Actual source code: ex6.c

```  2: /* Program usage:  ex3 [-help] [all PETSc options] */

4: static char help[] ="Solves a simple time-dependent linear PDE (the heat equation).\n\
5: Input parameters include:\n\
6:   -m <points>, where <points> = number of grid points\n\
7:   -time_dependent_rhs : Treat the problem as having a time-dependent right-hand side\n\
8:   -debug              : Activate debugging printouts\n\
9:   -nox                : Deactivate x-window graphics\n\n";

11: /*
12:    Concepts: TS^time-dependent linear problems
13:    Concepts: TS^heat equation
14:    Concepts: TS^diffusion equation
15:    Routines: TSCreate(); TSSetSolution(); TSSetRHSMatrix();
16:    Routines: TSSetInitialTimeStep(); TSSetDuration(); TSMonitorSet();
17:    Routines: TSSetFromOptions(); TSStep(); TSDestroy();
18:    Routines: TSSetTimeStep(); TSGetTimeStep();
19:    Processors: 1
20: */

22: /* ------------------------------------------------------------------------

24:    This program solves the one-dimensional heat equation (also called the
25:    diffusion equation),
26:        u_t = u_xx,
27:    on the domain 0 <= x <= 1, with the boundary conditions
28:        u(t,0) = 0, u(t,1) = 0,
29:    and the initial condition
30:        u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
31:    This is a linear, second-order, parabolic equation.

33:    We discretize the right-hand side using finite differences with
34:    uniform grid spacing h:
35:        u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
36:    We then demonstrate time evolution using the various TS methods by
37:    running the program via
38:        ex3 -ts_type <timestepping solver>

40:    We compare the approximate solution with the exact solution, given by
41:        u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
42:                       3*exp(-4*pi*pi*t) * sin(2*pi*x)

44:    Notes:
45:    This code demonstrates the TS solver interface to two variants of
46:    linear problems, u_t = f(u,t), namely
47:      - time-dependent f:   f(u,t) is a function of t
48:      - time-independent f: f(u,t) is simply f(u)

50:     The parallel version of this code is ts/examples/tutorials/ex4.c

52:   ------------------------------------------------------------------------- */

54: /*
55:    Include "ts.h" so that we can use TS solvers.  Note that this file
56:    automatically includes:
57:      petsc.h  - base PETSc routines   vec.h  - vectors
58:      sys.h    - system routines       mat.h  - matrices
59:      is.h     - index sets            ksp.h  - Krylov subspace methods
60:      viewer.h - viewers               pc.h   - preconditioners
61:      snes.h - nonlinear solvers
62: */

64:  #include petscts.h

66: /*
67:    User-defined application context - contains data needed by the
68:    application-provided call-back routines.
69: */
70: typedef struct {
71:   Vec         solution;          /* global exact solution vector */
72:   PetscInt    m;                 /* total number of grid points */
73:   PetscReal   h;                 /* mesh width h = 1/(m-1) */
74:   PetscTruth  debug;             /* flag (1 indicates activation of debugging printouts) */
75:   PetscViewer viewer1, viewer2;  /* viewers for the solution and error */
76:   PetscReal   norm_2, norm_max;  /* error norms */
77: } AppCtx;

79: /*
80:    User-defined routines
81: */

90: int main(int argc,char **argv)
91: {
92:   AppCtx         appctx;                 /* user-defined application context */
93:   TS             ts;                     /* timestepping context */
94:   Mat            A;                      /* matrix data structure */
95:   Vec            u;                      /* approximate solution vector */
96:   PetscReal      time_total_max = 100.0; /* default max total time */
97:   PetscInt       time_steps_max = 100;   /* default max timesteps */
98:   PetscDraw      draw;                   /* drawing context */
100:   PetscInt       steps, m;
101:   PetscMPIInt    size;
102:   PetscReal      dt;
103:   PetscReal      ftime;
104:   PetscTruth     flg;
105:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
106:      Initialize program and set problem parameters
107:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
108:
109:   PetscInitialize(&argc,&argv,(char*)0,help);
110:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
111:   if (size != 1) SETERRQ(1,"This is a uniprocessor example only!");

113:   m    = 60;
114:   PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
115:   PetscOptionsHasName(PETSC_NULL,"-debug",&appctx.debug);
116:   appctx.m        = m;
117:   appctx.h        = 1.0/(m-1.0);
118:   appctx.norm_2   = 0.0;
119:   appctx.norm_max = 0.0;
120:   PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");

122:   PetscOptionsGetInt(PETSC_NULL,"-time_steps_max",&time_steps_max,PETSC_NULL);

124:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
125:      Create vector data structures
126:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

128:   /*
129:      Create vector data structures for approximate and exact solutions
130:   */
131:   VecCreateSeq(PETSC_COMM_SELF,m,&u);
132:   VecDuplicate(u,&appctx.solution);

134:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
135:      Set up displays to show graphs of the solution and error
136:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

138:   PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
139:   PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
140:   PetscDrawSetDoubleBuffer(draw);
141:   PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
142:   PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
143:   PetscDrawSetDoubleBuffer(draw);

145:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
146:      Create timestepping solver context
147:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

149:   TSCreate(PETSC_COMM_SELF,&ts);
150:   TSSetProblemType(ts,TS_LINEAR);

152:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
153:      Set optional user-defined monitoring routine
154:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

156:   TSMonitorSet(ts,Monitor,&appctx,PETSC_NULL);

158:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

160:      Create matrix data structure; set matrix evaluation routine.
161:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

163:   MatCreate(PETSC_COMM_SELF,&A);
164:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
165:   MatSetFromOptions(A);

167:   PetscOptionsHasName(PETSC_NULL,"-time_dependent_rhs",&flg);
168:   if (flg) {
169:     /*
170:        For linear problems with a time-dependent f(u,t) in the equation
171:        u_t = f(u,t), the user provides the discretized right-hand-side
172:        as a time-dependent matrix.
173:     */
174:     TSSetRHSMatrix(ts,A,A,RHSMatrixHeat,&appctx);
175:   } else {
176:     /*
177:        For linear problems with a time-independent f(u) in the equation
178:        u_t = f(u), the user provides the discretized right-hand-side
179:        as a matrix only once, and then sets a null matrix evaluation
180:        routine.
181:     */
182:     MatStructure A_structure;
183:     RHSMatrixHeat(ts,0.0,&A,&A,&A_structure,&appctx);
184:     TSSetRHSMatrix(ts,A,A,PETSC_NULL,&appctx);
185:   }

187:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
188:      Set solution vector and initial timestep
189:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

191:   dt = appctx.h*appctx.h/2.0;
192:   TSSetInitialTimeStep(ts,0.0,dt);
193:   TSSetSolution(ts,u);

195:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
196:      Customize timestepping solver:
197:        - Set the solution method to be the Backward Euler method.
198:        - Set timestepping duration info
199:      Then set runtime options, which can override these defaults.
200:      For example,
201:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
202:      to override the defaults set by TSSetDuration().
203:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

205:   TSSetDuration(ts,time_steps_max,time_total_max);
206:   TSSetFromOptions(ts);

208:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
209:      Solve the problem
210:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

212:   /*
213:      Evaluate initial conditions
214:   */
215:   InitialConditions(u,&appctx);

217:   /*
218:      Run the timestepping solver
219:   */
220:   TSStep(ts,&steps,&ftime);

222:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
223:      View timestepping solver info
224:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

226:   PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %G, avg. error (max norm) = %G\n",
227:               appctx.norm_2/steps,appctx.norm_max/steps);
228:   TSView(ts,PETSC_VIEWER_STDOUT_SELF);

230:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
231:      Free work space.  All PETSc objects should be destroyed when they
232:      are no longer needed.
233:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

235:   TSDestroy(ts);
236:   MatDestroy(A);
237:   VecDestroy(u);
238:   PetscViewerDestroy(appctx.viewer1);
239:   PetscViewerDestroy(appctx.viewer2);
240:   VecDestroy(appctx.solution);

242:   /*
243:      Always call PetscFinalize() before exiting a program.  This routine
244:        - finalizes the PETSc libraries as well as MPI
245:        - provides summary and diagnostic information if certain runtime
246:          options are chosen (e.g., -log_summary).
247:   */
248:   PetscFinalize();
249:   return 0;
250: }
251: /* --------------------------------------------------------------------- */
254: /*
255:    InitialConditions - Computes the solution at the initial time.

257:    Input Parameter:
258:    u - uninitialized solution vector (global)
259:    appctx - user-defined application context

261:    Output Parameter:
262:    u - vector with solution at initial time (global)
263: */
264: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
265: {
266:   PetscScalar    *u_localptr;
267:   PetscInt       i;

270:   /*
271:     Get a pointer to vector data.
