Actual source code: ex2f.F

petsc-3.4.4 2014-03-13
  2: !
  3: !/*T
  4: !   Concepts: TS^time-dependent nonlinear problems
  5: !   Processors: n
  6: !T*/
  7: !
  8: !  ------------------------------------------------------------------------
  9: !
 10: !   This program solves a simple time-dependent nonlinear PDE using implicit
 11: !   timestepping:
 12: !                                    u * u_xx
 13: !                              u_t = ---------
 14: !                                    2*(t+1)^2
 15: !
 16: !             u(0,x) = 1 + x*x; u(t,0) = t + 1; u(t,1) = 2*t + 2
 17: !
 18: !   The exact solution is u(t,x) = (1 + x*x) * (1 + t).
 19: !
 20: !   Note that since the solution is linear in time and quadratic in x,
 21: !   the finite difference scheme actually computes the "exact" solution.
 22: !
 23: !   We use the backward Euler method.
 24: !
 25: !  --------------------------------------------------------------------------

 27:       program main
 28:       implicit none

 30: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 31: !                    Include files
 32: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 33: !
 34: !  Each routine within this program uses the include file 'ex2f.h',
 35: !  which itself includes the various PETSc include files as well as
 36: !  problem-specific data in several common blocks.
 37: !
 38: !  This program uses CPP for preprocessing, as indicated by the use of
 39: !  PETSc include files in the directory petsc/include/finclude.  This
 40: !  convention enables use of the CPP preprocessor, which allows the use
 41: !  of the #include statements that define PETSc objects and variables.
 42: !
 43: !  Use of the conventional Fortran include statements is also supported
 44: !  In this case, the PETsc include files are located in the directory
 45: !  petsc/include/foldinclude.
 46: !
 47: !  Since one must be very careful to include each file no more than once
 48: !  in a Fortran routine, application programmers must exlicitly list
 49: !  each file needed for the various PETSc components within their
 50: !  program (unlike the C/C++ interface).
 51: !
 52: !  See the Fortran section of the PETSc users manual for details.

 54: #include "ex2f.h"

 56: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 57: !                   Variable declarations
 58: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 59: !
 60: !  Variables:
 61: !     ts             - timestepping solver
 62: !     A              - Jacobian matrix context
 63: !     u_local        - local vector
 64: !     u              - approximate solution vector
 65: !     ftime          - final time
 66: !     time_total_max - default total max time
 67: !     time_steps_max - default max timesteps
 68: !
 69: !  Note that vectors are declared as PETSc "Vec" objects.  These vectors
 70: !  are mathematical objects that contain more than just an array of
 71: !  double precision numbers. I.e., vectors in PETSc are not just
 72: !        double precision x(*).
 73: !  However, local vector data can be easily accessed via VecGetArray().
 74: !  See the Fortran section of the PETSc users manual for details.

 76:       TS               ts
 77:       Vec              u
 78:       Mat              A
 79:       PetscBool        flg,fdjac,monitor
 80:       PetscErrorCode ierr
 81:       PetscInt  time_steps_max,i1,steps
 82:       PetscReal dt,time_total_max,ftime,norm

 84: !  Note: Any user-defined Fortran routines (such as RHSFunction)
 85: !  MUST be declared as external.

 87:       external MyMonitor,RHSFunction
 88:       external InitialConditions,RHSJacobian

 90: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 91: !                 Beginning of program
 92: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

 94:       call PetscInitialize(PETSC_NULL_CHARACTER,ierr)
 95:       comm = PETSC_COMM_WORLD
 96:       call MPI_Comm_size(comm,size,ierr)
 97:       call MPI_Comm_rank(comm,rank,ierr)

 99: !  Initialize problem parameters

101:       time_steps_max = 1000
102:       time_total_max = 100.0
103:       m              = 60
104:       debug          = PETSC_FALSE
105:       monitor        = PETSC_FALSE
106:       call PetscOptionsGetInt(PETSC_NULL_CHARACTER,'-m',M,flg,ierr)
107:       call PetscOptionsGetBool(PETSC_NULL_CHARACTER,'-debug',debug,flg, &
108:      &     ierr)
109:       call PetscOptionsGetBool(PETSC_NULL_CHARACTER,'-monitor',monitor, &
110:      &                         flg,ierr)
111:       one_d0  = 1.0d0
112:       two_d0  = 2.0d0
113:       four_d0 = 4.0d0
114:       h       = one_d0/(m-one_d0)

