[2696] | 1 | SUBROUTINE INTEGRATE( TIN, TOUT ) |
---|
| 2 | |
---|
| 3 | IMPLICIT NONE |
---|
| 4 | INCLUDE 'KPP_ROOT_Parameters.h' |
---|
| 5 | INCLUDE 'KPP_ROOT_Global.h' |
---|
| 6 | INTEGER Nstp, Nacc, Nrej, Nsng, IERR |
---|
| 7 | SAVE Nstp, Nacc, Nrej, Nsng |
---|
| 8 | |
---|
| 9 | ! TIN - Start Time |
---|
| 10 | KPP_REAL TIN |
---|
| 11 | ! TOUT - End Time |
---|
| 12 | KPP_REAL TOUT |
---|
| 13 | INTEGER i |
---|
| 14 | |
---|
| 15 | KPP_REAL RPAR(20) |
---|
| 16 | INTEGER IPAR(20) |
---|
| 17 | EXTERNAL FunTemplate, JacTemplate |
---|
| 18 | |
---|
| 19 | |
---|
| 20 | DO i=1,20 |
---|
| 21 | IPAR(i) = 0 |
---|
| 22 | RPAR(i) = 0.0d0 |
---|
| 23 | ENDDO |
---|
| 24 | |
---|
| 25 | |
---|
| 26 | IPAR(1) = 0 ! non-autonomous |
---|
| 27 | IPAR(2) = 1 ! vector tolerances |
---|
| 28 | RPAR(3) = STEPMIN ! starting step |
---|
| 29 | IPAR(4) = 5 ! choice of the method |
---|
| 30 | |
---|
| 31 | CALL Rosenbrock(VAR,TIN,TOUT, |
---|
| 32 | & ATOL,RTOL, |
---|
| 33 | & FunTemplate,JacTemplate, |
---|
| 34 | & RPAR,IPAR,IERR) |
---|
| 35 | |
---|
| 36 | |
---|
| 37 | Nstp = Nstp + IPAR(13) |
---|
| 38 | Nacc = Nacc + IPAR(14) |
---|
| 39 | Nrej = Nrej + IPAR(15) |
---|
| 40 | Nsng = Nsng + IPAR(18) |
---|
| 41 | PRINT*,'Step=',Nstp,' Acc=',Nacc,' Rej=',Nrej, |
---|
| 42 | & ' Singular=',Nsng |
---|
| 43 | |
---|
| 44 | |
---|
| 45 | IF (IERR.LT.0) THEN |
---|
| 46 | print *,'Rosenbrock: Unsucessful step at T=', |
---|
| 47 | & TIN,' (IERR=',IERR,')' |
---|
| 48 | ENDIF |
---|
| 49 | |
---|
| 50 | TIN = RPAR(11) ! Exit time |
---|
| 51 | STEPMIN = RPAR(12) |
---|
| 52 | |
---|
| 53 | RETURN |
---|
| 54 | END |
---|
| 55 | |
---|
| 56 | |
---|
| 57 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 58 | SUBROUTINE Rosenbrock(Y,Tstart,Tend, |
---|
| 59 | & AbsTol,RelTol, |
---|
| 60 | & ode_Fun,ode_Jac , |
---|
| 61 | & RPAR,IPAR,IERR) |
---|
| 62 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 63 | ! |
---|
| 64 | ! Solves the system y'=F(t,y) using a Rosenbrock method defined by: |
---|
| 65 | ! |
---|
| 66 | ! G = 1/(H*gamma(1)) - ode_Jac(t0,Y0) |
---|
| 67 | ! T_i = t0 + Alpha(i)*H |
---|
| 68 | ! Y_i = Y0 + \sum_{j=1}^{i-1} A(i,j)*K_j |
---|
| 69 | ! G * K_i = ode_Fun( T_i, Y_i ) + \sum_{j=1}^S C(i,j)/H * K_j + |
---|
| 70 | ! gamma(i)*dF/dT(t0, Y0) |
---|
| 71 | ! Y1 = Y0 + \sum_{j=1}^S M(j)*K_j |
---|
| 72 | ! |
---|
| 73 | ! For details on Rosenbrock methods and their implementation consult: |
---|
| 74 | ! E. Hairer and G. Wanner |
---|
| 75 | ! "Solving ODEs II. Stiff and differential-algebraic problems". |
---|
| 76 | ! Springer series in computational mathematics, Springer-Verlag, 1996. |
---|
| 77 | ! The codes contained in the book inspired this implementation. |
---|
| 78 | ! |
---|
| 79 | ! (C) Adrian Sandu, August 2004 |
---|
| 80 | ! Virginia Polytechnic Institute and State University |
---|
| 81 | ! Contact: sandu@cs.vt.edu |
---|
| 82 | ! This implementation is part of KPP - the Kinetic PreProcessor |
---|
| 83 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 84 | ! |
---|
| 85 | !~~~> INPUT ARGUMENTS: |
---|
| 86 | ! |
---|
| 87 | !- Y(NVAR) = vector of initial conditions (at T=Tstart) |
---|
| 88 | !- [Tstart,Tend] = time range of integration |
---|
| 89 | ! (if Tstart>Tend the integration is performed backwards in time) |
---|
| 90 | !- RelTol, AbsTol = user precribed accuracy |
---|
| 91 | !- SUBROUTINE ode_Fun( T, Y, Ydot ) = ODE function, |
---|
| 92 | ! returns Ydot = Y' = F(T,Y) |
---|
| 93 | !- SUBROUTINE ode_Fun( T, Y, Ydot ) = Jacobian of the ODE function, |
---|
| 94 | ! returns Jcb = dF/dY |
---|
| 95 | !- IPAR(1:10) = integer inputs parameters |
---|
| 96 | !- RPAR(1:10) = real inputs parameters |
---|
| 97 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 98 | ! |
---|
| 99 | !~~~> OUTPUT ARGUMENTS: |
---|
| 100 | ! |
---|
| 101 | !- Y(NVAR) -> vector of final states (at T->Tend) |
---|
| 102 | !- IPAR(11:20) -> integer output parameters |
---|
| 103 | !- RPAR(11:20) -> real output parameters |
---|
| 104 | !- IERR -> job status upon return |
---|
| 105 | ! - succes (positive value) or failure (negative value) - |
---|
| 106 | ! = 1 : Success |
---|
| 107 | ! = -1 : Improper value for maximal no of steps |
---|
| 108 | ! = -2 : Selected Rosenbrock method not implemented |
---|
| 109 | ! = -3 : Hmin/Hmax/Hstart must be positive |
---|
| 110 | ! = -4 : FacMin/FacMax/FacRej must be positive |
---|
| 111 | ! = -5 : Improper tolerance values |
---|
| 112 | ! = -6 : No of steps exceeds maximum bound |
---|
| 113 | ! = -7 : Step size too small |
---|
| 114 | ! = -8 : Matrix is repeatedly singular |
---|
| 115 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 116 | ! |
---|
| 117 | !~~~> INPUT PARAMETERS: |
---|
| 118 | ! |
---|
| 119 | ! Note: For input parameters equal to zero the default values of the |
---|
| 120 | ! corresponding variables are used. |
---|
| 121 | ! |
---|
| 122 | ! IPAR(1) = 1: F = F(y) Independent of T (AUTONOMOUS) |
---|
| 123 | ! = 0: F = F(t,y) Depends on T (NON-AUTONOMOUS) |
---|
| 124 | ! IPAR(2) = 0: AbsTol, RelTol are NVAR-dimensional vectors |
---|
| 125 | ! = 1: AbsTol, RelTol are scalars |
---|
| 126 | ! IPAR(3) -> maximum number of integration steps |
---|
| 127 | ! For IPAR(3)=0) the default value of 100000 is used |
---|
| 128 | ! |
---|
| 129 | ! IPAR(4) -> selection of a particular Rosenbrock method |
---|
| 130 | ! = 0 : default method is Rodas3 |
---|
| 131 | ! = 1 : method is Ros2 |
---|
| 132 | ! = 2 : method is Ros3 |
---|
| 133 | ! = 3 : method is Ros4 |
---|
| 134 | ! = 4 : method is Rodas3 |
---|
| 135 | ! = 5: method is Rodas4 |
---|
| 136 | ! |
---|
| 137 | ! RPAR(1) -> Hmin, lower bound for the integration step size |
---|
| 138 | ! It is strongly recommended to keep Hmin = ZERO |
---|
| 139 | ! RPAR(2) -> Hmax, upper bound for the integration step size |
---|
| 140 | ! RPAR(3) -> Hstart, starting value for the integration step size |
---|
| 141 | ! |
---|
| 142 | ! RPAR(4) -> FacMin, lower bound on step decrease factor (default=0.2) |
---|
| 143 | ! RPAR(5) -> FacMin,upper bound on step increase factor (default=6) |
---|
| 144 | ! RPAR(6) -> FacRej, step decrease factor after multiple rejections |
---|
| 145 | ! (default=0.1) |
---|
| 146 | ! RPAR(7) -> FacSafe, by which the new step is slightly smaller |
---|
| 147 | ! than the predicted value (default=0.9) |
---|
| 148 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 149 | ! |
---|
| 150 | !~~~> OUTPUT PARAMETERS: |
---|
| 151 | ! |
---|
| 152 | ! Note: each call to Rosenbrock adds the corrent no. of fcn calls |
---|
| 153 | ! to previous value of IPAR(11), and similar for the other params. |
---|
| 154 | ! Set IPAR(11:20) = 0 before call to avoid this accumulation. |
---|
| 155 | ! |
---|
| 156 | ! IPAR(11) = No. of function calls |
---|
| 157 | ! IPAR(12) = No. of jacobian calls |
---|
| 158 | ! IPAR(13) = No. of steps |
