Actual source code: ex2.c


  2: static char help[] = "Solves a time-dependent nonlinear PDE. Uses implicit\n\
  3: timestepping.  Runtime options include:\n\
  4:   -M <xg>, where <xg> = number of grid points\n\
  5:   -debug : Activate debugging printouts\n\
  6:   -nox   : Deactivate x-window graphics\n\n";

  8: /* ------------------------------------------------------------------------

 10:    This program solves the PDE

 12:                u * u_xx
 13:          u_t = ---------
 14:                2*(t+1)^2

 16:     on the domain 0 <= x <= 1, with boundary conditions
 17:          u(t,0) = t + 1,  u(t,1) = 2*t + 2,
 18:     and initial condition
 19:          u(0,x) = 1 + x*x.

 21:     The exact solution is:
 22:          u(t,x) = (1 + x*x) * (1 + t)

 24:     Note that since the solution is linear in time and quadratic in x,
 25:     the finite difference scheme actually computes the "exact" solution.

 27:     We use by default the backward Euler method.

 29:   ------------------------------------------------------------------------- */

 31: /*
 32:    Include "petscts.h" to use the PETSc timestepping routines. Note that
 33:    this file automatically includes "petscsys.h" and other lower-level
 34:    PETSc include files.

 36:    Include the "petscdmda.h" to allow us to use the distributed array data
 37:    structures to manage the parallel grid.
 38: */
 39: #include <petscts.h>
 40: #include <petscdm.h>
 41: #include <petscdmda.h>
 42: #include <petscdraw.h>

 44: /*
 45:    User-defined application context - contains data needed by the
 46:    application-provided callback routines.
 47: */
 48: typedef struct {
 49:   MPI_Comm  comm;      /* communicator */
 50:   DM        da;        /* distributed array data structure */
 51:   Vec       localwork; /* local ghosted work vector */
 52:   Vec       u_local;   /* local ghosted approximate solution vector */
 53:   Vec       solution;  /* global exact solution vector */
 54:   PetscInt  m;         /* total number of grid points */
 55:   PetscReal h;         /* mesh width: h = 1/(m-1) */
 56:   PetscBool debug;     /* flag (1 indicates activation of debugging printouts) */
 57: } AppCtx;

 59: /*
 60:    User-defined routines, provided below.
 61: */
 62: extern PetscErrorCode InitialConditions(Vec, AppCtx *);
 63: extern PetscErrorCode RHSFunction(TS, PetscReal, Vec, Vec, void *);
 64: extern PetscErrorCode RHSJacobian(TS, PetscReal, Vec, Mat, Mat, void *);
 65: extern PetscErrorCode Monitor(TS, PetscInt, PetscReal, Vec, void *);
 66: extern PetscErrorCode ExactSolution(PetscReal, Vec, AppCtx *);

 68: int main(int argc, char **argv)
 69: {
 70:   AppCtx    appctx;               /* user-defined application context */
 71:   TS        ts;                   /* timestepping context */
 72:   Mat       A;                    /* Jacobian matrix data structure */
 73:   Vec       u;                    /* approximate solution vector */
 74:   PetscInt  time_steps_max = 100; /* default max timesteps */
 75:   PetscReal dt;
 76:   PetscReal time_total_max = 100.0; /* default max total time */
 77:   PetscBool mymonitor      = PETSC_FALSE;
 78:   PetscReal bounds[]       = {1.0, 3.3};

 80:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 81:      Initialize program and set problem parameters
 82:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

 84:   PetscFunctionBeginUser;
 85:   PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));
 86:   PetscCall(PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 1, bounds));

 88:   appctx.comm = PETSC_COMM_WORLD;
 89:   appctx.m    = 60;

 91:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-M", &appctx.m, NULL));
 92:   PetscCall(PetscOptionsHasName(NULL, NULL, "-debug", &appctx.debug));
 93:   PetscCall(PetscOptionsHasName(NULL, NULL, "-mymonitor", &mymonitor));

 95:   appctx.h = 1.0 / (appctx.m - 1.0);

 97:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 98:      Create vector data structures
 99:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

