSTATUS(:) |
---|
| 90 | IF (PRESENT(Ierr_U)) Ierr_U = Ierr |
---|
| 91 | |
---|
| 92 | END SUBROUTINE INTEGRATE |
---|
| 93 | |
---|
| 94 | |
---|
| 95 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 96 | SUBROUTINE SDIRK(N, Tinitial, Tfinal, Y, RelTol, AbsTol, & |
---|
| 97 | RCNTRL, ICNTRL, RSTATUS, ISTATUS, Ierr) |
---|
| 98 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 99 | ! |
---|
| 100 | ! Solves the system y'=F(t,y) using a Singly-Diagonally-Implicit |
---|
| 101 | ! Runge-Kutta (SDIRK) method. |
---|
| 102 | ! |
---|
| 103 | ! This implementation is based on the book and the code Sdirk4: |
---|
| 104 | ! |
---|
| 105 | ! E. Hairer and G. Wanner |
---|
| 106 | ! "Solving ODEs II. Stiff and differential-algebraic problems". |
---|
| 107 | ! Springer series in computational mathematics, Springer-Verlag, 1996. |
---|
| 108 | ! This code is based on the SDIRK4 routine in the above book. |
---|
| 109 | ! |
---|
| 110 | ! Methods: |
---|
| 111 | ! * Sdirk 2a, 2b: L-stable, 2 stages, order 2 |
---|
| 112 | ! * Sdirk 3a: L-stable, 3 stages, order 2, adjoint-invariant |
---|
| 113 | ! * Sdirk 4a, 4b: L-stable, 5 stages, order 4 |
---|
| 114 | ! |
---|
| 115 | ! (C) Adrian Sandu, July 2005 |
---|
| 116 | ! Virginia Polytechnic Institute and State University |
---|
| 117 | ! Contact: sandu@cs.vt.edu |
---|
| 118 | ! Revised by Philipp Miehe and Adrian Sandu, May 2006 |
---|
| 119 | ! This implementation is part of KPP - the Kinetic PreProcessor |
---|
| 120 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 121 | ! |
---|
| 122 | !~~~> INPUT ARGUMENTS: |
---|
| 123 | ! |
---|
| 124 | !- Y(NVAR) = vector of initial conditions (at T=Tinitial) |
---|
| 125 | !- [Tinitial,Tfinal] = time range of integration |
---|
| 126 | ! (if Tinitial>Tfinal the integration is performed backwards in time) |
---|
| 127 | !- RelTol, AbsTol = user precribed accuracy |
---|
| 128 | !- SUBROUTINE ode_Fun( T, Y, Ydot ) = ODE function, |
---|
| 129 | ! returns Ydot = Y' = F(T,Y) |
---|
| 130 | !- SUBROUTINE ode_Fun( T, Y, Ydot ) = Jacobian of the ODE function, |
---|
| 131 | ! returns Jcb = dF/dY |
---|
| 132 | !- ICNTRL(1:20) = integer inputs parameters |
---|
| 133 | !- RCNTRL(1:20) = real inputs parameters |
---|
| 134 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 135 | ! |
---|
| 136 | !~~~> OUTPUT ARGUMENTS: |
---|
| 137 | ! |
---|
| 138 | !- Y(NVAR) -> vector of final states (at T->Tfinal) |
---|
| 139 | !- ISTATUS(1:20) -> integer output parameters |
---|
| 140 | !- RSTATUS(1:20) -> real output parameters |
---|
| 141 | !- Ierr -> job status upon return |
---|
| 142 | ! success (positive value) or |
---|
| 143 | ! failure (negative value) |
---|
| 144 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 145 | ! |
---|
| 146 | !~~~> INPUT PARAMETERS: |
---|
| 147 | ! |
---|
| 148 | ! Note: For input parameters equal to zero the default values of the |
---|
| 149 | ! corresponding variables are used. |
---|
| 150 | ! |
---|
| 151 | ! Note: For input parameters equal to zero the default values of the |
---|
| 152 | ! corresponding variables are used. |
---|
| 153 | !~~~> |
---|
| 154 | ! ICNTRL(1) = not used |
---|
| 155 | ! |
---|
| 156 | ! ICNTRL(2) = 0: AbsTol, RelTol are NVAR-dimensional vectors |
---|
| 157 | ! = 1: AbsTol, RelTol are scalars |
---|
| 158 | ! |
---|
| 159 | ! ICNTRL(3) = Method |
---|
| 160 | ! |
---|
| 161 | ! ICNTRL(4) -> maximum number of integration steps |
---|
| 162 | ! For ICNTRL(4)=0 the default value of 100000 is used |
---|
| 163 | ! |
---|
| 164 | ! ICNTRL(5) -> maximum number of Newton iterations |
---|
| 165 | ! For ICNTRL(5)=0 the default value of 8 is used |
---|
| 166 | ! |
---|
| 167 | ! ICNTRL(6) -> starting values of Newton iterations: |
---|
| 168 | ! ICNTRL(6)=0 : starting values are interpolated (the default) |
---|
| 169 | ! ICNTRL(6)=1 : starting values are zero |
---|
| 170 | ! |
---|
| 171 | !~~~> Real parameters |
---|
| 172 | ! |
---|
| 173 | ! RCNTRL(1) -> Hmin, lower bound for the integration step size |
---|
| 174 | ! It is strongly recommended to keep Hmin = ZERO |
---|
| 175 | ! RCNTRL(2) -> Hmax, upper bound for the integration step size |
---|
| 176 | ! RCNTRL(3) -> Hstart, starting value for the integration step size |
---|
| 177 | ! |
---|
| 178 | ! RCNTRL(4) -> FacMin, lower bound on step decrease factor (default=0.2) |
---|
| 179 | ! RCNTRL(5) -> FacMax, upper bound on step increase factor (default=6) |
---|
| 180 | ! RCNTRL(6) -> FacRej, step decrease factor after multiple rejections |
---|
| 181 | ! (default=0.1) |
---|
| 182 | ! RCNTRL(7) -> FacSafe, by which the new step is slightly smaller |
---|
| 183 | ! than the predicted value (default=0.9) |
---|
| 184 | ! RCNTRL(8) -> ThetaMin. If Newton convergence rate smaller |
---|
| 185 | ! than ThetaMin the Jacobian is not recomputed; |
---|
| 186 | ! (default=0.001) |
---|
| 187 | ! RCNTRL(9) -> NewtonTol, stopping criterion for Newton's method |
---|
| 188 | ! (default=0.03) |
---|
| 189 | ! RCNTRL(10) -> Qmin |
---|
| 190 | ! RCNTRL(11) -> Qmax. If Qmin < Hnew/Hold < Qmax, then the |
---|
| 191 | ! step size is kept constant and the LU factorization |
---|
| 192 | ! reused (default Qmin=1, Qmax=1.2) |
---|
| 193 | ! |
---|
| 194 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 195 | ! |
---|
| 196 | !~~~> OUTPUT PARAMETERS: |
---|
| 197 | ! |
---|
| 198 | ! Note: each call to Rosenbrock adds the current no. of fcn calls |
---|
| 199 | ! to previous value of ISTATUS(1), and similar for the other params. |
---|
| 200 | ! Set ISTATUS(1:10) = 0 before call to avoid this accumulation. |
---|
| 201 | ! |
---|
| 202 | ! ISTATUS(1) = No. of function calls |
---|
| 203 | ! ISTATUS(2) = No. of jacobian calls |
---|
| 204 | ! ISTATUS(3) = No. of steps |
---|
| 205 | ! ISTATUS(4) = No. of accepted steps |
---|
| 206 | ! ISTATUS(5) = No. of rejected steps (except at the beginning) |
---|
| 207 | ! ISTATUS(6) = No. of LU decompositions |
---|
| 208 | ! ISTATUS(7) = No. of forward/backward substitutions |
---|
| 209 | ! ISTATUS(8) = No. of singular matrix decompositions |
---|
| 210 | ! |
---|
| 211 | ! RSTATUS(1) -> Texit, the time corresponding to the |
---|
| 212 | ! computed Y upon return |
