!----------------------------------------------------------------------------------------------------------------------------------! !()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()! !----------------------------------------------------------------------------------------------------------------------------------! !> !! DSTOKA performs one step of the integration of an initial value !! problem for a system of Ordinary Differential Equations. !! !! This routine was derived from Subroutine DSTODPK in the DLSODPK !! package by the addition of automatic functional/Newton iteration !! switching and logic for re-use of Jacobian data. !! !! Note: DSTOKA is independent of the value of the iteration method !! indicator MITER, when this is .ne. 0, and hence is independent !! of the type of chord method used, or the Jacobian structure. !! !! Communication with DSTOKA is done with the following variables: !! !! NEQ !! !! : integer array containing problem size in NEQ(1), and !! passed as the NEQ argument in all calls to F and JAC. !! !! Y !! !! : an array of length .ge. N used as the Y argument in !! all calls to F and JAC. !! !! YH !! !! : an NYH by LMAX array containing the dependent variables !! and their approximate scaled derivatives, where !! LMAX = MAXORD + 1. YH(i,j+1) contains the approximate !! j-th derivative of y(i), scaled by H**j/factorial(j) !! (j = 0,1,...,NQ). On entry for the first step, the first !! two columns of YH must be set from the initial values. !! !! NYH !! !! : a constant integer .ge. N, the first dimension of YH. !! !! YH1 !! !! : a one-dimensional array occupying the same space as YH. !! !! EWT !! !! : an array of length N containing multiplicative weights !! for local error measurements. Local errors in y(i) are !! compared to 1.0/EWT(i) in various error tests. !! !! SAVF !! !! : an array of working storage, of length N. !! Also used for input of YH(*,MAXORD+2) when JSTART = -1 !! and MAXORD .lt. the current order NQ. !! !! SAVX !! !! : an array of working storage, of length N. !! !! ACOR !! !! : a work array of length N, used for the accumulated !! corrections. On a successful return, ACOR(i) contains !! the estimated one-step local error in y(i). !! !! WM,IWM !! !! : real and integer work arrays associated with matrix !! operations in chord iteration (MITER .ne. 0). !! !! CCMAX !! !! : maximum relative change in H*EL0 before DSETPK is called. !! !! H !! !! : the step size to be attempted on the next step. !! H is altered by the error control algorithm during the !! problem. H can be either positive or negative, but its !! sign must remain constant throughout the problem. !! !! HMIN !! !! : the minimum absolute value of the step size H to be used. !! !! HMXI !! !! : inverse of the maximum absolute value of H to be used. !! !! !! HMXI = 0.0 is allowed and corresponds to an infinite HMAX. !! !! !! HMIN and HMXI may be changed at any time, but will not !! take effect until the next change of H is considered. !! !! TN !! !! : the independent variable. TN is updated on each step taken. !! !! JSTART !! !! : an integer used for input only, with the following !! values and meanings: !! !! 0 perform the first step. !! .gt.0 take a new step continuing from the last. !! -1 take the next step with a new value of H, MAXORD, !! N, METH, MITER, and/or matrix parameters. !! -2 take the next step with a new value of H, !! but with other inputs unchanged. !! !! On return, JSTART is set to 1 to facilitate continuation. !! !! KFLAG !! !! : a completion code with the following meanings: !! !! 