solve T

This reverts commit 0d771077f6.

Conflicts:
	code/ysolve.f90

.gitlab-ci.yml

@@ -16,4 +16,6 @@ test:
   script:
     - cd ${CI_PROJECT_DIR}
     - cd code
-    - make test
+    - make clean && make
+    - cd sample/4pi-IC1
+    - ../../ex

code/makefile

@@ -1,17 +1,25 @@
 flags = -Wall -O3 -fdefault-integer-8 -fdefault-double-8 -fdefault-real-8  -march=native 
 
+ifdef CHECK
+flags += -fcheck=all
+endif
+
+ifdef DEBUG
+flags += -g
+endif
+
 compiler = gfortran
 
-ex  : test.o ysolve.mod compact.mod
+ex  : test.o ysolve.o Compact.o
 	${compiler} -o ex test.o ysolve.o Compact.o 
 
-test.o  : test.f90 ysolve.mod
+test.o  : test.f90 ysolve.mod compact.mod
 	${compiler} -c ${flags} test.f90
 
-ysolve.mod: ysolve.f90 compact.mod
+ysolve.o ysolve.mod : ysolve.f90 compact.mod
 	${compiler} -c ${flags} ysolve.f90
 
-compact.mod: Compact.f90
+Compact.o compact.mod : Compact.f90
 	${compiler} -c ${flags} Compact.f90
 
 test: ex

code/sample/4pi-IC1/itape

@@ -16,4 +16,5 @@
      c_ref     0.01  !0.003     ! 
     min_wr     0.        ! 5.0e-14
    prof_wr     1.0       
+     lewis     1.0
 

code/sample/itape.py

@@ -0,0 +1,83 @@
+#!/bin/python
+
+import os, sys
+
+import argparse
+
+parser = argparse.ArgumentParser()
+file_parser = argparse.ArgumentParser()
+
+param_dict = {
+"nx"		:512,
+"l_0"		:4.,
+"int_pr"	:400,
+"tar_lo"	:0.60,
+"dt"		:0.001,
+"sc"		:0.75,
+"vis"		:0.020,
+"pre"		:2.10E+4,
+"ac"		:26.7,
+"bc"		:3.,
+"u0"		:0.21,
+"tf"		:100.,
+"dt_uf"		:0.4,
+"ctmp"		:0.,
+"c_cut"		:0.001,
+"c_ref"		:0.01,
+"min_wr"	:0.,
+"prof_wr"	:1.0,
+"rvis"		:1.0,
+"cstar"		:1.0,
+"sigw"		:100.0,
+}
+
+file_parser.add_argument("--default", help="file containing override default values")
+
+file_args, others = file_parser.parse_known_args()
+
+if file_args.default is not None:
+    with open(file_args.default) as df:
+        try:
+            for line in df:
+                k, v = line.split()[:2]
+                if k in param_dict:
+                    param_dict[k] = v
+        except ValueError:
+          pass
+
+
+for k, v in param_dict.iteritems():
+    parser.add_argument("--"+k, default=v)
+
+args = parser.parse_args(others)
+
+params = vars(args)
+
+itape = '''        nx     {0[nx]}
+       l_0     {0[l_0]}
+    int_pr     {0[int_pr]}
+    tar_lo     {0[tar_lo]}
+        dt     {0[dt]}
+        sc     {0[sc]}
+       vis     {0[vis]}
+       pre     {0[pre]}
+        ac     {0[ac]}
+        bc     {0[bc]}
+        u0     {0[u0]}
+        tf     {0[tf]}
+     dt_uf     {0[dt_uf]}
+      ctmp     {0[ctmp]}
+     c_cut     {0[c_cut]}
+     c_ref     {0[c_ref]}
+    min_wr     {0[min_wr]}
+   prof_wr     {0[prof_wr]}
+      rvis     {0[rvis]}
+     cstar     {0[cstar]}
+      sigw     {0[sigw]} '''
+
+print itape.format( params )
+
+# for i in range(15,60,5):
+    # with open("itape", 'w') as itape:
+        # itape.write(format.format(float(i)/10.0))
+    # os.system("../../ex &>"+str(i)+".txt")

code/test.f90

@@ -6,5 +6,6 @@
     PROGRAM test
      USE ysolve
      IMPLICIT NONE
+     CALL parse
      CALL solve
     END PROGRAM test