---|
| 213 | ! RSTATUS(2) -> Hexit,last accepted step before return |
---|
| 214 | ! RSTATUS(3) -> Hnew, last predicted step before return |
---|
| 215 | ! For multiple restarts, use Hnew as Hstart in the following run |
---|
| 216 | ! |
---|
| 217 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 218 | IMPLICIT NONE |
---|
| 219 | |
---|
| 220 | ! Arguments |
---|
| 221 | INTEGER, INTENT(IN) :: N, ICNTRL(20) |
---|
| 222 | KPP_REAL, INTENT(IN) :: Tinitial, Tfinal, & |
---|
| 223 | RelTol(NVAR), AbsTol(NVAR), RCNTRL(20) |
---|
| 224 | KPP_REAL, INTENT(INOUT) :: Y(NVAR) |
---|
| 225 | INTEGER, INTENT(OUT) :: Ierr |
---|
| 226 | INTEGER, INTENT(INOUT) :: ISTATUS(20) |
---|
| 227 | KPP_REAL, INTENT(OUT) :: RSTATUS(20) |
---|
| 228 | |
---|
| 229 | !~~~> SDIRK method coefficients, up to 5 stages |
---|
| 230 | INTEGER, PARAMETER :: Smax = 5 |
---|
| 231 | INTEGER, PARAMETER :: S2A=1, S2B=2, S3A=3, S4A=4, S4B=5 |
---|
| 232 | KPP_REAL :: rkGamma, rkA(Smax,Smax), rkB(Smax), rkC(Smax), & |
---|
| 233 | rkD(Smax), rkE(Smax), rkBhat(Smax), rkELO, & |
---|
| 234 | rkAlpha(Smax,Smax), rkTheta(Smax,Smax) |
---|
| 235 | INTEGER :: sdMethod, rkS ! The number of stages |
---|
| 236 | ! Local variables |
---|
| 237 | LOGICAL :: StartNewton |
---|
| 238 | KPP_REAL :: Hmin, Hmax, Hstart, Roundoff, & |
---|
| 239 | FacMin, Facmax, FacSafe, FacRej, & |
---|
| 240 | ThetaMin, NewtonTol, Qmin, Qmax |
---|
| 241 | INTEGER :: ITOL, NewtonMaxit, Max_no_steps, i |
---|
| 242 | KPP_REAL, PARAMETER :: ZERO = 0.0d0, ONE = 1.0d0 |
---|
| 243 | |
---|
| 244 | |
---|
| 245 | ISTATUS(1:20) = 0 |
---|
| 246 | RSTATUS(1:20) = ZERO |
---|
| 247 | Ierr = 0 |
---|
| 248 | |
---|
| 249 | !~~~> For Scalar tolerances (ICNTRL(2).NE.0) the code uses AbsTol(1) and RelTol(1) |
---|
| 250 | ! For Vector tolerances (ICNTRL(2) == 0) the code uses AbsTol(1:NVAR) and RelTol(1:NVAR) |
---|
| 251 | IF (ICNTRL(2) == 0) THEN |
---|
| 252 | ITOL = 1 |
---|
| 253 | ELSE |
---|
| 254 | ITOL = 0 |
---|
| 255 | END IF |
---|
| 256 | |
---|
| 257 | !~~~> ICNTRL(3) - method selection |
---|
| 258 | SELECT CASE (ICNTRL(3)) |
---|
| 259 | CASE (0,1) |
---|
| 260 | CALL Sdirk2a |
---|
| 261 | CASE (2) |
---|
| 262 | CALL Sdirk2b |
---|
| 263 | CASE (3) |
---|
| 264 | CALL Sdirk3a |
---|
| 265 | CASE (4) |
---|
| 266 | CALL Sdirk4a |
---|
| 267 | CASE (5) |
---|
| 268 | CALL Sdirk4b |
---|
| 269 | CASE DEFAULT |
---|
| 270 | CALL Sdirk2a |
---|
| 271 | END SELECT |
---|
| 272 | |
---|
| 273 | !~~~> The maximum number of time steps admitted |
---|
| 274 | IF (ICNTRL(4) == 0) THEN |
---|
| 275 | Max_no_steps = 200000 |
---|
| 276 | ELSEIF (ICNTRL(4) > 0) THEN |
---|
| 277 | Max_no_steps=ICNTRL(4) |
---|
| 278 | ELSE |
---|
| 279 | PRINT * ,'User-selected ICNTRL(4)=',ICNTRL(4) |
---|
| 280 | CALL SDIRK_ErrorMsg(-1,Tinitial,ZERO,Ierr) |
---|
| 281 | END IF |
---|
| 282 | |
---|
| 283 | !~~~> The maximum number of Newton iterations admitted |
---|
| 284 | IF(ICNTRL(5) == 0)THEN |
---|
| 285 | NewtonMaxit=8 |
---|
| 286 | ELSE |
---|
| 287 | NewtonMaxit=ICNTRL(5) |
---|
| 288 | IF(NewtonMaxit <= 0)THEN |
---|
| 289 | PRINT * ,'User-selected ICNTRL(5)=',ICNTRL(5) |
---|
| 290 | CALL SDIRK_ErrorMsg(-2,Tinitial,ZERO,Ierr) |
---|
| 291 | END IF |
---|
| 292 | END IF |
---|
| 293 | |
---|
| 294 | !~~~> StartNewton: Use extrapolation for starting values of Newton iterations |
---|
| 295 | IF (ICNTRL(6) == 0) THEN |
---|
| 296 | StartNewton = .TRUE. |
---|
| 297 | ELSE |
---|
| 298 | StartNewton = .FALSE. |
---|
| 299 | END IF |
---|
| 300 | |
---|
| 301 | !~~~> Unit roundoff (1+Roundoff>1) |
---|
| 302 | Roundoff = WLAMCH('E') |
---|
| 303 | |
---|
| 304 | !~~~> Lower bound on the step size: (positive value) |
---|
| 305 | IF (RCNTRL(1) == ZERO) THEN |
---|
| 306 | Hmin = ZERO |
---|
| 307 | ELSEIF (RCNTRL(1) > ZERO) THEN |
---|
| 308 | Hmin = RCNTRL(1) |
---|
| 309 | ELSE |
---|
| 310 | PRINT * , 'User-selected RCNTRL(1)=', RCNTRL(1) |
---|
| 311 | CALL SDIRK_ErrorMsg(-3,Tinitial,ZERO,Ierr) |
---|
| 312 | END IF |
---|
| 313 | |
---|
| 314 | !~~~> Upper bound on the step size: (positive value) |
---|
| 315 | IF (RCNTRL(2) == ZERO) THEN |
---|
| 316 | Hmax = ABS(Tfinal-Tinitial) |
---|
| 317 | ELSEIF (RCNTRL(2) > ZERO) THEN |
---|
| 318 | Hmax = MIN(ABS(RCNTRL(2)),ABS(Tfinal-Tinitial)) |
---|
| 319 | ELSE |
---|
| 320 | PRINT * , 'User-selected RCNTRL(2)=', RCNTRL(2) |
---|
| 321 | CALL SDIRK_ErrorMsg(-3,Tinitial,ZERO,Ierr) |
---|
| 322 | END IF |
---|
| 323 | |
---|
| 324 | !~~~> Starting step size: (positive value) |
---|
| 325 | IF (RCNTRL(3) == ZERO) THEN |
---|
| 326 | Hstart = MAX(Hmin,Roundoff) |
---|
| 327 | ELSEIF (RCNTRL(3) > ZERO) THEN |
---|
| 328 | Hstart = MIN(ABS(RCNTRL(3)),ABS(Tfinal-Tinitial)) |
---|
| 329 | ELSE |
---|
| 330 | PRINT * , 'User-selected Hstart: RCNTRL(3)=', RCNTRL(3) |
---|
| 331 | CALL SDIRK_ErrorMsg(-3,Tinitial,ZERO,Ierr) |
---|
| 332 | END IF |
---|
| 333 | |
---|
| 334 | !~~~> Step size can be changed s.t. FacMin < Hnew/Hexit < FacMax |
---|
| 335 | IF (RCNTRL(4) == ZERO) THEN |
---|
| 336 | FacMin = 0.2_dp |
---|
| 337 | ELSEIF (RCNTRL(4) > ZERO) THEN |
---|
| 338 | FacMin = RCNTRL(4) |
---|
| 339 | ELSE |
---|
| 340 | PRINT * , 'User-selected FacMin: RCNTRL(4)=', RCNTRL(4) |
---|
| 341 | CALL SDIRK_ErrorMsg(-4,Tinitial,ZERO,Ierr) |
---|
| 342 | END IF |
---|
| 343 | IF (RCNTRL(5) == ZERO) THEN |
---|
| 344 | FacMax = 10.0_dp |
---|
| 345 | ELSEIF (RCNTRL(5) > ZERO) THEN |
---|
| 346 | FacMax = RCNTRL(5) |
---|
| 347 | ELSE |
---|
| 348 | PRINT * , 'User-selected FacMax: RCNTRL(5)=', RCNTRL(5) |
---|
| 349 | CALL SDIRK_ErrorMsg(-4,Tinitial,ZERO,Ierr) |
---|
| 350 | END IF |
---|
| 351 | !~~~> FacRej: Factor to decrease step after 2 succesive rejections |
---|
| 352 | IF (RCNTRL(6) == ZERO) THEN |
---|
| 353 | FacRej = 0.1_dp |
---|
| 354 | ELSEIF (RCNTRL(6) > ZERO) THEN |
---|
| 355 | FacRej = RCNTRL(6) |
---|
| 356 | ELSE |
---|
| 357 | PRINT * , 'User-selected FacRej: RCNTRL(6)=', RCNTRL(6) |
---|
| 358 | CALL SDIRK_ErrorMsg(-4,Tinitial,ZERO,Ierr) |
---|
| 359 | END IF |
---|
| 360 | !~~~> FacSafe: Safety Factor in the computation of new step size |
---|
| 361 | IF (RCNTRL(7) == ZERO) THEN |
---|
| 362 | FacSafe = 0.9_dp |
---|