---|
| 159 | ! IPAR(14) = No. of accepted steps |
---|
| 160 | ! IPAR(15) = No. of rejected steps (except at the beginning) |
---|
| 161 | ! IPAR(16) = No. of LU decompositions |
---|
| 162 | ! IPAR(17) = No. of forward/backward substitutions |
---|
| 163 | ! IPAR(18) = No. of singular matrix decompositions |
---|
| 164 | ! |
---|
| 165 | ! RPAR(11) -> Texit, the time corresponding to the |
---|
| 166 | ! computed Y upon return |
---|
| 167 | ! RPAR(12) -> Hexit, last accepted step before exit |
---|
| 168 | ! For multiple restarts, use Hexit as Hstart in the following run |
---|
| 169 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 170 | |
---|
| 171 | IMPLICIT NONE |
---|
| 172 | INCLUDE 'KPP_ROOT_Parameters.h' |
---|
| 173 | INCLUDE 'KPP_ROOT_Sparse.h' |
---|
| 174 | |
---|
| 175 | KPP_REAL Tstart,Tend |
---|
| 176 | KPP_REAL Y(KPP_NVAR),AbsTol(KPP_NVAR),RelTol(KPP_NVAR) |
---|
| 177 | INTEGER IPAR(20) |
---|
| 178 | KPP_REAL RPAR(20) |
---|
| 179 | INTEGER IERR |
---|
| 180 | !~~~> The method parameters |
---|
| 181 | INTEGER Smax |
---|
| 182 | PARAMETER (Smax = 6) |
---|
| 183 | INTEGER Method, ros_S |
---|
| 184 | KPP_REAL ros_M(Smax), ros_E(Smax) |
---|
| 185 | KPP_REAL ros_A(Smax*(Smax-1)/2), ros_C(Smax*(Smax-1)/2) |
---|
| 186 | KPP_REAL ros_Alpha(Smax), ros_Gamma(Smax), ros_ELO |
---|
| 187 | LOGICAL ros_NewF(Smax) |
---|
| 188 | CHARACTER*12 ros_Name |
---|
| 189 | !~~~> Local variables |
---|
| 190 | KPP_REAL Roundoff,FacMin,FacMax,FacRej,FacSafe |
---|
| 191 | KPP_REAL Hmin, Hmax, Hstart, Hexit |
---|
| 192 | KPP_REAL Texit |
---|
| 193 | INTEGER i, UplimTol, Max_no_steps |
---|
| 194 | LOGICAL Autonomous, VectorTol |
---|
| 195 | !~~~> Statistics on the work performed |
---|
| 196 | INTEGER Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng |
---|
| 197 | COMMON /Statistics/ Nfun,Njac,Nstp,Nacc,Nrej, |
---|
| 198 | & Ndec,Nsol,Nsng |
---|
| 199 | !~~~> Parameters |
---|
| 200 | KPP_REAL ZERO, ONE, DeltaMin |
---|
| 201 | PARAMETER (ZERO = 0.0d0) |
---|
| 202 | PARAMETER (ONE = 1.0d0) |
---|
| 203 | PARAMETER (DeltaMin = 1.0d-5) |
---|
| 204 | !~~~> Functions |
---|
| 205 | EXTERNAL ode_Fun, ode_Jac |
---|
| 206 | KPP_REAL WLAMCH, ros_ErrorNorm |
---|
| 207 | EXTERNAL WLAMCH, ros_ErrorNorm |
---|
| 208 | |
---|
| 209 | !~~~> Initialize statistics |
---|
| 210 | Nfun = IPAR(11) |
---|
| 211 | Njac = IPAR(12) |
---|
| 212 | Nstp = IPAR(13) |
---|
| 213 | Nacc = IPAR(14) |
---|
| 214 | Nrej = IPAR(15) |
---|
| 215 | Ndec = IPAR(16) |
---|
| 216 | Nsol = IPAR(17) |
---|
| 217 | Nsng = IPAR(18) |
---|
| 218 | |
---|
| 219 | !~~~> Autonomous or time dependent ODE. Default is time dependent. |
---|
| 220 | Autonomous = .NOT.(IPAR(1).EQ.0) |
---|
| 221 | |
---|
| 222 | !~~~> For Scalar tolerances (IPAR(2).NE.0) the code uses AbsTol(1) and RelTol(1) |
---|
| 223 | ! For Vector tolerances (IPAR(2).EQ.0) the code uses AbsTol(1:NVAR) and RelTol(1:NVAR) |
---|
| 224 | IF (IPAR(2).EQ.0) THEN |
---|
| 225 | VectorTol = .TRUE. |
---|
| 226 | UplimTol = KPP_NVAR |
---|
| 227 | ELSE |
---|
| 228 | VectorTol = .FALSE. |
---|
| 229 | UplimTol = 1 |
---|
| 230 | END IF |
---|
| 231 | |
---|
| 232 | !~~~> The maximum number of steps admitted |
---|
| 233 | IF (IPAR(3).EQ.0) THEN |
---|
| 234 | Max_no_steps = 100000 |
---|
| 235 | ELSEIF (Max_no_steps.GT.0) THEN |
---|
| 236 | Max_no_steps=IPAR(3) |
---|
| 237 | ELSE |
---|
| 238 | WRITE(6,*)'User-selected max no. of steps: IPAR(3)=',IPAR(3) |
---|
| 239 | CALL ros_ErrorMsg(-1,Tstart,ZERO,IERR) |
---|
| 240 | RETURN |
---|
| 241 | END IF |
---|
| 242 | |
---|
| 243 | !~~~> The particular Rosenbrock method chosen |
---|
| 244 | IF (IPAR(4).EQ.0) THEN |
---|
| 245 | Method = 3 |
---|
| 246 | ELSEIF ( (IPAR(4).GE.1).AND.(IPAR(4).LE.5) ) THEN |
---|
| 247 | Method = IPAR(4) |
---|
| 248 | ELSE |
---|
| 249 | WRITE (6,*) 'User-selected Rosenbrock method: IPAR(4)=', Method |
---|
| 250 | CALL ros_ErrorMsg(-2,Tstart,ZERO,IERR) |
---|
| 251 | RETURN |
---|
| 252 | END IF |
---|
| 253 | |
---|
| 254 | !~~~> Unit roundoff (1+Roundoff>1) |
---|
| 255 | Roundoff = WLAMCH('E') |
---|
| 256 | |
---|
| 257 | !~~~> Lower bound on the step size: (positive value) |
---|
| 258 | IF (RPAR(1).EQ.ZERO) THEN |
---|
| 259 | Hmin = ZERO |
---|
| 260 | ELSEIF (RPAR(1).GT.ZERO) THEN |
---|
| 261 | Hmin = RPAR(1) |
---|
| 262 | ELSE |
---|
| 263 | WRITE (6,*) 'User-selected Hmin: RPAR(1)=', RPAR(1) |
---|
| 264 | CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR) |
---|
| 265 | RETURN |
---|
| 266 | END IF |
---|
| 267 | !~~~> Upper bound on the step size: (positive value) |
---|
| 268 | IF (RPAR(2).EQ.ZERO) THEN |
---|
| 269 | Hmax = ABS(Tend-Tstart) |
---|
| 270 | ELSEIF (RPAR(2).GT.ZERO) THEN |
---|
| 271 | Hmax = MIN(ABS(RPAR(2)),ABS(Tend-Tstart)) |
---|
| 272 | ELSE |
---|
| 273 | WRITE (6,*) 'User-selected Hmax: RPAR(2)=', RPAR(2) |
---|
| 274 | CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR) |
---|
| 275 | RETURN |
---|
| 276 | END IF |
---|
| 277 | !~~~> Starting step size: (positive value) |
---|
| 278 | IF (RPAR(3).EQ.ZERO) THEN |
---|
| 279 | Hstart = MAX(Hmin,DeltaMin) |
---|
| 280 | ELSEIF (RPAR(3).GT.ZERO) THEN |
---|
| 281 | Hstart = MIN(ABS(RPAR(3)),ABS(Tend-Tstart)) |
---|
| 282 | ELSE |
---|
| 283 | WRITE (6,*) 'User-selected Hstart: RPAR(3)=', RPAR(3) |
---|
| 284 | CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR) |
---|
| 285 | RETURN |
---|
| 286 | END IF |
---|
| 287 | !~~~> Step size can be changed s.t. FacMin < Hnew/Hexit < FacMax |
---|
| 288 | IF (RPAR(4).EQ.ZERO) THEN |
---|
| 289 | FacMin = 0.2d0 |
---|
| 290 | ELSEIF (RPAR(4).GT.ZERO) THEN |
---|
| 291 | FacMin = RPAR(4) |
---|
| 292 | ELSE |
---|
| 293 | WRITE (6,*) 'User-selected FacMin: RPAR(4)=', RPAR(4) |
---|
| 294 | CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR) |
---|
| 295 | RETURN |
---|
| 296 | END IF |
---|
| 297 | IF (RPAR(5).EQ.ZERO) THEN |
---|
| 298 | FacMax = 6.0d0 |
---|
| 299 | ELSEIF (RPAR(5).GT.ZERO) THEN |
---|
| 300 | FacMax = RPAR(5) |
---|
| 301 | ELSE |
---|
| 302 | WRITE (6,*) 'User-selected FacMax: RPAR(5)=', RPAR(5) |
---|
| 303 | CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR) |
---|
| 304 | RETURN |
---|
| 305 | END IF |
---|
| 306 | !~~~> FacRej: Factor to decrease step after 2 succesive rejections |
---|
| 307 | IF (RPAR(6).EQ.ZERO) THEN |
---|
| 308 | FacRej = 0.1d0 |
---|
| 309 | ELSEIF (RPAR(6).GT.ZERO) THEN |
---|
| 310 | FacRej = RPAR(6) |
---|
| 311 | ELSE |
---|
| 312 | WRITE (6,*) 'User-selected FacRej: RPAR(6)=', RPAR(6) |
---|
| 313 | CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR) |
---|
| 314 | RETURN |
---|
| 315 | END IF |
---|
| 316 | !~~~> FacSafe: Safety Factor in the computation of new step size |
---|
| 317 | IF (RPAR(7).EQ.ZERO) THEN |
---|
| 318 | FacSafe = 0.9d0 |
---|
| 319 | ELSEIF (RPAR(7).GT.ZERO) THEN |
---|
| 320 | FacSafe = RPAR(7) |
---|
| 321 | ELSE |
---|
