!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~! ! RungeKutta - Fully Implicit 3-stage Runge-Kutta methods based on: ! ! * Radau-2A quadrature (order 5) ! ! * Radau-1A quadrature (order 5) ! ! * Lobatto-3C quadrature (order 4) ! ! * Gauss quadrature (order 6) ! ! By default the code employs the KPP sparse linear algebra routines ! ! Compile with -DFULL_ALGEBRA to use full linear algebra (LAPACK) ! ! ! ! (C) Adrian Sandu, August 2005 ! ! Virginia Polytechnic Institute and State University ! ! Contact: sandu@cs.vt.edu ! ! Revised by Philipp Miehe and Adrian Sandu, May 2006 ! ! This implementation is part of KPP - the Kinetic PreProcessor ! !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~! MODULE KPP_ROOT_Integrator USE KPP_ROOT_Precision USE KPP_ROOT_Parameters, ONLY: NVAR, NSPEC, NFIX, LU_NONZERO USE KPP_ROOT_Global, ONLY: FIX, RCONST, TIME USE KPP_ROOT_Jacobian, ONLY: LU_DIAG USE KPP_ROOT_LinearAlgebra IMPLICIT NONE PUBLIC SAVE !~~~> Statistics on the work performed by the Runge-Kutta method INTEGER, PARAMETER :: Nfun=1, Njac=2, Nstp=3, Nacc=4, & Nrej=5, Ndec=6, Nsol=7, Nsng=8, Ntexit=1, Nhacc=2, Nhnew=3 CONTAINS ! ************************************************************************** SUBROUTINE INTEGRATE( TIN, TOUT, & ICNTRL_U, RCNTRL_U, ISTATUS_U, RSTATUS_U, IERR_U ) USE KPP_ROOT_Parameters, ONLY: NVAR USE KPP_ROOT_Global, ONLY: ATOL,RTOL,VAR IMPLICIT NONE KPP_REAL :: TIN ! TIN - Start Time KPP_REAL :: TOUT ! TOUT - End Time INTEGER, INTENT(IN), OPTIONAL :: ICNTRL_U(20) KPP_REAL, INTENT(IN), OPTIONAL :: RCNTRL_U(20) INTEGER, INTENT(OUT), OPTIONAL :: ISTATUS_U(20) KPP_REAL, INTENT(OUT), OPTIONAL :: RSTATUS_U(20) INTEGER, INTENT(OUT), OPTIONAL :: IERR_U INTEGER :: IERR KPP_REAL :: RCNTRL(20), RSTATUS(20), T1, T2 INTEGER :: ICNTRL(20), ISTATUS(20) INTEGER, SAVE :: Ntotal = 0 RCNTRL(1:20) = 0.0_dp ICNTRL(1:20) = 0 !~~~> fine-tune the integrator: ICNTRL(2) = 0 ! 0=vector tolerances, 1=scalar tolerances ICNTRL(5) = 8 ! Max no. of Newton iterations ICNTRL(6) = 0 ! Starting values for Newton are interpolated (0) or zero (1) ICNTRL(10) = 1 ! 0 - classic or 1 - SDIRK error estimation ICNTRL(11) = 0 ! Gustaffson (0) or classic(1) controller !~~~> if optional parameters are given, and if they are >0, ! then use them to overwrite default settings IF (PRESENT(ICNTRL_U)) THEN WHERE(ICNTRL_U(:) > 0) ICNTRL(:) = ICNTRL_U(:) END IF IF (PRESENT(RCNTRL_U)) THEN WHERE(RCNTRL_U(:) > 0) RCNTRL(:) = RCNTRL_U(:) END IF T1 = TIN; T2 = TOUT CALL RungeKutta( NVAR, T1, T2, VAR, RTOL, ATOL, & RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR ) Ntotal = Ntotal + ISTATUS(Nstp) PRINT*,'NSTEPS=',ISTATUS(Nstp),' (',Ntotal,')',' O3=', VAR(ind_O3) ! if optional parameters are given for output ! use them to store information in them IF (PRESENT(ISTATUS_U)) ISTATUS_U(:) = ISTATUS(:) IF (PRESENT(RSTATUS_U)) RSTATUS_U(:) = RSTATUS(:) IF (PRESENT(IERR_U)) IERR_U = IERR IF (IERR < 0) THEN PRINT *,'Runge-Kutta: Unsuccessful exit at T=', TIN,' (IERR=',IERR,')' ENDIF END SUBROUTINE INTEGRATE !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE RungeKutta( N,T,Tend,Y,RelTol,AbsTol, & RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR ) !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ! ! This implementation is based on the book and the code Radau5: ! ! E. HAIRER AND G. WANNER ! "SOLVING ORDINARY DIFFERENTIAL EQUATIONS II. ! STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS." ! SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS 14, ! SPRINGER-VERLAG (1991) ! ! UNIVERSITE DE GENEVE, DEPT. DE MATHEMATIQUES ! CH-1211 GENEVE 24, SWITZERLAND ! E-MAIL: HAIRER@DIVSUN.UNIGE.CH, WANNER@DIVSUN.UNIGE.CH ! ! Methods: ! * Radau-2A quadrature (order 5) ! * Radau-1A quadrature (order 5) ! * Lobatto-3C quadrature (order 4) ! * Gauss quadrature (order 6) ! ! (C) Adrian Sandu, August 2005 ! Virginia Polytechnic Institute and State University ! Contact: sandu@cs.vt.edu ! Revised by Philipp Miehe and Adrian Sandu, May 2006 ! This implementation is part of KPP - the Kinetic PreProcessor ! !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ! !~~~> INPUT ARGUMENTS: ! ---------------- ! ! Note: For input parameters equal to zero the default values of the ! corresponding variables are used. ! ! N Dimension of the system ! T Initial time value ! ! Tend Final T value (Tend-T may be positive or negative) ! ! Y(N) Initial values for Y ! ! RelTol,AbsTol Relative and absolute error tolerances. ! for ICNTRL(2) = 0: AbsTol, RelTol are N-dimensional vectors ! = 1: AbsTol, RelTol are scalars ! !~~~> Integer input parameters: ! ! ICNTRL(1) = not used ! ! ICNTRL(2) = 0: AbsTol, RelTol are NVAR-dimensional vectors ! = 1: AbsTol, RelTol are scalars ! ! ICNTRL(3) = RK method selection ! = 1: Radau-2A (the default) ! = 2: Lobatto-3C ! = 3: Gauss ! = 4: Radau-1A ! = 5: Lobatto-3A (not yet implemented) ! ! ICNTRL(4) -> maximum number of integration steps ! For ICNTRL(4)=0 the default value of 10000 is used ! ! ICNTRL(5) -> maximum number of Newton iterations ! For ICNTRL(5)=0 the default value of 8 is used ! ! ICNTRL(6) -> starting values of Newton iterations: ! ICNTRL(6)=0 : starting values are obtained from ! the extrapolated collocation solution ! (the default) ! ICNTRL(6)=1 : starting values are zero ! ! ICNTRL(10) -> switch for error estimation strategy ! ICNTRL(10) = 0: one additional stage at c=0, ! see Hairer (default) ! ICNTRL(10) = 1: two additional stages at c=0 ! and SDIRK at c=1, stiffly accurate ! ! ICNTRL(11) -> switch for step size strategy ! ICNTRL(11)=0: mod. predictive controller (Gustafsson, default) ! ICNTRL(11)=1: classical step size control ! the choice 1 seems to produce safer results; ! for simple problems, the choice 2 produces ! often slightly faster runs ! !~~~> Real input parameters: ! ! RCNTRL(1) -> Hmin, lower bound for the integration step size ! (highly recommended to keep Hmin = ZERO, the default) ! ! RCNTRL(2) -> Hmax, upper bound for the integration step size ! ! RCNTRL(3) -> Hstart, the starting step size ! ! RCNTRL(4) -> FacMin, lower bound on step decrease factor (default=0.2) ! ! RCNTRL(5) -> FacMax, upper bound on step increase factor (default=6) ! ! RCNTRL(6) -> FacRej, step decrease factor after multiple rejections ! (default=0.1) ! ! RCNTRL(7) -> FacSafe, by which the new step is slightly smaller ! than the predicted value (default=0.9) ! ! RCNTRL(8) -> ThetaMin. If Newton convergence rate smaller ! than ThetaMin the Jacobian is not recomputed; ! (default=0.001) ! ! RCNTRL(9) -> NewtonTol, stopping criterion for Newton's method ! (default=0.03) ! ! RCNTRL(10) -> Qmin ! ! RCNTRL(11) -> Qmax. If Qmin < Hnew/Hold < Qmax, then the ! step size is kept constant and the LU factorization ! reused (default Qmin=1, Qmax=1.2) !