| 363 | ELSEIF (RCNTRL(7) > ZERO) THEN |
---|
| 364 | FacSafe = RCNTRL(7) |
---|
| 365 | ELSE |
---|
| 366 | PRINT * , 'User-selected FacSafe: RCNTRL(7)=', RCNTRL(7) |
---|
| 367 | CALL SDIRK_ErrorMsg(-4,Tinitial,ZERO,Ierr) |
---|
| 368 | END IF |
---|
| 369 | |
---|
| 370 | !~~~> ThetaMin: decides whether the Jacobian should be recomputed |
---|
| 371 | IF(RCNTRL(8) == 0.D0)THEN |
---|
| 372 | ThetaMin = 1.0d-3 |
---|
| 373 | ELSE |
---|
| 374 | ThetaMin = RCNTRL(8) |
---|
| 375 | END IF |
---|
| 376 | |
---|
| 377 | !~~~> Stopping criterion for Newton's method |
---|
| 378 | IF(RCNTRL(9) == ZERO)THEN |
---|
| 379 | NewtonTol = 3.0d-2 |
---|
| 380 | ELSE |
---|
| 381 | NewtonTol = RCNTRL(9) |
---|
| 382 | END IF |
---|
| 383 | |
---|
| 384 | !~~~> Qmin, Qmax: IF Qmin < Hnew/Hold < Qmax, STEP SIZE = CONST. |
---|
| 385 | IF(RCNTRL(10) == ZERO)THEN |
---|
| 386 | Qmin=ONE |
---|
| 387 | ELSE |
---|
| 388 | Qmin=RCNTRL(10) |
---|
| 389 | END IF |
---|
| 390 | IF(RCNTRL(11) == ZERO)THEN |
---|
| 391 | Qmax=1.2D0 |
---|
| 392 | ELSE |
---|
| 393 | Qmax=RCNTRL(11) |
---|
| 394 | END IF |
---|
| 395 | |
---|
| 396 | !~~~> Check if tolerances are reasonable |
---|
| 397 | IF (ITOL == 0) THEN |
---|
| 398 | IF (AbsTol(1) <= ZERO .OR. RelTol(1) <= 10.D0*Roundoff) THEN |
---|
| 399 | PRINT * , ' Scalar AbsTol = ',AbsTol(1) |
---|
| 400 | PRINT * , ' Scalar RelTol = ',RelTol(1) |
---|
| 401 | CALL SDIRK_ErrorMsg(-5,Tinitial,ZERO,Ierr) |
---|
| 402 | END IF |
---|
| 403 | ELSE |
---|
| 404 | DO i=1,N |
---|
| 405 | IF (AbsTol(i) <= 0.D0.OR.RelTol(i) <= 10.D0*Roundoff) THEN |
---|
| 406 | PRINT * , ' AbsTol(',i,') = ',AbsTol(i) |
---|
| 407 | PRINT * , ' RelTol(',i,') = ',RelTol(i) |
---|
| 408 | CALL SDIRK_ErrorMsg(-5,Tinitial,ZERO,Ierr) |
---|
| 409 | END IF |
---|
| 410 | END DO |
---|
| 411 | END IF |
---|
| 412 | |
---|
| 413 | IF (Ierr < 0) RETURN |
---|
| 414 | |
---|
| 415 | CALL SDIRK_Integrator( N,Tinitial,Tfinal,Y,Ierr ) |
---|
| 416 | |
---|
| 417 | |
---|
| 418 | !END SUBROUTINE SDIRK |
---|
| 419 | |
---|
| 420 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 421 | CONTAINS ! PROCEDURES INTERNAL TO SDIRK |
---|
| 422 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 423 | |
---|
| 424 | |
---|
| 425 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 426 | SUBROUTINE SDIRK_Integrator( N,Tinitial,Tfinal,Y,Ierr ) |
---|
| 427 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 428 | |
---|
| 429 | USE KPP_ROOT_Parameters |
---|
| 430 | IMPLICIT NONE |
---|
| 431 | |
---|
| 432 | !~~~> Arguments: |
---|
| 433 | INTEGER, INTENT(IN) :: N |
---|
| 434 | KPP_REAL, INTENT(INOUT) :: Y(NVAR) |
---|
| 435 | KPP_REAL, INTENT(IN) :: Tinitial, Tfinal |
---|
| 436 | INTEGER, INTENT(OUT) :: Ierr |
---|
| 437 | |
---|
| 438 | !~~~> Local variables: |
---|
| 439 | KPP_REAL :: Z(NVAR,Smax), G(NVAR), TMP(NVAR), & |
---|
| 440 | NewtonRate, SCAL(NVAR), RHS(NVAR), & |
---|
| 441 | T, H, Theta, Hratio, NewtonPredictedErr, & |
---|
| 442 | Qnewton, Err, Fac, Hnew,Tdirection, & |
---|
| 443 | NewtonIncrement, NewtonIncrementOld |
---|
| 444 | INTEGER :: j, IER, istage, NewtonIter, IP(NVAR) |
---|
| 445 | LOGICAL :: Reject, FirstStep, SkipJac, SkipLU, NewtonDone |
---|
| 446 | |
---|
| 447 | #ifdef FULL_ALGEBRA |
---|
| 448 | KPP_REAL FJAC(NVAR,NVAR), E(NVAR,NVAR) |
---|
| 449 | #else |
---|
| 450 | KPP_REAL FJAC(LU_NONZERO), E(LU_NONZERO) |
---|
| 451 | #endif |
---|
| 452 | KPP_REAL, PARAMETER :: ZERO = 0.0d0, ONE = 1.0d0 |
---|
| 453 | |
---|
| 454 | |
---|
| 455 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 456 | !~~~> Initializations |
---|
| 457 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 458 | |
---|
| 459 | T = Tinitial |
---|
| 460 | Tdirection = SIGN(ONE,Tfinal-Tinitial) |
---|
| 461 | H = MAX(ABS(Hmin),ABS(Hstart)) |
---|
| 462 | IF (ABS(H) <= 10.D0*Roundoff) H=1.0D-6 |
---|
| 463 | H=MIN(ABS(H),Hmax) |
---|
| 464 | H=SIGN(H,Tdirection) |
---|
| 465 | SkipLU =.FALSE. |
---|
| 466 | SkipJac = .FALSE. |
---|
| 467 | Reject=.FALSE. |
---|
| 468 | FirstStep=.TRUE. |
---|
| 469 | |
---|
| 470 | CALL SDIRK_ErrorScale(N, ITOL, AbsTol, RelTol, Y, SCAL) |
---|
| 471 | |
---|
| 472 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 473 | !~~~> Time loop begins |
---|
| 474 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 475 | Tloop: DO WHILE ( (Tfinal-T)*Tdirection - Roundoff > ZERO ) |
---|
| 476 | |
---|
| 477 | |
---|
| 478 | !~~~> Compute E = 1/(h*gamma)-Jac and its LU decomposition |
---|
| 479 | IF ( .NOT.SkipLU ) THEN ! This time around skip the Jac update and LU |
---|
| 480 | CALL SDIRK_PrepareMatrix ( H, T, Y, FJAC, & |
---|
| 481 | SkipJac, SkipLU, E, IP, Reject, IER ) |
---|
| 482 | IF (IER /= 0) THEN |
---|
| 483 | CALL SDIRK_ErrorMsg(-8,T,H,Ierr); RETURN |
---|
| 484 | END IF |
---|
| 485 | END IF |
---|
| 486 | |
---|
| 487 | IF (ISTATUS(Nstp) > Max_no_steps) THEN |
---|
| 488 | CALL SDIRK_ErrorMsg(-6,T,H,Ierr); RETURN |
---|
| 489 | END IF |
---|
| 490 | IF ( (T+0.1d0*H == T) .OR. (ABS(H) <= Roundoff) ) THEN |
---|
| 491 | CALL SDIRK_ErrorMsg(-7,T,H,Ierr); RETURN |
---|
| 492 | END IF |
---|
| 493 | |
---|
| 494 | stages:DO istage = 1, rkS |
---|
| 495 | |
---|
| 496 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 497 | !~~~> Simplified Newton iterations |
---|
| 498 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 499 | |
---|
| 500 | !~~~> Starting values for Newton iterations |
---|
| 501 | CALL Set2zero(N,Z(1,istage)) |
---|
| 502 | |
---|
| 503 | !~~~> Prepare the loop-independent part of the right-hand side |
---|
| 504 | CALL Set2zero(N,G) |
---|
| 505 | IF (istage > 1) THEN |
---|
| 506 | DO j = 1, istage-1 |
---|
| 507 | ! Gj(:) = sum_j Theta(i,j)*Zj(:) = H * sum_j A(i,j)*Fun(Zj) |
---|
| 508 | CALL WAXPY(N,rkTheta(istage,j),Z(1,j),1,G,1) |
---|
| 509 | ! Zi(:) = sum_j Alpha(i,j)*Zj(:) |
---|
| 510 | IF (StartNewton) THEN |
---|
| 511 | CALL WAXPY(N,rkAlpha(istage,j),Z(1,j),1,Z(1,istage),1) |
---|
| 512 | END IF |
---|
| 513 | END DO |
---|
| 514 | END IF |
---|
| 515 | |
---|
| 516 | !~~~> Initializations for Newton iteration |
---|
| 517 | NewtonDone = .FALSE. |
---|
| 518 | Fac = 0.5d0 ! Step reduction factor if too many iterations |
---|
| 519 | |