101:   /*
102:      Create distributed array (DMDA) to manage parallel grid and vectors
103:      and to set up the ghost point communication pattern.  There are M
104:      total grid values spread equally among all the processors.
105:   */
106:   PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, appctx.m, 1, 1, NULL, &appctx.da));
107:   PetscCall(DMSetFromOptions(appctx.da));
108:   PetscCall(DMSetUp(appctx.da));

110:   /*
111:      Extract global and local vectors from DMDA; we use these to store the
112:      approximate solution.  Then duplicate these for remaining vectors that
113:      have the same types.
114:   */
115:   PetscCall(DMCreateGlobalVector(appctx.da, &u));
116:   PetscCall(DMCreateLocalVector(appctx.da, &appctx.u_local));

118:   /*
119:      Create local work vector for use in evaluating right-hand-side function;
120:      create global work vector for storing exact solution.
121:   */
122:   PetscCall(VecDuplicate(appctx.u_local, &appctx.localwork));
123:   PetscCall(VecDuplicate(u, &appctx.solution));

125:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
126:      Create timestepping solver context; set callback routine for
127:      right-hand-side function evaluation.
128:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

130:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
131:   PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
132:   PetscCall(TSSetRHSFunction(ts, NULL, RHSFunction, &appctx));

134:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
135:      Set optional user-defined monitoring routine
136:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

138:   if (mymonitor) PetscCall(TSMonitorSet(ts, Monitor, &appctx, NULL));

140:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
141:      For nonlinear problems, the user can provide a Jacobian evaluation
142:      routine (or use a finite differencing approximation).

144:      Create matrix data structure; set Jacobian evaluation routine.
145:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

147:   PetscCall(MatCreate(PETSC_COMM_WORLD, &A));
148:   PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, appctx.m, appctx.m));
149:   PetscCall(MatSetFromOptions(A));
150:   PetscCall(MatSetUp(A));
151:   PetscCall(TSSetRHSJacobian(ts, A, A, RHSJacobian, &appctx));

153:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
154:      Set solution vector and initial timestep
155:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

157:   dt = appctx.h / 2.0;
158:   PetscCall(TSSetTimeStep(ts, dt));

160:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
161:      Customize timestepping solver:
162:        - Set the solution method to be the Backward Euler method.
163:        - Set timestepping duration info
164:      Then set runtime options, which can override these defaults.
165:      For example,
166:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
167:      to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
168:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

170:   PetscCall(TSSetType(ts, TSBEULER));
171:   PetscCall(TSSetMaxSteps(ts, time_steps_max));
172:   PetscCall(TSSetMaxTime(ts, time_total_max));
173:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
174:   PetscCall(TSSetFromOptions(ts));

176:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
177:      Solve the problem
178:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

180:   /*
181:      Evaluate initial conditions
182:   */
183:   PetscCall(InitialConditions(u, &appctx));

185:   /*
186:      Run the timestepping solver
187:   */
188:   PetscCall(TSSolve(ts, u));

190:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
191:      Free work space.  All PETSc objects should be destroyed when they
192:      are no longer needed.
193:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

195:   PetscCall(TSDestroy(&ts));
196:   PetscCall(VecDestroy(&u));
197:   PetscCall(MatDestroy(&A));
198:   PetscCall(DMDestroy(&appctx.da));
199:   PetscCall(VecDestroy(&appctx.localwork));
200:   PetscCall(VecDestroy(&appctx.solution));
201:   PetscCall(VecDestroy(&appctx.u_local));

203:   /*
204:      Always call PetscFinalize() before exiting a program.  This routine
205:        - finalizes the PETSc libraries as well as MPI
206:        - provides summary and diagnostic information if certain runtime
207:          options are chosen (e.g., -log_view).
208:   */
209:   PetscCall(PetscFinalize());
210:   return 0;
211: }
212: /* --------------------------------------------------------------------- */
213: /*
214:    InitialConditions - Computes the solution at the initial time.