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ! ! ! OUTPUT ARGUMENTS: ! ----------------- ! ! T -> T value for which the solution has been computed ! (after successful return T=Tend). ! ! Y(N) -> Numerical solution at T ! ! IERR -> Reports on successfulness upon return: ! = 1 for success ! < 0 for error (value equals error code) ! ! ISTATUS(1) -> No. of function calls ! ISTATUS(2) -> No. of Jacobian calls ! ISTATUS(3) -> No. of steps ! ISTATUS(4) -> No. of accepted steps ! ISTATUS(5) -> No. of rejected steps (except at very beginning) ! ISTATUS(6) -> No. of LU decompositions ! ISTATUS(7) -> No. of forward/backward substitutions ! ISTATUS(8) -> No. of singular matrix decompositions ! ! RSTATUS(1) -> Texit, the time corresponding to the ! computed Y upon return ! RSTATUS(2) -> Hexit, last accepted step before exit ! RSTATUS(3) -> Hnew, last predicted step (not yet taken) ! For multiple restarts, use Hnew as Hstart ! in the subsequent run ! !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE INTEGER :: N KPP_REAL :: Y(N),AbsTol(N),RelTol(N),RCNTRL(20),RSTATUS(20) INTEGER :: ICNTRL(20), ISTATUS(20) LOGICAL :: StartNewton, Gustafsson, SdirkError INTEGER :: IERR, ITOL KPP_REAL :: T,Tend !~~~> Control arguments INTEGER :: Max_no_steps, NewtonMaxit, rkMethod KPP_REAL :: Hmin,Hmax,Hstart,Qmin,Qmax KPP_REAL :: Roundoff, ThetaMin, NewtonTol KPP_REAL :: FacSafe,FacMin,FacMax,FacRej ! Runge-Kutta method parameters INTEGER, PARAMETER :: R2A=1, R1A=2, L3C=3, GAU=4, L3A=5 KPP_REAL :: rkT(3,3), rkTinv(3,3), rkTinvAinv(3,3), rkAinvT(3,3), & rkA(0:3,0:3), rkB(0:3), rkC(0:3), rkD(0:3), rkE(0:3), & rkBgam(0:4), rkBhat(0:4), rkTheta(0:3), rkF(0:4), & rkGamma, rkAlpha, rkBeta, rkELO !~~~> Local variables INTEGER :: i KPP_REAL, PARAMETER :: ZERO = 0.0d0, ONE = 1.0d0 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ! SETTING THE PARAMETERS !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IERR = 0 ISTATUS(1:20) = 0 RSTATUS(1:20) = ZERO !~~~> ICNTRL(1) - autonomous system - not used !~~~> ITOL: 1 for vector and 0 for scalar AbsTol/RelTol IF (ICNTRL(2) == 0) THEN ITOL = 1 ELSE ITOL = 0 END IF !~~~> Error control selection IF (ICNTRL(10) == 0) THEN SdirkError = .FALSE. ELSE SdirkError = .TRUE. END IF !~~~> Method selection SELECT CASE (ICNTRL(3)) CASE (0,1) CALL Radau2A_Coefficients CASE (2) CALL Lobatto3C_Coefficients CASE (3) CALL Gauss_Coefficients CASE (4) CALL Radau1A_Coefficients CASE (5) CALL Lobatto3A_Coefficients CASE DEFAULT WRITE(6,*) 'ICNTRL(3)=',ICNTRL(3) CALL RK_ErrorMsg(-13,T,ZERO,IERR) END SELECT !~~~> Max_no_steps: the maximal number of time steps IF (ICNTRL(4) == 0) THEN Max_no_steps = 200000 ELSE Max_no_steps=ICNTRL(4) IF (Max_no_steps <= 0) THEN WRITE(6,*) 'ICNTRL(4)=',ICNTRL(4) CALL RK_ErrorMsg(-1,T,ZERO,IERR) END IF END IF !~~~> NewtonMaxit maximal number of Newton iterations IF (ICNTRL(5) == 0) THEN NewtonMaxit = 8 ELSE NewtonMaxit=ICNTRL(5) IF (NewtonMaxit <= 0) THEN WRITE(6,*) 'ICNTRL(5)=',ICNTRL(5) CALL RK_ErrorMsg(-2,T,ZERO,IERR) END IF END IF !~~~> StartNewton: Use extrapolation for starting values of Newton iterations IF (ICNTRL(6) == 0) THEN StartNewton = .TRUE. ELSE StartNewton = .FALSE. END IF !~~~> Gustafsson: step size controller IF(ICNTRL(11) == 0)THEN Gustafsson = .TRUE. ELSE Gustafsson = .FALSE. END IF !~~~> Roundoff: smallest number s.t. 1.0 + Roundoff > 1.0 Roundoff=WLAMCH('E'); !~~~> Hmin = minimal step size IF (RCNTRL(1) == ZERO) THEN Hmin = ZERO ELSE Hmin = MIN(ABS(RCNTRL(1)),ABS(Tend-T)) END IF !~~~> Hmax = maximal step size IF (RCNTRL(2) == ZERO) THEN Hmax = ABS(Tend-T) ELSE Hmax = MIN(ABS(RCNTRL(2)),ABS(Tend-T)) END IF !~~~> Hstart = starting step size IF (RCNTRL(3) == ZERO) THEN Hstart = ZERO ELSE Hstart = MIN(ABS(RCNTRL(3)),ABS(Tend-T)) END IF !~~~> FacMin: lower bound on step decrease factor IF(RCNTRL(4) == ZERO)THEN FacMin = 0.2d0 ELSE FacMin = RCNTRL(4) END IF !~~~> FacMax: upper bound on step increase factor IF(RCNTRL(5) == ZERO)THEN FacMax = 8.D0 ELSE FacMax = RCNTRL(5) END IF !~~~> FacRej: step decrease factor after 2 consecutive rejections IF(RCNTRL(6) == ZERO)THEN FacRej = 0.1d0 ELSE FacRej = RCNTRL(6) END IF !~~~> FacSafe: by which the new step is slightly smaller ! than the predicted value IF (RCNTRL(7) == ZERO) THEN FacSafe=0.9d0 ELSE FacSafe=RCNTRL(7) END IF IF ( (FacMax < ONE) .OR. (FacMin > ONE) .OR. & (FacSafe <= 1.0d-3) .OR. (FacSafe >= ONE) ) THEN WRITE(6,*)'RCNTRL(4:7)=',RCNTRL(4:7) CALL RK_ErrorMsg(-4,T,ZERO,IERR) END IF !~~~> ThetaMin: decides whether the Jacobian should be recomputed IF (RCNTRL(8) == ZERO) THEN ThetaMin = 1.0d-3 ELSE ThetaMin=RCNTRL(8) IF (ThetaMin <= 0.0d0 .OR. ThetaMin >= 1.0d0) THEN WRITE(6,*) 'RCNTRL(8)=', RCNTRL(8) CALL RK_ErrorMsg(-5,T,ZERO,IERR) END IF END IF !~~~> NewtonTol: stopping crierion for Newton's method IF (RCNTRL(9) == ZERO) THEN NewtonTol = 3.0d-2 ELSE NewtonTol = RCNTRL(9) IF (NewtonTol <= Roundoff) THEN WRITE(6,*) 'RCNTRL(9)=',RCNTRL(9) CALL RK_ErrorMsg(-6,T,ZERO,IERR) END IF END IF !~~~> Qmin AND Qmax: IF Qmin < Hnew/Hold < Qmax then step size = const. IF (RCNTRL(10) == ZERO) THEN Qmin=1.D0 ELSE Qmin=RCNTRL(10) END IF IF (RCNTRL(11) == ZERO) THEN Qmax=1.2D0 ELSE Qmax=RCNTRL(11) END IF IF (Qmin > ONE .OR. Qmax < ONE) THEN WRITE(6,*) 'RCNTRL(10:11)=',Qmin,Qmax CALL RK_ErrorMsg(-7,T,ZERO,IERR) END IF !~~~> Check if tolerances are reasonable IF (ITOL == 0) THEN IF (AbsTol(1) <= ZERO.OR.RelTol(1) <= 10.d0*Roundoff) THEN WRITE (6,*) 'AbsTol/RelTol=',AbsTol,RelTol CALL RK_ErrorMsg(-8,T,ZERO,IERR) END IF ELSE DO i=1,N IF (AbsTol(i) <= ZERO.OR.RelTol(i) <= 10.d0*Roundoff) THEN WRITE (6,*) 'AbsTol/RelTol(',i,')=',AbsTol(i),RelTol(i) CALL RK_ErrorMsg(-8,T,ZERO,IERR) END IF END DO END IF !~~~> Parameters are wrong IF (IERR < 0) RETURN !~~~> Call the core method CALL RK_Integrator( N,T,Tend,Y,IERR ) CONTAINS ! Internal procedures to RungeKutta !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE RK_Integrator( N,T,Tend,Y,IERR ) !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE !~~~> Arguments INTEGER, INTENT(IN) :: N KPP_REAL, INTENT(IN) :: Tend KPP_REAL, INTENT(INOUT) :: T, Y(NVAR) INTEGER, INTENT(OUT) :: IERR !~~~> Local variables #ifdef FULL_ALGEBRA KPP_REAL :: FJAC(NVAR,NVAR), E1(NVAR,NVAR) COMPLEX(kind=dp) :: E2(NVAR,NVAR) #else KPP_REAL :: FJAC(LU_NONZERO), E1(LU_NONZERO) COMPLEX(kind=dp) :: E2(LU_NONZERO) #endif KPP_REAL, DIMENSION(NVAR) :: Z1,Z2,Z3,Z4,SCAL,DZ1,DZ2,DZ3,DZ4, & G,TMP,F0 KPP_REAL :: CONT(NVAR,3), Tdirection, H, Hacc, Hnew, Hold, Fac, & FacGus, Theta, Err, ErrOld, NewtonRate, NewtonIncrement, & Hratio, Qnewton, NewtonPredictedErr,NewtonIncrementOld, ThetaSD INTEGER :: IP1(NVAR),IP2(NVAR),NewtonIter, ISING, Nconsecutive LOGICAL :: Reject, FirstStep, SkipJac, NewtonDone, SkipLU !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ !~~~> Initial setting !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Tdirection = SIGN(ONE,Tend-T) H = MIN( MAX(ABS(Hmin),ABS(Hstart)) , Hmax ) IF (ABS(H) <= 10.d0*Roundoff) H = 1.0d-6 H = SIGN(H,Tdirection) Hold = H Reject = .FALSE. FirstStep = .TRUE. SkipJac = .FALSE. SkipLU = .FALSE. IF ((T+H*1.0001D0-Tend)*Tdirection >= ZERO) THEN H = Tend-T END IF Nconsecutive = 0 CALL RK_ErrorScale(N,ITOL,AbsTol,RelTol,Y,SCAL) !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ !~~~> Time loop begins !