0 the step was succesful. !! -1 the requested error could not be achieved. !! -2 corrector convergence could not be achieved. !! -3 fatal error in DSETPK or DSOLPK. !! !! A return with KFLAG = -1 or -2 means either !! ABS(H) = HMIN or 10 consecutive failures occurred. !! On a return with KFLAG negative, the values of TN and !! the YH array are as of the beginning of the last !! step, and H is the last step size attempted. !! !! MAXORD !! !! : the maximum order of integration method to be allowed. !! !! MAXCOR !! !! : the maximum number of corrector iterations allowed. !! !! MSBP !! !! : maximum number of steps between DSETPK calls (MITER .gt. 0). !! !! MXNCF !! !! : maximum number of convergence failures allowed. !! METH/MITER = the method flags. See description in driver. !! !! N !! !! : the number of first-order differential equations. !----------------------------------------------------------------------- subroutine dstoka(Neq,Y,Yh,Nyh,Yh1,Ewt,Savf,Savx,Acor,Wm,Iwm,f,jac,psol) integer, dimension(*) :: Neq real(kind=dp), dimension(*) :: Y integer, intent(in) :: Nyh real(kind=dp), intent(inout), dimension(Nyh,*) :: Yh real(kind=dp), intent(inout), dimension(*) :: Yh1 real(kind=dp), dimension(*) :: Ewt real(kind=dp), intent(inout), dimension(*) :: Savf real(kind=dp), intent(inout), dimension(*) :: Savx real(kind=dp), intent(inout), dimension(*) :: Acor real(kind=dp), dimension(*) :: Wm integer, dimension(*) :: Iwm external f external jac external psol real(kind=dp) :: dcon, ddn, del, delp, dfnrm, drc, dsm, dup, exdn, exsm, exup, r, rh, rhdn, rhsm, rhup, roc, stiff, told integer :: i, i1, iredo, iret, j, jb, jok, m, ncf, newq, nslow dls1%kflag = 0 told = dls1%tn ncf = 0 dls1%ierpj = 0 dls1%iersl = 0 dls1%jcur = 0 dls1%icf = 0 delp = 0.0D0 if ( dls1%jstart>0 ) goto 400 if ( dls1%jstart==-1 ) then !----------------------------------------------------------------------- ! The following block handles preliminaries needed when JSTART = -1. ! IPUP is set to MITER to force a matrix update. ! If an order increase is about to be considered (IALTH = 1), ! IALTH is reset to 2 to postpone consideration one more step. ! If the caller has changed METH, DCFODE is called to reset ! the coefficients of the method. ! If the caller has changed MAXORD to a value less than the current ! order NQ, NQ is reduced to MAXORD, and a new H chosen accordingly. ! If H is to be changed, YH must be rescaled. ! If H or METH is being changed, IALTH is reset to L = NQ + 1 ! to prevent further changes in H for that many steps. !----------------------------------------------------------------------- dls1%ipup = dls1%miter dls1%lmax = dls1%maxord + 1 if ( dls1%ialth==1 ) dls1%ialth = 2 if ( dls1%meth/=dls1%meo ) then call dcfode(dls1%meth,dls1%elco,dls1%tesco) dls1%meo = dls1%meth if ( dls1%nq<=dls1%maxord ) then dls1%ialth = dls1%l iret = 1 goto 100 endif elseif ( dls1%nq<=dls1%maxord ) then goto 200 endif dls1%nq = dls1%maxord dls1%l = dls1%lmax do i = 1, dls1%l dls1%el(i) = dls1%elco(i,dls1%nq) enddo dls1%nqnyh = dls1%nq*Nyh dls1%rc = dls1%rc*dls1%el(1)/dls1%el0 dls1%el0 = dls1%el(1) dls1%conit = 0.5D0/(dls1%nq+2) dlpk%epcon = dls1%conit*dls1%tesco(2,dls1%nq) ddn = dvnorm(dls1%n,Savf,Ewt)/dls1%tesco(1,dls1%l) exdn = 1.0D0/dls1%l rhdn = 1.0D0/(1.3D0*ddn**exdn+0.0000013D0) rh = min(rhdn,1.0D0) iredo = 3 if ( dls1%h==dls1%hold ) then rh = max(rh,dls1%hmin/abs(dls1%h)) else rh = min(rh,abs(dls1%h/dls1%hold)) dls1%h = dls1%hold endif goto 300 else if ( dls1%jstart==-2 ) goto 200 !