---|
| 520 | NewtonLoop:DO NewtonIter = 1, NewtonMaxit |
---|
| 521 | |
---|
| 522 | !~~~> Prepare the loop-dependent part of the right-hand side |
---|
| 523 | CALL WADD(N,Y,Z(1,istage),TMP) ! TMP <- Y + Zi |
---|
| 524 | CALL FUN_CHEM(T+rkC(istage)*H,TMP,RHS) ! RHS <- Fun(Y+Zi) |
---|
| 525 | ISTATUS(Nfun) = ISTATUS(Nfun) + 1 |
---|
| 526 | ! RHS(1:N) = G(1:N) - Z(1:N,istage) + (H*rkGamma)*RHS(1:N) |
---|
| 527 | CALL WSCAL(N, H*rkGamma, RHS, 1) |
---|
| 528 | CALL WAXPY (N, -ONE, Z(1,istage), 1, RHS, 1) |
---|
| 529 | CALL WAXPY (N, ONE, G,1, RHS,1) |
---|
| 530 | |
---|
| 531 | !~~~> Solve the linear system |
---|
| 532 | CALL SDIRK_Solve ( H, N, E, IP, IER, RHS ) |
---|
| 533 | |
---|
| 534 | !~~~> Check convergence of Newton iterations |
---|
| 535 | CALL SDIRK_ErrorNorm(N, RHS, SCAL, NewtonIncrement) |
---|
| 536 | IF ( NewtonIter == 1 ) THEN |
---|
| 537 | Theta = ABS(ThetaMin) |
---|
| 538 | NewtonRate = 2.0d0 |
---|
| 539 | ELSE |
---|
| 540 | Theta = NewtonIncrement/NewtonIncrementOld |
---|
| 541 | IF (Theta < 0.99d0) THEN |
---|
| 542 | NewtonRate = Theta/(ONE-Theta) |
---|
| 543 | ! Predict error at the end of Newton process |
---|
| 544 | NewtonPredictedErr = NewtonIncrement & |
---|
| 545 | *Theta**(NewtonMaxit-NewtonIter)/(ONE-Theta) |
---|
| 546 | IF (NewtonPredictedErr >= NewtonTol) THEN |
---|
| 547 | ! Non-convergence of Newton: predicted error too large |
---|
| 548 | Qnewton = MIN(10.0d0,NewtonPredictedErr/NewtonTol) |
---|
| 549 | Fac = 0.8d0*Qnewton**(-ONE/(1+NewtonMaxit-NewtonIter)) |
---|
| 550 | EXIT NewtonLoop |
---|
| 551 | END IF |
---|
| 552 | ELSE ! Non-convergence of Newton: Theta too large |
---|
| 553 | EXIT NewtonLoop |
---|
| 554 | END IF |
---|
| 555 | END IF |
---|
| 556 | NewtonIncrementOld = NewtonIncrement |
---|
| 557 | ! Update solution: Z(:) <-- Z(:)+RHS(:) |
---|
| 558 | CALL WAXPY(N,ONE,RHS,1,Z(1,istage),1) |
---|
| 559 | |
---|
| 560 | ! Check error in Newton iterations |
---|
| 561 | NewtonDone = (NewtonRate*NewtonIncrement <= NewtonTol) |
---|
| 562 | IF (NewtonDone) EXIT NewtonLoop |
---|
| 563 | |
---|
| 564 | END DO NewtonLoop |
---|
| 565 | |
---|
| 566 | IF (.NOT.NewtonDone) THEN |
---|
| 567 | !CALL RK_ErrorMsg(-12,T,H,Ierr); |
---|
| 568 | H = Fac*H; Reject=.TRUE. |
---|
| 569 | SkipJac = .TRUE.; SkipLU = .FALSE. |
---|
| 570 | CYCLE Tloop |
---|
| 571 | END IF |
---|
| 572 | |
---|
| 573 | !~~~> End of implified Newton iterations |
---|
| 574 | |
---|
| 575 | |
---|
| 576 | END DO stages |
---|
| 577 | |
---|
| 578 | |
---|
| 579 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 580 | !~~~> Error estimation |
---|
| 581 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 582 | ISTATUS(Nstp) = ISTATUS(Nstp) + 1 |
---|
| 583 | CALL Set2zero(N,TMP) |
---|
| 584 | DO i = 1,rkS |
---|
| 585 | IF (rkE(i)/=ZERO) CALL WAXPY(N,rkE(i),Z(1,i),1,TMP,1) |
---|
| 586 | END DO |
---|
| 587 | |
---|
| 588 | CALL SDIRK_Solve( H, N, E, IP, IER, TMP ) |
---|
| 589 | CALL SDIRK_ErrorNorm(N, TMP, SCAL, Err) |
---|
| 590 | |
---|
| 591 | !~~~> Computation of new step size Hnew |
---|
| 592 | Fac = FacSafe*(Err)**(-ONE/rkELO) |
---|
| 593 | Fac = MAX(FacMin,MIN(FacMax,Fac)) |
---|
| 594 | Hnew = H*Fac |
---|
| 595 | |
---|
| 596 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 597 | !~~~> Accept/Reject step |
---|
| 598 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 599 | accept: IF ( Err < ONE ) THEN !~~~> Step is accepted |
---|
| 600 | |
---|
| 601 | FirstStep=.FALSE. |
---|
| 602 | ISTATUS(Nacc) = ISTATUS(Nacc) + 1 |
---|
| 603 | |
---|
| 604 | !~~~> Update time and solution |
---|
| 605 | T = T + H |
---|
| 606 | ! Y(:) <-- Y(:) + Sum_j rkD(j)*Z_j(:) |
---|
| 607 | DO i = 1,rkS |
---|
| 608 | IF (rkD(i)/=ZERO) CALL WAXPY(N,rkD(i),Z(1,i),1,Y,1) |
---|
| 609 | END DO |
---|
| 610 | |
---|
| 611 | !~~~> Update scaling coefficients |
---|
| 612 | CALL SDIRK_ErrorScale(N, ITOL, AbsTol, RelTol, Y, SCAL) |
---|
| 613 | |
---|
| 614 | !~~~> Next time step |
---|
| 615 | Hnew = Tdirection*MIN(ABS(Hnew),Hmax) |
---|
| 616 | ! Last T and H |
---|
| 617 | RSTATUS(Ntexit) = T |
---|
| 618 | RSTATUS(Nhexit) = H |
---|
| 619 | RSTATUS(Nhnew) = Hnew |
---|
| 620 | ! No step increase after a rejection |
---|
| 621 | IF (Reject) Hnew = Tdirection*MIN(ABS(Hnew),ABS(H)) |
---|
| 622 | Reject = .FALSE. |
---|
| 623 | IF ((T+Hnew/Qmin-Tfinal)*Tdirection > ZERO) THEN |
---|
| 624 | H = Tfinal-T |
---|
| 625 | ELSE |
---|
| 626 | Hratio=Hnew/H |
---|
| 627 | ! If step not changed too much keep Jacobian and reuse LU |
---|
| 628 | SkipLU = ( (Theta <= ThetaMin) .AND. (Hratio >= Qmin) & |
---|
| 629 | .AND. (Hratio <= Qmax) ) |
---|
| 630 | IF (.NOT.SkipLU) H = Hnew |
---|
| 631 | END IF |
---|
| 632 | ! If convergence is fast enough, do not update Jacobian |
---|
| 633 | ! SkipJac = (Theta <= ThetaMin) |
---|
| 634 | SkipJac = .FALSE. |
---|
| 635 | |
---|
| 636 | ELSE accept !~~~> Step is rejected |
---|
| 637 | |
---|
| 638 | IF (FirstStep .OR. Reject) THEN |
---|
| 639 | H = FacRej*H |
---|
| 640 | ELSE |
---|
| 641 | H = Hnew |
---|
| 642 | END IF |
---|
| 643 | Reject = .TRUE. |
---|
| 644 | SkipJac = .TRUE. |
---|
| 645 | SkipLU = .FALSE. |
---|
| 646 | IF (ISTATUS(Nacc) >= 1) ISTATUS(Nrej) = ISTATUS(Nrej) + 1 |
---|
| 647 | |
---|
| 648 | END IF accept |
---|
| 649 | |
---|
| 650 | END DO Tloop |
---|
| 651 | |
---|
| 652 | ! Successful return |
---|
| 653 | Ierr = 1 |
---|
| 654 | |
---|
| 655 | END SUBROUTINE SDIRK_Integrator |
---|
| 656 | |
---|
| 657 | |
---|
| 658 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 659 | SUBROUTINE SDIRK_ErrorScale(N, ITOL, AbsTol, RelTol, Y, SCAL) |
---|
| 660 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 661 | IMPLICIT NONE |
---|
| 662 | INTEGER :: i, N, ITOL |
---|
| 663 | KPP_REAL :: AbsTol(NVAR), RelTol(NVAR), & |
---|
| 664 | Y(NVAR), SCAL(NVAR) |
---|
| 665 | IF (ITOL == 0) THEN |
---|
| 666 | DO i=1,NVAR |
---|
| 667 | SCAL(i) = ONE / ( AbsTol(1)+RelTol(1)*ABS(Y(i)) ) |
---|
| 668 | END DO |
---|
| 669 | ELSE |
---|
| 670 | DO i=1,NVAR |
---|
| 671 | SCAL(i) = ONE / ( AbsTol(i)+RelTol(i)*ABS(Y(i)) ) |
---|
| 672 | END DO |
---|
| 673 | END IF |
---|
| 674 | END SUBROUTINE SDIRK_ErrorScale |
---|
| 675 | |