code/test.sh

@@ -1,4 +1,4 @@
 cd sample/4pi-IC1
 ls
 ../../ex
-diff sfield.dat sfield.diff
+diff -y sfield.dat sfield.diff

code/ysolve.f90

@@ -4,12 +4,15 @@
      PRIVATE
      REAL, PARAMETER :: pi=3.14159265358979323846
      REAL :: hx,dt,vis,sc,diff,pre,ac,bc,tf,t_now,t_uf,dt_uf
+     REAL :: lewis
      REAL :: c_cut,c_ref,refwr,minf,tar_lo,u0,ctmp,l_0,lo_flm=0.
      REAL :: er_lof=0., erdot=0.,min_wr,prof_wr
      REAL :: pflame,pflold,oldu
      INTEGER :: ncyc=0,int_pr,nx
-     REAL, DIMENSION(:,:,:), ALLOCATABLE :: u,y1,y2,yf
+     REAL, DIMENSION(:), ALLOCATABLE :: u
+     REAL, DIMENSION(:,:), ALLOCATABLE :: y1,y2,yf
      REAL, DIMENSION(:), ALLOCATABLE :: uxt,duxt
+     REAL, DIMENSION(:), ALLOCATABLE :: dm
 
      INTEGER :: istage
      REAL, DIMENSION(5) :: a=(/ 970286171893./4311952581923.,    &
@@ -23,8 +26,38 @@
                                 2114624349019./3568978502595.,   &
                                 5198255086312./14908931495163. /)
 
-     PUBLIC :: solve
+     LOGICAL :: read_itape, read_stdin
+     CHARACTER(100) :: itape_name
+     PUBLIC :: solve, parse
     CONTAINS
+!------------------------------------------------------------------------
+     SUBROUTINE parse
+        CHARACTER(100) :: num1char
+
+        !First, make sure the right number of inputs have been provided
+        IF(COMMAND_ARGUMENT_COUNT().EQ.0)THEN
+          ! read from itape
+          read_itape = .true.
+        ELSE IF(COMMAND_ARGUMENT_COUNT().EQ.1)THEN
+          CALL GET_COMMAND_ARGUMENT(1,num1char)   !first, read in the two values
+        ELSE
+          WRITE(*,*)'ERROR, TOO MANY COMMAND-LINE ARGUMENTS(MORE THAN ONE). STOPPING'
+          STOP
+        ENDIF
+
+        read_stdin = (num1char=="-")
+
+        IF(read_itape)THEN
+          itape_name = "itape"
+          WRITE(*,*) "Read from itape"
+        ELSE IF(read_stdin)THEN
+          WRITE(*,*) "Read from STDIN"
+        ELSE 
+          itape_name = num1char
+          WRITE(*,*) "Read from " // num1char
+        END IF
+
+     END SUBROUTINE parse
 !------------------------------------------------------------------------
      SUBROUTINE solve
       INTEGER :: i,j,k,savenum
@@ -46,17 +79,17 @@
         pflold=pflame
         pflame=0.
         DO i=1,nx
-         pflame=pflame+y1(i,1,1)*hx
+         pflame=pflame+y1(i,1)*hx
-         uxt(i)=y1(i,1,1)
+         uxt(i)=y1(i,1)
         ENDDO
         CALL dfnonp(nx,hx,uxt,duxt,1,1)
         delf=1./MAXVAL(ABS(duxt))
-        oldu=u(1,1,1)
+        oldu=u(1)
         u=u+0.5*(hx*REAL(nx)*tar_lo-pflame)+0.5*(pflold-pflame)
         t_uf=0.
         WRITE(*,'(a3,f8.3,a10,f6.3,a10,f6.3,a10,f7.4,a10,f7.4,a8,f7.4)') &
          ' T:',t_now,' // Tar_L:',l_0*tar_lo,' // cur_L:',pflame/hx/REAL(nx)*l_0, &
-         ' // Old_U:',oldu,' // New_U:',u(1,1,1),' // L_f:',delf
+         ' // Old_U:',oldu,' // New_U:',u(1),' // L_f:',delf
 
 !        WRITE(*,'(a7,f7.4,a7,f7.4,a10,f7.4)') ' cur_U:',oldu,' // dU:',u(1,1,1)-oldu,' // new_U:',u(1,1,1)
 !        WRITE(*,*)
@@ -81,9 +114,9 @@
      END SUBROUTINE solve
 