| 322 | WRITE (6,*) 'User-selected FacSafe: RPAR(7)=', RPAR(7) |
---|
| 323 | CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR) |
---|
| 324 | RETURN |
---|
| 325 | END IF |
---|
| 326 | !~~~> Check if tolerances are reasonable |
---|
| 327 | DO i=1,UplimTol |
---|
| 328 | IF ( (AbsTol(i).LE.ZERO) .OR. (RelTol(i).LE.10.d0*Roundoff) |
---|
| 329 | & .OR. (RelTol(i).GE.1.0d0) ) THEN |
---|
| 330 | WRITE (6,*) ' AbsTol(',i,') = ',AbsTol(i) |
---|
| 331 | WRITE (6,*) ' RelTol(',i,') = ',RelTol(i) |
---|
| 332 | CALL ros_ErrorMsg(-5,Tstart,ZERO,IERR) |
---|
| 333 | RETURN |
---|
| 334 | END IF |
---|
| 335 | END DO |
---|
| 336 | |
---|
| 337 | |
---|
| 338 | !~~~> Initialize the particular Rosenbrock method |
---|
| 339 | |
---|
| 340 | IF (Method .EQ. 1) THEN |
---|
| 341 | CALL Ros2(ros_S, ros_A, ros_C, ros_M, ros_E, |
---|
| 342 | & ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name) |
---|
| 343 | ELSEIF (Method .EQ. 2) THEN |
---|
| 344 | CALL Ros3(ros_S, ros_A, ros_C, ros_M, ros_E, |
---|
| 345 | & ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name) |
---|
| 346 | ELSEIF (Method .EQ. 3) THEN |
---|
| 347 | CALL Ros4(ros_S, ros_A, ros_C, ros_M, ros_E, |
---|
| 348 | & ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name) |
---|
| 349 | ELSEIF (Method .EQ. 4) THEN |
---|
| 350 | CALL Rodas3(ros_S, ros_A, ros_C, ros_M, ros_E, |
---|
| 351 | & ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name) |
---|
| 352 | ELSEIF (Method .EQ. 5) THEN |
---|
| 353 | CALL Rodas4(ros_S, ros_A, ros_C, ros_M, ros_E, |
---|
| 354 | & ros_Alpha, ros_Gamma, ros_NewF, ros_ELO, ros_Name) |
---|
| 355 | ELSE |
---|
| 356 | WRITE (6,*) 'Unknown Rosenbrock method: IPAR(4)=', Method |
---|
| 357 | CALL ros_ErrorMsg(-2,Tstart,ZERO,IERR) |
---|
| 358 | RETURN |
---|
| 359 | END IF |
---|
| 360 | |
---|
| 361 | !~~~> CALL Rosenbrock method |
---|
| 362 | CALL RosenbrockIntegrator(Y,Tstart,Tend,Texit, |
---|
| 363 | & AbsTol,RelTol, |
---|
| 364 | & ode_Fun,ode_Jac , |
---|
| 365 | ! Rosenbrock method coefficients |
---|
| 366 | & ros_S, ros_M, ros_E, ros_A, ros_C, |
---|
| 367 | & ros_Alpha, ros_Gamma, ros_ELO, ros_NewF, |
---|
| 368 | ! Integration parameters |
---|
| 369 | & Autonomous, VectorTol, Max_no_steps, |
---|
| 370 | & Roundoff, Hmin, Hmax, Hstart, Hexit, |
---|
| 371 | & FacMin, FacMax, FacRej, FacSafe, |
---|
| 372 | ! Error indicator |
---|
| 373 | & IERR) |
---|
| 374 | |
---|
| 375 | |
---|
| 376 | !~~~> Collect run statistics |
---|
| 377 | IPAR(11) = Nfun |
---|
| 378 | IPAR(12) = Njac |
---|
| 379 | IPAR(13) = Nstp |
---|
| 380 | IPAR(14) = Nacc |
---|
| 381 | IPAR(15) = Nrej |
---|
| 382 | IPAR(16) = Ndec |
---|
| 383 | IPAR(17) = Nsol |
---|
| 384 | IPAR(18) = Nsng |
---|
| 385 | !~~~> Last T and H |
---|
| 386 | RPAR(11) = Texit |
---|
| 387 | RPAR(12) = Hexit |
---|
| 388 | |
---|
| 389 | RETURN |
---|
| 390 | END ! SUBROUTINE Rosenbrock |
---|
| 391 | |
---|
| 392 | |
---|
| 393 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 394 | SUBROUTINE RosenbrockIntegrator(Y,Tstart,Tend,T, |
---|
| 395 | & AbsTol,RelTol, |
---|
| 396 | & ode_Fun,ode_Jac , |
---|
| 397 | !~~~> Rosenbrock method coefficients |
---|
| 398 | & ros_S, ros_M, ros_E, ros_A, ros_C, |
---|
| 399 | & ros_Alpha, ros_Gamma, ros_ELO, ros_NewF, |
---|
| 400 | !~~~> Integration parameters |
---|
| 401 | & Autonomous, VectorTol, Max_no_steps, |
---|
| 402 | & Roundoff, Hmin, Hmax, Hstart, Hexit, |
---|
| 403 | & FacMin, FacMax, FacRej, FacSafe, |
---|
| 404 | !~~~> Error indicator |
---|
| 405 | & IERR) |
---|
| 406 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 407 | ! Template for the implementation of a generic Rosenbrock method |
---|
| 408 | ! defined by ros_S (no of stages) |
---|
| 409 | ! and its coefficients ros_{A,C,M,E,Alpha,Gamma} |
---|
| 410 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 411 | |
---|
| 412 | IMPLICIT NONE |
---|
| 413 | INCLUDE 'KPP_ROOT_Parameters.h' |
---|
| 414 | INCLUDE 'KPP_ROOT_Sparse.h' |
---|
| 415 | |
---|
| 416 | !~~~> Input: the initial condition at Tstart; Output: the solution at T |
---|
| 417 | KPP_REAL Y(KPP_NVAR) |
---|
| 418 | !~~~> Input: integration interval |
---|
| 419 | KPP_REAL Tstart,Tend |
---|
| 420 | !~~~> Output: time at which the solution is returned (T=Tend if success) |
---|
| 421 | KPP_REAL T |
---|
| 422 | !~~~> Input: tolerances |
---|
| 423 | KPP_REAL AbsTol(KPP_NVAR), RelTol(KPP_NVAR) |
---|
| 424 | !~~~> Input: ode function and its Jacobian |
---|
| 425 | EXTERNAL ode_Fun, ode_Jac |
---|
| 426 | !~~~> Input: The Rosenbrock method parameters |
---|
| 427 | INTEGER ros_S |
---|
| 428 | KPP_REAL ros_M(ros_S), ros_E(ros_S) |
---|
| 429 | KPP_REAL ros_A(ros_S*(ros_S-1)/2), ros_C(ros_S*(ros_S-1)/2) |
---|
| 430 | KPP_REAL ros_Alpha(ros_S), ros_Gamma(ros_S), ros_ELO |
---|
| 431 | LOGICAL ros_NewF(ros_S) |
---|
| 432 | !~~~> Input: integration parameters |
---|
| 433 | LOGICAL Autonomous, VectorTol |
---|
| 434 | KPP_REAL Hstart, Hmin, Hmax |
---|
| 435 | INTEGER Max_no_steps |
---|
| 436 | KPP_REAL Roundoff, FacMin, FacMax, FacRej, FacSafe |
---|
| 437 | !~~~> Output: last accepted step |
---|
| 438 | KPP_REAL Hexit |
---|
| 439 | !~~~> Output: Error indicator |
---|
| 440 | INTEGER IERR |
---|
| 441 | ! ~~~~ Local variables |
---|
| 442 | KPP_REAL Ynew(KPP_NVAR), Fcn0(KPP_NVAR), Fcn(KPP_NVAR), |
---|
| 443 | & K(KPP_NVAR*ros_S), dFdT(KPP_NVAR), |
---|
| 444 | & Jac0(KPP_LU_NONZERO), Ghimj(KPP_LU_NONZERO) |
---|
| 445 | KPP_REAL H, Hnew, HC, HG, Fac, Tau |
---|
| 446 | KPP_REAL Err, Yerr(KPP_NVAR) |
---|
| 447 | INTEGER Pivot(KPP_NVAR), Direction, ioffset, j, istage |
---|
| 448 | LOGICAL RejectLastH, RejectMoreH, Singular |
---|
| 449 | !~~~> Local parameters |
---|
| 450 | KPP_REAL ZERO, ONE, DeltaMin |
---|
| 451 | PARAMETER (ZERO = 0.0d0) |
---|
| 452 | PARAMETER (ONE = 1.0d0) |
---|
| 453 | PARAMETER (DeltaMin = 1.0d-5) |
---|
| 454 | !~~~> Locally called functions |
---|
| 455 | KPP_REAL WLAMCH, ros_ErrorNorm |
---|
| 456 | EXTERNAL WLAMCH, ros_ErrorNorm |
---|
| 457 | !~~~> Statistics on the work performed |
---|
| 458 | INTEGER Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng |
---|
| 459 | COMMON /Statistics/ Nfun,Njac,Nstp,Nacc,Nrej, |
---|
| 460 | & Ndec,Nsol,Nsng |
---|
| 461 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 462 | |
---|
| 463 | |
---|
| 464 | !~~~> INITIAL PREPARATIONS |
---|
| 465 | T = Tstart |
---|
| 466 | Hexit = 0.0d0 |
---|
| 467 | H = MIN(Hstart,Hmax) |
---|
| 468 | IF (ABS(H).LE.10.d0*Roundoff) THEN |
---|
| 469 | H = DeltaMin |
---|
| 470 | END IF |
---|
| 471 | |
---|
| 472 | IF (Tend .GE. Tstart) THEN |
---|
| 473 | Direction = +1 |
---|
| 474 | ELSE |
---|
| 475 | Direction = -1 |
---|
| 476 | END IF |
---|
| 477 | |
---|
| 478 | RejectLastH=.FALSE. |
---|
| 479 | RejectMoreH=.FALSE. |
---|
| 480 | |
---|
| 481 | !~~~> Time loop begins below |
---|
| 482 | |
---|