216:    Input Parameters:
217:    u - uninitialized solution vector (global)
218:    appctx - user-defined application context

220:    Output Parameter:
221:    u - vector with solution at initial time (global)
222: */
223: PetscErrorCode InitialConditions(Vec u, AppCtx *appctx)
224: {
225:   PetscScalar *u_localptr, h = appctx->h, x;
226:   PetscInt     i, mybase, myend;

228:   PetscFunctionBeginUser;
229:   /*
230:      Determine starting point of each processor's range of
231:      grid values.
232:   */
233:   PetscCall(VecGetOwnershipRange(u, &mybase, &myend));

235:   /*
236:     Get a pointer to vector data.
237:     - For default PETSc vectors, VecGetArray() returns a pointer to
238:       the data array.  Otherwise, the routine is implementation dependent.
239:     - You MUST call VecRestoreArray() when you no longer need access to
240:       the array.
241:     - Note that the Fortran interface to VecGetArray() differs from the
242:       C version.  See the users manual for details.
243:   */
244:   PetscCall(VecGetArray(u, &u_localptr));

246:   /*
247:      We initialize the solution array by simply writing the solution
248:      directly into the array locations.  Alternatively, we could use
249:      VecSetValues() or VecSetValuesLocal().
250:   */
251:   for (i = mybase; i < myend; i++) {
252:     x                      = h * (PetscReal)i; /* current location in global grid */
253:     u_localptr[i - mybase] = 1.0 + x * x;
254:   }

256:   /*
257:      Restore vector
258:   */
259:   PetscCall(VecRestoreArray(u, &u_localptr));

261:   /*
262:      Print debugging information if desired
263:   */
264:   if (appctx->debug) {
265:     PetscCall(PetscPrintf(appctx->comm, "initial guess vector\n"));
266:     PetscCall(VecView(u, PETSC_VIEWER_STDOUT_WORLD));
267:   }

269:   PetscFunctionReturn(PETSC_SUCCESS);
270: }
271: /* --------------------------------------------------------------------- */
272: /*
273:    ExactSolution - Computes the exact solution at a given time.

275:    Input Parameters:
276:    t - current time
277:    solution - vector in which exact solution will be computed
278:    appctx - user-defined application context

280:    Output Parameter:
281:    solution - vector with the newly computed exact solution
282: */
283: PetscErrorCode ExactSolution(PetscReal t, Vec solution, AppCtx *appctx)
284: {
285:   PetscScalar *s_localptr, h = appctx->h, x;
286:   PetscInt     i, mybase, myend;

288:   PetscFunctionBeginUser;
289:   /*
290:      Determine starting and ending points of each processor's
291:      range of grid values
292:   */
293:   PetscCall(VecGetOwnershipRange(solution, &mybase, &myend));

295:   /*
296:      Get a pointer to vector data.
297:   */
298:   PetscCall(VecGetArray(solution, &s_localptr));

300:   /*
301:      Simply write the solution directly into the array locations.
302:      Alternatively, we could use VecSetValues() or VecSetValuesLocal().
303:   */
304:   for (i = mybase; i < myend; i++) {
305:     x                      = h * (PetscReal)i;
306:     s_localptr[i - mybase] = (t + 1.0) * (1.0 + x * x);
307:   }

309:   /*
310:      Restore vector
311:   */
312:   PetscCall(VecRestoreArray(solution, &s_localptr));
313:   PetscFunctionReturn(PETSC_SUCCESS);
314: }
315: /* --------------------------------------------------------------------- */
316: /*
317:    Monitor - User-provided routine to monitor the solution computed at
318:    each timestep.  This example plots the solution and computes the
319:    error in two different norms.

321:    Input Parameters:
322:    ts     - the timestep context
323:    step   - the count of the current step (with 0 meaning the
324:             initial condition)
325:    time   - the current time
326:    u      - the solution at this timestep
327:    ctx    - the user-provided context for this monitoring routine.
328:             In this case we use the application context which contains
329:             information about the problem size, workspace and the exact
330:             solution.
331: */
332: PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal time, Vec u, void *ctx)
333: {
334:   AppCtx   *appctx = (AppCtx *)ctx; /* user-defined application context */
335:   PetscReal en2, en2s, enmax;
336:   PetscDraw draw;

338:   PetscFunctionBeginUser;
339:   /*
340:      We use the default X Windows viewer
341:              PETSC_VIEWER_DRAW_(appctx->comm)
342:      that is associated with the current communicator. This saves
343:      the effort of calling PetscViewerDrawOpen() to create the window.
344:      Note that if we wished to plot several items in separate windows we
345:      would create each viewer with PetscViewerDrawOpen() and store them in
346:      the application context, appctx.