      SUBROUTINE write_sd
-      REAL :: c,yr,wrate,dely,sdr,sdd,sd,uu,onelw,dd
+      REAL :: c,yr,theta,wrate,dely,sdr,sdd,sd,uu,onelw,dd
       INTEGER :: i,j,k,nd
-      REAL, DIMENSION(1,nx) :: ux,dux,d2ux
+      REAL, DIMENSION(2,nx) :: ux,dux,d2ux
       REAL, DIMENSION(10,nx) :: sav
         
       sav=0.
@@ -97,21 +130,22 @@
       WRITE(500,*) ' "Sdd","Sd","(1/C)/(dC/dx)","DIV(rho*Dmu*Gra(C))"'
 
       DO i=1,nx
-       ux(1,i)=y1(i,1,1)                               ! Yr
+       ux(1,i)=y1(i,1)                               ! Yr
+       ux(2,i)=y1(i,2)                               ! T
        IF (ux(1,i).gt.1.) ux(1,i)=1.                                  
       ENDDO
           
-      nd=1
+      nd=2
-      CALL dfnonp(nx,hx,ux(1,:),dux(1,:),nd,1)  
+      CALL dfnonp(nx,hx,ux(:,:),dux(:,:),nd,1)
-      nd=1 
+      CALL d2fnonp(nx,hx,ux(:,:),d2ux(:,:),nd,1)
-      CALL d2fnonp(nx,hx,ux(1,:),d2ux(1,:),nd,1) 
 
       DO i=1,nx
        yr=ux(1,i)
+       theta=ux(2,i)
        c=1.-yr
        IF (c.lt.0.) c=0.
 
-       wrate=pre*yr*exp(-ac/(1.+bc*(1.-yr)))   !wrate
+       wrate=pre*yr*exp(-ac/(1.+bc*(theta)))   !wrate
 
 !       IF (c.le.c_cut) THEN
 !        wrate=min_wr
@@ -119,22 +153,22 @@
 !         ((exp((c-c_cut)*prof_wr)-minf)/(1.-minf)*(refwr-min_wr))+min_wr
 !       ENDIF
 
-       IF (c.le.c_ref) THEN
+       IF (theta.le.c_ref) THEN
          wrate=min_wr
-        IF (c.gt.c_cut) wrate=((refwr-min_wr)*exp(prof_wr*(c-c_ref))+ &
+        IF (theta.gt.c_cut) wrate=((refwr-min_wr)*exp(prof_wr*(theta-c_ref))+ &
         min_wr-refwr*exp(prof_wr*(c_cut-c_ref)))/(1.-exp(prof_wr*(c_cut-c_ref)))
        ENDIF
 
        dely=ABS(dux(1,i))
        sdr=wrate/dely
-       sdd=-diff*d2ux(1,i)/dely
+       sdd=-dm(i)*d2ux(1,i)/dely
        IF (dely.eq.0.) THEN
         sdr=0.; sdd=0.
        ENDIF
        sd=sdr+sdd
-       uu=u(1,1,1)
+       uu=u(1)
        onelw=(-dux(1,i))/c
-       dd=-diff*d2ux(1,i)
+       dd=-dm(i)*d2ux(1,i)
        if (c.eq.0.) onelw=0.
 
        sav(1,i)=sav(1,i)+yr
@@ -159,38 +193,39 @@
 
       OPEN (305,FILE='sfield.bin',form='unformatted',status='unknown')
       DO i=1,nx
-       WRITE (305) y1(i,1,1)
+       WRITE (305) y1(i,1)
       ENDDO
       CLOSE (305)
 
       OPEN (305,FILE='sfield.dat')
       DO i=1,nx
-       WRITE (305,'(e30.20)') y1(i,1,1)
+       WRITE (305,'(2e30.20)') y1(i,1), y1(i,2)
       ENDDO
       CLOSE (305)
 
      END SUBROUTINE save_final_field
 
      SUBROUTINE write_pre
-      REAL :: yr,c,dy,maxdy=0.,del_f
+      REAL :: theta,yr,c,dy,maxdy=0.,del_f
       REAL :: S_L=0.,wrate
       INTEGER :: i
       REAL, DIMENSION(1,nx) :: ux, dux
       