| 483 | DO WHILE ( (Direction.GT.0).AND.((T-Tend)+Roundoff.LE.ZERO) |
---|
| 484 | & .OR. (Direction.LT.0).AND.((Tend-T)+Roundoff.LE.ZERO) ) |
---|
| 485 | |
---|
| 486 | IF ( Nstp.GT.Max_no_steps ) THEN ! Too many steps |
---|
| 487 | CALL ros_ErrorMsg(-6,T,H,IERR) |
---|
| 488 | RETURN |
---|
| 489 | END IF |
---|
| 490 | IF ( ((T+0.1d0*H).EQ.T).OR.(H.LE.Roundoff) ) THEN ! Step size too small |
---|
| 491 | CALL ros_ErrorMsg(-7,T,H,IERR) |
---|
| 492 | RETURN |
---|
| 493 | END IF |
---|
| 494 | |
---|
| 495 | !~~~> Limit H if necessary to avoid going beyond Tend |
---|
| 496 | Hexit = H |
---|
| 497 | H = MIN(H,ABS(Tend-T)) |
---|
| 498 | |
---|
| 499 | !~~~> Compute the function at current time |
---|
| 500 | CALL ode_Fun(T,Y,Fcn0) |
---|
| 501 | |
---|
| 502 | !~~~> Compute the function derivative with respect to T |
---|
| 503 | IF (.NOT.Autonomous) THEN |
---|
| 504 | CALL ros_FunTimeDerivative ( T, Roundoff, Y, |
---|
| 505 | & Fcn0, ode_Fun, dFdT ) |
---|
| 506 | END IF |
---|
| 507 | |
---|
| 508 | !~~~> Compute the Jacobian at current time |
---|
| 509 | CALL ode_Jac(T,Y,Jac0) |
---|
| 510 | |
---|
| 511 | !~~~> Repeat step calculation until current step accepted |
---|
| 512 | DO WHILE (.TRUE.) ! WHILE STEP NOT ACCEPTED |
---|
| 513 | |
---|
| 514 | |
---|
| 515 | CALL ros_PrepareMatrix(H,Direction,ros_Gamma(1), |
---|
| 516 | & Jac0,Ghimj,Pivot,Singular) |
---|
| 517 | IF (Singular) THEN ! More than 5 consecutive failed decompositions |
---|
| 518 | CALL ros_ErrorMsg(-8,T,H,IERR) |
---|
| 519 | RETURN |
---|
| 520 | END IF |
---|
| 521 | |
---|
| 522 | !~~~> Compute the stages |
---|
| 523 | DO istage = 1, ros_S |
---|
| 524 | |
---|
| 525 | ! Current istage offset. Current istage vector is K(ioffset+1:ioffset+KPP_NVAR) |
---|
| 526 | ioffset = KPP_NVAR*(istage-1) |
---|
| 527 | |
---|
| 528 | ! For the 1st istage the function has been computed previously |
---|
| 529 | IF ( istage.EQ.1 ) THEN |
---|
| 530 | CALL WCOPY(KPP_NVAR,Fcn0,1,Fcn,1) |
---|
| 531 | ! istage>1 and a new function evaluation is needed at the current istage |
---|
| 532 | ELSEIF ( ros_NewF(istage) ) THEN |
---|
| 533 | CALL WCOPY(KPP_NVAR,Y,1,Ynew,1) |
---|
| 534 | DO j = 1, istage-1 |
---|
| 535 | CALL WAXPY(KPP_NVAR,ros_A((istage-1)*(istage-2)/2+j), |
---|
| 536 | & K(KPP_NVAR*(j-1)+1),1,Ynew,1) |
---|
| 537 | END DO |
---|
| 538 | Tau = T + ros_Alpha(istage)*Direction*H |
---|
| 539 | CALL ode_Fun(Tau,Ynew,Fcn) |
---|
| 540 | END IF ! if istage.EQ.1 elseif ros_NewF(istage) |
---|
| 541 | CALL WCOPY(KPP_NVAR,Fcn,1,K(ioffset+1),1) |
---|
| 542 | DO j = 1, istage-1 |
---|
| 543 | HC = ros_C((istage-1)*(istage-2)/2+j)/(Direction*H) |
---|
| 544 | CALL WAXPY(KPP_NVAR,HC,K(KPP_NVAR*(j-1)+1),1,K(ioffset+1),1) |
---|
| 545 | END DO |
---|
| 546 | IF ((.NOT. Autonomous).AND.(ros_Gamma(istage).NE.ZERO)) THEN |
---|
| 547 | HG = Direction*H*ros_Gamma(istage) |
---|
| 548 | CALL WAXPY(KPP_NVAR,HG,dFdT,1,K(ioffset+1),1) |
---|
| 549 | END IF |
---|
| 550 | CALL SolveTemplate(Ghimj, Pivot, K(ioffset+1)) |
---|
| 551 | |
---|
| 552 | END DO ! istage |
---|
| 553 | |
---|
| 554 | |
---|
| 555 | !~~~> Compute the new solution |
---|
| 556 | CALL WCOPY(KPP_NVAR,Y,1,Ynew,1) |
---|
| 557 | DO j=1,ros_S |
---|
| 558 | CALL WAXPY(KPP_NVAR,ros_M(j),K(KPP_NVAR*(j-1)+1),1,Ynew,1) |
---|
| 559 | END DO |
---|
| 560 | |
---|
| 561 | !~~~> Compute the error estimation |
---|
| 562 | CALL WSCAL(KPP_NVAR,ZERO,Yerr,1) |
---|
| 563 | DO j=1,ros_S |
---|
| 564 | CALL WAXPY(KPP_NVAR,ros_E(j),K(KPP_NVAR*(j-1)+1),1,Yerr,1) |
---|
| 565 | END DO |
---|
| 566 | Err = ros_ErrorNorm ( Y, Ynew, Yerr, AbsTol, RelTol, VectorTol ) |
---|
| 567 | |
---|
| 568 | !~~~> New step size is bounded by FacMin <= Hnew/H <= FacMax |
---|
| 569 | Fac = MIN(FacMax,MAX(FacMin,FacSafe/Err**(ONE/ros_ELO))) |
---|
| 570 | Hnew = H*Fac |
---|
| 571 | |
---|
| 572 | !~~~> Check the error magnitude and adjust step size |
---|
| 573 | Nstp = Nstp+1 |
---|
| 574 | IF ( (Err.LE.ONE).OR.(H.LE.Hmin) ) THEN !~~~> Accept step |
---|
| 575 | Nacc = Nacc+1 |
---|
| 576 | CALL WCOPY(KPP_NVAR,Ynew,1,Y,1) |
---|
| 577 | T = T + Direction*H |
---|
| 578 | Hnew = MAX(Hmin,MIN(Hnew,Hmax)) |
---|
| 579 | IF (RejectLastH) THEN ! No step size increase after a rejected step |
---|
| 580 | Hnew = MIN(Hnew,H) |
---|
| 581 | END IF |
---|
| 582 | RejectLastH = .FALSE. |
---|
| 583 | RejectMoreH = .FALSE. |
---|
| 584 | H = Hnew |
---|
| 585 | GOTO 101 ! EXIT THE LOOP: WHILE STEP NOT ACCEPTED |
---|
| 586 | ELSE !~~~> Reject step |
---|
| 587 | IF (RejectMoreH) THEN |
---|
| 588 | Hnew=H*FacRej |
---|
| 589 | END IF |
---|
| 590 | RejectMoreH = RejectLastH |
---|
| 591 | RejectLastH = .TRUE. |
---|
| 592 | H = Hnew |
---|
| 593 | IF (Nacc.GE.1) THEN |
---|
| 594 | Nrej = Nrej+1 |
---|
| 595 | END IF |
---|
| 596 | END IF ! Err <= 1 |
---|
| 597 | |
---|
| 598 | END DO ! LOOP: WHILE STEP NOT ACCEPTED |
---|
| 599 | |
---|
| 600 | 101 CONTINUE |
---|
| 601 | |
---|
| 602 | END DO ! Time loop |
---|
| 603 | |
---|
| 604 | !~~~> Succesful exit |
---|
| 605 | IERR = 1 !~~~> The integration was successful |
---|
| 606 | |
---|
| 607 | RETURN |
---|
| 608 | END ! SUBROUTINE RosenbrockIntegrator |
---|
| 609 | |
---|
| 610 | |
---|
| 611 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 612 | KPP_REAL FUNCTION ros_ErrorNorm ( Y, Ynew, Yerr, |
---|
| 613 | & AbsTol, RelTol, VectorTol ) |
---|
| 614 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 615 | !~~~> Computes the "scaled norm" of the error vector Yerr |
---|
| 616 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 617 | IMPLICIT NONE |
---|
| 618 | INCLUDE 'KPP_ROOT_Parameters.h' |
---|
| 619 | |
---|
| 620 | ! Input arguments |
---|
| 621 | KPP_REAL Y(KPP_NVAR), Ynew(KPP_NVAR), Yerr(KPP_NVAR) |
---|
| 622 | KPP_REAL AbsTol(KPP_NVAR), RelTol(KPP_NVAR) |
---|
| 623 | LOGICAL VectorTol |
---|
| 624 | ! Local variables |
---|
| 625 | KPP_REAL Err, Scale, Ymax, ZERO |
---|
| 626 | INTEGER i |
---|
| 627 | PARAMETER (ZERO = 0.0d0) |
---|
| 628 | |
---|
| 629 | Err = ZERO |
---|
| 630 | DO i=1,KPP_NVAR |
---|
| 631 | Ymax = MAX(ABS(Y(i)),ABS(Ynew(i))) |
---|
| 632 | IF (VectorTol) THEN |
---|
| 633 | Scale = AbsTol(i)+RelTol(i)*Ymax |
---|
| 634 | ELSE |
---|
| 635 | Scale = AbsTol(1)+RelTol(1)*Ymax |
---|
| 636 | END IF |
---|
| 637 | Err = Err+(Yerr(i)/Scale)**2 |
---|
| 638 | END DO |
---|
| 639 | Err = SQRT(Err/KPP_NVAR) |
---|
| 640 | |
---|
| 641 | ros_ErrorNorm = Err |
---|
| 642 | |
---|
| 643 | RETURN |
---|
| 644 | END ! FUNCTION ros_ErrorNorm |
---|
| 645 | |
---|
| 646 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 647 | SUBROUTINE ros_FunTimeDerivative ( T, Roundoff, Y, |
---|
| 648 | & Fcn0, ode_Fun, dFdT ) |
---|
| 649 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 650 | !~~~> The time partial derivative of the function by finite differences |
---|
| 651 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 652 | IMPLICIT NONE |