----------------------------------------------------------------------- ! On the first call, the order is set to 1, and other variables are ! initialized. RMAX is the maximum ratio by which H can be increased ! in a single step. It is initially 1.E4 to compensate for the small ! initial H, but then is normally equal to 10. If a failure ! occurs (in corrector convergence or error test), RMAX is set at 2 ! for the next increase. !----------------------------------------------------------------------- dls1%lmax = dls1%maxord + 1 dls1%nq = 1 dls1%l = 2 dls1%ialth = 2 dls1%rmax = 10000.0D0 dls1%rc = 0.0D0 dls1%el0 = 1.0D0 dls1%crate = 0.7D0 dls1%hold = dls1%h dls1%meo = dls1%meth dls1%nslp = 0 dls%nslj = 0 dls1%ipup = 0 iret = 3 dls%newt = 0 dls%stifr = 0.0D0 !----------------------------------------------------------------------- ! DCFODE is called to get all the integration coefficients for the ! current METH. Then the EL vector and related constants are reset ! whenever the order NQ is changed, or at the start of the problem. !----------------------------------------------------------------------- call dcfode(dls1%meth,dls1%elco,dls1%tesco) endif 100 continue do i = 1, dls1%l dls1%el(i) = dls1%elco(i,dls1%nq) enddo dls1%nqnyh = dls1%nq*Nyh dls1%rc = dls1%rc*dls1%el(1)/dls1%el0 dls1%el0 = dls1%el(1) dls1%conit = 0.5D0/(dls1%nq+2) dlpk%epcon = dls1%conit*dls1%tesco(2,dls1%nq) select case (iret) case (2) rh = max(rh,dls1%hmin/abs(dls1%h)) goto 300 case (3) goto 400 case default endselect !----------------------------------------------------------------------- ! If H is being changed, the H ratio RH is checked against ! RMAX, HMIN, and HMXI, and the YH array rescaled. IALTH is set to ! L = NQ + 1 to prevent a change of H for that many steps, unless ! forced by a convergence or error test failure. !----------------------------------------------------------------------- 200 continue if ( dls1%h==dls1%hold ) goto 400 rh = dls1%h/dls1%hold dls1%h = dls1%hold iredo = 3 300 continue rh = min(rh,dls1%rmax) rh = rh/max(1.0D0,abs(dls1%h)*dls1%hmxi*rh) r = 1.0D0 do j = 2, dls1%l r = r*rh do i = 1, dls1%n Yh(i,j) = Yh(i,j)*r enddo enddo dls1%h = dls1%h*rh dls1%rc = dls1%rc*rh dls1%ialth = dls1%l if ( iredo==0 ) then dls1%rmax = 10.0D0 goto 1200 endif !----------------------------------------------------------------------- ! This section computes the predicted values by effectively ! multiplying the YH array by the Pascal triangle matrix. ! The flag IPUP is set according to whether matrix data is involved ! (NEWT .gt. 0 .and. JACFLG .ne. 0) or not, to trigger a call to DSETPK. ! IPUP is set to MITER when RC differs from 1 by more than CCMAX, ! and at least every MSBP steps, when JACFLG = 1. ! RC is the ratio of new to old values of the coefficient H*EL(1). !----------------------------------------------------------------------- 400 continue if ( dls%newt==0 .or. dlpk%jacflg==0 ) then drc = 0.0D0 dls1%ipup = 0 dls1%crate = 0.7D0 else drc = abs(dls1%rc-1.0D0) if ( drc>dls1%ccmax ) dls1%ipup = dls1%miter if ( dls1%nst>=dls1%nslp+dls1%msbp ) dls1%ipup = dls1%miter endif dls1%tn = dls1%tn + dls1%h i1 = dls1%nqnyh + 1 do jb = 1, dls1%nq i1 = i1 - Nyh ! DIR$ IVDEP do i = i1, dls1%nqnyh Yh1(i) = Yh1(i) + Yh1(i+Nyh) enddo enddo !----------------------------------------------------------------------- ! Up to MAXCOR corrector iterations are taken. A convergence test is ! made on the RMS-norm of each correction, weighted by the error ! weight vector EWT. The sum of the corrections is accumulated in the ! vector ACOR(i). The YH array is not altered in the corrector loop. ! Within the corrector loop, an estimated rate of convergence (ROC) ! and a stiffness ratio estimate (STIFF) are kept. Corresponding ! global estimates are kept as CRATE and dls%stifr. !