       DO i=1,nx
-       yr=y1(i,1,1)
+       yr=y1(i,1)
+       theta=y1(i,2)
        c=1.-yr
        ux(1,i)=yr
 
-       wrate=pre*yr*exp(-ac/(1.+bc*(1.-yr)))   !wrate
+       wrate=pre*yr*exp(-ac/(1.+bc*theta))   !wrate
 !       IF (c.le.c_cut) THEN
 !        wrate=min_wr
 !        IF (c.gt.c_ref) wrate= &
 !         ((exp((c-c_cut)*prof_wr)-minf)/(1.-minf)*(refwr-min_wr))+min_wr
 !       ENDIF
-       IF (c.le.c_ref) THEN
+       IF (theta.le.c_ref) THEN
          wrate=min_wr
-        IF (c.gt.c_cut) wrate=((refwr-min_wr)*exp(prof_wr*(c-c_ref))+ &
+        IF (theta.gt.c_cut) wrate=((refwr-min_wr)*exp(prof_wr*(theta-c_ref))+ &
         min_wr-refwr*exp(prof_wr*(c_cut-c_ref)))/(1.-exp(prof_wr*(c_cut-c_ref)))
        ENDIF
 
@@ -207,7 +242,7 @@
       ENDDO
       del_f=1./maxdy
        
-      WRITE(*,'(a13,e14.8,a25,e14.8)') ' Grid size : ',hx,' / Laminar flame speed : ',u(1,1,1)
+      WRITE(*,'(a13,e14.8,a25,e14.8)') ' Grid size : ',hx,' / Laminar flame speed : ',u(1)
       WRITE(*,'(a19,e14.8,a3,f9.5,a20)') &
        ' Flame thickness : ',del_f,' / ',del_f/hx,' grids in the flame.'
       WRITE(*,*)
@@ -215,7 +250,7 @@
 
      SUBROUTINE SET_BC
       
-      y1(1,1,1)=1.-ctmp
+      y1(1,1)=1.-ctmp
      END SUBROUTINE SET_BC
 
      SUBROUTINE SET_IC
@@ -236,12 +271,14 @@
        ENDIF
 
 !       y1(i,1,1)=(1.-xi)*1.
-       y1(i,1,1)=(1.-xi)   ! reactant mass fraction
+       y1(i,1)=(1.-xi)   ! reactant mass fraction
       ENDDO
 
+      y1(:,2) = 1.0 - y1(:,1)
+
       pflame=0.
       DO i=1,nx
-       pflame=pflame+y1(i,1,1)*hx
+       pflame=pflame+y1(i,1)*hx
       ENDDO
 
      END SUBROUTINE SET_IC
@@ -249,9 +286,14 @@
      SUBROUTINE READ_INTRO
        CHARACTER(LEN=8) :: cdum
        INTEGER :: itape=300,otape=301,ierr
+       INTEGER :: i
 
+       IF (read_stdin) THEN
+         itape=5
+       ELSE
+         OPEN(itape,FILE=itape_name)
+       END IF
 
-       OPEN(itape,FILE='itape')
        OPEN(otape,FILE='otape')
         READ(itape,*) cdum,nx
         WRITE(otape,*) cdum,nx
@@ -291,8 +333,12 @@
         WRITE(otape,*) cdum,min_wr
         READ(itape,*) cdum,prof_wr
         WRITE(otape,*) cdum,prof_wr
+        READ(itape,*) cdum,lewis
+        WRITE(otape,*) cdum,lewis
 
-       CLOSE(itape)
+       IF (.not.read_stdin) THEN
+         CLOSE(itape)
+       END IF
 
 !       hx=l_0*pi/REAL(nx)
        hx=l_0*pi/REAL(nx-1)
@@ -312,12 +358,13 @@
        refwr=pre*1.*exp(-ac/(1.+bc*c_ref))          
        l_0=l_0*pi
 
-       ALLOCATE(u(nx,1,1),STAT=ierr)   ; u=0.
+       ALLOCATE( u(nx),STAT=ierr)    ; u=0.
-       ALLOCATE(y1(nx,1,1),STAT=ierr)  ; y1=0.
+       ALLOCATE(y1(nx,2),STAT=ierr)  ; y1=0.
-       ALLOCATE(y2(nx,1,1),STAT=ierr)  ; y2=0.
+       ALLOCATE(y2(nx,2),STAT=ierr)  ; y2=0.
-       ALLOCATE(yf(nx,1,1),STAT=ierr)  ; yf=0.
+       ALLOCATE(yf(nx,2),STAT=ierr)  ; yf=0.
        ALLOCATE(uxt(nx),STAT=ierr)     ; uxt=0.
        ALLOCATE(duxt(nx),STAT=ierr)    ; duxt=0.
+       ALLOCATE(dm(nx),STAT=ierr)      ; dm=diff
 
