Actual source code: ex2f.F

petsc-dev 2014-07-25
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  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
 83:       PetscReal zero

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

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

 91: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 92: !                 Beginning of program
 93: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

100: !  Initialize problem parameters

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

117: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
118: !  Create vector data structures
119: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

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

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

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

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

140: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
141: !  Create timestepping solver context; set callback routine for
142: !  right-hand-side function evaluation.
143: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

150: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
151: !  Set optional user-defined monitoring routine
152: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -


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

160: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
161: !  For nonlinear problems, the user can provide a Jacobian evaluation
162: !  routine (or use a finite differencing approximation).
163: !
164: !  Create matrix data structure; set Jacobian evaluation routine.
165: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

173: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
174: !  Set solution vector and initial timestep
175: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

177:       dt = h/two_d0
178:       zero = 0.0
179:       call TSSetInitialTimeStep(ts,zero,dt,ierr)

181: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
182: !  Customize timestepping solver:
183: !     - Set the solution method to be the Backward Euler method.
184: !     - Set timestepping duration info
185: !  Then set runtime options, which can override these defaults.
186: !  For example,
187: !     -ts_max_steps <maxsteps> -ts_final_time <maxtime>
188: !  to override the defaults set by TSSetDuration().
189: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

196: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
197: !  Solve the problem
198: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

200: !  Evaluate initial conditions

202:       call InitialConditions(u)

204: !  Run the timestepping solver

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

211:       write(6,100) steps,ftime,norm
212:  100  format('Number of pseudo time-steps ',i5,' final time ',1pe9.2,   &
213:      & ' solution norm ',1pe9.2)

215: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
216: !  Free work space.  All PETSc objects should be destroyed when they
217: !  are no longer needed.
218: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

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

233:       call PetscFinalize(ierr)
234:       end

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

251: !  Input/output parameters:
252:       Vec     u

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

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

264:       call VecGetOwnershipRange(u,mybase,myend,ierr)
265:       mybase = mybase + 1

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

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

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

281:       do 10, i=mybase,myend
282: !       x - current location in global grid
283:         x = h*(i-1)
284:         localptr(i-mybase+idx+1) = one_d0 + x*x
285:  10   continue

287: !  Restore vector

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

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

297:       return
298:       end

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

317: !  Input/output parameters:
318:       PetscReal  t
319:       PetscScalar  x
320:       PetscErrorCode          ierr

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

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

331:       call VecGetOwnershipRange(solution,mybase,myend,ierr)
332:       mybase = mybase + 1

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

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

340:       do 10, i=mybase,myend
341: !       x - current location in global grid
342:         x = h*(i-1)
343:         localptr(i-mybase+idx+1) = (t + one_d0)*(one_d0 + x*x)
344:  10   continue

346: !  Restore vector

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

350:       return
351:       end

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

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

380: !  Local variables:
381:       PetscScalar en2,en2s,enmax
382:       PetscScalar mone
383:       PetscDraw   draw

385:       mone = -1.0d0

387: !  We use the default X windows viewer
388: !       PETSC_VIEWER_DRAW_WORLD
389: !  that is associated with the PETSC_COMM_WORLD communicator. This
390: !  saves the effort of calling PetscViewerDrawOpen() to create the window.
391: !  Note that if we wished to plot several items in separate windows we
392: !  would create each viewer with PetscViewerDrawOpen() and store them in
393: !  the application context, appctx.
394: !
395: !  Double buffering makes graphics look better.

397:       call PetscViewerDrawGetDraw(PETSC_VIEWER_DRAW_WORLD,0,draw,ierr)
398:       call PetscDrawSetDoubleBuffer(draw,ierr)
399:       call VecView(u,PETSC_VIEWER_DRAW_WORLD,ierr)

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

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

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

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

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

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

433:       return
434:       end

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

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

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

469: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
470: !  Get ready for local function computations
471: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

478:       call DMGlobalToLocalBegin(da,global_in,INSERT_VALUES,u_local,ierr)
479:       call DMGlobalToLocalEnd(da,global_in,INSERT_VALUES,u_local,ierr)

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

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

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

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

492: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
493: !  Compute entries for the locally owned part
494: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

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

507: !  Handle the interior nodes where the PDE is replace by finite
508: !  difference operators.

510:       do 10, i=2,localsize-1
511:          il = i + idx_l
512:          copyptr(i+idx_c) =  localptr(il) * sc *                        &
513:      &     (localptr(il+1) + localptr(il-1) - two_d0*localptr(il))
514:  10   continue

516:       call VecRestoreArray(u_local,localptr,idx_l,ierr)
517:       call VecRestoreArray(localwork,copyptr,idx_c,ierr)

519: !  Return the values from our local OUTPUT vector into our global
520: !  output vector.

522:       call DMLocalToGlobalBegin(da,localwork,INSERT_VALUES,global_out   &
523:      &                          ,ierr)
524:       call DMLocalToGlobalEnd(da,localwork,INSERT_VALUES,global_out,    &
525:      &                          ierr)

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

533:       return
534:       end

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

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

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

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

585: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
586: !  Get ready for local Jacobian computations
587: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

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

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

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

605: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
606: !  Compute entries for the locally owned part of the Jacobian.
607: !   - Currently, all PETSc parallel matrix formats are partitioned by
608: !     contiguous chunks of rows across the processors.
609: !   - Each processor needs to insert only elements that it owns
610: !     locally (but any non-local elements will be sent to the
611: !     appropriate processor during matrix assembly).
612: !   - Here, we set all entries for a particular row at once.
613: !   - We can set matrix entries either using either
614: !     MatSetValuesLocal() or MatSetValues().
615: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

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

634:       sc = one_d0/(h*h*two_d0*(one_d0+t)*(one_d0+t))
635:       do 10, i=mstart(1),mend(1)-1
636:          col(1) = i-1
637:          col(2) = i
638:          col(3) = i+1
639:          is     = i - mstart(1) + 1 +idx + 1
640:          v(1)   = sc*localptr(is)
641:          v(2)   = sc*(localptr(is+1) + localptr(is-1) -                 &
642:      &                four_d0*localptr(is))
643:          v(3)   = sc*localptr(is)
644:         call MatSetValues(A,i1,i,i3,col,v,INSERT_VALUES,ierr)
645:  10   continue

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

650: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
651: !  Complete the matrix assembly process and set some options
652: ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

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


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

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

668:       return
669:       end