---|
| 676 | |
---|
| 677 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 678 | SUBROUTINE SDIRK_ErrorNorm(N, Y, SCAL, Err) |
---|
| 679 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 680 | ! |
---|
| 681 | INTEGER :: i, N |
---|
| 682 | KPP_REAL :: Y(N), SCAL(N), Err |
---|
| 683 | Err = ZERO |
---|
| 684 | DO i=1,N |
---|
| 685 | Err = Err+(Y(i)*SCAL(i))**2 |
---|
| 686 | END DO |
---|
| 687 | Err = MAX( SQRT(Err/DBLE(N)), 1.0d-10 ) |
---|
| 688 | ! |
---|
| 689 | END SUBROUTINE SDIRK_ErrorNorm |
---|
| 690 | |
---|
| 691 | |
---|
| 692 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 693 | SUBROUTINE SDIRK_ErrorMsg(Code,T,H,Ierr) |
---|
| 694 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 695 | ! Handles all error messages |
---|
| 696 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 697 | |
---|
| 698 | KPP_REAL, INTENT(IN) :: T, H |
---|
| 699 | INTEGER, INTENT(IN) :: Code |
---|
| 700 | INTEGER, INTENT(OUT) :: Ierr |
---|
| 701 | |
---|
| 702 | Ierr = Code |
---|
| 703 | PRINT * , & |
---|
| 704 | 'Forced exit from SDIRK due to the following error:' |
---|
| 705 | |
---|
| 706 | SELECT CASE (Code) |
---|
| 707 | CASE (-1) |
---|
| 708 | PRINT * , '--> Improper value for maximal no of steps' |
---|
| 709 | CASE (-2) |
---|
| 710 | PRINT * , '--> Improper value for maximal no of Newton iterations' |
---|
| 711 | CASE (-3) |
---|
| 712 | PRINT * , '--> Hmin/Hmax/Hstart must be positive' |
---|
| 713 | CASE (-4) |
---|
| 714 | PRINT * , '--> FacMin/FacMax/FacRej must be positive' |
---|
| 715 | CASE (-5) |
---|
| 716 | PRINT * , '--> Improper tolerance values' |
---|
| 717 | CASE (-6) |
---|
| 718 | PRINT * , '--> No of steps exceeds maximum bound' |
---|
| 719 | CASE (-7) |
---|
| 720 | PRINT * , '--> Step size too small: T + 10*H = T', & |
---|
| 721 | ' or H < Roundoff' |
---|
| 722 | CASE (-8) |
---|
| 723 | PRINT * , '--> Matrix is repeatedly singular' |
---|
| 724 | CASE DEFAULT |
---|
| 725 | PRINT *, 'Unknown Error code: ', Code |
---|
| 726 | END SELECT |
---|
| 727 | |
---|
| 728 | PRINT *, "T=", T, "and H=", H |
---|
| 729 | |
---|
| 730 | END SUBROUTINE SDIRK_ErrorMsg |
---|
| 731 | |
---|
| 732 | |
---|
| 733 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 734 | SUBROUTINE SDIRK_PrepareMatrix ( H, T, Y, FJAC, & |
---|
| 735 | SkipJac, SkipLU, E, IP, Reject, ISING ) |
---|
| 736 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 737 | !~~~> Compute the matrix E = 1/(H*GAMMA)*Jac, and its decomposition |
---|
| 738 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 739 | |
---|
| 740 | IMPLICIT NONE |
---|
| 741 | |
---|
| 742 | KPP_REAL, INTENT(INOUT) :: H |
---|
| 743 | KPP_REAL, INTENT(IN) :: T, Y(NVAR) |
---|
| 744 | LOGICAL, INTENT(INOUT) :: SkipJac,SkipLU,Reject |
---|
| 745 | INTEGER, INTENT(OUT) :: ISING, IP(NVAR) |
---|
| 746 | #ifdef FULL_ALGEBRA |
---|
| 747 | KPP_REAL, INTENT(INOUT) :: FJAC(NVAR,NVAR) |
---|
| 748 | KPP_REAL, INTENT(OUT) :: E(NVAR,NVAR) |
---|
| 749 | #else |
---|
| 750 | KPP_REAL, INTENT(INOUT) :: FJAC(LU_NONZERO) |
---|
| 751 | KPP_REAL, INTENT(OUT) :: E(LU_NONZERO) |
---|
| 752 | #endif |
---|
| 753 | KPP_REAL :: HGammaInv |
---|
| 754 | INTEGER :: i, j, ConsecutiveSng |
---|
| 755 | |
---|
| 756 | ConsecutiveSng = 0 |
---|
| 757 | ISING = 1 |
---|
| 758 | |
---|
| 759 | Hloop: DO WHILE (ISING /= 0) |
---|
| 760 | |
---|
| 761 | HGammaInv = ONE/(H*rkGamma) |
---|
| 762 | |
---|
| 763 | !~~~> Compute the Jacobian |
---|
| 764 | ! IF (SkipJac) THEN |
---|
| 765 | ! SkipJac = .FALSE. |
---|
| 766 | ! ELSE |
---|
| 767 | IF (.NOT. SkipJac) THEN |
---|
| 768 | CALL JAC_CHEM( T, Y, FJAC ) |
---|
| 769 | ISTATUS(Njac) = ISTATUS(Njac) + 1 |
---|
| 770 | END IF |
---|
| 771 | |
---|
| 772 | #ifdef FULL_ALGEBRA |
---|
| 773 | DO j=1,NVAR |
---|
| 774 | DO i=1,NVAR |
---|
| 775 | E(i,j) = -FJAC(i,j) |
---|
| 776 | END DO |
---|
| 777 | E(j,j) = E(j,j)+HGammaInv |
---|
| 778 | END DO |
---|
| 779 | CALL DGETRF( NVAR, NVAR, E, NVAR, IP, ISING ) |
---|
| 780 | #else |
---|
| 781 | DO i = 1,LU_NONZERO |
---|
| 782 | E(i) = -FJAC(i) |
---|
| 783 | END DO |
---|
| 784 | DO i = 1,NVAR |
---|
| 785 | j = LU_DIAG(i); E(j) = E(j) + HGammaInv |
---|
| 786 | END DO |
---|
| 787 | CALL KppDecomp ( E, ISING) |
---|
| 788 | IP(1) = 1 |
---|
| 789 | #endif |
---|
| 790 | ISTATUS(Ndec) = ISTATUS(Ndec) + 1 |
---|
| 791 | |
---|
| 792 | IF (ISING /= 0) THEN |
---|
| 793 | WRITE (6,*) ' MATRIX IS SINGULAR, ISING=',ISING,' T=',T,' H=',H |
---|
| 794 | ISTATUS(Nsng) = ISTATUS(Nsng) + 1; ConsecutiveSng = ConsecutiveSng + 1 |
---|
| 795 | IF (ConsecutiveSng >= 6) RETURN ! Failure |
---|
| 796 | H = 0.5d0*H |
---|
| 797 | SkipJac = .TRUE. |
---|
| 798 | SkipLU = .FALSE. |
---|
| 799 | Reject = .TRUE. |
---|
| 800 | END IF |
---|
| 801 | |
---|
| 802 | END DO Hloop |
---|
| 803 | |
---|
| 804 | END SUBROUTINE SDIRK_PrepareMatrix |
---|
| 805 | |
---|
| 806 | |
---|
| 807 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 808 | SUBROUTINE SDIRK_Solve ( H, N, E, IP, ISING, RHS ) |
---|
| 809 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 810 | !~~~> Solves the system (H*Gamma-Jac)*x = RHS |
---|
| 811 | ! using the LU decomposition of E = I - 1/(H*Gamma)*Jac |
---|
| 812 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 813 | IMPLICIT NONE |
---|
| 814 | INTEGER, INTENT(IN) :: N, IP(N), ISING |
---|
| 815 | KPP_REAL, INTENT(IN) :: H |
---|
| 816 | #ifdef FULL_ALGEBRA |
---|
| 817 | KPP_REAL, INTENT(IN) :: E(NVAR,NVAR) |
---|
| 818 | #else |
---|
| 819 | KPP_REAL, INTENT(IN) :: E(LU_NONZERO) |
---|
| 820 | #endif |
---|
| 821 | KPP_REAL, INTENT(INOUT) :: RHS(N) |
---|
| 822 | KPP_REAL :: HGammaInv |
---|
| 823 | |
---|
| 824 | HGammaInv = ONE/(H*rkGamma) |
---|
| 825 | CALL WSCAL(N,HGammaInv,RHS,1) |
---|
| 826 | #ifdef FULL_ALGEBRA |
---|
| 827 | CALL DGETRS( 'N', N, 1, E, N, IP, RHS, N, ISING ) |
---|
| 828 | #else |
---|
| 829 | CALL KppSolve(E, RHS) |
---|
| 830 | #endif |
---|
| 831 | ISTATUS(Nsol) = ISTATUS(Nsol) + 1 |
---|
| 832 | |
---|
| 833 | END SUBROUTINE SDIRK_Solve |
---|
| 834 | |
---|
| 835 | |
---|
| 836 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 837 | SUBROUTINE Sdirk4a |
---|