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Tloop: DO WHILE ( (Tend-T)*Tdirection - Roundoff > ZERO ) !IF ( .NOT.Reject ) THEN CALL FUN_CHEM(T,Y,F0) ISTATUS(Nfun) = ISTATUS(Nfun) + 1 !END IF IF ( .NOT.SkipLU ) THEN ! This time around skip the Jac update and LU !~~~> Compute the Jacobian matrix IF ( .NOT.SkipJac ) THEN CALL JAC_CHEM(T,Y,FJAC) ISTATUS(Njac) = ISTATUS(Njac) + 1 END IF !~~~> Compute the matrices E1 and E2 and their decompositions CALL RK_Decomp(N,H,FJAC,E1,IP1,E2,IP2,ISING) IF (ISING /= 0) THEN ISTATUS(Nsng) = ISTATUS(Nsng) + 1; Nconsecutive = Nconsecutive + 1 IF (Nconsecutive >= 5) THEN CALL RK_ErrorMsg(-12,T,H,IERR); RETURN END IF H=H*0.5d0; Reject=.TRUE.; SkipJac = .TRUE.; SkipLU = .FALSE. CYCLE Tloop ELSE Nconsecutive = 0 END IF END IF ! SkipLU ISTATUS(Nstp) = ISTATUS(Nstp) + 1 IF (ISTATUS(Nstp) > Max_no_steps) THEN PRINT*,'Max number of time steps is ',Max_no_steps CALL RK_ErrorMsg(-9,T,H,IERR); RETURN END IF IF (0.1D0*ABS(H) <= ABS(T)*Roundoff) THEN CALL RK_ErrorMsg(-10,T,H,IERR); RETURN END IF !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ !~~~> Loop for the simplified Newton iterations !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ !~~~> Starting values for Newton iteration IF ( FirstStep .OR. (.NOT.StartNewton) ) THEN CALL Set2zero(N,Z1) CALL Set2zero(N,Z2) CALL Set2zero(N,Z3) ELSE ! Evaluate quadratic polynomial CALL RK_Interpolate('eval',N,H,Hold,Z1,Z2,Z3,CONT) END IF !~~~> Initializations for Newton iteration NewtonDone = .FALSE. Fac = 0.5d0 ! Step reduction if too many iterations NewtonLoop:DO NewtonIter = 1, NewtonMaxit !~~~> Prepare the right-hand side CALL RK_PrepareRHS(N,T,H,Y,F0,Z1,Z2,Z3,DZ1,DZ2,DZ3) !~~~> Solve the linear systems CALL RK_Solve( N,H,E1,IP1,E2,IP2,DZ1,DZ2,DZ3,ISING ) NewtonIncrement = SQRT( ( RK_ErrorNorm(N,SCAL,DZ1)**2 + & RK_ErrorNorm(N,SCAL,DZ2)**2 + & RK_ErrorNorm(N,SCAL,DZ3)**2 )/3.0d0 ) IF ( NewtonIter == 1 ) THEN Theta = ABS(ThetaMin) NewtonRate = 2.0d0 ELSE Theta = NewtonIncrement/NewtonIncrementOld IF (Theta < 0.99d0) THEN NewtonRate = Theta/(ONE-Theta) ELSE ! Non-convergence of Newton: Theta too large EXIT NewtonLoop END IF IF ( NewtonIter < NewtonMaxit ) THEN ! Predict error at the end of Newton process NewtonPredictedErr = NewtonIncrement & *Theta**(NewtonMaxit-NewtonIter)/(ONE-Theta) IF (NewtonPredictedErr >= NewtonTol) THEN ! Non-convergence of Newton: predicted error too large Qnewton = MIN(10.0d0,NewtonPredictedErr/NewtonTol) Fac=0.8d0*Qnewton**(-ONE/(1+NewtonMaxit-NewtonIter)) EXIT NewtonLoop END IF END IF END IF NewtonIncrementOld = MAX(NewtonIncrement,Roundoff) ! Update solution CALL WAXPY(N,-ONE,DZ1,1,Z1,1) ! Z1 <- Z1 - DZ1 CALL WAXPY(N,-ONE,DZ2,1,Z2,1) ! Z2 <- Z2 - DZ2 CALL WAXPY(N,-ONE,DZ3,1,Z3,1) ! Z3 <- Z3 - DZ3 ! Check error in Newton iterations NewtonDone = (NewtonRate*NewtonIncrement <= NewtonTol) IF (NewtonDone) EXIT NewtonLoop IF (NewtonIter == NewtonMaxit) THEN PRINT*, 'Slow or no convergence in Newton Iteration: Max no. of', & 'Newton iterations reached' END IF END DO NewtonLoop IF (.NOT.NewtonDone) THEN !CALL RK_ErrorMsg(-12,T,H,IERR); H = Fac*H; Reject=.TRUE.; SkipJac = .TRUE.; SkipLU = .FALSE. CYCLE Tloop END IF !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ !~~~> SDIRK Stage !~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IF (SdirkError) THEN !~~~> Starting values for Newton iterations Z4(1:N) = Z3(1:N) !~~~> Prepare the loop-independent part of the right-hand side ! G = H*rkBgam(0)*F0 + rkTheta(1)*Z1 + rkTheta(2)*Z2 + rkTheta(3)*Z3 CALL Set2Zero(N, G) IF (rkMethod/=L3A) CALL WAXPY(N,rkBgam(0)*H, F0,1,G,1) CALL WAXPY(N,rkTheta(1),Z1,1,G,1) CALL WAXPY(N,rkTheta(2),Z2,1,G,1) CALL WAXPY(N,rkTheta(3),Z3,1,G,1) !~~~> Initializations for Newton iteration NewtonDone = .FALSE. Fac = 0.5d0 ! Step reduction factor if too many iterations SDNewtonLoop:DO NewtonIter = 1, NewtonMaxit !~~~> Prepare the loop-dependent part of the right-hand side CALL WADD(N,Y,Z4,TMP) ! TMP <- Y + Z4 CALL FUN_CHEM(T+H,TMP,DZ4) ! DZ4 <- Fun(Y+Z4) ISTATUS(Nfun) = ISTATUS(Nfun) + 1 ! DZ4(1:N) = (G(1:N)-Z4(1:N))*(rkGamma/H) + DZ4(1:N) CALL WAXPY (N, -ONE*rkGamma/H, Z4, 1, DZ4, 1) CALL WAXPY (N, rkGamma/H, G,1, DZ4,1) !~~~> Solve the linear system #ifdef FULL_ALGEBRA CALL DGETRS( 'N', N, 1, E1, N, IP1, DZ4, N, ISING ) #else CALL KppSolve(E1, DZ4) #endif !~~~> Check convergence of Newton iterations NewtonIncrement = RK_ErrorNorm(N,SCAL,DZ4) IF ( NewtonIter == 1 ) THEN ThetaSD = ABS(ThetaMin) NewtonRate = 2.0d0 ELSE ThetaSD = NewtonIncrement/NewtonIncrementOld IF (ThetaSD < 0.99d0) THEN NewtonRate = ThetaSD/(ONE-ThetaSD) ! Predict error at the end of Newton process NewtonPredictedErr = NewtonIncrement & *ThetaSD**(NewtonMaxit-NewtonIter)/(ONE-ThetaSD) IF (NewtonPredictedErr >= NewtonTol) THEN ! Non-convergence of Newton: predicted error too large !PRINT*,'Error too large: ', NewtonPredictedErr Qnewton = MIN(10.0d0,NewtonPredictedErr/NewtonTol) Fac = 0.8d0*Qnewton**(-ONE/(1+NewtonMaxit-NewtonIter)) EXIT SDNewtonLoop END IF ELSE ! Non-convergence of Newton: Theta too large !PRINT*,'Theta too large: ',ThetaSD EXIT SDNewtonLoop END IF END IF NewtonIncrementOld = NewtonIncrement ! Update solution: Z4 <-- Z4 + DZ4 CALL WAXPY(N,ONE,DZ4,1,Z4,1) ! Check error in Newton iterations NewtonDone = (NewtonRate*NewtonIncrement <= NewtonTol) IF (NewtonDone) EXIT SDNewtonLoop END DO SDNewtonLoop IF (.NOT.NewtonDone) THEN H = Fac*H; Reject=.TRUE.; SkipJac = .TRUE.; SkipLU = .FALSE. CYCLE Tloop END IF END IF !~~~> End of implified SDIRK Newton iterations !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ !~~~> Error estimation !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IF (SdirkError) THEN CALL Set2Zero(N, DZ4) IF (rkMethod==L3A) THEN DZ4(1:N) = H*rkF(0)*F0(1:N) IF (rkF(1) /= ZERO) CALL WAXPY(N, rkF(1), Z1, 1, DZ4, 1) IF (rkF(2) /= ZERO) CALL WAXPY(N, rkF(2), Z2, 1, DZ4, 1) IF (rkF(3) /= ZERO) CALL WAXPY(N, rkF(3), Z3, 1, DZ4, 1) TMP = Y + Z4 CALL FUN_CHEM(T+H,TMP,DZ1) CALL WAXPY(N, H*rkBgam(4), DZ1, 1, DZ4, 1) ELSE ! DZ4(1:N) = rkD(1)*Z1 + rkD(2)*Z2 + rkD(3)*Z3 - Z4 IF (rkD(1) /= ZERO) CALL WAXPY(N, rkD(1), Z1, 1, DZ4, 1) IF (rkD(2) /= ZERO) CALL WAXPY(N, rkD(2), Z2, 1, DZ4, 1) IF (rkD(3) /= ZERO) CALL WAXPY(N, rkD(3), Z3, 1, DZ4, 1) CALL WAXPY(N, -ONE, Z4, 1, DZ4, 1) END IF Err = RK_ErrorNorm(N,SCAL,DZ4) ELSE CALL RK_ErrorEstimate(N,H,T,Y,F0, & E1,IP1,Z1,Z2,Z3,SCAL,Err,FirstStep,Reject) END IF !~~~> Computation of new step size Hnew Fac = Err**(-ONE/rkELO)* & MIN(FacSafe,(ONE+2*NewtonMaxit)/(NewtonIter+2*NewtonMaxit)) Fac = MIN(FacMax,MAX(FacMin,Fac)) Hnew = Fac*H !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ !~~~> Accept/reject step !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ accept:IF (Err < ONE) THEN !~~~> STEP IS ACCEPTED FirstStep=.FALSE. ISTATUS(Nacc) = ISTATUS(Nacc) + 1 IF (Gustafsson) THEN !