272:     - For default PETSc vectors, VecGetArray() returns a pointer to
273:       the data array.  Otherwise, the routine is implementation dependent.
274:     - You MUST call VecRestoreArray() when you no longer need access to
275:       the array.
276:     - Note that the Fortran interface to VecGetArray() differs from the
277:       C version.  See the users manual for details.
278:   */
279:   VecGetArray(u,&u_localptr);

281:   /*
282:      We initialize the solution array by simply writing the solution
283:      directly into the array locations.  Alternatively, we could use
284:      VecSetValues() or VecSetValuesLocal().
285:   */
286:   for (i=0; i<appctx->m; i++) {
287:     u_localptr[i] = sin(PETSC_PI*i*6.*appctx->h) + 3.*sin(PETSC_PI*i*2.*appctx->h);
288:   }

290:   /*
291:      Restore vector
292:   */
293:   VecRestoreArray(u,&u_localptr);

295:   /*
296:      Print debugging information if desired
297:   */
298:   if (appctx->debug) {
299:      printf("initial guess vector\n");
300:      VecView(u,PETSC_VIEWER_STDOUT_SELF);
301:   }

303:   return 0;
304: }
305: /* --------------------------------------------------------------------- */
308: /*
309:    ExactSolution - Computes the exact solution at a given time.

311:    Input Parameters:
312:    t - current time
313:    solution - vector in which exact solution will be computed
314:    appctx - user-defined application context

316:    Output Parameter:
317:    solution - vector with the newly computed exact solution
318: */
319: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
320: {
321:   PetscScalar    *s_localptr, h = appctx->h, ex1, ex2, sc1, sc2;
322:   PetscInt       i;

325:   /*
326:      Get a pointer to vector data.
327:   */
328:   VecGetArray(solution,&s_localptr);

330:   /*
331:      Simply write the solution directly into the array locations.
332:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
333:   */
334:   ex1 = exp(-36.*PETSC_PI*PETSC_PI*t); ex2 = exp(-4.*PETSC_PI*PETSC_PI*t);
335:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
336:   for (i=0; i<appctx->m; i++) {
337:     s_localptr[i] = sin(PetscRealPart(sc1)*(PetscReal)i)*ex1 + 3.*sin(PetscRealPart(sc2)*(PetscReal)i)*ex2;
338:   }

340:   /*
341:      Restore vector
342:   */
343:   VecRestoreArray(solution,&s_localptr);
344:   return 0;
345: }
346: /* --------------------------------------------------------------------- */
349: /*
350:    Monitor - User-provided routine to monitor the solution computed at
351:    each timestep.  This example plots the solution and computes the
352:    error in two different norms.

354:    This example also demonstrates changing the timestep via TSSetTimeStep().

356:    Input Parameters:
357:    ts     - the timestep context
358:    step   - the count of the current step (with 0 meaning the
359:              initial condition)
360:    crtime  - the current time
361:    u      - the solution at this timestep
362:    ctx    - the user-provided context for this monitoring routine.
363:             In this case we use the application context which contains
364:             information about the problem size, workspace and the exact
365:             solution.
366: */
367: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal crtime,Vec u,void *ctx)
368: {
369:   AppCtx         *appctx = (AppCtx*) ctx;   /* user-defined application context */
371:   PetscReal      norm_2, norm_max, dt, dttol;
372:   PetscTruth     flg;

374:   /*
375:      View a graph of the current iterate
376:   */
377:   VecView(u,appctx->viewer2);

379:   /*
380:      Compute the exact solution
381:   */
382:   ExactSolution(crtime,appctx->solution,appctx);

384:   /*
385:      Print debugging information if desired
386:   */
387:   if (appctx->debug) {
388:      printf("Computed solution vector\n");
389:      VecView(u,PETSC_VIEWER_STDOUT_SELF);
390:      printf("Exact solution vector\n");
391:      VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
392:   }

394:   /*
395:      Compute the 2-norm and max-norm of the error
396:   */
397:   VecAXPY(appctx->solution,-1.0,u);
398:   VecNorm(appctx->solution,NORM_2,&norm_2);
399:   norm_2 = sqrt(appctx->h)*norm_2;
400:   VecNorm(appctx->solution,NORM_MAX,&norm_max);

402:   TSGetTimeStep(ts,&dt);
403:   printf("Timestep %d: step size = %G, time = %G, 2-norm error = %G, max norm error = %G\n",
404:          (int)step,dt,crtime,norm_2,norm_max);
405:   appctx->norm_2   += norm_2;
406:   appctx->norm_max += norm_max;

408:   dttol = .0001;
409:   PetscOptionsGetReal(PETSC_NULL,"-dttol",&dttol,&flg);
410:   if (dt < dttol) {
411:     dt *= .999;
412:     TSSetTimeStep(ts,dt);
413:   }

415:   /*
416:      View a graph of the error
417:   */
418:   VecView(appctx->solution,appctx->viewer1);

420:   /*
421:      Print debugging information if desired
422:   */
423:   if (appctx->debug) {
424:      printf("Error vector\n");
425:      VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
426:   }

428:   return 0;
429: }
430: /* --------------------------------------------------------------------- */
433: /*
434:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
435:    matrix for the heat equation.