---|
| 653 | INCLUDE 'KPP_ROOT_Parameters.h' |
---|
| 654 | |
---|
| 655 | !~~~> Input arguments |
---|
| 656 | KPP_REAL T, Roundoff, Y(KPP_NVAR), Fcn0(KPP_NVAR) |
---|
| 657 | EXTERNAL ode_Fun |
---|
| 658 | !~~~> Output arguments |
---|
| 659 | KPP_REAL dFdT(KPP_NVAR) |
---|
| 660 | !~~~> Global variables |
---|
| 661 | INTEGER Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng |
---|
| 662 | COMMON /Statistics/ Nfun,Njac,Nstp,Nacc,Nrej, |
---|
| 663 | & Ndec,Nsol,Nsng |
---|
| 664 | !~~~> Local variables |
---|
| 665 | KPP_REAL Delta, DeltaMin, ONE |
---|
| 666 | PARAMETER ( DeltaMin = 1.0d-6 ) |
---|
| 667 | PARAMETER ( ONE = 1.0d0 ) |
---|
| 668 | |
---|
| 669 | Delta = SQRT(Roundoff)*MAX(DeltaMin,ABS(T)) |
---|
| 670 | CALL ode_Fun(T+Delta,Y,dFdT) |
---|
| 671 | CALL WAXPY(KPP_NVAR,(-ONE),Fcn0,1,dFdT,1) |
---|
| 672 | CALL WSCAL(KPP_NVAR,(ONE/Delta),dFdT,1) |
---|
| 673 | |
---|
| 674 | RETURN |
---|
| 675 | END ! SUBROUTINE ros_FunTimeDerivative |
---|
| 676 | |
---|
| 677 | |
---|
| 678 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 679 | SUBROUTINE ros_PrepareMatrix ( H, Direction, gam, |
---|
| 680 | & Jac0, Ghimj, Pivot, Singular ) |
---|
| 681 | ! --- --- --- --- --- --- --- --- --- --- --- --- --- |
---|
| 682 | ! Prepares the LHS matrix for stage calculations |
---|
| 683 | ! 1. Construct Ghimj = 1/(H*ham) - Jac0 |
---|
| 684 | ! "(Gamma H) Inverse Minus Jacobian" |
---|
| 685 | ! 2. Repeat LU decomposition of Ghimj until successful. |
---|
| 686 | ! -half the step size if LU decomposition fails and retry |
---|
| 687 | ! -exit after 5 consecutive fails |
---|
| 688 | ! --- --- --- --- --- --- --- --- --- --- --- --- --- |
---|
| 689 | IMPLICIT NONE |
---|
| 690 | INCLUDE 'KPP_ROOT_Parameters.h' |
---|
| 691 | INCLUDE 'KPP_ROOT_Sparse.h' |
---|
| 692 | |
---|
| 693 | !~~~> Input arguments |
---|
| 694 | KPP_REAL gam, Jac0(KPP_LU_NONZERO) |
---|
| 695 | INTEGER Direction |
---|
| 696 | !~~~> Output arguments |
---|
| 697 | KPP_REAL Ghimj(KPP_LU_NONZERO) |
---|
| 698 | LOGICAL Singular |
---|
| 699 | INTEGER Pivot(KPP_NVAR) |
---|
| 700 | !~~~> Inout arguments |
---|
| 701 | KPP_REAL H ! step size is decreased when LU fails |
---|
| 702 | !~~~> Global variables |
---|
| 703 | INTEGER Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng |
---|
| 704 | COMMON /Statistics/ Nfun,Njac,Nstp,Nacc,Nrej, |
---|
| 705 | & Ndec,Nsol,Nsng |
---|
| 706 | !~~~> Local variables |
---|
| 707 | INTEGER i, ising, Nconsecutive |
---|
| 708 | KPP_REAL ghinv, ONE, HALF |
---|
| 709 | PARAMETER ( ONE = 1.0d0 ) |
---|
| 710 | PARAMETER ( HALF = 0.5d0 ) |
---|
| 711 | |
---|
| 712 | Nconsecutive = 0 |
---|
| 713 | Singular = .TRUE. |
---|
| 714 | |
---|
| 715 | DO WHILE (Singular) |
---|
| 716 | |
---|
| 717 | !~~~> Construct Ghimj = 1/(H*ham) - Jac0 |
---|
| 718 | CALL WCOPY(KPP_LU_NONZERO,Jac0,1,Ghimj,1) |
---|
| 719 | CALL WSCAL(KPP_LU_NONZERO,(-ONE),Ghimj,1) |
---|
| 720 | ghinv = ONE/(Direction*H*gam) |
---|
| 721 | DO i=1,KPP_NVAR |
---|
| 722 | Ghimj(LU_DIAG(i)) = Ghimj(LU_DIAG(i))+ghinv |
---|
| 723 | END DO |
---|
| 724 | !~~~> Compute LU decomposition |
---|
| 725 | CALL DecompTemplate( Ghimj, Pivot, ising ) |
---|
| 726 | IF (ising .EQ. 0) THEN |
---|
| 727 | !~~~> If successful done |
---|
| 728 | Singular = .FALSE. |
---|
| 729 | ELSE ! ising .ne. 0 |
---|
| 730 | !~~~> If unsuccessful half the step size; if 5 consecutive fails then return |
---|
| 731 | Nsng = Nsng+1 |
---|
| 732 | Nconsecutive = Nconsecutive+1 |
---|
| 733 | Singular = .TRUE. |
---|
| 734 | PRINT*,'Warning: LU Decomposition returned ising = ',ising |
---|
| 735 | IF (Nconsecutive.LE.5) THEN ! Less than 5 consecutive failed decompositions |
---|
| 736 | H = H*HALF |
---|
| 737 | ELSE ! More than 5 consecutive failed decompositions |
---|
| 738 | RETURN |
---|
| 739 | END IF ! Nconsecutive |
---|
| 740 | END IF ! ising |
---|
| 741 | |
---|
| 742 | END DO ! WHILE Singular |
---|
| 743 | |
---|
| 744 | RETURN |
---|
| 745 | END ! SUBROUTINE ros_PrepareMatrix |
---|
| 746 | |
---|
| 747 | |
---|
| 748 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 749 | SUBROUTINE ros_ErrorMsg(Code,T,H,IERR) |
---|
| 750 | KPP_REAL T, H |
---|
| 751 | INTEGER IERR, Code |
---|
| 752 | |
---|
| 753 | IERR = Code |
---|
| 754 | WRITE(6,*) |
---|
| 755 | & 'Forced exit from Rosenbrock due to the following error:' |
---|
| 756 | |
---|
| 757 | IF (Code .EQ. -1) THEN |
---|
| 758 | WRITE(6,*) '--> Improper value for maximal no of steps' |
---|
| 759 | ELSEIF (Code .EQ. -2) THEN |
---|
| 760 | WRITE(6,*) '--> Selected Rosenbrock method not implemented' |
---|
| 761 | ELSEIF (Code .EQ. -3) THEN |
---|
| 762 | WRITE(6,*) '--> Hmin/Hmax/Hstart must be positive' |
---|
| 763 | ELSEIF (Code .EQ. -4) THEN |
---|
| 764 | WRITE(6,*) '--> FacMin/FacMax/FacRej must be positive' |
---|
| 765 | ELSEIF (Code .EQ. -5) THEN |
---|
| 766 | WRITE(6,*) '--> Improper tolerance values' |
---|
| 767 | ELSEIF (Code .EQ. -6) THEN |
---|
| 768 | WRITE(6,*) '--> No of steps exceeds maximum bound' |
---|
| 769 | ELSEIF (Code .EQ. -7) THEN |
---|
| 770 | WRITE(6,*) '--> Step size too small: T + 10*H = T', |
---|
| 771 | & ' or H < Roundoff' |
---|
| 772 | ELSEIF (Code .EQ. -8) THEN |
---|
| 773 | WRITE(6,*) '--> Matrix is repeatedly singular' |
---|
| 774 | ELSE |
---|
| 775 | WRITE(6,102) 'Unknown Error code: ',Code |
---|
| 776 | END IF |
---|
| 777 | |
---|
| 778 | 102 FORMAT(' ',A,I4) |
---|
| 779 | WRITE(6,103) T, H |
---|
| 780 | |
---|
| 781 | 103 FORMAT(' T=',E15.7,' and H=',E15.7) |
---|
| 782 | |
---|
| 783 | RETURN |
---|
| 784 | END |
---|
| 785 | |
---|
| 786 | |
---|
| 787 | |
---|
| 788 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 789 | SUBROUTINE Ros2 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha, |
---|
| 790 | & ros_Gamma,ros_NewF,ros_ELO,ros_Name) |
---|
| 791 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 792 | ! --- AN L-STABLE METHOD, 2 stages, order 2 |
---|
| 793 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 794 | IMPLICIT NONE |
---|
| 795 | INTEGER S |
---|
| 796 | PARAMETER (S=2) |
---|
| 797 | INTEGER ros_S |
---|
| 798 | KPP_REAL ros_M(S), ros_E(S), ros_A(S*(S-1)/2), ros_C(S*(S-1)/2) |
---|
| 799 | KPP_REAL ros_Alpha(S), ros_Gamma(S), ros_ELO |
---|
| 800 | LOGICAL ros_NewF(S) |
---|
| 801 | CHARACTER*12 ros_Name |
---|
| 802 | DOUBLE PRECISION g |
---|
| 803 | |
---|
| 804 | g = 1.0d0 + 1.0d0/SQRT(2.0d0) |
---|
| 805 | |
---|
| 806 | !~~~> Name of the method |
---|
| 807 | ros_Name = 'ROS-2' |
---|
| 808 | !~~~> Number of stages |
---|
| 809 | ros_S = 2 |
---|
| 810 | |
---|
| 811 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 812 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 813 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 814 | ! The general mapping formula is: |
---|