116: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
117: !  Create vector data structures
118: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

120: !  Create distributed array (DMDA) to manage parallel grid and vectors
121: !  Set up the ghost point communication pattern.  There are m total
122: !  grid values spread equally among all the processors.
123:       i1 = 1
124:       call DMDACreate1d(PETSC_COMM_WORLD,DMDA_BOUNDARY_NONE,m,i1,i1,    &
125:      &     PETSC_NULL_INTEGER,da,ierr)

127: !  Extract global and local vectors from DMDA; then duplicate for remaining
128: !  vectors that are the same types.

130:       call DMCreateGlobalVector(da,u,ierr)
131:       call DMCreateLocalVector(da,u_local,ierr)

133: !  Make local work vector for evaluating right-hand-side function
134:       call VecDuplicate(u_local,localwork,ierr)

136: !  Make global work vector for storing exact solution
137:       call VecDuplicate(u,solution,ierr)

139: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
140: !  Create timestepping solver context; set callback routine for
141: !  right-hand-side function evaluation.
142: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

144:       call TSCreate(comm,ts,ierr)
145:       call TSSetProblemType(ts,TS_NONLINEAR,ierr)
146:       call TSSetRHSFunction(ts,PETSC_NULL_OBJECT,RHSFunction,           &
147:      &                      PETSC_NULL_OBJECT,ierr)

149: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
150: !  Set optional user-defined monitoring routine
151: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -


154:       if (monitor) then
155:         call TSMonitorSet(ts,MyMonitor,PETSC_NULL_OBJECT,               &
156:      &                    PETSC_NULL_FUNCTION,ierr)
157:       endif

159: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
160: !  For nonlinear problems, the user can provide a Jacobian evaluation
161: !  routine (or use a finite differencing approximation).
162: !
163: !  Create matrix data structure; set Jacobian evaluation routine.
164: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

166:       call MatCreate(comm,A,ierr)
167:       call MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m,ierr)
168:       call MatSetFromOptions(A,ierr)
169:       call MatSetUp(A,ierr)
170:       call TSSetRHSJacobian(ts,A,A,RHSJacobian,PETSC_NULL_OBJECT,ierr)

172: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
173: !  Set solution vector and initial timestep
174: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

176:       dt = h/two_d0
177:       call TSSetInitialTimeStep(ts,0.d0,dt,ierr)

179: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
180: !  Customize timestepping solver:
181: !     - Set the solution method to be the Backward Euler method.
182: !     - Set timestepping duration info
183: !  Then set runtime options, which can override these defaults.
184: !  For example,
185: !     -ts_max_steps <maxsteps> -ts_final_time <maxtime>
186: !  to override the defaults set by TSSetDuration().
187: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

189:       call TSSetType(ts,TSTHETA,ierr)
190:       call TSThetaSetTheta(ts,one_d0,ierr)
191:       call TSSetDuration(ts,time_steps_max,time_total_max,ierr)
192:       call TSSetFromOptions(ts,ierr)

194: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
195: !  Solve the problem
196: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

198: !  Evaluate initial conditions

200:       call InitialConditions(u)

202: !  Run the timestepping solver

204:       call TSSolve(ts,u,ierr)
205:       call TSGetSolveTime(ts,ftime,ierr)
206:       call TSGetTimeStepNumber(ts,steps,ierr)
207:       call VecNorm(u,NORM_2,norm,ierr)

209:       write(6,100) steps,ftime,norm
210:  100  format('Number of pseudo time-steps ',i5,' final time ',1pe9.2,   &
211:      & ' solution norm ',1pe9.2)

213: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
214: !  Free work space.  All PETSc objects should be destroyed when they
215: !  are no longer needed.
216: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

218:       call TSDestroy(ts,ierr)
219:       call VecDestroy(localwork,ierr)
220:       call VecDestroy(solution,ierr)
221:       call VecDestroy(u_local,ierr)
222:       call VecDestroy(u,ierr)
223:       call DMDestroy(da,ierr)
224:       call MatDestroy(A,ierr)

226: !  Always call PetscFinalize() before exiting a program.  This routine
227: !    - finalizes the PETSc libraries as well as MPI
228: !    - provides summary and diagnostic information if certain runtime
229: !      options are chosen (e.g., -log_summary).