      END SUBROUTINE READ_INTRO
 
@@ -330,8 +377,8 @@
      END SUBROUTINE RK4
 !------------------------------------------------------------------------
      SUBROUTINE substep(ri,r1,r2,f)
-      REAL, INTENT(INOUT),DIMENSION(:,:,:) :: ri,r1,r2
+      REAL, INTENT(INOUT),DIMENSION(:,:) :: ri,r1,r2
-      REAL, INTENT(OUT),DIMENSION(:,:,:) :: f
+      REAL, INTENT(OUT),DIMENSION(:,:) :: f
 
       INTEGER :: i,j,k
       REAL :: at,bt
@@ -340,118 +387,66 @@
        IF(istage<5) THEN
         at=a(istage)*dt
         bt=(b(istage)-a(istage))*dt
-        DO k=1,1  ! nz
+        r1=r1+at*f
-        DO j=1,1  ! ny
+        r2=r1+bt*f
-        DO i=1,nx
-        r1(i,j,k)=r1(i,j,k)+at*f(i,j,k)
-        r2(i,j,k)=r1(i,j,k)+bt*f(i,j,k)
-        ENDDO
-        ENDDO
-        ENDDO
        ELSE
         bt=b(istage)*dt
-        DO k=1,1  ! nz
+        r1=r1+bt*f
-        DO j=1,1  ! ny
-        DO i=1,nx
-        r1(i,j,k)=r1(i,j,k)+bt*f(i,j,k)
-        ENDDO
-        ENDDO
-        ENDDO
        ENDIF
      END SUBROUTINE substep
 !------------------------------------------------------------------------
      SUBROUTINE fns(r1,f)
-      REAL, INTENT(IN),DIMENSION(:,:,:) :: r1
+      REAL, INTENT(IN),DIMENSION(:,:) :: r1
-      REAL, INTENT(OUT),DIMENSION(:,:,:) :: f
+      REAL, INTENT(OUT),DIMENSION(:,:) :: f
-      REAL, DIMENSION(3,nx) :: ux,dux,d2ux
+      REAL, DIMENSION(2,nx) :: ux, dux, d2ux
       INTEGER :: i,j,k
-      REAL :: wrate,Ly,Dy
+      REAL :: wrate,Ly,Dy,Lt,Dt
 
-      DO k=1,1   !nz
 ! x-direction
-       DO j=1,1   !ny
         DO i=1,nx
-         ux(1,i)=u(i,j,k)*r1(i,j,k)                     ! u*Y
+         ux(1,i)=r1(i,1)                              ! Y
-         ux(2,i)=u(i,j,k)                               ! u
+         ux(2,i)=r1(i,2)                              ! T
-         ux(3,i)=r1(i,j,k)                              ! Y
         ENDDO
 
-        CALL dfnonp(nx,hx,ux(1:3,:),dux(1:3,:),3,1)
+        CALL dfnonp(nx,hx,ux(:,:),dux(:,:),2,1)
-        CALL d2fnonp(nx,hx,ux(3:3,:),d2ux(1,:),1,1)
+
+        CALL d2fnonp(nx,hx,ux(:,:),d2ux(:,:),2,1)
 
         DO i=1,nx
-         wrate=pre*ux(3,i)*exp(-ac/(1.+bc*(1.-ux(3,i))))   !wrate
+         wrate=pre*ux(1,i)*exp(-ac/(1.+bc*(ux(2,i))))   !wrate
-!         IF ((1.-ux(3,i)).le.c_cut) THEN
+
-!          wrate=min_wr
+         IF ((ux(2,i)).le.c_ref) THEN
-!          IF ((1.-ux(3,i)).gt.c_ref) wrate= &
-!           ((exp(((1.-ux(3,i))-c_cut)*prof_wr)-minf)/(1.-minf)*(refwr-min_wr))+min_wr
-!         ENDIF
-         IF ((1.-ux(3,i)).le.c_ref) THEN
            wrate=min_wr
-          IF ((1.-ux(3,i)).gt.c_cut) wrate=((refwr-min_wr)*exp(prof_wr*(1.-ux(3,i)-c_ref))+ &
+          IF ((ux(2,i)).gt.c_cut) wrate=((refwr-min_wr)*exp(prof_wr*(ux(2,i)-c_ref))+ &
           min_wr-refwr*exp(prof_wr*(c_cut-c_ref)))/(1.-exp(prof_wr*(c_cut-c_ref)))
          ENDIF
 