| 815 | ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 816 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 817 | |
---|
| 818 | ros_A(1) = (1.d0)/g |
---|
| 819 | ros_C(1) = (-2.d0)/g |
---|
| 820 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 821 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 822 | ros_NewF(1) = .TRUE. |
---|
| 823 | ros_NewF(2) = .TRUE. |
---|
| 824 | !~~~> M_i = Coefficients for new step solution |
---|
| 825 | ros_M(1)= (3.d0)/(2.d0*g) |
---|
| 826 | ros_M(2)= (1.d0)/(2.d0*g) |
---|
| 827 | ! E_i = Coefficients for error estimator |
---|
| 828 | ros_E(1) = 1.d0/(2.d0*g) |
---|
| 829 | ros_E(2) = 1.d0/(2.d0*g) |
---|
| 830 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 831 | ! main and the embedded scheme orders plus one |
---|
| 832 | ros_ELO = 2.0d0 |
---|
| 833 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 834 | ros_Alpha(1) = 0.0d0 |
---|
| 835 | ros_Alpha(2) = 1.0d0 |
---|
| 836 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 837 | ros_Gamma(1) = g |
---|
| 838 | ros_Gamma(2) =-g |
---|
| 839 | |
---|
| 840 | RETURN |
---|
| 841 | END ! SUBROUTINE Ros2 |
---|
| 842 | |
---|
| 843 | |
---|
| 844 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 845 | SUBROUTINE Ros3 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha, |
---|
| 846 | & ros_Gamma,ros_NewF,ros_ELO,ros_Name) |
---|
| 847 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 848 | ! --- AN L-STABLE METHOD, 3 stages, order 3, 2 function evaluations |
---|
| 849 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 850 | IMPLICIT NONE |
---|
| 851 | INTEGER S |
---|
| 852 | PARAMETER (S=3) |
---|
| 853 | INTEGER ros_S |
---|
| 854 | KPP_REAL ros_M(S), ros_E(S), ros_A(S*(S-1)/2), ros_C(S*(S-1)/2) |
---|
| 855 | KPP_REAL ros_Alpha(S), ros_Gamma(S), ros_ELO |
---|
| 856 | LOGICAL ros_NewF(S) |
---|
| 857 | CHARACTER*12 ros_Name |
---|
| 858 | |
---|
| 859 | !~~~> Name of the method |
---|
| 860 | ros_Name = 'ROS-3' |
---|
| 861 | !~~~> Number of stages |
---|
| 862 | ros_S = 3 |
---|
| 863 | |
---|
| 864 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 865 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 866 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 867 | ! The general mapping formula is: |
---|
| 868 | ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 869 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 870 | |
---|
| 871 | ros_A(1)= 1.d0 |
---|
| 872 | ros_A(2)= 1.d0 |
---|
| 873 | ros_A(3)= 0.d0 |
---|
| 874 | |
---|
| 875 | ros_C(1) = -0.10156171083877702091975600115545d+01 |
---|
| 876 | ros_C(2) = 0.40759956452537699824805835358067d+01 |
---|
| 877 | ros_C(3) = 0.92076794298330791242156818474003d+01 |
---|
| 878 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 879 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 880 | ros_NewF(1) = .TRUE. |
---|
| 881 | ros_NewF(2) = .TRUE. |
---|
| 882 | ros_NewF(3) = .FALSE. |
---|
| 883 | !~~~> M_i = Coefficients for new step solution |
---|
| 884 | ros_M(1) = 0.1d+01 |
---|
| 885 | ros_M(2) = 0.61697947043828245592553615689730d+01 |
---|
| 886 | ros_M(3) = -0.42772256543218573326238373806514d+00 |
---|
| 887 | ! E_i = Coefficients for error estimator |
---|
| 888 | ros_E(1) = 0.5d+00 |
---|
| 889 | ros_E(2) = -0.29079558716805469821718236208017d+01 |
---|
| 890 | ros_E(3) = 0.22354069897811569627360909276199d+00 |
---|
| 891 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 892 | ! main and the embedded scheme orders plus 1 |
---|
| 893 | ros_ELO = 3.0d0 |
---|
| 894 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 895 | ros_Alpha(1)= 0.0d+00 |
---|
| 896 | ros_Alpha(2)= 0.43586652150845899941601945119356d+00 |
---|
| 897 | ros_Alpha(3)= 0.43586652150845899941601945119356d+00 |
---|
| 898 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 899 | ros_Gamma(1)= 0.43586652150845899941601945119356d+00 |
---|
| 900 | ros_Gamma(2)= 0.24291996454816804366592249683314d+00 |
---|
| 901 | ros_Gamma(3)= 0.21851380027664058511513169485832d+01 |
---|
| 902 | RETURN |
---|
| 903 | END ! SUBROUTINE Ros3 |
---|
| 904 | |
---|
| 905 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 906 | |
---|
| 907 | |
---|
| 908 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 909 | SUBROUTINE Ros4 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha, |
---|
| 910 | & ros_Gamma,ros_NewF,ros_ELO,ros_Name) |
---|
| 911 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 912 | ! L-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 4 STAGES |
---|
| 913 | ! L-STABLE EMBEDDED ROSENBROCK METHOD OF ORDER 3 |
---|
| 914 | ! |
---|
| 915 | ! E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL |
---|
| 916 | ! EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS. |
---|
| 917 | ! SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS, |
---|
| 918 | ! SPRINGER-VERLAG (1990) |
---|
| 919 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 920 | |
---|
| 921 | IMPLICIT NONE |
---|
| 922 | INTEGER S |
---|
| 923 | PARAMETER (S=4) |
---|
| 924 | INTEGER ros_S |
---|
| 925 | KPP_REAL ros_M(S), ros_E(S), ros_A(S*(S-1)/2), ros_C(S*(S-1)/2) |
---|
| 926 | KPP_REAL ros_Alpha(S), ros_Gamma(S), ros_ELO |
---|
| 927 | LOGICAL ros_NewF(S) |
---|
| 928 | CHARACTER*12 ros_Name |
---|
| 929 | |
---|
| 930 | !~~~> Name of the method |
---|
| 931 | ros_Name = 'ROS-4' |
---|
| 932 | !~~~> Number of stages |
---|
| 933 | ros_S = 4 |
---|
| 934 | |
---|
| 935 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 936 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 937 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 938 | ! The general mapping formula is: |
---|
| 939 | ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 940 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 941 | |
---|
| 942 | ros_A(1) = 0.2000000000000000d+01 |
---|
| 943 | ros_A(2) = 0.1867943637803922d+01 |
---|
| 944 | ros_A(3) = 0.2344449711399156d+00 |
---|
| 945 | ros_A(4) = ros_A(2) |
---|
| 946 | ros_A(5) = ros_A(3) |
---|
| 947 | ros_A(6) = 0.0D0 |
---|
| 948 | |
---|
| 949 | ros_C(1) =-0.7137615036412310d+01 |
---|
| 950 | ros_C(2) = 0.2580708087951457d+01 |
---|
| 951 | ros_C(3) = 0.6515950076447975d+00 |
---|
| 952 | ros_C(4) =-0.2137148994382534d+01 |
---|
| 953 | ros_C(5) =-0.3214669691237626d+00 |
---|
| 954 | ros_C(6) =-0.6949742501781779d+00 |
---|
| 955 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 956 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 957 | ros_NewF(1) = .TRUE. |
---|
| 958 | ros_NewF(2) = .TRUE. |
---|
| 959 | ros_NewF(3) = .TRUE. |
---|
| 960 | ros_NewF(4) = .FALSE. |
---|
| 961 | !~~~> M_i = Coefficients for new step solution |
---|
| 962 | ros_M(1) = 0.2255570073418735d+01 |
---|
| 963 | ros_M(2) = 0.2870493262186792d+00 |
---|
| 964 | ros_M(3) = 0.4353179431840180d+00 |
---|
| 965 | ros_M(4) = 0.1093502252409163d+01 |
---|
| 966 | !~~~> E_i = Coefficients for error estimator |
---|