~~~> Predictive controller of Gustafsson IF (ISTATUS(Nacc) > 1) THEN FacGus=FacSafe*(H/Hacc)*(Err**2/ErrOld)**(-0.25d0) FacGus=MIN(FacMax,MAX(FacMin,FacGus)) Fac=MIN(Fac,FacGus) Hnew = Fac*H END IF Hacc=H ErrOld=MAX(1.0d-2,Err) END IF Hold = H T = T+H ! Update solution: Y <- Y + sum(d_i Z_i) IF (rkD(1) /= ZERO) CALL WAXPY(N,rkD(1),Z1,1,Y,1) IF (rkD(2) /= ZERO) CALL WAXPY(N,rkD(2),Z2,1,Y,1) IF (rkD(3) /= ZERO) CALL WAXPY(N,rkD(3),Z3,1,Y,1) ! Construct the solution quadratic interpolant Q(c_i) = Z_i, i=1:3 IF (StartNewton) CALL RK_Interpolate('make',N,H,Hold,Z1,Z2,Z3,CONT) CALL RK_ErrorScale(N,ITOL,AbsTol,RelTol,Y,SCAL) RSTATUS(Ntexit) = T RSTATUS(Nhnew) = Hnew RSTATUS(Nhacc) = H Hnew = Tdirection*MIN( MAX(ABS(Hnew),Hmin) , Hmax ) IF (Reject) Hnew = Tdirection*MIN(ABS(Hnew),ABS(H)) Reject = .FALSE. IF ((T+Hnew/Qmin-Tend)*Tdirection >= ZERO) THEN H = Tend-T ELSE Hratio=Hnew/H ! Reuse the LU decomposition SkipLU = (Theta<=ThetaMin) .AND. (Hratio>=Qmin) .AND. (Hratio<=Qmax) IF (.NOT.SkipLU) H=Hnew END IF ! If convergence is fast enough, do not update Jacobian ! SkipJac = (Theta <= ThetaMin) SkipJac = .FALSE. ELSE accept !~~~> Step is rejected IF (FirstStep .OR. Reject) THEN H = FacRej*H ELSE H = Hnew END IF Reject = .TRUE. SkipJac = .TRUE. ! Skip if rejected - Jac is independent of H SkipLU = .FALSE. IF (ISTATUS(Nacc) >= 1) ISTATUS(Nrej) = ISTATUS(Nrej) + 1 END IF accept END DO Tloop ! Successful exit IERR = 1 END SUBROUTINE RK_Integrator !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE RK_ErrorMsg(Code,T,H,IERR) !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ! Handles all error messages !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE KPP_REAL, INTENT(IN) :: T, H INTEGER, INTENT(IN) :: Code INTEGER, INTENT(OUT) :: IERR IERR = Code PRINT * , & 'Forced exit from RungeKutta due to the following error:' SELECT CASE (Code) CASE (-1) PRINT * , '--> Improper value for maximal no of steps' CASE (-2) PRINT * , '--> Improper value for maximal no of Newton iterations' CASE (-3) PRINT * , '--> Hmin/Hmax/Hstart must be positive' CASE (-4) PRINT * , '--> Improper values for FacMin/FacMax/FacSafe/FacRej' CASE (-5) PRINT * , '--> Improper value for ThetaMin' CASE (-6) PRINT * , '--> Newton stopping tolerance too small' CASE (-7) PRINT * , '--> Improper values for Qmin, Qmax' CASE (-8) PRINT * , '--> Tolerances are too small' CASE (-9) PRINT * , '--> No of steps exceeds maximum bound' CASE (-10) PRINT * , '--> Step size too small: T + 10*H = T', & ' or H < Roundoff' CASE (-11) PRINT * , '--> Matrix is repeatedly singular' CASE (-12) PRINT * , '--> Non-convergence of Newton iterations' CASE (-13) PRINT * , '--> Requested RK method not implemented' CASE DEFAULT PRINT *, 'Unknown Error code: ', Code END SELECT WRITE(6,FMT="(5X,'T=',E12.5,' H=',E12.5)") T, H END SUBROUTINE RK_ErrorMsg !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE RK_ErrorScale(N,ITOL,AbsTol,RelTol,Y,SCAL) !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ! Handles all error messages !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE INTEGER, INTENT(IN) :: N, ITOL KPP_REAL, INTENT(IN) :: AbsTol(*), RelTol(*), Y(N) KPP_REAL, INTENT(OUT) :: SCAL(N) INTEGER :: i IF (ITOL==0) THEN DO i=1,N SCAL(i)= ONE/(AbsTol(1)+RelTol(1)*ABS(Y(i))) END DO ELSE DO i=1,N SCAL(i)=ONE/(AbsTol(i)+RelTol(i)*ABS(Y(i))) END DO END IF END SUBROUTINE RK_ErrorScale !!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ! SUBROUTINE RK_Transform(N,Tr,Z1,Z2,Z3,W1,W2,W3) !!~~~> W <-- Tr x Z !!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ! IMPLICIT NONE ! INTEGER :: N, i ! KPP_REAL :: Tr(3,3),Z1(N),Z2(N),Z3(N),W1(N),W2(N),W3(N) ! KPP_REAL :: x1, x2, x3 ! DO i=1,N ! x1 = Z1(i); x2 = Z2(i); x3 = Z3(i) ! W1(i) = Tr(1,1)*x1 + Tr(1,2)*x2 + Tr(1,3)*x3 ! W2(i) = Tr(2,1)*x1 + Tr(2,2)*x2 + Tr(2,3)*x3 ! W3(i) = Tr(3,1)*x1 + Tr(3,2)*x2 + Tr(3,3)*x3 ! END DO ! END SUBROUTINE RK_Transform !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE RK_Interpolate(action,N,H,Hold,Z1,Z2,Z3,CONT) !~~~> Constructs or evaluates a quadratic polynomial ! that interpolates the Z solution at current step ! and provides starting values for the next step !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ INTEGER :: N, i KPP_REAL :: H,Hold,Z1(N),Z2(N),Z3(N),CONT(N,3) KPP_REAL :: r, x1, x2, x3, den CHARACTER(LEN=4) :: action SELECT CASE (action) CASE ('make') ! Construct the solution quadratic interpolant Q(c_i) = Z_i, i=1:3 den = (rkC(3)-rkC(2))*(rkC(2)-rkC(1))*(rkC(1)-rkC(3)) DO i=1,N CONT(i,1)=(-rkC(3)**2*rkC(2)*Z1(i)+Z3(i)*rkC(2)*rkC(1)**2 & +rkC(2)**2*rkC(3)*Z1(i)-rkC(2)**2*rkC(1)*Z3(i) & +rkC(3)**2*rkC(1)*Z2(i)-Z2(i)*rkC(3)*rkC(1)**2)& /den-Z3(i) CONT(i,2)= -( rkC(1)**2*(Z3(i)-Z2(i)) + rkC(2)**2*(Z1(i) & -Z3(i)) +rkC(3)**2*(Z2(i)-Z1(i)) )/den CONT(i,3)= ( rkC(1)*(Z3(i)-Z2(i)) + rkC(2)*(Z1(i)-Z3(i)) & +rkC(3)*(Z2(i)-Z1(i)) )/den END DO CASE ('eval') ! Evaluate quadratic polynomial r = H/Hold x1 = ONE + rkC(1)*r x2 = ONE + rkC(2)*r x3 = ONE + rkC(3)*r DO i=1,N Z1(i) = CONT(i,1)+x1*(CONT(i,2)+x1*CONT(i,3)) Z2(i) = CONT(i,1)+x2*(CONT(i,2)+x2*CONT(i,3)) Z3(i) = CONT(i,1)+x3*(CONT(i,2)+x3*CONT(i,3)) END DO END SELECT END SUBROUTINE RK_Interpolate !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE RK_PrepareRHS(N,T,H,Y,F0,Z1,Z2,Z3,R1,R2,R3) !~~~> Prepare the right-hand side for Newton iterations ! R = Z - hA x F !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE INTEGER :: N KPP_REAL :: T, H KPP_REAL, DIMENSION(N) :: Y,Z1,Z2,Z3,F0,F,R1,R2,R3,TMP CALL WCOPY(N,Z1,1,R1,1) ! R1 <- Z1 CALL WCOPY(N,Z2,1,R2,1) ! R2 <- Z2 CALL WCOPY(N,Z3,1,R3,1) ! R3 <- Z3 IF (rkMethod==L3A) THEN CALL WAXPY(N,-H*rkA(1,0),F0,1,R1,1) ! R1 <- R1 - h*A_10*F0 CALL WAXPY(N,-H*rkA(2,0),F0,1,R2,1) ! R2 <- R2 - h*A_20*F0 CALL WAXPY(N,-H*rkA(3,0),F0,1,R3,1) ! R3 <- R3 - h*A_30*F0 END IF CALL WADD(N,Y,Z1,TMP) ! TMP <- Y + Z1 CALL FUN_CHEM(T+rkC(1)*H,TMP,F) ! F1 <- Fun(Y+Z1) CALL WAXPY(N,-H*rkA(1,1),F,1,R1,1) ! R1 <- R1 - h*A_11*F1 CALL WAXPY(N,-H*rkA(2,1),F,1,R2,1) ! R2 <- R2 - h*A_21*F1 CALL WAXPY(N,-H*rkA(3,1),F,1,R3,1) ! R3 <- R3 - h*A_31*F1 CALL WADD(N,Y,Z2,TMP) ! TMP <- Y + Z2 CALL FUN_CHEM(T+rkC(2)*H,TMP,F) ! F2 <- Fun(Y+Z2) CALL WAXPY(N,-H*rkA(1,2),F,1,R1,1) ! R1 <- R1 - h*A_12*F2 CALL WAXPY(N,-H*rkA(2,2),F,1,R2,1) ! R2 <- R2 - h*A_22*F2 CALL WAXPY(N,-H*rkA(3,2),F,1,R3,1) ! R3 <- R3 - h*A_32*F2 CALL WADD(N,Y,Z3,TMP) ! TMP <- Y + Z3 CALL FUN_CHEM(T+rkC(3)*H,TMP,F) ! F3 <- Fun(Y+Z3) CALL WAXPY(N,-H*rkA(1,3),F,1,R1,1) ! R1 <- R1 - h*A_13*F3 CALL WAXPY(N,-H*rkA(2,3),F,1,R2,1) ! R2 <- R2 - h*A_23*F3 CALL WAXPY(N,-H*rkA(3,3),F,1,R3,1) ! R3 <- R3 - h*A_33*F3 END SUBROUTINE RK_PrepareRHS !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE RK_Decomp(N,H,FJAC,E1,IP1,E2,IP2,ISING) !~~~> Compute the matrices E1 and E2 and their decompositions !