231:       call PetscFinalize(ierr)
232:       end

234: !  ------------------------------------------------------------------------
235: !
236: !  InitialConditions - Computes the solution at the initial time.
237: !
238: !  Input Parameters:
239: !     u - uninitialized solution vector (global)
240: !     appctx - user-defined application context
241: !
242: !  Output Parameter:
243: !     u - vector with solution at initial time (global)
244: !
245:       subroutine InitialConditions(u)
246:       implicit none
247: #include "ex2f.h"

249: !  Input/output parameters:
250:       Vec     u

252: !  Local variables:
253:       PetscScalar localptr(1),x
254:       PetscInt    i,mybase,myend
255:       PetscErrorCode ierr
256:       PetscOffset idx

258: !  Determine starting and ending point of each processor's range of
259: !  grid values.  Note that we shift by 1 to convert from the 0-based
260: !  C convention of starting indices to the 1-based Fortran convention.

262:       call VecGetOwnershipRange(u,mybase,myend,ierr)
263:       mybase = mybase + 1

265: !  Get a pointer to vector data.
266: !    - For default PETSc vectors, VecGetArray() returns a pointer to
267: !      the data array.  Otherwise, the routine is implementation dependent.
268: !    - You MUST call VecRestoreArray() when you no longer need access to
269: !      the array.
270: !    - Note that the Fortran interface to VecGetArray() differs from the
271: !      C version.  See the users manual for details.

273:       call VecGetArray(u,localptr,idx,ierr)

275: !     We initialize the solution array by simply writing the solution
276: !     directly into the array locations.  Alternatively, we could use
277: !     VecSetValues() or VecSetValuesLocal().

279:       do 10, i=mybase,myend
280: !       x - current location in global grid
281:         x = h*(i-1)
282:         localptr(i-mybase+idx+1) = one_d0 + x*x
283:  10   continue

285: !  Restore vector

287:       call VecRestoreArray(u,localptr,idx,ierr)

289: !  Print debugging information if desired
290:       if (debug) then
291:         if (rank .eq. 0) write(6,*) 'initial guess vector'
292:         call VecView(u,PETSC_VIEWER_STDOUT_WORLD,ierr)
293:       endif

295:       return
296:       end

298: !  ------------------------------------------------------------------------
299: !
300: !  ExactSolution - Computes the exact solution at a given time.
301: !
302: !  Input Parameters:
303: !    t - current time
304: !    appctx - user-defined application context
305: !
306: !  Output Parameter:
307: !    ierr - error code
308: !
309: !  Note: The solution vector is stored in the common block!
310: !
311:       subroutine ExactSolution(t,ierr)
312:       implicit none
313: #include "ex2f.h"

315: !  Input/output parameters:
316:       PetscReal  t
317:       PetscScalar  x
318:       PetscErrorCode          ierr

320: !  Local variables:
321:       PetscScalar localptr(1)
322:       PetscInt    i,mybase,myend
323:       PetscOffset idx

325: !  Determine starting and ending point of each processors range of
326: !  grid values.  Note that we shift by 1 to convert from the 0-based
327: !  C convention of starting indices to the 1-based Fortran convention.

329:       call VecGetOwnershipRange(solution,mybase,myend,ierr)
330:       mybase = mybase + 1

332: !  Get a pointer to vector data
333:       call VecGetArray(solution,localptr,idx,ierr)

335: !  Simply write the solution directly into the array locations
336: !  Alternatively, we could use VecSetValues() or VecSetValuesLocal().