| 838 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 839 | sdMethod = S4A |
---|
| 840 | ! Number of stages |
---|
| 841 | rkS = 5 |
---|
| 842 | |
---|
| 843 | ! Method coefficients |
---|
| 844 | rkGamma = .2666666666666666666666666666666667d0 |
---|
| 845 | |
---|
| 846 | rkA(1,1) = .2666666666666666666666666666666667d0 |
---|
| 847 | rkA(2,1) = .5000000000000000000000000000000000d0 |
---|
| 848 | rkA(2,2) = .2666666666666666666666666666666667d0 |
---|
| 849 | rkA(3,1) = .3541539528432732316227461858529820d0 |
---|
| 850 | rkA(3,2) = -.5415395284327323162274618585298197d-1 |
---|
| 851 | rkA(3,3) = .2666666666666666666666666666666667d0 |
---|
| 852 | rkA(4,1) = .8515494131138652076337791881433756d-1 |
---|
| 853 | rkA(4,2) = -.6484332287891555171683963466229754d-1 |
---|
| 854 | rkA(4,3) = .7915325296404206392428857585141242d-1 |
---|
| 855 | rkA(4,4) = .2666666666666666666666666666666667d0 |
---|
| 856 | rkA(5,1) = 2.100115700566932777970612055999074d0 |
---|
| 857 | rkA(5,2) = -.7677800284445976813343102185062276d0 |
---|
| 858 | rkA(5,3) = 2.399816361080026398094746205273880d0 |
---|
| 859 | rkA(5,4) = -2.998818699869028161397714709433394d0 |
---|
| 860 | rkA(5,5) = .2666666666666666666666666666666667d0 |
---|
| 861 | |
---|
| 862 | rkB(1) = 2.100115700566932777970612055999074d0 |
---|
| 863 | rkB(2) = -.7677800284445976813343102185062276d0 |
---|
| 864 | rkB(3) = 2.399816361080026398094746205273880d0 |
---|
| 865 | rkB(4) = -2.998818699869028161397714709433394d0 |
---|
| 866 | rkB(5) = .2666666666666666666666666666666667d0 |
---|
| 867 | |
---|
| 868 | rkBhat(1)= 2.885264204387193942183851612883390d0 |
---|
| 869 | rkBhat(2)= -.1458793482962771337341223443218041d0 |
---|
| 870 | rkBhat(3)= 2.390008682465139866479830743628554d0 |
---|
| 871 | rkBhat(4)= -4.129393538556056674929560012190140d0 |
---|
| 872 | rkBhat(5)= 0.d0 |
---|
| 873 | |
---|
| 874 | rkC(1) = .2666666666666666666666666666666667d0 |
---|
| 875 | rkC(2) = .7666666666666666666666666666666667d0 |
---|
| 876 | rkC(3) = .5666666666666666666666666666666667d0 |
---|
| 877 | rkC(4) = .3661315380631796996374935266701191d0 |
---|
| 878 | rkC(5) = 1.d0 |
---|
| 879 | |
---|
| 880 | ! Ynew = Yold + h*Sum_i {rkB_i*k_i} = Yold + Sum_i {rkD_i*Z_i} |
---|
| 881 | rkD(1) = 0.d0 |
---|
| 882 | rkD(2) = 0.d0 |
---|
| 883 | rkD(3) = 0.d0 |
---|
| 884 | rkD(4) = 0.d0 |
---|
| 885 | rkD(5) = 1.d0 |
---|
| 886 | |
---|
| 887 | ! Err = h * Sum_i {(rkB_i-rkBhat_i)*k_i} = Sum_i {rkE_i*Z_i} |
---|
| 888 | rkE(1) = -.6804000050475287124787034884002302d0 |
---|
| 889 | rkE(2) = 1.558961944525217193393931795738823d0 |
---|
| 890 | rkE(3) = -13.55893003128907927748632408763868d0 |
---|
| 891 | rkE(4) = 15.48522576958521253098585004571302d0 |
---|
| 892 | rkE(5) = 1.d0 |
---|
| 893 | |
---|
| 894 | ! Local order of Err estimate |
---|
| 895 | rkElo = 4 |
---|
| 896 | |
---|
| 897 | ! h*Sum_j {rkA_ij*k_j} = Sum_j {rkTheta_ij*Z_j} |
---|
| 898 | rkTheta(2,1) = 1.875000000000000000000000000000000d0 |
---|
| 899 | rkTheta(3,1) = 1.708847304091539528432732316227462d0 |
---|
| 900 | rkTheta(3,2) = -.2030773231622746185852981969486824d0 |
---|
| 901 | rkTheta(4,1) = .2680325578937783958847157206823118d0 |
---|
| 902 | rkTheta(4,2) = -.1828840955527181631794050728644549d0 |
---|
| 903 | rkTheta(4,3) = .2968246986151577397160821594427966d0 |
---|
| 904 | rkTheta(5,1) = .9096171815241460655379433581446771d0 |
---|
| 905 | rkTheta(5,2) = -3.108254967778352416114774430509465d0 |
---|
| 906 | rkTheta(5,3) = 12.33727431701306195581826123274001d0 |
---|
| 907 | rkTheta(5,4) = -11.24557012450885560524143016037523d0 |
---|
| 908 | |
---|
| 909 | ! Starting value for Newton iterations: Z_i^0 = Sum_j {rkAlpha_ij*Z_j} |
---|
| 910 | rkAlpha(2,1) = 2.875000000000000000000000000000000d0 |
---|
| 911 | rkAlpha(3,1) = .8500000000000000000000000000000000d0 |
---|
| 912 | rkAlpha(3,2) = .4434782608695652173913043478260870d0 |
---|
| 913 | rkAlpha(4,1) = .7352046091658870564637910527807370d0 |
---|
| 914 | rkAlpha(4,2) = -.9525565003057343527941920657462074d-1 |
---|
| 915 | rkAlpha(4,3) = .4290111305453813852259481840631738d0 |
---|
| 916 | rkAlpha(5,1) = -16.10898993405067684831655675112808d0 |
---|
| 917 | rkAlpha(5,2) = 6.559571569643355712998131800797873d0 |
---|
| 918 | rkAlpha(5,3) = -15.90772144271326504260996815012482d0 |
---|
| 919 | rkAlpha(5,4) = 25.34908987169226073668861694892683d0 |
---|
| 920 | |
---|
| 921 | !~~~> Coefficients for continuous solution |
---|
| 922 | ! rkD(1,1)= 24.74416644927758d0 |
---|
| 923 | ! rkD(1,2)= -4.325375951824688d0 |
---|
| 924 | ! rkD(1,3)= 41.39683763286316d0 |
---|
| 925 | ! rkD(1,4)= -61.04144619901784d0 |
---|
| 926 | ! rkD(1,5)= -3.391332232917013d0 |
---|
| 927 | ! rkD(2,1)= -51.98245719616925d0 |
---|
| 928 | ! rkD(2,2)= 10.52501981094525d0 |
---|
| 929 | ! rkD(2,3)= -154.2067922191855d0 |
---|
| 930 | ! rkD(2,4)= 214.3082125319825d0 |
---|
| 931 | ! rkD(2,5)= 14.71166018088679d0 |
---|
| 932 | ! rkD(3,1)= 33.14347947522142d0 |
---|
| 933 | ! rkD(3,2)= -19.72986789558523d0 |
---|
| 934 | ! rkD(3,3)= 230.4878502285804d0 |
---|
| 935 | ! rkD(3,4)= -287.6629744338197d0 |
---|
| 936 | ! rkD(3,5)= -18.99932366302254d0 |
---|
| 937 | ! rkD(4,1)= -5.905188728329743d0 |
---|
| 938 | ! rkD(4,2)= 13.53022403646467d0 |
---|
| 939 | ! rkD(4,3)= -117.6778956422581d0 |
---|
| 940 | ! rkD(4,4)= 134.3962081008550d0 |
---|
| 941 | ! rkD(4,5)= 8.678995715052762d0 |
---|
| 942 | ! |
---|
| 943 | ! DO i=1,4 ! CONTi <-- Sum_j rkD(i,j)*Zj |
---|
| 944 | ! CALL Set2zero(N,CONT(1,i)) |
---|
| 945 | ! DO j = 1,rkS |
---|
| 946 | ! CALL WAXPY(N,rkD(i,j),Z(1,j),1,CONT(1,i),1) |
---|
| 947 | ! END DO |
---|
| 948 | ! END DO |
---|
| 949 | |
---|
| 950 | rkELO = 4.0d0 |
---|
| 951 | |
---|
| 952 | END SUBROUTINE Sdirk4a |
---|
| 953 | |
---|
| 954 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 955 | SUBROUTINE Sdirk4b |
---|
| 956 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 957 | |
---|
| 958 | sdMethod = S4B |
---|
| 959 | ! Number of stages |
---|
| 960 | rkS = 5 |
---|
| 961 | |
---|
| 962 | ! Method coefficients |
---|
| 963 | rkGamma = .25d0 |
---|
| 964 | |
---|
| 965 | rkA(1,1) = 0.25d0 |
---|
| 966 | rkA(2,1) = 0.5d00 |
---|
| 967 | rkA(2,2) = 0.25d0 |
---|