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE INTEGER :: N, ISING KPP_REAL :: H, Alpha, Beta, Gamma #ifdef FULL_ALGEBRA KPP_REAL :: FJAC(NVAR,NVAR),E1(NVAR,NVAR) COMPLEX(kind=dp) :: E2(N,N) #else KPP_REAL :: FJAC(LU_NONZERO),E1(LU_NONZERO) COMPLEX(kind=dp) :: E2(LU_NONZERO) #endif INTEGER :: IP1(N), IP2(N), i, j Gamma = rkGamma/H Alpha = rkAlpha/H Beta = rkBeta /H #ifdef FULL_ALGEBRA DO j=1,N DO i=1,N E1(i,j)=-FJAC(i,j) END DO E1(j,j)=E1(j,j)+Gamma END DO CALL DGETRF(N,N,E1,N,IP1,ISING) #else DO i=1,LU_NONZERO E1(i)=-FJAC(i) END DO DO i=1,NVAR j=LU_DIAG(i); E1(j)=E1(j)+Gamma END DO CALL KppDecomp(E1,ISING) #endif IF (ISING /= 0) THEN ISTATUS(Ndec) = ISTATUS(Ndec) + 1 RETURN END IF #ifdef FULL_ALGEBRA DO j=1,N DO i=1,N E2(i,j) = DCMPLX( -FJAC(i,j), ZERO ) END DO E2(j,j) = E2(j,j) + CMPLX( Alpha, Beta ) END DO CALL ZGETRF(N,N,E2,N,IP2,ISING) #else DO i=1,LU_NONZERO E2(i) = DCMPLX( -FJAC(i), ZERO ) END DO DO i=1,NVAR j=LU_DIAG(i); E2(j)=E2(j) + CMPLX( Alpha, Beta ) END DO CALL KppDecompCmplx(E2,ISING) #endif ISTATUS(Ndec) = ISTATUS(Ndec) + 1 END SUBROUTINE RK_Decomp !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE RK_Solve(N,H,E1,IP1,E2,IP2,R1,R2,R3,ISING) !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE INTEGER :: N,IP1(NVAR),IP2(NVAR),ISING #ifdef FULL_ALGEBRA KPP_REAL :: E1(NVAR,NVAR) COMPLEX(kind=dp) :: E2(NVAR,NVAR) #else KPP_REAL :: E1(LU_NONZERO) COMPLEX(kind=dp) :: E2(LU_NONZERO) #endif KPP_REAL :: R1(N),R2(N),R3(N) KPP_REAL :: H, x1, x2, x3 COMPLEX(kind=dp) :: BC(N) INTEGER :: i ! ! Z <- h^{-1) T^{-1) A^{-1) x Z DO i=1,N x1 = R1(i)/H; x2 = R2(i)/H; x3 = R3(i)/H R1(i) = rkTinvAinv(1,1)*x1 + rkTinvAinv(1,2)*x2 + rkTinvAinv(1,3)*x3 R2(i) = rkTinvAinv(2,1)*x1 + rkTinvAinv(2,2)*x2 + rkTinvAinv(2,3)*x3 R3(i) = rkTinvAinv(3,1)*x1 + rkTinvAinv(3,2)*x2 + rkTinvAinv(3,3)*x3 END DO #ifdef FULL_ALGEBRA CALL DGETRS ('N',N,1,E1,N,IP1,R1,N,0) #else CALL KppSolve (E1,R1) #endif ! DO i=1,N BC(i) = DCMPLX(R2(i),R3(i)) END DO #ifdef FULL_ALGEBRA CALL ZGETRS ('N',N,1,E2,N,IP2,BC,N,0) #else CALL KppSolveCmplx (E2,BC) #endif DO i=1,N R2(i) = DBLE( BC(i) ) R3(i) = AIMAG( BC(i) ) END DO ! Z <- T x Z DO i=1,N x1 = R1(i); x2 = R2(i); x3 = R3(i) R1(i) = rkT(1,1)*x1 + rkT(1,2)*x2 + rkT(1,3)*x3 R2(i) = rkT(2,1)*x1 + rkT(2,2)*x2 + rkT(2,3)*x3 R3(i) = rkT(3,1)*x1 + rkT(3,2)*x2 + rkT(3,3)*x3 END DO ISTATUS(Nsol) = ISTATUS(Nsol) + 1 END SUBROUTINE RK_Solve !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE RK_ErrorEstimate(N,H,T,Y,F0, & E1,IP1,Z1,Z2,Z3,SCAL,Err, & FirstStep,Reject) !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE INTEGER :: N #ifdef FULL_ALGEBRA KPP_REAL :: E1(NVAR,NVAR) #else KPP_REAL :: E1(LU_NONZERO) #endif KPP_REAL :: SCAL(N),Z1(N),Z2(N),Z3(N),F1(N),F2(N), & F0(N),Y(N),TMP(N),T,H INTEGER :: IP1(N), i LOGICAL FirstStep,Reject KPP_REAL :: HrkE1,HrkE2,HrkE3,Err HrkE1 = rkE(1)/H HrkE2 = rkE(2)/H HrkE3 = rkE(3)/H DO i=1,N F2(i) = HrkE1*Z1(i)+HrkE2*Z2(i)+HrkE3*Z3(i) TMP(i) = rkE(0)*F0(i) + F2(i) END DO #ifdef FULL_ALGEBRA CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) IF ((rkMethod==R1A).OR.(rkMethod==GAU).OR.(rkMethod==L3A)) CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) IF (rkMethod==GAU) CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) #else CALL KppSolve (E1, TMP) IF ((rkMethod==R1A).OR.(rkMethod==GAU).OR.(rkMethod==L3A)) CALL KppSolve (E1,TMP) IF (rkMethod==GAU) CALL KppSolve (E1,TMP) #endif Err = RK_ErrorNorm(N,SCAL,TMP) ! IF (Err < ONE) RETURN firej:IF (FirstStep.OR.Reject) THEN DO i=1,N TMP(i)=Y(i)+TMP(i) END DO CALL FUN_CHEM(T,TMP,F1) ISTATUS(Nfun) = ISTATUS(Nfun) + 1 DO i=1,N TMP(i)=F1(i)+F2(i) END DO #ifdef FULL_ALGEBRA CALL DGETRS ('N',N,1,E1,N,IP1,TMP,N,0) #else CALL KppSolve (E1, TMP) #endif Err = RK_ErrorNorm(N,SCAL,TMP) END IF firej END SUBROUTINE RK_ErrorEstimate !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ KPP_REAL FUNCTION RK_ErrorNorm(N,SCAL,DY) !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE INTEGER :: N KPP_REAL :: SCAL(N),DY(N) INTEGER :: i RK_ErrorNorm = ZERO DO i=1,N RK_ErrorNorm = RK_ErrorNorm + (DY(i)*SCAL(i))**2 END DO RK_ErrorNorm = MAX( SQRT(RK_ErrorNorm/N), 1.0d-10 ) END FUNCTION RK_ErrorNorm !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE Radau2A_Coefficients ! The coefficients of the 3-stage Radau-2A method ! (given to ~30 accurate digits) !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE ! The coefficients of the Radau2A method KPP_REAL :: b0 ! b0 = 1.0d0 IF (SdirkError) THEN b0 = 0.2d-1 ELSE b0 = 0.5d-1 END IF ! The coefficients of the Radau2A method rkMethod = R2A rkA(1,1) = 1.968154772236604258683861429918299d-1 rkA(1,2) = -6.55354258501983881085227825696087d-2 rkA(1,3) = 2.377097434822015242040823210718965d-2 rkA(2,1) = 3.944243147390872769974116714584975d-1 rkA(2,2) = 2.920734116652284630205027458970589d-1 rkA(2,3) = -4.154875212599793019818600988496743d-2 rkA(3,1) = 3.764030627004672750500754423692808d-1 rkA(3,2) = 5.124858261884216138388134465196080d-1 rkA(3,3) = 1.111111111111111111111111111111111d-1 rkB(1) = 3.764030627004672750500754423692808d-1 rkB(2) = 5.124858261884216138388134465196080d-1 rkB(3) = 1.111111111111111111111111111111111d-1 rkC(1) = 1.550510257216821901802715925294109d-1 rkC(2) = 6.449489742783178098197284074705891d-1 rkC(3) = 1.0d0 ! New solution: H* Sum B_j*f(Z_j) = Sum D_j*Z_j rkD(1) = 0.0d0 rkD(2) = 0.0d0 rkD(3) = 1.0d0 ! Classical error estimator: ! H* Sum (B_j-Bhat_j)*f(Z_j) = H*E(0)*f(0) + Sum E_j*Z_j rkE(0) = 1.0d0*b0 rkE(1) = -10.04880939982741556246032950764708d0*b0 rkE(2) = 1.382142733160748895793662840980412d0*b0 rkE(3) = -.3333333333333333333333333333333333d0*b0 ! Sdirk error estimator rkBgam(0) = b0 rkBgam(1) = .3764030627004672750500754423692807d0-1.558078204724922382431975370686279d0*b0 rkBgam(2) = .8914115380582557157653087040196118d0*b0+.5124858261884216138388134465196077d0 rkBgam(3) = -.1637777184845662566367174924883037d0-.3333333333333333333333333333333333d0*b0 rkBgam(4) = .2748888295956773677478286035994148d0 ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j rkTheta(1) = -1.520677486405081647234271944611547d0-10.04880939982741556246032950764708d0*b0 rkTheta(2) = 2.070455145596436382729929151810376d0+1.382142733160748895793662840980413d0*b0 rkTheta(3) = -.3333333333333333333333333333333333d0*b0-.3744441479783868387391430179970741d0 ! Local order of error estimator IF (b0==0.0d0) THEN rkELO = 6.0d0 ELSE rkELO = 4.0d0 END IF !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ !~~~> Diagonalize the RK matrix: ! rkTinv * inv(rkA) * rkT = ! | rkGamma 0 0 | ! | 0 rkAlpha -rkBeta | ! | 0 rkBeta rkAlpha | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ rkGamma = 3.637834252744495732208418513577775d0 rkAlpha = 2.681082873627752133895790743211112d0 rkBeta = 3.050430199247410569426377624787569d0 rkT(1,1) = 9.443876248897524148749007950641664d-2 rkT(1,2) = -1.412552950209542084279903838077973d-1 rkT(1,3) = -3.00291941051474244918611170890539d-2 rkT(2,1) = 2.502131229653333113765090675125018d-1 rkT(2,2) = 2.041293522937999319959908102983381d-1 rkT(2,3) = 3.829421127572619377954382335998733d-1 rkT(3,1) = 1.0d0 rkT(3,2) = 1.0d0 rkT(3,3) = 0.0d0 rkTinv(1,1) = 4.178718591551904727346462658512057d0 rkTinv(1,2) = 3.27682820761062387082533272429617d-1 rkTinv(1,3) = 5.233764454994495480399309159089876d-1 rkTinv(2,1) = -4.178718591551904727346462658512057d0 rkTinv(2,2) = -3.27682820761062387082533272429617d-1 rkTinv(2,3) = 4.766235545005504519600690840910124d-1 rkTinv(3,1) = -5.02872634945786875951247343139544d-1 rkTinv(3,2) = 2.571926949855605429186785353601676d0 