348:      PetscReal buffering makes graphics look better.
349:   */
350:   PetscCall(PetscViewerDrawGetDraw(PETSC_VIEWER_DRAW_(appctx->comm), 0, &draw));
351:   PetscCall(PetscDrawSetDoubleBuffer(draw));
352:   PetscCall(VecView(u, PETSC_VIEWER_DRAW_(appctx->comm)));

354:   /*
355:      Compute the exact solution at this timestep
356:   */
357:   PetscCall(ExactSolution(time, appctx->solution, appctx));

359:   /*
360:      Print debugging information if desired
361:   */
362:   if (appctx->debug) {
363:     PetscCall(PetscPrintf(appctx->comm, "Computed solution vector\n"));
364:     PetscCall(VecView(u, PETSC_VIEWER_STDOUT_WORLD));
365:     PetscCall(PetscPrintf(appctx->comm, "Exact solution vector\n"));
366:     PetscCall(VecView(appctx->solution, PETSC_VIEWER_STDOUT_WORLD));
367:   }

369:   /*
370:      Compute the 2-norm and max-norm of the error
371:   */
372:   PetscCall(VecAXPY(appctx->solution, -1.0, u));
373:   PetscCall(VecNorm(appctx->solution, NORM_2, &en2));
374:   en2s = PetscSqrtReal(appctx->h) * en2; /* scale the 2-norm by the grid spacing */
375:   PetscCall(VecNorm(appctx->solution, NORM_MAX, &enmax));

377:   /*
378:      PetscPrintf() causes only the first processor in this
379:      communicator to print the timestep information.
380:   */
381:   PetscCall(PetscPrintf(appctx->comm, "Timestep %" PetscInt_FMT ": time = %g 2-norm error = %g  max norm error = %g\n", step, (double)time, (double)en2s, (double)enmax));

383:   /*
384:      Print debugging information if desired
385:   */
386:   if (appctx->debug) {
387:     PetscCall(PetscPrintf(appctx->comm, "Error vector\n"));
388:     PetscCall(VecView(appctx->solution, PETSC_VIEWER_STDOUT_WORLD));
389:   }
390:   PetscFunctionReturn(PETSC_SUCCESS);
391: }
392: /* --------------------------------------------------------------------- */
393: /*
394:    RHSFunction - User-provided routine that evalues the right-hand-side
395:    function of the ODE.  This routine is set in the main program by
396:    calling TSSetRHSFunction().  We compute:
397:           global_out = F(global_in)

399:    Input Parameters:
400:    ts         - timesteping context
401:    t          - current time
402:    global_in  - vector containing the current iterate
403:    ctx        - (optional) user-provided context for function evaluation.
404:                 In this case we use the appctx defined above.

406:    Output Parameter:
407:    global_out - vector containing the newly evaluated function
408: */
409: PetscErrorCode RHSFunction(TS ts, PetscReal t, Vec global_in, Vec global_out, void *ctx)
410: {
411:   AppCtx            *appctx    = (AppCtx *)ctx;     /* user-defined application context */
412:   DM                 da        = appctx->da;        /* distributed array */
413:   Vec                local_in  = appctx->u_local;   /* local ghosted input vector */
414:   Vec                localwork = appctx->localwork; /* local ghosted work vector */
415:   PetscInt           i, localsize;
416:   PetscMPIInt        rank, size;
417:   PetscScalar       *copyptr, sc;
418:   const PetscScalar *localptr;

420:   PetscFunctionBeginUser;
421:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
422:      Get ready for local function computations
423:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
424:   /*
425:      Scatter ghost points to local vector, using the 2-step process
426:         DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
427:      By placing code between these two statements, computations can be
428:      done while messages are in transition.
429:   */
430:   PetscCall(DMGlobalToLocalBegin(da, global_in, INSERT_VALUES, local_in));
431:   PetscCall(DMGlobalToLocalEnd(da, global_in, INSERT_VALUES, local_in));

433:   /*
434:       Access directly the values in our local INPUT work array
435:   */
436:   PetscCall(VecGetArrayRead(local_in, &localptr));

438:   /*
439:       Access directly the values in our local OUTPUT work array
440:   */
441:   PetscCall(VecGetArray(localwork, &copyptr));