| 968 | rkA(3,1) = 0.34d0 |
---|
| 969 | rkA(3,2) =-0.40d-1 |
---|
| 970 | rkA(3,3) = 0.25d0 |
---|
| 971 | rkA(4,1) = 0.2727941176470588235294117647058824d0 |
---|
| 972 | rkA(4,2) =-0.5036764705882352941176470588235294d-1 |
---|
| 973 | rkA(4,3) = 0.2757352941176470588235294117647059d-1 |
---|
| 974 | rkA(4,4) = 0.25d0 |
---|
| 975 | rkA(5,1) = 1.041666666666666666666666666666667d0 |
---|
| 976 | rkA(5,2) =-1.020833333333333333333333333333333d0 |
---|
| 977 | rkA(5,3) = 7.812500000000000000000000000000000d0 |
---|
| 978 | rkA(5,4) =-7.083333333333333333333333333333333d0 |
---|
| 979 | rkA(5,5) = 0.25d0 |
---|
| 980 | |
---|
| 981 | rkB(1) = 1.041666666666666666666666666666667d0 |
---|
| 982 | rkB(2) = -1.020833333333333333333333333333333d0 |
---|
| 983 | rkB(3) = 7.812500000000000000000000000000000d0 |
---|
| 984 | rkB(4) = -7.083333333333333333333333333333333d0 |
---|
| 985 | rkB(5) = 0.250000000000000000000000000000000d0 |
---|
| 986 | |
---|
| 987 | rkBhat(1)= 1.069791666666666666666666666666667d0 |
---|
| 988 | rkBhat(2)= -0.894270833333333333333333333333333d0 |
---|
| 989 | rkBhat(3)= 7.695312500000000000000000000000000d0 |
---|
| 990 | rkBhat(4)= -7.083333333333333333333333333333333d0 |
---|
| 991 | rkBhat(5)= 0.212500000000000000000000000000000d0 |
---|
| 992 | |
---|
| 993 | rkC(1) = 0.25d0 |
---|
| 994 | rkC(2) = 0.75d0 |
---|
| 995 | rkC(3) = 0.55d0 |
---|
| 996 | rkC(4) = 0.50d0 |
---|
| 997 | rkC(5) = 1.00d0 |
---|
| 998 | |
---|
| 999 | ! Ynew = Yold + h*Sum_i {rkB_i*k_i} = Yold + Sum_i {rkD_i*Z_i} |
---|
| 1000 | rkD(1) = 0.0d0 |
---|
| 1001 | rkD(2) = 0.0d0 |
---|
| 1002 | rkD(3) = 0.0d0 |
---|
| 1003 | rkD(4) = 0.0d0 |
---|
| 1004 | rkD(5) = 1.0d0 |
---|
| 1005 | |
---|
| 1006 | ! Err = h * Sum_i {(rkB_i-rkBhat_i)*k_i} = Sum_i {rkE_i*Z_i} |
---|
| 1007 | rkE(1) = 0.5750d0 |
---|
| 1008 | rkE(2) = 0.2125d0 |
---|
| 1009 | rkE(3) = -4.6875d0 |
---|
| 1010 | rkE(4) = 4.2500d0 |
---|
| 1011 | rkE(5) = 0.1500d0 |
---|
| 1012 | |
---|
| 1013 | ! Local order of Err estimate |
---|
| 1014 | rkElo = 4 |
---|
| 1015 | |
---|
| 1016 | ! h*Sum_j {rkA_ij*k_j} = Sum_j {rkTheta_ij*Z_j} |
---|
| 1017 | rkTheta(2,1) = 2.d0 |
---|
| 1018 | rkTheta(3,1) = 1.680000000000000000000000000000000d0 |
---|
| 1019 | rkTheta(3,2) = -.1600000000000000000000000000000000d0 |
---|
| 1020 | rkTheta(4,1) = 1.308823529411764705882352941176471d0 |
---|
| 1021 | rkTheta(4,2) = -.1838235294117647058823529411764706d0 |
---|
| 1022 | rkTheta(4,3) = 0.1102941176470588235294117647058824d0 |
---|
| 1023 | rkTheta(5,1) = -3.083333333333333333333333333333333d0 |
---|
| 1024 | rkTheta(5,2) = -4.291666666666666666666666666666667d0 |
---|
| 1025 | rkTheta(5,3) = 34.37500000000000000000000000000000d0 |
---|
| 1026 | rkTheta(5,4) = -28.33333333333333333333333333333333d0 |
---|
| 1027 | |
---|
| 1028 | ! Starting value for Newton iterations: Z_i^0 = Sum_j {rkAlpha_ij*Z_j} |
---|
| 1029 | rkAlpha(2,1) = 3. |
---|
| 1030 | rkAlpha(3,1) = .8800000000000000000000000000000000d0 |
---|
| 1031 | rkAlpha(3,2) = .4400000000000000000000000000000000d0 |
---|
| 1032 | rkAlpha(4,1) = .1666666666666666666666666666666667d0 |
---|
| 1033 | rkAlpha(4,2) = -.8333333333333333333333333333333333d-1 |
---|
| 1034 | rkAlpha(4,3) = .9469696969696969696969696969696970d0 |
---|
| 1035 | rkAlpha(5,1) = -6.d0 |
---|
| 1036 | rkAlpha(5,2) = 9.d0 |
---|
| 1037 | rkAlpha(5,3) = -56.81818181818181818181818181818182d0 |
---|
| 1038 | rkAlpha(5,4) = 54.d0 |
---|
| 1039 | |
---|
| 1040 | END SUBROUTINE Sdirk4b |
---|
| 1041 | |
---|
| 1042 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1043 | SUBROUTINE Sdirk2a |
---|
| 1044 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1045 | |
---|
| 1046 | sdMethod = S2A |
---|
| 1047 | ! Number of stages |
---|
| 1048 | rkS = 2 |
---|
| 1049 | |
---|
| 1050 | ! Method coefficients |
---|
| 1051 | rkGamma = .2928932188134524755991556378951510d0 |
---|
| 1052 | |
---|
| 1053 | rkA(1,1) = .2928932188134524755991556378951510d0 |
---|
| 1054 | rkA(2,1) = .7071067811865475244008443621048490d0 |
---|
| 1055 | rkA(2,2) = .2928932188134524755991556378951510d0 |
---|
| 1056 | |
---|
| 1057 | rkB(1) = .7071067811865475244008443621048490d0 |
---|
| 1058 | rkB(2) = .2928932188134524755991556378951510d0 |
---|
| 1059 | |
---|
| 1060 | rkBhat(1)= .6666666666666666666666666666666667d0 |
---|
| 1061 | rkBhat(2)= .3333333333333333333333333333333333d0 |
---|
| 1062 | |
---|
| 1063 | rkC(1) = 0.292893218813452475599155637895151d0 |
---|
| 1064 | rkC(2) = 1.0d0 |
---|
| 1065 | |
---|
| 1066 | ! Ynew = Yold + h*Sum_i {rkB_i*k_i} = Yold + Sum_i {rkD_i*Z_i} |
---|
| 1067 | rkD(1) = 0.0d0 |
---|
| 1068 | rkD(2) = 1.0d0 |
---|
| 1069 | |
---|
| 1070 | ! Err = h * Sum_i {(rkB_i-rkBhat_i)*k_i} = Sum_i {rkE_i*Z_i} |
---|
| 1071 | rkE(1) = 0.4714045207910316829338962414032326d0 |
---|
| 1072 | rkE(2) = -0.1380711874576983496005629080698993d0 |
---|
| 1073 | |
---|
| 1074 | ! Local order of Err estimate |
---|
| 1075 | rkElo = 2 |
---|
| 1076 | |
---|
| 1077 | ! h*Sum_j {rkA_ij*k_j} = Sum_j {rkTheta_ij*Z_j} |
---|
| 1078 | rkTheta(2,1) = 2.414213562373095048801688724209698d0 |
---|
| 1079 | |
---|
| 1080 | ! Starting value for Newton iterations: Z_i^0 = Sum_j {rkAlpha_ij*Z_j} |
---|
| 1081 | rkAlpha(2,1) = 3.414213562373095048801688724209698d0 |
---|
| 1082 | |
---|
| 1083 | END SUBROUTINE Sdirk2a |
---|
| 1084 | |
---|
| 1085 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1086 | SUBROUTINE Sdirk2b |
---|
| 1087 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1088 | |
---|
| 1089 | sdMethod = S2B |
---|
| 1090 | ! Number of stages |
---|
| 1091 | rkS = 2 |
---|
| 1092 | |
---|
| 1093 | ! Method coefficients |
---|
| 1094 | rkGamma = 1.707106781186547524400844362104849d0 |
---|
| 1095 | |
---|
| 1096 | rkA(1,1) = 1.707106781186547524400844362104849d0 |
---|
| 1097 | rkA(2,1) = -.707106781186547524400844362104849d0 |
---|
| 1098 | rkA(2,2) = 1.707106781186547524400844362104849d0 |
---|
| 1099 | |
---|
| 1100 | rkB(1) = -.707106781186547524400844362104849d0 |
---|
| 1101 | rkB(2) = 1.707106781186547524400844362104849d0 |
---|
| 1102 | |
---|
| 1103 | rkBhat(1)= .6666666666666666666666666666666667d0 |
---|
| 1104 | rkBhat(2)= .3333333333333333333333333333333333d0 |