rkTinv(3,3) = -5.960392048282249249688219110993024d-1 rkTinvAinv(1,1) = 1.520148562492775501049204957366528d+1 rkTinvAinv(1,2) = 1.192055789400527921212348994770778d0 rkTinvAinv(1,3) = 1.903956760517560343018332287285119d0 rkTinvAinv(2,1) = -9.669512977505946748632625374449567d0 rkTinvAinv(2,2) = -8.724028436822336183071773193986487d0 rkTinvAinv(2,3) = 3.096043239482439656981667712714881d0 rkTinvAinv(3,1) = -1.409513259499574544876303981551774d+1 rkTinvAinv(3,2) = 5.895975725255405108079130152868952d0 rkTinvAinv(3,3) = -1.441236197545344702389881889085515d-1 rkAinvT(1,1) = .3435525649691961614912493915818282d0 rkAinvT(1,2) = -.4703191128473198422370558694426832d0 rkAinvT(1,3) = .3503786597113668965366406634269080d0 rkAinvT(2,1) = .9102338692094599309122768354288852d0 rkAinvT(2,2) = 1.715425895757991796035292755937326d0 rkAinvT(2,3) = .4040171993145015239277111187301784d0 rkAinvT(3,1) = 3.637834252744495732208418513577775d0 rkAinvT(3,2) = 2.681082873627752133895790743211112d0 rkAinvT(3,3) = -3.050430199247410569426377624787569d0 END SUBROUTINE Radau2A_Coefficients !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE Lobatto3C_Coefficients ! The coefficients of the 3-stage Lobatto-3C method ! (given to ~30 accurate digits) ! The parameter b0 can be chosen to tune the error estimator !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE KPP_REAL :: b0 rkMethod = L3C ! b0 = 1.0d0 IF (SdirkError) THEN b0 = 0.2d0 ELSE b0 = 0.5d0 END IF ! The coefficients of the Lobatto3C method rkA(1,1) = .1666666666666666666666666666666667d0 rkA(1,2) = -.3333333333333333333333333333333333d0 rkA(1,3) = .1666666666666666666666666666666667d0 rkA(2,1) = .1666666666666666666666666666666667d0 rkA(2,2) = .4166666666666666666666666666666667d0 rkA(2,3) = -.8333333333333333333333333333333333d-1 rkA(3,1) = .1666666666666666666666666666666667d0 rkA(3,2) = .6666666666666666666666666666666667d0 rkA(3,3) = .1666666666666666666666666666666667d0 rkB(1) = .1666666666666666666666666666666667d0 rkB(2) = .6666666666666666666666666666666667d0 rkB(3) = .1666666666666666666666666666666667d0 rkC(1) = 0.0d0 rkC(2) = 0.5d0 rkC(3) = 1.0d0 ! Classical error estimator, embedded solution: rkBhat(0) = b0 rkBhat(1) = .16666666666666666666666666666666667d0-b0 rkBhat(2) = .66666666666666666666666666666666667d0 rkBhat(3) = .16666666666666666666666666666666667d0 ! New solution: h Sum_j b_j f(Z_j) = sum d_j Z_j rkD(1) = 0.0d0 rkD(2) = 0.0d0 rkD(3) = 1.0d0 ! Classical error estimator: ! H* Sum (B_j-Bhat_j)*f(Z_j) = H*E(0)*f(0) + Sum E_j*Z_j rkE(0) = .3808338772072650364017425226487022*b0 rkE(1) = -1.142501631621795109205227567946107*b0 rkE(2) = -1.523335508829060145606970090594809*b0 rkE(3) = .3808338772072650364017425226487022*b0 ! Sdirk error estimator rkBgam(0) = b0 rkBgam(1) = .1666666666666666666666666666666667d0-1.d0*b0 rkBgam(2) = .6666666666666666666666666666666667d0 rkBgam(3) = -.2141672105405983697350758559820354d0 rkBgam(4) = .3808338772072650364017425226487021d0 ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j rkTheta(1) = -3.d0*b0-.3808338772072650364017425226487021d0 rkTheta(2) = -4.d0*b0+1.523335508829060145606970090594808d0 rkTheta(3) = -.142501631621795109205227567946106d0+b0 ! Local order of error estimator IF (b0==0.0d0) THEN rkELO = 5.0d0 ELSE rkELO = 4.0d0 END IF !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ !~~~> Diagonalize the RK matrix: ! rkTinv * inv(rkA) * rkT = ! | rkGamma 0 0 | ! | 0 rkAlpha -rkBeta | ! | 0 rkBeta rkAlpha | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ rkGamma = 2.625816818958466716011888933765284d0 rkAlpha = 1.687091590520766641994055533117359d0 rkBeta = 2.508731754924880510838743672432351d0 rkT(1,1) = 1.d0 rkT(1,2) = 1.d0 rkT(1,3) = 0.d0 rkT(2,1) = .4554100411010284672111720348287483d0 rkT(2,2) = -.6027050205505142336055860174143743d0 rkT(2,3) = -.4309321229203225731070721341350346d0 rkT(3,1) = 2.195823345445647152832799205549709d0 rkT(3,2) = -1.097911672722823576416399602774855d0 rkT(3,3) = .7850032632435902184104551358922130d0 rkTinv(1,1) = .4205559181381766909344950150991349d0 rkTinv(1,2) = .3488903392193734304046467270632057d0 rkTinv(1,3) = .1915253879645878102698098373933487d0 rkTinv(2,1) = .5794440818618233090655049849008650d0 rkTinv(2,2) = -.3488903392193734304046467270632057d0 rkTinv(2,3) = -.1915253879645878102698098373933487d0 rkTinv(3,1) = -.3659705575742745254721332009249516d0 rkTinv(3,2) = -1.463882230297098101888532803699806d0 rkTinv(3,3) = .4702733607340189781407813565524989d0 rkTinvAinv(1,1) = 1.104302803159744452668648155627548d0 rkTinvAinv(1,2) = .916122120694355522658740710823143d0 rkTinvAinv(1,3) = .5029105849749601702795812241441172d0 rkTinvAinv(2,1) = 1.895697196840255547331351844372453d0 rkTinvAinv(2,2) = 3.083877879305644477341259289176857d0 rkTinvAinv(2,3) = -1.502910584974960170279581224144117d0 rkTinvAinv(3,1) = .8362439183082935036129145574774502d0 rkTinvAinv(3,2) = -3.344975673233174014451658229909802d0 rkTinvAinv(3,3) = .312908409479233358005944466882642d0 rkAinvT(1,1) = 2.625816818958466716011888933765282d0 rkAinvT(1,2) = 1.687091590520766641994055533117358d0 rkAinvT(1,3) = -2.508731754924880510838743672432351d0 rkAinvT(2,1) = 1.195823345445647152832799205549710d0 rkAinvT(2,2) = -2.097911672722823576416399602774855d0 rkAinvT(2,3) = .7850032632435902184104551358922130d0 rkAinvT(3,1) = 5.765829871932827589653709477334136d0 rkAinvT(3,2) = .1170850640335862051731452613329320d0 rkAinvT(3,3) = 4.078738281412060947659653944216779d0 END SUBROUTINE Lobatto3C_Coefficients !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE Gauss_Coefficients ! The coefficients of the 3-stage Gauss method ! (given to ~30 accurate digits) ! The parameter b3 can be chosen by the user ! to tune the error estimator !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE KPP_REAL :: b0 ! The coefficients of the Gauss method rkMethod = GAU ! b0 = 4.0d0 b0 = 0.1d0 ! The coefficients of the Gauss method rkA(1,1) = .1388888888888888888888888888888889d0 rkA(1,2) = -.359766675249389034563954710966045d-1 rkA(1,3) = .97894440153083260495800422294756d-2 rkA(2,1) = .3002631949808645924380249472131556d0 rkA(2,2) = .2222222222222222222222222222222222d0 rkA(2,3) = -.224854172030868146602471694353778d-1 rkA(3,1) = .2679883337624694517281977355483022d0 rkA(3,2) = .4804211119693833479008399155410489d0 rkA(3,3) = .1388888888888888888888888888888889d0 rkB(1) = .2777777777777777777777777777777778d0 rkB(2) = .4444444444444444444444444444444444d0 rkB(3) = .2777777777777777777777777777777778d0 rkC(1) = .1127016653792583114820734600217600d0 rkC(2) = .5000000000000000000000000000000000d0 rkC(3) = .8872983346207416885179265399782400d0 ! Classical error estimator, embedded solution: rkBhat(0) = b0 rkBhat(1) =-1.4788305577012361475298775666303999d0*b0 & +.27777777777777777777777777777777778d0 rkBhat(2) = .44444444444444444444444444444444444d0 & +.66666666666666666666666666666666667d0*b0 rkBhat(3) = -.18783610896543051913678910003626672d0*b0 & +.27777777777777777777777777777777778d0 ! New solution: h Sum_j b_j f(Z_j) = sum d_j Z_j rkD(1) = .1666666666666666666666666666666667d1 rkD(2) = -.1333333333333333333333333333333333d1 rkD(3) = .1666666666666666666666666666666667d1 ! Classical error estimator: ! H* Sum (B_j-Bhat_j)*f(Z_j) = H*E(0)*f(0) + Sum E_j*Z_j rkE(0) = .2153144231161121782447335303806954d0*b0 rkE(1) = -2.825278112319014084275808340593191d0*b0 rkE(2) = .2870858974881495709929780405075939d0*b0 rkE(3) = -.4558086256248162565397206448274867d-1*b0 ! Sdirk error estimator rkBgam(0) = 0.d0 rkBgam(1) = .2373339543355109188382583162660537d0 rkBgam(2) = .5879873931885192299409334646982414d0 rkBgam(3) = -.4063577064014232702392531134499046d-1 rkBgam(4) = .2153144231161121782447335303806955d0 ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j rkTheta(1) = -2.594040933093095272574031876464493d0 rkTheta(2) = 1.824611539036311947589425112250199d0 rkTheta(3) = .1856563166634371860478043996459493d0 ! ELO = local order of classical error estimator rkELO = 4.0d0 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ !~~~> Diagonalize the RK matrix: ! rkTinv * inv(rkA) * rkT = ! | rkGamma 0 0 | ! | 0 rkAlpha -rkBeta | ! | 0 rkBeta rkAlpha | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ rkGamma = 4.644370709252171185822941421408064d0 rkAlpha = 3.677814645373914407088529289295970d0 rkBeta = 3.508761919567443321903661209182446d0 rkT(1,1) = .7215185205520017032081769924397664d-1 rkT(1,2) = -.8224123057363067064866206597516454d-1 rkT(1,3) = -.6012073861930850173085948921439054d-1 rkT(2,1) = .1188325787412778070708888193730294d0 rkT(2,2) = .5306509074206139504614411373957448d-1 rkT(2,3) = .3162050511322915732224862926182701d0 rkT(3,1) = 1.d0 rkT(3,2) = 1.d0 rkT(3,3) = 0.d0 rkTinv(1,1) = 5.991698084937800775649580743981285d0 rkTinv(1,2) = 1.139214295155735444567002236934009d0 rkTinv(1,3) = .4323121137838583855696375901180497d0 rkTinv(2,1) = -5.991698084937800775649580743981285d0 rkTinv(2,2) = -1.139214295155735444567002236934009d0 rkTinv(2,3) = .5676878862161416144303624098819503d0 rkTinv(3,1) = -1.246213273586231410815571640493082d0 rkTinv(3,2) = 2.925559646192313662599230367054972d0 rkTinv(3,3) = -.2577352012734324923468722836888244d0 rkTinvAinv(1,1) = 27.82766708436744962047620566703329d0 rkTinvAinv(1,2) = 5.290933503982655311815946575100597d0 rkTinvAinv(1,3) = 2.007817718512643701322151051660114d0 rkTinvAinv(2,1) = -17.66368928942422710690385180065675d0 rkTinvAinv(2,2) = -14.45491129892587782538830044147713d0 rkTinvAinv(2,3) = 2.992182281487356298677848948339886d0 rkTinvAinv(3,1) = -25.60678350282974256072419392007303d0 rkTinvAinv(3,2) = 6.762434375611708328910623303779923d0 rkTinvAinv(3,3) = 1.043979339483109825041215970036771d0 rkAinvT(1,1) = .3350999483034677402618981153470483d0 rkAinvT(1,2) = -.5134173605009692329246186488441294d0 rkAinvT(1,3) = .6745196507033116204327635673208923d-1 rkAinvT(2,1) = .5519025480108928886873752035738885d0 rkAinvT(2,2) = 1.304651810077110066076640761092008d0 rkAinvT(2,3) = .9767507983414134987545585703726984d0 rkAinvT(3,1) = 4.644370709252171185822941421408064d0 rkAinvT(3,2) = 3.677814645373914407088529289295970d0 rkAinvT(3,3) = -3.508761919567443321903661209182446d0 END SUBROUTINE Gauss_Coefficients !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE Radau1A_Coefficients ! The coefficients of the 3-stage Gauss method ! (given to ~30 accurate digits) ! The parameter b3 can be chosen by the user ! to tune the error estimator !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE ! KPP_REAL :: b0 = 0.3d0 KPP_REAL :: b0 = 0.1d0 ! The coefficients of the Radau1A method rkMethod = R1A rkA(1,1) = .1111111111111111111111111111111111d0 rkA(1,2) = -.1916383190435098943442935597058829d0 rkA(1,3) = .8052720793239878323318244859477174d-1 rkA(2,1) = .1111111111111111111111111111111111d0 rkA(2,2) = .2920734116652284630205027458970589d0 rkA(2,3) = -.481334970546573839513422644787591d-1 rkA(3,1) = .1111111111111111111111111111111111d0 rkA(3,2) = .5370223859435462728402311533676479d0 rkA(3,3) = .1968154772236604258683861429918299d0 rkB(1) = .1111111111111111111111111111111111d0 rkB(2) = .5124858261884216138388134465196080d0 rkB(3) = .3764030627004672750500754423692808d0 rkC(1) = 0.d0 rkC(2) = .3550510257216821901802715925294109d0 rkC(3) = .8449489742783178098197284074705891d0 ! Classical error estimator, embedded solution: rkBhat(0) = b0 rkBhat(1) = .11111111111111111111111111111111111d0-b0 rkBhat(2) = .51248582618842161383881344651960810d0 rkBhat(3) = .37640306270046727505007544236928079d0 ! New solution: H* Sum B_j*f(Z_j) = Sum D_j*Z_j rkD(1) = .3333333333333333333333333333333333d0 rkD(2) = -.8914115380582557157653087040196127d0 rkD(3) = .1558078204724922382431975370686279d1 ! Classical error estimator: ! H* Sum (b_j-bhat_j) f(Z_j) = H*E(0)*F(0) + Sum E_j Z_j rkE(0) = .2748888295956773677478286035994148d0*b0 rkE(1) = -1.374444147978386838739143017997074d0*b0 rkE(2) = -1.335337922441686804550326197041126d0*b0 rkE(3) = .235782604058977333559011782643466d0*b0 ! Sdirk error estimator rkBgam(0) = 0.0d0 rkBgam(1) = .1948150124588532186183490991130616d-1 rkBgam(2) = .7575249005733381398986810981093584d0 rkBgam(3) = -.518952314149008295083446116200793d-1 rkBgam(4) = .2748888295956773677478286035994148d0 ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j rkTheta(1) = -1.224370034375505083904362087063351d0 rkTheta(2) = .9340045331532641409047527962010133d0 rkTheta(3) = .4656990124352088397561234800640929d0 ! ELO = local order of classical error estimator rkELO = 4.0d0 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ !~~~> Diagonalize the RK matrix: ! rkTinv * inv(rkA) * rkT = ! | rkGamma 0 0 | ! | 0 rkAlpha -rkBeta | ! | 0 rkBeta rkAlpha | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ rkGamma = 3.637834252744495732208418513577775d0 rkAlpha = 2.681082873627752133895790743211112d0 rkBeta = 3.050430199247410569426377624787569d0 rkT(1,1) = .424293819848497965354371036408369d0 rkT(1,2) = -.3235571519651980681202894497035503d0 rkT(1,3) = -.522137786846287839586599927945048d0 rkT(2,1) = .57594609499806128896291585429339d-1 rkT(2,2) = .3148663231849760131614374283783d-2 rkT(2,3) = .452429247674359778577728510381731d0 rkT(3,1) = 1.d0 rkT(3,2) = 1.d0 rkT(3,3) = 0.d0 rkTinv(1,1) = 1.233523612685027760114769983066164d0 rkTinv(1,2) = 1.423580134265707095505388133369554d0 rkTinv(1,3) = .3946330125758354736049045150429624d0 rkTinv(2,1) = -1.233523612685027760114769983066164d0 rkTinv(2,2) = -1.423580134265707095505388133369554d0 rkTinv(2,3) = .6053669874241645263950954849570376d0 rkTinv(3,1) = -.1484438963257383124456490049673414d0 rkTinv(3,2) = 2.038974794939896109682070471785315d0 rkTinv(3,3) = -.544501292892686735299355831692542d-1 rkTinvAinv(1,1) = 4.487354449794728738538663081025420d0 rkTinvAinv(1,2) = 5.178748573958397475446442544234494d0 rkTinvAinv(1,3) = 1.435609490412123627047824222335563d0 rkTinvAinv(2,1) = -2.854361287939276673073807031221493d0 rkTinvAinv(2,2) = -1.003648660720543859000994063139137d+1 rkTinvAinv(2,3) = 1.789135380979465422050817815017383d0 rkTinvAinv(3,1) = -4.160768067752685525282947313530352d0 rkTinvAinv(3,2) = 1.124128569859216916690209918405860d0 rkTinvAinv(3,3) = 1.700644430961823796581896350418417d0 rkAinvT(1,1) = 1.543510591072668287198054583233180d0 rkAinvT(1,2) = -2.460228411937788329157493833295004d0 rkAinvT(1,3) = -.412906170450356277003910443520499d0 rkAinvT(2,1) = .209519643211838264029272585946993d0 rkAinvT(2,2) = 1.388545667194387164417459732995766d0 rkAinvT(2,3) = 1.20339553005832004974976023130002d0 rkAinvT(3,1) = 3.637834252744495732208418513577775d0 rkAinvT(3,2) = 2.681082873627752133895790743211112d0 rkAinvT(3,3) = -3.050430199247410569426377624787569d0 END SUBROUTINE Radau1A_Coefficients !