443:   sc = 1.0 / (appctx->h * appctx->h * 2.0 * (1.0 + t) * (1.0 + t));

445:   /*
446:       Evaluate our function on the nodes owned by this processor
447:   */
448:   PetscCall(VecGetLocalSize(local_in, &localsize));

450:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
451:      Compute entries for the locally owned part
452:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

454:   /*
455:      Handle boundary conditions: This is done by using the boundary condition
456:         u(t,boundary) = g(t,boundary)
457:      for some function g. Now take the derivative with respect to t to obtain
458:         u_{t}(t,boundary) = g_{t}(t,boundary)

460:      In our case, u(t,0) = t + 1, so that u_{t}(t,0) = 1
461:              and  u(t,1) = 2t+ 2, so that u_{t}(t,1) = 2
462:   */
463:   PetscCallMPI(MPI_Comm_rank(appctx->comm, &rank));
464:   PetscCallMPI(MPI_Comm_size(appctx->comm, &size));
465:   if (rank == 0) copyptr[0] = 1.0;
466:   if (rank == size - 1) copyptr[localsize - 1] = 2.0;

468:   /*
469:      Handle the interior nodes where the PDE is replace by finite
470:      difference operators.
471:   */
472:   for (i = 1; i < localsize - 1; i++) copyptr[i] = localptr[i] * sc * (localptr[i + 1] + localptr[i - 1] - 2.0 * localptr[i]);

474:   /*
475:      Restore vectors
476:   */
477:   PetscCall(VecRestoreArrayRead(local_in, &localptr));
478:   PetscCall(VecRestoreArray(localwork, &copyptr));

480:   /*
481:      Insert values from the local OUTPUT vector into the global
482:      output vector
483:   */
484:   PetscCall(DMLocalToGlobalBegin(da, localwork, INSERT_VALUES, global_out));
485:   PetscCall(DMLocalToGlobalEnd(da, localwork, INSERT_VALUES, global_out));

487:   /* Print debugging information if desired */
488:   if (appctx->debug) {
489:     PetscCall(PetscPrintf(appctx->comm, "RHS function vector\n"));
490:     PetscCall(VecView(global_out, PETSC_VIEWER_STDOUT_WORLD));
491:   }

493:   PetscFunctionReturn(PETSC_SUCCESS);
494: }
495: /* --------------------------------------------------------------------- */
496: /*
497:    RHSJacobian - User-provided routine to compute the Jacobian of
498:    the nonlinear right-hand-side function of the ODE.

500:    Input Parameters:
501:    ts - the TS context
502:    t - current time
503:    global_in - global input vector
504:    dummy - optional user-defined context, as set by TSetRHSJacobian()

506:    Output Parameters:
507:    AA - Jacobian matrix
508:    BB - optionally different preconditioning matrix
509:    str - flag indicating matrix structure

511:   Notes:
512:   RHSJacobian computes entries for the locally owned part of the Jacobian.
513:    - Currently, all PETSc parallel matrix formats are partitioned by
514:      contiguous chunks of rows across the processors.
515:    - Each processor needs to insert only elements that it owns
516:      locally (but any non-local elements will be sent to the
517:      appropriate processor during matrix assembly).
518:    - Always specify global row and columns of matrix entries when
519:      using MatSetValues().
520:    - Here, we set all entries for a particular row at once.
521:    - Note that MatSetValues() uses 0-based row and column numbers
522:      in Fortran as well as in C.
523: */
524: PetscErrorCode RHSJacobian(TS ts, PetscReal t, Vec global_in, Mat AA, Mat BB, void *ctx)
525: {
526:   AppCtx            *appctx   = (AppCtx *)ctx;   /* user-defined application context */
527:   Vec                local_in = appctx->u_local; /* local ghosted input vector */
528:   DM                 da       = appctx->da;      /* distributed array */
529:   PetscScalar        v[3], sc;
530:   const PetscScalar *localptr;
531:   PetscInt           i, mstart, mend, mstarts, mends, idx[3], is;