----------------------------------------------------------------------- 500 continue m = 0 dlpk%mnewt = 0 stiff = 0.0D0 roc = 0.05D0 nslow = 0 do i = 1, dls1%n Y(i) = Yh(i,1) enddo call f(Neq,dls1%tn,Y,Savf) dls1%nfe = dls1%nfe + 1 if ( dls%newt/=0 .and. dls1%ipup>0 ) then !----------------------------------------------------------------------- ! If indicated, DSETPK is called to update any matrix data needed, ! before starting the corrector iteration. ! JOK is set to indicate if the matrix data need not be recomputed. ! IPUP is set to 0 as an indicator that the matrix data is up to date. !----------------------------------------------------------------------- jok = 1 if ( dls1%nst==0 .or. dls1%nst>dls%nslj+50 ) jok = -1 if ( dls1%icf==1 .and. drc<0.2D0 ) jok = -1 if ( dls1%icf==2 ) jok = -1 if ( jok==-1 ) then dls%nslj = dls1%nst dls%njev = dls%njev + 1 endif call dsetpk(Neq,Y,Yh1,Ewt,Acor,Savf,jok,Wm,Iwm,f,jac) dls1%ipup = 0 dls1%rc = 1.0D0 drc = 0.0D0 dls1%nslp = dls1%nst dls1%crate = 0.7D0 if ( dls1%ierpj/=0 ) goto 800 endif do i = 1, dls1%n Acor(i) = 0.0D0 enddo 600 continue if ( dls%newt/=0 ) then !----------------------------------------------------------------------- ! In the case of the chord method, compute the corrector error, ! and solve the linear system with that as right-hand side and ! P as coefficient matrix. STIFF is set to the ratio of the norms ! of the residual and the correction vector. !----------------------------------------------------------------------- do i = 1, dls1%n Savx(i) = dls1%h*Savf(i) - (Yh(i,2)+Acor(i)) enddo dfnrm = dvnorm(dls1%n,Savx,Ewt) call dsolpk(Neq,Y,Savf,Savx,Ewt,Wm,Iwm,f,psol) if ( dls1%iersl<0 ) goto 800 if ( dls1%iersl>0 ) goto 700 del = dvnorm(dls1%n,Savx,Ewt) if ( del>1.0D-8 ) stiff = max(stiff,dfnrm/del) do i = 1, dls1%n Acor(i) = Acor(i) + Savx(i) Y(i) = Yh(i,1) + dls1%el(1)*Acor(i) enddo else !----------------------------------------------------------------------- ! In the case of functional iteration, update Y directly from ! the result of the last function evaluation, and STIFF is set to 1.0. !----------------------------------------------------------------------- do i = 1, dls1%n Savf(i) = dls1%h*Savf(i) - Yh(i,2) Y(i) = Savf(i) - Acor(i) enddo del = dvnorm(dls1%n,Y,Ewt) do i = 1, dls1%n Y(i) = Yh(i,1) + dls1%el(1)*Savf(i) Acor(i) = Savf(i) enddo stiff = 1.0D0 endif !----------------------------------------------------------------------- ! Test for convergence. If M .gt. 0, an estimate of the convergence ! rate constant is made for the iteration switch, and is also used ! in the convergence test. If the iteration seems to be diverging or ! converging at a slow rate (.gt. 0.8 more than once), it is stopped. !----------------------------------------------------------------------- if ( m/=0 ) then roc = max(0.05D0,del/delp) dls1%crate = max(0.2D0*dls1%crate,roc) endif dcon = del*min(1.0D0,1.5D0*dls1%crate)/dlpk%epcon if ( dcon<=1.0D0 ) then !----------------------------------------------------------------------- ! The corrector has converged. JCUR is set to 0 to signal that the ! preconditioner involved may need updating later. ! The stiffness ratio STIFR is updated using the latest STIFF value. ! The local error test is made and control passes to statement 500 ! if it fails. !----------------------------------------------------------------------- dls1%jcur = 0 if ( dls%newt>0 ) dls%stifr = 0.5D0*(dls%stifr+stiff) if ( m==0 ) dsm = del/dls1%tesco(2,dls1%nq) if ( m>0 ) dsm = dvnorm(dls1%n,Acor,Ewt)/dls1%tesco(2,dls1%nq) if ( dsm>1.0D0 ) then !