338:       do 10, i=mybase,myend
339: !       x - current location in global grid
340:         x = h*(i-1)
341:         localptr(i-mybase+idx+1) = (t + one_d0)*(one_d0 + x*x)
342:  10   continue

344: !  Restore vector

346:       call VecRestoreArray(solution,localptr,idx,ierr)

348:       return
349:       end

351: !  ------------------------------------------------------------------------
352: !
353: !   MyMonitor - A user-provided routine to monitor the solution computed at
354: !   each time-step.  This example plots the solution and computes the
355: !   error in two different norms.
356: !
357: !   Input Parameters:
358: !   ts     - the time-step context
359: !   step   - the count of the current step (with 0 meaning the
360: !            initial condition)
361: !   time   - the current time
362: !   u      - the solution at this timestep
363: !   dummy  - optional user-provided context for this monitoring
364: !            routine (not used here)
365: !
366:       subroutine MyMonitor(ts,step,time,u,dummy,ierr)
367:       implicit none
368: #include "ex2f.h"

370: !  Input/output parameters:
371:       TS       ts
372:       PetscInt step
373:       PetscObject dummy
374:       PetscReal time
375:       Vec      u
376:       PetscErrorCode ierr

378: !  Local variables:
379:       PetscScalar en2,en2s,enmax
380:       PetscScalar mone
381:       PetscDraw   draw

383:       mone = -1.0d0

385: !  We use the default X windows viewer
386: !       PETSC_VIEWER_DRAW_WORLD
387: !  that is associated with the PETSC_COMM_WORLD communicator. This
388: !  saves the effort of calling PetscViewerDrawOpen() to create the window.
389: !  Note that if we wished to plot several items in separate windows we
390: !  would create each viewer with PetscViewerDrawOpen() and store them in
391: !  the application context, appctx.
392: !
393: !  Double buffering makes graphics look better.

395:       call PetscViewerDrawGetDraw(PETSC_VIEWER_DRAW_WORLD,0,draw,ierr)
396:       call PetscDrawSetDoubleBuffer(draw,ierr)
397:       call VecView(u,PETSC_VIEWER_DRAW_WORLD,ierr)

399: !  Compute the exact solution at this time-step.  Note that the
400: !  solution vector is passed via the user-defined common block.
401:       call ExactSolution(time,ierr)

403: !  Print debugging information if desired
404:       if (debug) then
405:         if (rank .eq. 0) write(6,*) 'Computed solution vector'
406:         call VecView(u,PETSC_VIEWER_STDOUT_WORLD,ierr)
407:         if (rank .eq. 0) write(6,*) 'Exact solution vector'
408:         call VecView(solution,PETSC_VIEWER_STDOUT_WORLD,ierr)
409:       endif

411: !  Compute the 2-norm and max-norm of the error
412:       call VecAXPY(solution,mone,u,ierr)
413:       call VecNorm(solution,NORM_2,en2,ierr)

415: !  Scale the 2-norm by the grid spacing
416:       en2s = sqrt(dble(h))*en2
417:       call VecNorm(solution,NORM_MAX,enmax,ierr)

419: !  Print only from processor 0
420:       if (rank .eq. 0) write(6,100) step,time,en2s,enmax
421:  100  format('Timestep = ',i5,',time = ',f8.3,                          &
422:      &       ' sec, error [2-norm] = ',e10.3,                            &
423:      &       ', error [max-norm] = ',e10.3)

425: !  Print debugging information if desired
426:       if (debug) then
427:         if (rank .eq. 0) write(6,*) 'Error vector'
428:         call VecView(solution,PETSC_VIEWER_STDOUT_WORLD,ierr)
429:       endif

431:       return
432:       end

434: !  ------------------------------------------------------------------------
435: !
436: !   RHSFunction - User-provided routine that evalues the RHS function
437: !   in the ODE.  This routine is set in the main program by calling
438: !   TSSetRHSFunction().  We compute:
439: !         global_out = F(global_in)
440: !
441: !   Input Parameters:
442: !      ts         - timestep context
443: !      t          - current time
444: !      global_in  - input vector to function
445: !      dummy      - (optional) user-provided context for function evaluation
446: !                   (not used here because we use a common block instead)
447: !
448: !   Output Parameter:
449: !      global_out - value of function
450: !
451:       subroutine RHSFunction(ts,t,global_in,global_out,dummy)
452:       implicit none
453: #include "ex2f.h"

455: !  Input/output parameters:
456:       TS        ts
457:       PetscReal t
458:       Vec       global_in,global_out
459:       integer   dummy

461: !  Local variables:
462:       PetscScalar localptr(1),copyptr(1),sc
463:       PetscErrorCode ierr
464:       PetscInt    i,localsize
465:       PetscOffset idx_c,idx_l,il

467: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
468: !  Get ready for local function computations
469: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

471: !  Scatter ghost points to local vector, using the 2-step process
472: !     DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
473: !  By placing code between these two statements, computations can be
474: !  done while messages are in transition.