| 967 | ros_E(1) =-0.2815431932141155d+00 |
---|
| 968 | ros_E(2) =-0.7276199124938920d-01 |
---|
| 969 | ros_E(3) =-0.1082196201495311d+00 |
---|
| 970 | ros_E(4) =-0.1093502252409163d+01 |
---|
| 971 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 972 | ! main and the embedded scheme orders plus 1 |
---|
| 973 | ros_ELO = 4.0d0 |
---|
| 974 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 975 | ros_Alpha(1) = 0.D0 |
---|
| 976 | ros_Alpha(2) = 0.1145640000000000d+01 |
---|
| 977 | ros_Alpha(3) = 0.6552168638155900d+00 |
---|
| 978 | ros_Alpha(4) = ros_Alpha(3) |
---|
| 979 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 980 | ros_Gamma(1) = 0.5728200000000000d+00 |
---|
| 981 | ros_Gamma(2) =-0.1769193891319233d+01 |
---|
| 982 | ros_Gamma(3) = 0.7592633437920482d+00 |
---|
| 983 | ros_Gamma(4) =-0.1049021087100450d+00 |
---|
| 984 | RETURN |
---|
| 985 | END ! SUBROUTINE Ros4 |
---|
| 986 | |
---|
| 987 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 988 | SUBROUTINE Rodas3 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha, |
---|
| 989 | & ros_Gamma,ros_NewF,ros_ELO,ros_Name) |
---|
| 990 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 991 | ! --- A STIFFLY-STABLE METHOD, 4 stages, order 3 |
---|
| 992 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 993 | IMPLICIT NONE |
---|
| 994 | INTEGER S |
---|
| 995 | PARAMETER (S=4) |
---|
| 996 | INTEGER ros_S |
---|
| 997 | KPP_REAL ros_M(S), ros_E(S), ros_A(S*(S-1)/2), ros_C(S*(S-1)/2) |
---|
| 998 | KPP_REAL ros_Alpha(S), ros_Gamma(S), ros_ELO |
---|
| 999 | LOGICAL ros_NewF(S) |
---|
| 1000 | CHARACTER*12 ros_Name |
---|
| 1001 | |
---|
| 1002 | !~~~> Name of the method |
---|
| 1003 | ros_Name = 'RODAS-3' |
---|
| 1004 | !~~~> Number of stages |
---|
| 1005 | ros_S = 4 |
---|
| 1006 | |
---|
| 1007 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 1008 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 1009 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 1010 | ! The general mapping formula is: |
---|
| 1011 | ! A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 1012 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 1013 | |
---|
| 1014 | ros_A(1) = 0.0d+00 |
---|
| 1015 | ros_A(2) = 2.0d+00 |
---|
| 1016 | ros_A(3) = 0.0d+00 |
---|
| 1017 | ros_A(4) = 2.0d+00 |
---|
| 1018 | ros_A(5) = 0.0d+00 |
---|
| 1019 | ros_A(6) = 1.0d+00 |
---|
| 1020 | |
---|
| 1021 | ros_C(1) = 4.0d+00 |
---|
| 1022 | ros_C(2) = 1.0d+00 |
---|
| 1023 | ros_C(3) =-1.0d+00 |
---|
| 1024 | ros_C(4) = 1.0d+00 |
---|
| 1025 | ros_C(5) =-1.0d+00 |
---|
| 1026 | ros_C(6) =-(8.0d+00/3.0d+00) |
---|
| 1027 | |
---|
| 1028 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 1029 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 1030 | ros_NewF(1) = .TRUE. |
---|
| 1031 | ros_NewF(2) = .FALSE. |
---|
| 1032 | ros_NewF(3) = .TRUE. |
---|
| 1033 | ros_NewF(4) = .TRUE. |
---|
| 1034 | !~~~> M_i = Coefficients for new step solution |
---|
| 1035 | ros_M(1) = 2.0d+00 |
---|
| 1036 | ros_M(2) = 0.0d+00 |
---|
| 1037 | ros_M(3) = 1.0d+00 |
---|
| 1038 | ros_M(4) = 1.0d+00 |
---|
| 1039 | !~~~> E_i = Coefficients for error estimator |
---|
| 1040 | ros_E(1) = 0.0d+00 |
---|
| 1041 | ros_E(2) = 0.0d+00 |
---|
| 1042 | ros_E(3) = 0.0d+00 |
---|
| 1043 | ros_E(4) = 1.0d+00 |
---|
| 1044 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 1045 | ! main and the embedded scheme orders plus 1 |
---|
| 1046 | ros_ELO = 3.0d+00 |
---|
| 1047 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 1048 | ros_Alpha(1) = 0.0d+00 |
---|
| 1049 | ros_Alpha(2) = 0.0d+00 |
---|
| 1050 | ros_Alpha(3) = 1.0d+00 |
---|
| 1051 | ros_Alpha(4) = 1.0d+00 |
---|
| 1052 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 1053 | ros_Gamma(1) = 0.5d+00 |
---|
| 1054 | ros_Gamma(2) = 1.5d+00 |
---|
| 1055 | ros_Gamma(3) = 0.0d+00 |
---|
| 1056 | ros_Gamma(4) = 0.0d+00 |
---|
| 1057 | RETURN |
---|
| 1058 | END ! SUBROUTINE Rodas3 |
---|
| 1059 | |
---|
| 1060 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1061 | SUBROUTINE Rodas4 (ros_S,ros_A,ros_C,ros_M,ros_E,ros_Alpha, |
---|
| 1062 | & ros_Gamma,ros_NewF,ros_ELO,ros_Name) |
---|
| 1063 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1064 | ! STIFFLY-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 6 STAGES |
---|
| 1065 | ! |
---|
| 1066 | ! E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL |
---|
| 1067 | ! EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS. |
---|
| 1068 | ! SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS, |
---|
| 1069 | ! SPRINGER-VERLAG (1996) |
---|
| 1070 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1071 | |
---|
| 1072 | IMPLICIT NONE |
---|
| 1073 | INTEGER S |
---|
| 1074 | PARAMETER (S=6) |
---|
| 1075 | INTEGER ros_S |
---|
| 1076 | KPP_REAL ros_M(S), ros_E(S), ros_A(S*(S-1)/2), ros_C(S*(S-1)/2) |
---|
| 1077 | KPP_REAL ros_Alpha(S), ros_Gamma(S), ros_ELO |
---|
| 1078 | LOGICAL ros_NewF(S) |
---|
| 1079 | CHARACTER*12 ros_Name |
---|
| 1080 | |
---|
| 1081 | !~~~> Name of the method |
---|
| 1082 | ros_Name = 'RODAS-4' |
---|
| 1083 | !~~~> Number of stages |
---|
| 1084 | ros_S = 6 |
---|
| 1085 | |
---|
| 1086 | !~~~> Y_stage_i ~ Y( T + H*Alpha_i ) |
---|
| 1087 | ros_Alpha(1) = 0.000d0 |
---|
| 1088 | ros_Alpha(2) = 0.386d0 |
---|
| 1089 | ros_Alpha(3) = 0.210d0 |
---|
| 1090 | ros_Alpha(4) = 0.630d0 |
---|
| 1091 | ros_Alpha(5) = 1.000d0 |
---|
| 1092 | ros_Alpha(6) = 1.000d0 |
---|
| 1093 | |
---|
| 1094 | !~~~> Gamma_i = \sum_j gamma_{i,j} |
---|
| 1095 | ros_Gamma(1) = 0.2500000000000000d+00 |
---|
| 1096 | ros_Gamma(2) =-0.1043000000000000d+00 |
---|
| 1097 | ros_Gamma(3) = 0.1035000000000000d+00 |
---|
| 1098 | ros_Gamma(4) =-0.3620000000000023d-01 |
---|
| 1099 | ros_Gamma(5) = 0.0d0 |
---|
| 1100 | ros_Gamma(6) = 0.0d0 |
---|
| 1101 | |
---|
| 1102 | !~~~> The coefficient matrices A and C are strictly lower triangular. |
---|
| 1103 | ! The lower triangular (subdiagonal) elements are stored in row-wise order: |
---|
| 1104 | ! A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc. |
---|
| 1105 | ! The general mapping formula is: A(i,j) = ros_A( (i-1)*(i-2)/2 + j ) |
---|
| 1106 | ! C(i,j) = ros_C( (i-1)*(i-2)/2 + j ) |
---|
| 1107 | |
---|
| 1108 | ros_A(1) = 0.1544000000000000d+01 |
---|
| 1109 | ros_A(2) = 0.9466785280815826d+00 |
---|
| 1110 | ros_A(3) = 0.2557011698983284d+00 |
---|
| 1111 | ros_A(4) = 0.3314825187068521d+01 |
---|
| 1112 | ros_A(5) = 0.2896124015972201d+01 |
---|
| 1113 | ros_A(6) = 0.9986419139977817d+00 |
---|
| 1114 | ros_A(7) = 0.1221224509226641d+01 |
---|
| 1115 | ros_A(8) = 0.6019134481288629d+01 |
---|
| 1116 | ros_A(9) = 0.1253708332932087d+02 |
---|
| 1117 | ros_A(10) =-0.6878860361058950d+00 |
---|
| 1118 | ros_A(11) = ros_A(7) |
---|
| 1119 | ros_A(12) = ros_A(8) |
---|
| 1120 | ros_A(13) = ros_A(9) |
---|
| 1121 | ros_A(14) = ros_A(10) |
---|
| 1122 | ros_A(15) = 1.0d+00 |