----------------------------------------------------------------------- ! The error test failed. KFLAG keeps track of multiple failures. ! Restore TN and the YH array to their previous values, and prepare ! to try the step again. Compute the optimum step size for this or ! one lower order. After 2 or more failures, H is forced to decrease ! by a factor of 0.2 or less. !----------------------------------------------------------------------- dls1%kflag = dls1%kflag - 1 dls1%tn = told i1 = dls1%nqnyh + 1 do jb = 1, dls1%nq i1 = i1 - Nyh ! DIR$ IVDEP do i = i1, dls1%nqnyh Yh1(i) = Yh1(i) - Yh1(i+Nyh) enddo enddo dls1%rmax = 2.0D0 if ( abs(dls1%h)<=dls1%hmin*1.00001D0 ) then !----------------------------------------------------------------------- ! All returns are made through this section. H is saved in HOLD ! to allow the caller to change H on the next step. !----------------------------------------------------------------------- dls1%kflag = -1 goto 1300 elseif ( dls1%kflag<=-3 ) then !----------------------------------------------------------------------- ! Control reaches this section if 3 or more failures have occured. ! If 10 failures have occurred, exit with KFLAG = -1. ! It is assumed that the derivatives that have accumulated in the ! YH array have errors of the wrong order. Hence the first ! derivative is recomputed, and the order is set to 1. Then ! H is reduced by a factor of 10, and the step is retried, ! until it succeeds or H reaches HMIN. !----------------------------------------------------------------------- if ( dls1%kflag==-10 ) then dls1%kflag = -1 goto 1300 else rh = 0.1D0 rh = max(dls1%hmin/abs(dls1%h),rh) dls1%h = dls1%h*rh do i = 1, dls1%n Y(i) = Yh(i,1) enddo call f(Neq,dls1%tn,Y,Savf) dls1%nfe = dls1%nfe + 1 do i = 1, dls1%n Yh(i,2) = dls1%h*Savf(i) enddo dls1%ipup = dls1%miter dls1%ialth = 5 if ( dls1%nq==1 ) goto 400 dls1%nq = 1 dls1%l = 2 iret = 3 goto 100 endif else iredo = 2 rhup = 0.0D0 goto 900 endif else !----------------------------------------------------------------------- ! After a successful step, update the YH array. ! If Newton iteration is being done and STIFR is less than 1.5, ! then switch to functional iteration. ! Consider changing H if IALTH = 1. Otherwise decrease IALTH by 1. ! If IALTH is then 1 and NQ .lt. MAXORD, then ACOR is saved for ! use in a possible order increase on the next step. ! If a change in H is considered, an increase or decrease in order ! by one is considered also. A change in H is made only if it is by a ! factor of at least 1.1. If not, IALTH is set to 3 to prevent ! testing for that many steps. !----------------------------------------------------------------------- dls1%kflag = 0 iredo = 0 dls1%nst = dls1%nst + 1 if ( dls%newt==0 ) dls%nsfi = dls%nsfi + 1 if ( dls%newt>0 .and. dls%stifr<1.5D0 ) dls%newt = 0 dls1%hu = dls1%h dls1%nqu = dls1%nq do j = 1, dls1%l do i = 1, dls1%n Yh(i,j) = Yh(i,j) + dls1%el(j)*Acor(i) enddo enddo dls1%ialth = dls1%ialth - 1 if ( dls1%ialth==0 ) then !----------------------------------------------------------------------- ! Regardless of the success or failure of the step, factors ! RHDN, RHSM, and RHUP are computed, by which H could be multiplied ! at order NQ - 1, order NQ, or order NQ + 1, respectively. ! in the case of failure, RHUP = 0.0 to avoid an order increase. ! the largest of these is determined and the new order chosen ! accordingly. If the order is to be increased, we compute one ! additional scaled derivative. !