476:       call DMGlobalToLocalBegin(da,global_in,INSERT_VALUES,u_local,ierr)
477:       call DMGlobalToLocalEnd(da,global_in,INSERT_VALUES,u_local,ierr)

479: !  Access directly the values in our local INPUT work vector
480:       call VecGetArray(u_local,localptr,idx_l,ierr)

482: !  Access directly the values in our local OUTPUT work vector
483:       call VecGetArray(localwork,copyptr,idx_c,ierr)

485:       sc = one_d0/(h*h*two_d0*(one_d0+t)*(one_d0+t))

487: !  Evaluate our function on the nodes owned by this processor
488:       call VecGetLocalSize(u_local,localsize,ierr)

490: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
491: !  Compute entries for the locally owned part
492: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

494: !  Handle boundary conditions: This is done by using the boundary condition
495: !        u(t,boundary) = g(t,boundary)
496: !  for some function g. Now take the derivative with respect to t to obtain
497: !        u_{t}(t,boundary) = g_{t}(t,boundary)
498: !
499: !  In our case, u(t,0) = t + 1, so that u_{t}(t,0) = 1
500: !          and  u(t,1) = 2t+ 1, so that u_{t}(t,1) = 2

502:        if (rank .eq. 0)      copyptr(1+idx_c) = one_d0
503:        if (rank .eq. size-1) copyptr(localsize+idx_c) = two_d0

505: !  Handle the interior nodes where the PDE is replace by finite
506: !  difference operators.

508:       do 10, i=2,localsize-1
509:          il = i + idx_l
510:          copyptr(i+idx_c) =  localptr(il) * sc *                        &
511:      &     (localptr(il+1) + localptr(il-1) - two_d0*localptr(il))
512:  10   continue

514:       call VecRestoreArray(u_local,localptr,idx_l,ierr)
515:       call VecRestoreArray(localwork,copyptr,idx_c,ierr)

517: !  Return the values from our local OUTPUT vector into our global
518: !  output vector.

520:       call DMLocalToGlobalBegin(da,localwork,INSERT_VALUES,global_out   &
521:      &                          ,ierr)
522:       call DMLocalToGlobalEnd(da,localwork,INSERT_VALUES,global_out,    &
523:      &                          ierr)

525: !  Print debugging information if desired
526:       if (debug) then
527:         if (rank .eq. 0) write(6,*) 'RHS function vector'
528:         call VecView(global_out,PETSC_VIEWER_STDOUT_WORLD,ierr)
529:       endif

531:       return
532:       end

534: !  ------------------------------------------------------------------------
535: !
536: !  RHSJacobian - User-provided routine to compute the Jacobian of the
537: !                right-hand-side function.
538: !
539: !  Input Parameters:
540: !     ts - the TS context
541: !     t - current time
542: !     global_in - global input vector
543: !     dummy - optional user-defined context, as set by TSetRHSJacobian()
544: !
545: !  Output Parameters:
546: !     A - Jacobian matrix
547: !     B - optionally different preconditioning matrix
548: !     str - flag indicating matrix structure
549: !
550: !  Notes:
551: !  RHSJacobian computes entries for the locally owned part of the Jacobian.
552: !   - Currently, all PETSc parallel matrix formats are partitioned by
553: !     contiguous chunks of rows across the processors. The "grow"
554: !     parameter computed below specifies the global row number
555: !     corresponding to each local grid point.
556: !   - Each processor needs to insert only elements that it owns
557: !     locally (but any non-local elements will be sent to the
558: !     appropriate processor during matrix assembly).
559: !   - Always specify global row and columns of matrix entries.
560: !   - Here, we set all entries for a particular row at once.
561: !   - Note that MatSetValues() uses 0-based row and column numbers
562: !     in Fortran as well as in C.