 ! -0.5*( d(u*Y)/dx + u*dY/dx + Y*du/dx ) + D*d2Y/d2x
-         f(i,j,k)=-0.5*( dux(1,i) + ux(2,i)*dux(3,i) + &
+         f(i,1) = - ( u(i)*dux(1,i) ) + diff * d2ux(1,i) - wrate
-          ux(3,i)*dux(2,i) ) &
+
-                  + diff*d2ux(1,i) - wrate
+! -  u*dT/dx + D*d2T/d2x
+         f(i,2) = - ( u(i)*dux(2,i) ) + (lewis * diff) * d2ux(2,i) + wrate
+
          IF (i.eq.nx) THEN
-          Ly=ux(2,nx)*dux(3,nx)                           ! Ly = u*dYr/dx
+          Ly=u(nx)*dux(1,nx)                           ! Ly = u*dYr/dx
-          f(nx,1,1)=-wrate
+          f(nx,1)=-wrate
+
+          Lt=u(nx)*dux(2,nx)                           ! Ly = u*dYr/dx
+          f(nx,2)=wrate
          ENDIF
         ENDDO
-       ENDDO
-!! y-direction
-!       DO i=1,nx
-!        DO j=1,ny
-!         uy(1,j)=v(i,j,k)*r1(i,j,k)                     ! v*Y
-!         uy(2,j)=v(i,j,k)                               ! v
-!         uy(3,j)=r1(i,j,k)                              ! Y
-!        ENDDO
-!        CALL dfyz3(ny,hy,uy(1:3,:),duy)
-!        CALL d2fyz1(ny,hy,uy(3:3,:),d2uy)
-!        DO j=1,ny
-!        f(i,j,k)=f(i,j,k) + &
-!        -0.5*( duy(1,j) + uy(2,j)*duy(3,j) + uy(3,j)*duy(2,j) ) &
-!                   + diff*d2uy(j)
-!        ENDDO
-!       ENDDO
-      ENDDO
-!      f(1,1,1)=1.0
-!      f(nx,1,1)=-0.5*( dux(1,nx) + ux(2,nx)*dux(3,nx) + ux(3,nx)*dux(2,nx) )
-!      f(nx,1,1)=0.0
-!! z-direction
-!      DO j=1,ny
-!       DO i=1,nx
-!        DO k=1,nz
-!         uy(1,k)=w(i,j,k)*r1(i,j,k)                     ! w*Y
-!         uy(2,k)=w(i,j,k)                               ! w
-!         uy(3,k)=r1(i,j,k)                              ! Y
-!        ENDDO
-!        CALL dfyz3(ny,hy,uy(1:3,:),duy)
-!        CALL d2fyz1(ny,hy,uy(3:3,:),d2uy)
-!        DO k=1,nz
-!        f(i,j,k)=f(i,j,k) + &
-!        -0.5*( duy(1,k) + uy(2,k)*duy(3,k) + uy(3,k)*duy(2,k) ) &
-!                        + diff*d2uy(k)
-!        ENDDO
-!       ENDDO
-!      ENDDO
-
-!      DO j=1,ny
-!       DO k=1,nz
-!         f(1,j,k)=0.0
-!         f(nx,j,k)=-0.5*(dux(1,nx)+ux(2,nx)*dux(3,nx)+ux(3,nx)*&
-!           dux(2,nx))
-!       ENDDO
-!      ENDDO
 
 !    Boundary conditionS
-      f(1,1,1)=0.
+      f(1,1)=0.
 
       Dy=Ly
-      f(nx,1,1)=f(nx,1,1)-Dy
+      f(nx,1)=f(nx,1)-Dy
+
+
+      f(1,2)=0.
+
+      Dt=Lt
+      f(nx,2)=f(nx,2)-Dt
 
      END SUBROUTINE fns
 !------------------------------------------------------------------------