----------------------------------------------------------------------- rhup = 0.0D0 if ( dls1%l/=dls1%lmax ) then do i = 1, dls1%n Savf(i) = Acor(i) - Yh(i,dls1%lmax) enddo dup = dvnorm(dls1%n,Savf,Ewt)/dls1%tesco(3,dls1%nq) exup = 1.0D0/(dls1%l+1) rhup = 1.0D0/(1.4D0*dup**exup+0.0000014D0) endif goto 900 else if ( dls1%ialth<=1 ) then if ( dls1%l/=dls1%lmax ) then do i = 1, dls1%n Yh(i,dls1%lmax) = Acor(i) enddo endif endif goto 1200 endif endif else m = m + 1 if ( m/=dls1%maxcor ) then if ( m<2 .or. del<=2.0D0*delp ) then if ( roc<=10.0D0 ) then if ( roc>0.8D0 ) nslow = nslow + 1 if ( nslow<2 ) then dlpk%mnewt = m delp = del call f(Neq,dls1%tn,Y,Savf) dls1%nfe = dls1%nfe + 1 goto 600 endif endif endif endif endif !----------------------------------------------------------------------- ! The corrector iteration failed to converge. ! If functional iteration is being done (NEWT = 0) and MITER .gt. 0 ! (and this is not the first step), then switch to Newton ! (NEWT = MITER), and retry the step. (Setting STIFR = 1023 insures ! that a switch back will not occur for 10 step attempts.) ! If Newton iteration is being done, but using a preconditioner that ! is out of date (JACFLG .ne. 0 .and. JCUR = 0), then signal for a ! re-evalutation of the preconditioner, and retry the step. ! In all other cases, the YH array is retracted to its values ! before prediction, and H is reduced, if possible. If H cannot be ! reduced or MXNCF failures have occurred, exit with KFLAG = -2. !----------------------------------------------------------------------- 700 continue dls1%icf = 1 if ( dls%newt==0 ) then if ( dls1%nst==0 ) goto 800 if ( dls1%miter==0 ) goto 800 dls%newt = dls1%miter dls%stifr = 1023.0D0 dls1%ipup = dls1%miter goto 500 endif if ( dls1%jcur/=1 .and. dlpk%jacflg/=0 ) then dls1%ipup = dls1%miter goto 500 endif 800 continue dls1%icf = 2 ncf = ncf + 1 dlpk%ncfn = dlpk%ncfn + 1 dls1%rmax = 2.0D0 dls1%tn = told i1 = dls1%nqnyh + 1 do jb = 1, dls1%nq i1 = i1 - Nyh ! DIR$ IVDEP do i = i1, dls1%nqnyh Yh1(i) = Yh1(i) - Yh1(i+Nyh) enddo enddo if ( dls1%ierpj<0 .or. dls1%iersl<0 ) then dls1%kflag = -3 goto 1300 elseif ( abs(dls1%h)<=dls1%hmin*1.00001D0 ) then dls1%kflag = -2 goto 1300 elseif ( ncf==dls1%mxncf ) then dls1%kflag = -2 goto 1300 else rh = 0.5D0 dls1%ipup = dls1%miter iredo = 1 rh = max(rh,dls1%hmin/abs(dls1%h)) goto 300 endif 900 continue exsm = 1.0D0/dls1%l rhsm = 1.0D0/(1.2D0*dsm**exsm+0.0000012D0) rhdn = 0.0D0 if ( dls1%nq/=1 ) then ddn = dvnorm(dls1%n,Yh(1,dls1%l),Ewt)/dls1%tesco(1,dls1%nq) exdn = 1.0D0/dls1%nq rhdn = 1.0D0/(1.3D0*ddn**exdn+0.0000013D0) endif if ( rhsm>=rhup ) then if ( rhsm>=rhdn ) then newq = dls1%nq rh = rhsm goto 1000 endif elseif ( rhup>rhdn ) then newq = dls1%l rh = rhup if ( rh<1.1D0 ) then dls1%ialth = 3 goto 1200 else r = dls1%el(dls1%l)/dls1%l do i = 1, dls1%n Yh(i,newq+1) = Acor(i)*r enddo goto 1100 endif endif newq = dls1%nq - 1 rh = rhdn if ( dls1%kflag<0 .and. rh>1.0D0 ) rh = 1.0D0 1000 continue if ( (dls1%kflag==0) .and. (rh<1.1D0) ) then dls1%ialth = 3 goto 1200 else if ( dls1%kflag<=-2 ) rh = min(rh,0.2D0) !----------------------------------------------------------------------- ! If there is a change of order, reset NQ, L, and the coefficients. ! In any case H is reset according to RH and the YH array is rescaled. ! Then exit from 690 if the step was OK, or redo the step otherwise. !----------------------------------------------------------------------- if ( newq==dls1%nq ) then rh = max(rh,dls1%hmin/abs(dls1%h)) goto 300 endif endif 1100 continue dls1%nq = newq dls1%l = dls1%nq + 1 iret = 2 goto 100 1200 continue r = 1.0D0/dls1%tesco(2,dls1%nqu) do i = 1, dls1%n Acor(i) = Acor(i)*r enddo 1300 continue dls1%hold = dls1%h dls1%jstart = 1 end subroutine dstoka