---|
| 1105 | |
---|
| 1106 | rkC(1) = 1.707106781186547524400844362104849d0 |
---|
| 1107 | rkC(2) = 1.0d0 |
---|
| 1108 | |
---|
| 1109 | ! Ynew = Yold + h*Sum_i {rkB_i*k_i} = Yold + Sum_i {rkD_i*Z_i} |
---|
| 1110 | rkD(1) = 0.0d0 |
---|
| 1111 | rkD(2) = 1.0d0 |
---|
| 1112 | |
---|
| 1113 | ! Err = h * Sum_i {(rkB_i-rkBhat_i)*k_i} = Sum_i {rkE_i*Z_i} |
---|
| 1114 | rkE(1) = -.4714045207910316829338962414032326d0 |
---|
| 1115 | rkE(2) = .8047378541243650162672295747365659d0 |
---|
| 1116 | |
---|
| 1117 | ! Local order of Err estimate |
---|
| 1118 | rkElo = 2 |
---|
| 1119 | |
---|
| 1120 | ! h*Sum_j {rkA_ij*k_j} = Sum_j {rkTheta_ij*Z_j} |
---|
| 1121 | rkTheta(2,1) = -.414213562373095048801688724209698d0 |
---|
| 1122 | |
---|
| 1123 | ! Starting value for Newton iterations: Z_i^0 = Sum_j {rkAlpha_ij*Z_j} |
---|
| 1124 | rkAlpha(2,1) = .5857864376269049511983112757903019d0 |
---|
| 1125 | |
---|
| 1126 | END SUBROUTINE Sdirk2b |
---|
| 1127 | |
---|
| 1128 | |
---|
| 1129 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1130 | SUBROUTINE Sdirk3a |
---|
| 1131 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1132 | |
---|
| 1133 | sdMethod = S3A |
---|
| 1134 | ! Number of stages |
---|
| 1135 | rkS = 3 |
---|
| 1136 | |
---|
| 1137 | ! Method coefficients |
---|
| 1138 | rkGamma = .2113248654051871177454256097490213d0 |
---|
| 1139 | |
---|
| 1140 | rkA(1,1) = .2113248654051871177454256097490213d0 |
---|
| 1141 | rkA(2,1) = .2113248654051871177454256097490213d0 |
---|
| 1142 | rkA(2,2) = .2113248654051871177454256097490213d0 |
---|
| 1143 | rkA(3,1) = .2113248654051871177454256097490213d0 |
---|
| 1144 | rkA(3,2) = .5773502691896257645091487805019573d0 |
---|
| 1145 | rkA(3,3) = .2113248654051871177454256097490213d0 |
---|
| 1146 | |
---|
| 1147 | rkB(1) = .2113248654051871177454256097490213d0 |
---|
| 1148 | rkB(2) = .5773502691896257645091487805019573d0 |
---|
| 1149 | rkB(3) = .2113248654051871177454256097490213d0 |
---|
| 1150 | |
---|
| 1151 | rkBhat(1)= .2113248654051871177454256097490213d0 |
---|
| 1152 | rkBhat(2)= .6477918909913548037576239837516312d0 |
---|
| 1153 | rkBhat(3)= .1408832436034580784969504064993475d0 |
---|
| 1154 | |
---|
| 1155 | rkC(1) = .2113248654051871177454256097490213d0 |
---|
| 1156 | rkC(2) = .4226497308103742354908512194980427d0 |
---|
| 1157 | rkC(3) = 1.d0 |
---|
| 1158 | |
---|
| 1159 | ! Ynew = Yold + h*Sum_i {rkB_i*k_i} = Yold + Sum_i {rkD_i*Z_i} |
---|
| 1160 | rkD(1) = 0.d0 |
---|
| 1161 | rkD(2) = 0.d0 |
---|
| 1162 | rkD(3) = 1.d0 |
---|
| 1163 | |
---|
| 1164 | ! Err = h * Sum_i {(rkB_i-rkBhat_i)*k_i} = Sum_i {rkE_i*Z_i} |
---|
| 1165 | rkE(1) = 0.9106836025229590978424821138352906d0 |
---|
| 1166 | rkE(2) = -1.244016935856292431175815447168624d0 |
---|
| 1167 | rkE(3) = 0.3333333333333333333333333333333333d0 |
---|
| 1168 | |
---|
| 1169 | ! Local order of Err estimate |
---|
| 1170 | rkElo = 2 |
---|
| 1171 | |
---|
| 1172 | ! h*Sum_j {rkA_ij*k_j} = Sum_j {rkTheta_ij*Z_j} |
---|
| 1173 | rkTheta(2,1) = 1.0d0 |
---|
| 1174 | rkTheta(3,1) = -1.732050807568877293527446341505872d0 |
---|
| 1175 | rkTheta(3,2) = 2.732050807568877293527446341505872d0 |
---|
| 1176 | |
---|
| 1177 | ! Starting value for Newton iterations: Z_i^0 = Sum_j {rkAlpha_ij*Z_j} |
---|
| 1178 | rkAlpha(2,1) = 2.0d0 |
---|
| 1179 | rkAlpha(3,1) = -12.92820323027550917410978536602349d0 |
---|
| 1180 | rkAlpha(3,2) = 8.83012701892219323381861585376468d0 |
---|
| 1181 | |
---|
| 1182 | END SUBROUTINE Sdirk3a |
---|
| 1183 | |
---|
| 1184 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1185 | END SUBROUTINE SDIRK ! AND ALL ITS INTERNAL PROCEDURES |
---|
| 1186 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1187 | |
---|
| 1188 | |
---|
| 1189 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1190 | SUBROUTINE FUN_CHEM( T, Y, P ) |
---|
| 1191 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1192 | |
---|
| 1193 | USE KPP_ROOT_Parameters, ONLY: NVAR, LU_NONZERO |
---|
| 1194 | USE KPP_ROOT_Global, ONLY: FIX, RCONST, TIME |
---|
| 1195 | USE KPP_ROOT_Function, ONLY: Fun |
---|
| 1196 | USE KPP_ROOT_Rates, ONLY: Update_SUN, Update_RCONST, Update_PHOTO |
---|
| 1197 | IMPLICIT NONE |
---|
| 1198 | |
---|
| 1199 | KPP_REAL :: T, Told |
---|
| 1200 | KPP_REAL :: Y(NVAR), P(NVAR) |
---|
| 1201 | |
---|
| 1202 | Told = TIME |
---|
| 1203 | TIME = T |
---|
| 1204 | CALL Update_SUN() |
---|
| 1205 | CALL Update_RCONST() |
---|
| 1206 | |
---|
| 1207 | CALL Fun( Y, FIX, RCONST, P ) |
---|
| 1208 | |
---|
| 1209 | TIME = Told |
---|
| 1210 | |
---|
| 1211 | END SUBROUTINE FUN_CHEM |
---|
| 1212 | |
---|
| 1213 | |
---|
| 1214 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1215 | SUBROUTINE JAC_CHEM( T, Y, JV ) |
---|
| 1216 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1217 | |
---|
| 1218 | USE KPP_ROOT_Parameters, ONLY: NVAR, LU_NONZERO |
---|
| 1219 | USE KPP_ROOT_Global, ONLY: FIX, RCONST, TIME |
---|
| 1220 | USE KPP_ROOT_Jacobian |
---|
| 1221 | USE KPP_ROOT_Jacobian, ONLY: Jac_SP |
---|
| 1222 | USE KPP_ROOT_Rates, ONLY: Update_SUN, Update_RCONST, Update_PHOTO |
---|
| 1223 | IMPLICIT NONE |
---|
| 1224 | |
---|
| 1225 | KPP_REAL :: T, Told |
---|
| 1226 | KPP_REAL :: Y(NVAR) |
---|
| 1227 | #ifdef FULL_ALGEBRA |
---|
| 1228 | KPP_REAL :: JS(LU_NONZERO), JV(NVAR,NVAR) |
---|
| 1229 | INTEGER :: i, j |
---|
| 1230 | #else |
---|
| 1231 | KPP_REAL :: JV(LU_NONZERO) |
---|
| 1232 | #endif |
---|
| 1233 | |
---|
| 1234 | Told = TIME |
---|
| 1235 | TIME = T |
---|
| 1236 | CALL Update_SUN() |
---|
| 1237 | CALL Update_RCONST() |
---|
| 1238 | |
---|
| 1239 | #ifdef FULL_ALGEBRA |
---|
| 1240 | CALL Jac_SP(Y, FIX, RCONST, JS) |
---|
| 1241 | DO j=1,NVAR |
---|
| 1242 | DO i=1,NVAR |
---|
| 1243 | JV(i,j) = 0.0D0 |
---|
| 1244 | END DO |
---|
| 1245 | END DO |
---|
| 1246 | DO i=1,LU_NONZERO |
---|
| 1247 | JV(LU_IROW(i),LU_ICOL(i)) = JS(i) |
---|
| 1248 | END DO |
---|
| 1249 | #else |
---|
| 1250 | CALL Jac_SP(Y, FIX, RCONST, JV) |
---|
| 1251 | #endif |
---|
| 1252 | TIME = Told |
---|
| 1253 | |
---|
| 1254 | END SUBROUTINE JAC_CHEM |
---|
| 1255 | |
---|
| 1256 | END MODULE KPP_ROOT_Integrator |
---|
| 1257 | |
---|
| 1258 | |
---|