533:   PetscFunctionBeginUser;
534:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
535:      Get ready for local Jacobian computations
536:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
537:   /*
538:      Scatter ghost points to local vector, using the 2-step process
539:         DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
540:      By placing code between these two statements, computations can be
541:      done while messages are in transition.
542:   */
543:   PetscCall(DMGlobalToLocalBegin(da, global_in, INSERT_VALUES, local_in));
544:   PetscCall(DMGlobalToLocalEnd(da, global_in, INSERT_VALUES, local_in));

546:   /*
547:      Get pointer to vector data
548:   */
549:   PetscCall(VecGetArrayRead(local_in, &localptr));

551:   /*
552:      Get starting and ending locally owned rows of the matrix
553:   */
554:   PetscCall(MatGetOwnershipRange(BB, &mstarts, &mends));
555:   mstart = mstarts;
556:   mend   = mends;

558:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
559:      Compute entries for the locally owned part of the Jacobian.
560:       - Currently, all PETSc parallel matrix formats are partitioned by
561:         contiguous chunks of rows across the processors.
562:       - Each processor needs to insert only elements that it owns
563:         locally (but any non-local elements will be sent to the
564:         appropriate processor during matrix assembly).
565:       - Here, we set all entries for a particular row at once.
566:       - We can set matrix entries either using either
567:         MatSetValuesLocal() or MatSetValues().
568:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

570:   /*
571:      Set matrix rows corresponding to boundary data
572:   */
573:   if (mstart == 0) {
574:     v[0] = 0.0;
575:     PetscCall(MatSetValues(BB, 1, &mstart, 1, &mstart, v, INSERT_VALUES));
576:     mstart++;
577:   }
578:   if (mend == appctx->m) {
579:     mend--;
580:     v[0] = 0.0;
581:     PetscCall(MatSetValues(BB, 1, &mend, 1, &mend, v, INSERT_VALUES));
582:   }

584:   /*
585:      Set matrix rows corresponding to interior data.  We construct the
586:      matrix one row at a time.
587:   */
588:   sc = 1.0 / (appctx->h * appctx->h * 2.0 * (1.0 + t) * (1.0 + t));
589:   for (i = mstart; i < mend; i++) {
590:     idx[0] = i - 1;
591:     idx[1] = i;
592:     idx[2] = i + 1;
593:     is     = i - mstart + 1;
594:     v[0]   = sc * localptr[is];
595:     v[1]   = sc * (localptr[is + 1] + localptr[is - 1] - 4.0 * localptr[is]);
596:     v[2]   = sc * localptr[is];
597:     PetscCall(MatSetValues(BB, 1, &i, 3, idx, v, INSERT_VALUES));
598:   }

600:   /*
601:      Restore vector
602:   */
603:   PetscCall(VecRestoreArrayRead(local_in, &localptr));

605:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
606:      Complete the matrix assembly process and set some options
607:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
608:   /*
609:      Assemble matrix, using the 2-step process:
610:        MatAssemblyBegin(), MatAssemblyEnd()
611:      Computations can be done while messages are in transition
612:      by placing code between these two statements.
613:   */
614:   PetscCall(MatAssemblyBegin(BB, MAT_FINAL_ASSEMBLY));
615:   PetscCall(MatAssemblyEnd(BB, MAT_FINAL_ASSEMBLY));
616:   if (BB != AA) {
617:     PetscCall(MatAssemblyBegin(AA, MAT_FINAL_ASSEMBLY));
618:     PetscCall(MatAssemblyEnd(AA, MAT_FINAL_ASSEMBLY));
619:   }

621:   /*
622:      Set and option to indicate that we will never add a new nonzero location
623:      to the matrix. If we do, it will generate an error.
624:   */
625:   PetscCall(MatSetOption(BB, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));

627:   PetscFunctionReturn(PETSC_SUCCESS);
628: }

630: /*TEST

632:     test:
633:       args: -nox -ts_dt 10 -mymonitor
634:       nsize: 2
635:       requires: !single

637:     test:
638:       suffix: tut_1
639:       nsize: 1
640:       args: -ts_max_steps 10 -ts_monitor

642:     test:
643:       suffix: tut_2
644:       nsize: 4
645:       args: -ts_max_steps 10 -ts_monitor -snes_monitor -ksp_monitor

647:     test:
648:       suffix: tut_3
649:       nsize: 4
650:       args: -ts_max_steps 10 -ts_monitor -M 128

652: TEST*/