564:       subroutine RHSJacobian(ts,t,global_in,A,B,str,dummy)
565:       implicit none
566: #include "ex2f.h"

568: !  Input/output parameters:
569:       TS               ts
570:       PetscReal t
571:       Vec              global_in
572:       Mat              A,B
573:       MatStructure     str
574:       integer          dummy

576: !  Local variables:
577:       PetscScalar localptr(1),sc,v(3)
578:       PetscInt    i,col(3),i1,i3
579:       PetscErrorCode ierr
580:       PetscInt    mstart(1),mend(1)
581:       PetscInt    mstarts,mends
582:       PetscOffset idx,is

584: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
585: !  Get ready for local Jacobian computations
586: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

588: !  Scatter ghost points to local vector, using the 2-step process
589: !     DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
590: !  By placing code between these two statements, computations can be
591: !  done while messages are in transition.

593:       call DMGlobalToLocalBegin(da,global_in,INSERT_VALUES,u_local,ierr)
594:       call DMGlobalToLocalEnd(da,global_in,INSERT_VALUES,u_local,ierr)

596: !  Get pointer to vector data
597:       call VecGetArray(u_local,localptr,idx,ierr)

599: !  Get starting and ending locally owned rows of the matrix
600:       call MatGetOwnershipRange(A,mstarts,mends,ierr)
601:       mstart(1) = mstarts
602:       mend(1)   = mends

604: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
605: !  Compute entries for the locally owned part of the Jacobian.
606: !   - Currently, all PETSc parallel matrix formats are partitioned by
607: !     contiguous chunks of rows across the processors.
608: !   - Each processor needs to insert only elements that it owns
609: !     locally (but any non-local elements will be sent to the
610: !     appropriate processor during matrix assembly).
611: !   - Here, we set all entries for a particular row at once.
612: !   - We can set matrix entries either using either
613: !     MatSetValuesLocal() or MatSetValues().
614: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

616: !  Set matrix rows corresponding to boundary data
617:       i1 = 1
618:       i3 = 3
619:       if (mstart(1) .eq. 0) then
620:         v(1) = zero_d0
621:         call MatSetValues(A,i1,mstart,i1,mstart,v,INSERT_VALUES,ierr)
622:         mstart(1) = mstart(1) + 1
623:       endif
624:       if (mend(1) .eq. m) then
625:         mend(1) = mend(1) - 1
626:         v(1) = zero_d0
627:         call MatSetValues(A,i1,mend,i1,mend,v,INSERT_VALUES,ierr)
628:       endif

630: !  Set matrix rows corresponding to interior data.
631: !  We construct the matrix one row at a time.

633:       sc = one_d0/(h*h*two_d0*(one_d0+t)*(one_d0+t))
634:       do 10, i=mstart(1),mend(1)-1
635:          col(1) = i-1
636:          col(2) = i
637:          col(3) = i+1
638:          is     = i - mstart(1) + 1 +idx + 1
639:          v(1)   = sc*localptr(is)
640:          v(2)   = sc*(localptr(is+1) + localptr(is-1) -                 &
641:      &                four_d0*localptr(is))
642:          v(3)   = sc*localptr(is)
643:         call MatSetValues(A,i1,i,i3,col,v,INSERT_VALUES,ierr)
644:  10   continue

646: !  Restore vector
647:       call VecRestoreArray(u_local,localptr,idx,ierr)

649: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
650: !  Complete the matrix assembly process and set some options
651: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

653: !  Assemble matrix, using the 2-step process:
654: !       MatAssemblyBegin(), MatAssemblyEnd()
655: !  Computations can be done while messages are in transition,
656: !  by placing code between these two statements.

658:       call MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY,ierr)
659:       call MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY,ierr)

661: !  Set flag to indicate that the Jacobian matrix retains an identical
662: !  nonzero structure throughout all timestepping iterations (although the
663: !  values of the entries change). Thus, we can save some work in setting
664: !  up the preconditioner (e.g., no need to redo symbolic factorization for
665: !  ILU/ICC preconditioners).
666: !   - If the nonzero structure of the matrix is different during
667: !     successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
668: !     must be used instead.  If you are unsure whether the matrix
669: !     structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
670: !   - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc
671: !     believes your assertion and does not check the structure
672: !     of the matrix.  If you erroneously claim that the structure
673: !     is the same when it actually is not, the new preconditioner
674: !     will not function correctly.  Thus, use this optimization
675: !     feature with caution!

677:       str = SAME_NONZERO_PATTERN

679: !  Set and option to indicate that we will never add a new nonzero location
680: !  to the matrix. If we do, it will generate an error.

682:       call MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE,ierr)

684:       return
685:       end