---|
| 1123 | |
---|
| 1124 | ros_C(1) =-0.5668800000000000d+01 |
---|
| 1125 | ros_C(2) =-0.2430093356833875d+01 |
---|
| 1126 | ros_C(3) =-0.2063599157091915d+00 |
---|
| 1127 | ros_C(4) =-0.1073529058151375d+00 |
---|
| 1128 | ros_C(5) =-0.9594562251023355d+01 |
---|
| 1129 | ros_C(6) =-0.2047028614809616d+02 |
---|
| 1130 | ros_C(7) = 0.7496443313967647d+01 |
---|
| 1131 | ros_C(8) =-0.1024680431464352d+02 |
---|
| 1132 | ros_C(9) =-0.3399990352819905d+02 |
---|
| 1133 | ros_C(10) = 0.1170890893206160d+02 |
---|
| 1134 | ros_C(11) = 0.8083246795921522d+01 |
---|
| 1135 | ros_C(12) =-0.7981132988064893d+01 |
---|
| 1136 | ros_C(13) =-0.3152159432874371d+02 |
---|
| 1137 | ros_C(14) = 0.1631930543123136d+02 |
---|
| 1138 | ros_C(15) =-0.6058818238834054d+01 |
---|
| 1139 | |
---|
| 1140 | !~~~> M_i = Coefficients for new step solution |
---|
| 1141 | ros_M(1) = ros_A(7) |
---|
| 1142 | ros_M(2) = ros_A(8) |
---|
| 1143 | ros_M(3) = ros_A(9) |
---|
| 1144 | ros_M(4) = ros_A(10) |
---|
| 1145 | ros_M(5) = 1.0d+00 |
---|
| 1146 | ros_M(6) = 1.0d+00 |
---|
| 1147 | |
---|
| 1148 | !~~~> E_i = Coefficients for error estimator |
---|
| 1149 | ros_E(1) = 0.0d+00 |
---|
| 1150 | ros_E(2) = 0.0d+00 |
---|
| 1151 | ros_E(3) = 0.0d+00 |
---|
| 1152 | ros_E(4) = 0.0d+00 |
---|
| 1153 | ros_E(5) = 0.0d+00 |
---|
| 1154 | ros_E(6) = 1.0d+00 |
---|
| 1155 | |
---|
| 1156 | !~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE) |
---|
| 1157 | ! or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE) |
---|
| 1158 | ros_NewF(1) = .TRUE. |
---|
| 1159 | ros_NewF(2) = .TRUE. |
---|
| 1160 | ros_NewF(3) = .TRUE. |
---|
| 1161 | ros_NewF(4) = .TRUE. |
---|
| 1162 | ros_NewF(5) = .TRUE. |
---|
| 1163 | ros_NewF(6) = .TRUE. |
---|
| 1164 | |
---|
| 1165 | !~~~> ros_ELO = estimator of local order - the minimum between the |
---|
| 1166 | ! main and the embedded scheme orders plus 1 |
---|
| 1167 | ros_ELO = 4.0d0 |
---|
| 1168 | |
---|
| 1169 | RETURN |
---|
| 1170 | END ! SUBROUTINE Rodas4 |
---|
| 1171 | |
---|
| 1172 | |
---|
| 1173 | |
---|
| 1174 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1175 | SUBROUTINE DecompTemplate( A, Pivot, ising ) |
---|
| 1176 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1177 | ! Template for the LU decomposition |
---|
| 1178 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1179 | INCLUDE 'KPP_ROOT_Parameters.h' |
---|
| 1180 | INCLUDE 'KPP_ROOT_Global.h' |
---|
| 1181 | !~~~> Inout variables |
---|
| 1182 | KPP_REAL A(KPP_LU_NONZERO) |
---|
| 1183 | !~~~> Output variables |
---|
| 1184 | INTEGER Pivot(KPP_NVAR), ising |
---|
| 1185 | !~~~> Collect statistics |
---|
| 1186 | INTEGER Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng |
---|
| 1187 | COMMON /Statistics/ Nfun,Njac,Nstp,Nacc,Nrej, |
---|
| 1188 | & Ndec,Nsol,Nsng |
---|
| 1189 | |
---|
| 1190 | CALL KppDecomp ( A, ising ) |
---|
| 1191 | !~~~> Note: for a full matrix use Lapack: |
---|
| 1192 | ! CALL DGETRF( KPP_NVAR, KPP_NVAR, A, KPP_NVAR, Pivot, ising ) |
---|
| 1193 | |
---|
| 1194 | Ndec = Ndec + 1 |
---|
| 1195 | |
---|
| 1196 | END ! SUBROUTINE DecompTemplate |
---|
| 1197 | |
---|
| 1198 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1199 | SUBROUTINE SolveTemplate( A, Pivot, b ) |
---|
| 1200 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1201 | ! Template for the forward/backward substitution (using pre-computed LU decomposition) |
---|
| 1202 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1203 | INCLUDE 'KPP_ROOT_Parameters.h' |
---|
| 1204 | INCLUDE 'KPP_ROOT_Global.h' |
---|
| 1205 | !~~~> Input variables |
---|
| 1206 | KPP_REAL A(KPP_LU_NONZERO) |
---|
| 1207 | INTEGER Pivot(KPP_NVAR) |
---|
| 1208 | !~~~> InOut variables |
---|
| 1209 | KPP_REAL b(KPP_NVAR) |
---|
| 1210 | !~~~> Collect statistics |
---|
| 1211 | INTEGER Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng |
---|
| 1212 | COMMON /Statistics/ Nfun,Njac,Nstp,Nacc,Nrej, |
---|
| 1213 | & Ndec,Nsol,Nsng |
---|
| 1214 | |
---|
| 1215 | CALL KppSolve( A, b ) |
---|
| 1216 | !~~~> Note: for a full matrix use Lapack: |
---|
| 1217 | ! NRHS = 1 |
---|
| 1218 | ! CALL DGETRS( 'N', KPP_NVAR , NRHS, A, KPP_NVAR, Pivot, b, KPP_NVAR, INFO ) |
---|
| 1219 | |
---|
| 1220 | Nsol = Nsol+1 |
---|
| 1221 | |
---|
| 1222 | END ! SUBROUTINE SolveTemplate |
---|
| 1223 | |
---|
| 1224 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1225 | SUBROUTINE FunTemplate( T, Y, Ydot ) |
---|
| 1226 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1227 | ! Template for the ODE function call. |
---|
| 1228 | ! Updates the rate coefficients (and possibly the fixed species) at each call |
---|
| 1229 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1230 | INCLUDE 'KPP_ROOT_Parameters.h' |
---|
| 1231 | INCLUDE 'KPP_ROOT_Global.h' |
---|
| 1232 | !~~~> Input variables |
---|
| 1233 | KPP_REAL T, Y(KPP_NVAR) |
---|
| 1234 | !~~~> Output variables |
---|
| 1235 | KPP_REAL Ydot(KPP_NVAR) |
---|
| 1236 | !~~~> Local variables |
---|
| 1237 | KPP_REAL Told |
---|
| 1238 | !~~~> Collect statistics |
---|
| 1239 | INTEGER Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng |
---|
| 1240 | COMMON /Statistics/ Nfun,Njac,Nstp,Nacc,Nrej, |
---|
| 1241 | & Ndec,Nsol,Nsng |
---|
| 1242 | |
---|
| 1243 | Told = TIME |
---|
| 1244 | TIME = T |
---|
| 1245 | CALL Update_SUN() |
---|
| 1246 | CALL Update_RCONST() |
---|
| 1247 | CALL Fun( Y, FIX, RCONST, Ydot ) |
---|
| 1248 | TIME = Told |
---|
| 1249 | |
---|
| 1250 | Nfun = Nfun+1 |
---|
| 1251 | |
---|
| 1252 | RETURN |
---|
| 1253 | END ! SUBROUTINE FunTemplate |
---|
| 1254 | |
---|
| 1255 | |
---|
| 1256 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1257 | SUBROUTINE JacTemplate( T, Y, Jcb ) |
---|
| 1258 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1259 | ! Template for the ODE Jacobian call. |
---|
| 1260 | ! Updates the rate coefficients (and possibly the fixed species) at each call |
---|
| 1261 | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |
---|
| 1262 | INCLUDE 'KPP_ROOT_Parameters.h' |
---|
| 1263 | INCLUDE 'KPP_ROOT_Global.h' |
---|
| 1264 | !~~~> Input variables |
---|
| 1265 | KPP_REAL T, Y(KPP_NVAR) |
---|
| 1266 | !~~~> Output variables |
---|
| 1267 | KPP_REAL Jcb(KPP_LU_NONZERO) |
---|
| 1268 | !~~~> Local variables |
---|
| 1269 | KPP_REAL Told |
---|
| 1270 | !~~~> Collect statistics |
---|
| 1271 | INTEGER Nfun,Njac,Nstp,Nacc,Nrej,Ndec,Nsol,Nsng |
---|
| 1272 | COMMON /Statistics/ Nfun,Njac,Nstp,Nacc,Nrej, |
---|
| 1273 | & Ndec,Nsol,Nsng |
---|
| 1274 | |
---|
| 1275 | Told = TIME |
---|
| 1276 | TIME = T |
---|
| 1277 | CALL Update_SUN() |
---|
| 1278 | CALL Update_RCONST() |
---|
| 1279 | CALL Jac_SP( Y, FIX, RCONST, Jcb ) |
---|
| 1280 | TIME = Told |
---|
| 1281 | |
---|
| 1282 | Njac = Njac+1 |
---|
| 1283 | |
---|
| 1284 | RETURN |
---|
| 1285 | END ! SUBROUTINE JacTemplate |
---|
| 1286 | |
---|