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE Lobatto3A_Coefficients ! The coefficients of the 4-stage Lobatto-3A method ! (given to ~30 accurate digits) !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IMPLICIT NONE ! The coefficients of the Lobatto-3A method rkMethod = L3A rkA(0,0) = 0.0d0 rkA(0,1) = 0.0d0 rkA(0,2) = 0.0d0 rkA(0,3) = 0.0d0 rkA(1,0) = .11030056647916491413674311390609397d0 rkA(1,1) = .1896994335208350858632568860939060d0 rkA(1,2) = -.339073642291438837776604807792215d-1 rkA(1,3) = .1030056647916491413674311390609397d-1 rkA(2,0) = .73032766854168419196590219427239365d-1 rkA(2,1) = .4505740308958105504443271474458881d0 rkA(2,2) = .2269672331458315808034097805727606d0 rkA(2,3) = -.2696723314583158080340978057276063d-1 rkA(3,0) = .83333333333333333333333333333333333d-1 rkA(3,1) = .4166666666666666666666666666666667d0 rkA(3,2) = .4166666666666666666666666666666667d0 rkA(3,3) = .8333333333333333333333333333333333d-1 rkB(0) = .83333333333333333333333333333333333d-1 rkB(1) = .4166666666666666666666666666666667d0 rkB(2) = .4166666666666666666666666666666667d0 rkB(3) = .8333333333333333333333333333333333d-1 rkC(0) = 0.0d0 rkC(1) = .2763932022500210303590826331268724d0 rkC(2) = .7236067977499789696409173668731276d0 rkC(3) = 1.0d0 ! New solution: H*Sum B_j*f(Z_j) = Sum D_j*Z_j rkD(0) = 0.0d0 rkD(1) = 0.0d0 rkD(2) = 0.0d0 rkD(3) = 1.0d0 ! Classical error estimator, embedded solution: rkBhat(0) = .90909090909090909090909090909090909d-1 rkBhat(1) = .39972675774621371442114262372173276d0 rkBhat(2) = .43360657558711961891219070961160058d0 rkBhat(3) = .15151515151515151515151515151515152d-1 ! Classical error estimator: ! H* Sum (B_j-Bhat_j)*f(Z_j) = H*E(0)*f(0) + Sum E_j*Z_j rkE(0) = .1957403846510110711315759367097231d-1 rkE(1) = -.1986820345632580910316020806676438d0 rkE(2) = .1660586371214229125096727578826900d0 rkE(3) = -.9787019232550553556578796835486154d-1 ! Sdirk error estimator: rkF(0) = 0.0d0 rkF(1) = -.66535815876916686607437314126436349d0 rkF(2) = 1.7419302743497277572980407931678409d0 rkF(3) = -1.2918865386966730694684011822841728d0 ! ELO = local order of classical error estimator rkELO = 4.0d0 ! Sdirk error estimator: rkBgam(0) = .2950472755430528877214995073815946d-1 rkBgam(1) = .5370310883226113978352873633882769d0 rkBgam(2) = .2963022450107219354980459699450564d0 rkBgam(3) = -.7815248400375080035021681445218837d-1 rkBgam(4) = .2153144231161121782447335303806956d0 ! H* Sum Bgam_j*f(Z_j) = H*Bgam(0)*f(0) + Sum Theta_j*Z_j rkTheta(0) = 0.0d0 rkTheta(1) = -.6653581587691668660743731412643631d0 rkTheta(2) = 1.741930274349727757298040793167842d0 rkTheta(3) = -.291886538696673069468401182284174d0 !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ !~~~> Diagonalize the RK matrix: ! rkTinv * inv(rkA) * rkT = ! | rkGamma 0 0 | ! | 0 rkAlpha -rkBeta | ! | 0 rkBeta rkAlpha | !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ rkGamma = 4.644370709252171185822941421408063d0 rkAlpha = 3.677814645373914407088529289295968d0 rkBeta = 3.508761919567443321903661209182446d0 rkT(1,1) = .5303036326129938105898786144870856d-1 rkT(1,2) = -.7776129960563076320631956091016914d-1 rkT(1,3) = .6043307469475508514468017399717112d-2 rkT(2,1) = .2637242522173698467283726114649606d0 rkT(2,2) = .2193839918662961493126393244533346d0 rkT(2,3) = .3198765142300936188514264752235344d0 rkT(3,1) = 1.d0 rkT(3,2) = 1.d0 rkT(3,3) = 0.d0 rkTinv(1,1) = 7.695032983257654470769069079238553d0 rkTinv(1,2) = -.1453793830957233720334601186354032d0 rkTinv(1,3) = .6302696746849084900422461036874826d0 rkTinv(2,1) = -7.695032983257654470769069079238553d0 rkTinv(2,2) = .1453793830957233720334601186354032d0 rkTinv(2,3) = .3697303253150915099577538963125174d0 rkTinv(3,1) = -1.066660885401270392058552736086173d0 rkTinv(3,2) = 3.146358406832537460764521760668932d0 rkTinv(3,3) = -.7732056038202974770406168510664738d0 rkTinvAinv(1,1) = 35.73858579417120341641749040405149d0 rkTinvAinv(1,2) = -.675195748578927863668368190236025d0 rkTinvAinv(1,3) = 2.927206016036483646751158874041632d0 rkTinvAinv(2,1) = -24.55824590667225493437162206039511d0 rkTinvAinv(2,2) = -10.50514413892002061837750015342036 rkTinvAinv(2,3) = 4.072793983963516353248841125958369d0 rkTinvAinv(3,1) = -30.92301972744621647251975054630589d0 rkTinvAinv(3,2) = 12.08182467154052413351908559269928d0 rkTinvAinv(3,3) = -1.546411207640594954081233702132946d0 rkAinvT(1,1) = .2462926658317812882584158369803835d0 rkAinvT(1,2) = -.2647871194157644619747121197289574d0 rkAinvT(1,3) = .2950720515900466654896406799284586d0 rkAinvT(2,1) = 1.224833192317784474576995878738004d0 rkAinvT(2,2) = 1.929224190340981580557006261869763d0 rkAinvT(2,3) = .4066803323234419988910915619080306d0 rkAinvT(3,1) = 4.644370709252171185822941421408064d0 rkAinvT(3,2) = 3.677814645373914407088529289295968d0 rkAinvT(3,3) = -3.508761919567443321903661209182446d0 END SUBROUTINE Lobatto3A_Coefficients !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ END SUBROUTINE RungeKutta ! and all its internal procedures !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE FUN_CHEM(T, V, FCT) !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ USE KPP_ROOT_Parameters USE KPP_ROOT_Global USE KPP_ROOT_Function, ONLY: Fun USE KPP_ROOT_Rates, ONLY: Update_SUN, Update_RCONST, Update_PHOTO IMPLICIT NONE KPP_REAL :: V(NVAR), FCT(NVAR) KPP_REAL :: T, Told Told = TIME TIME = T CALL Update_SUN() CALL Update_RCONST() CALL Update_PHOTO() TIME = Told CALL Fun(V, FIX, RCONST, FCT) END SUBROUTINE FUN_CHEM !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SUBROUTINE JAC_CHEM (T, V, JF) !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ USE KPP_ROOT_Parameters USE KPP_ROOT_Global USE KPP_ROOT_JacobianSP USE KPP_ROOT_Jacobian, ONLY: Jac_SP USE KPP_ROOT_Rates, ONLY: Update_SUN, Update_RCONST, Update_PHOTO IMPLICIT NONE KPP_REAL :: V(NVAR), T , Told #ifdef FULL_ALGEBRA KPP_REAL :: JV(LU_NONZERO), JF(NVAR,NVAR) INTEGER :: i, j #else KPP_REAL :: JF(LU_NONZERO) #endif Told = TIME TIME = T CALL Update_SUN() CALL Update_RCONST() CALL Update_PHOTO() TIME = Told #ifdef FULL_ALGEBRA CALL Jac_SP(V, FIX, RCONST, JV) DO j=1,NVAR DO i=1,NVAR JF(i,j) = 0.0d0 END DO END DO DO i=1,LU_NONZERO JF(LU_IROW(i),LU_ICOL(i)) = JV(i) END DO #else CALL Jac_SP(V, FIX, RCONST, JF) #endif END SUBROUTINE JAC_CHEM !~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ END MODULE KPP_ROOT_Integrator