437:    Input Parameters:
438:    ts - the TS context
439:    t - current time
440:    global_in - global input vector
441:    dummy - optional user-defined context, as set by TSetRHSJacobian()

443:    Output Parameters:
444:    AA - Jacobian matrix
445:    BB - optionally different preconditioning matrix
446:    str - flag indicating matrix structure

448:    Notes:
449:    Recall that MatSetValues() uses 0-based row and column numbers
450:    in Fortran as well as in C.
451: */
452: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
453: {
454:   Mat            A = *AA;                      /* Jacobian matrix */
455:   AppCtx         *appctx = (AppCtx *) ctx;     /* user-defined application context */
456:   PetscInt       mstart = 0;
457:   PetscInt       mend = appctx->m;
459:   PetscInt       i, idx[3];
460:   PetscScalar    v[3], stwo = -2./(appctx->h*appctx->h), sone = -.5*stwo;

462:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
463:      Compute entries for the locally owned part of the matrix
464:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
465:   /*
466:      Set matrix rows corresponding to boundary data
467:   */

469:   mstart = 0;
470:   v[0] = 1.0;
471:   MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
472:   mstart++;

474:   mend--;
475:   v[0] = 1.0;
476:   MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);

478:   /*
479:      Set matrix rows corresponding to interior data.  We construct the
480:      matrix one row at a time.
481:   */
482:   v[0] = sone; v[1] = stwo; v[2] = sone;
483:   for ( i=mstart; i<mend; i++ ) {
484:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
485:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
486:   }

488:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
489:      Complete the matrix assembly process and set some options
490:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
491:   /*
492:      Assemble matrix, using the 2-step process:
493:        MatAssemblyBegin(), MatAssemblyEnd()
494:      Computations can be done while messages are in transition
495:      by placing code between these two statements.
496:   */
497:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
498:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

500:   /*
501:      Set flag to indicate that the Jacobian matrix retains an identical
502:      nonzero structure throughout all timestepping iterations (although the
503:      values of the entries change). Thus, we can save some work in setting
504:      up the preconditioner (e.g., no need to redo symbolic factorization for
505:      ILU/ICC preconditioners).
506:       - If the nonzero structure of the matrix is different during
507:         successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
508:         must be used instead.  If you are unsure whether the matrix
509:         structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
510:       - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc
511:         believes your assertion and does not check the structure
512:         of the matrix.  If you erroneously claim that the structure
513:         is the same when it actually is not, the new preconditioner
514:         will not function correctly.  Thus, use this optimization
515:         feature with caution!
516:   */
517:   *str = SAME_NONZERO_PATTERN;

519:   /*
520:      Set and option to indicate that we will never add a new nonzero location
521:      to the matrix. If we do, it will generate an error.
522:   */
523:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR);

525:   return 0;
526: }
527: /* --------------------------------------------------------------------- */
530: /*
531:    Input Parameters:
532:    ts - the TS context
533:    t - current time
534:    f - function
535:    ctx - optional user-defined context, as set by TSetBCFunction()
536:  */
537: PetscErrorCode MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
538: {
539:   AppCtx         *appctx = (AppCtx *) ctx;     /* user-defined application context */
541:   PetscInt       m = appctx->m;
542:   PetscScalar    *fa;

544:   VecGetArray(f,&fa);
545:   fa[0] = 0.0;
546:   fa[m-1] = 1.0;
547:   VecRestoreArray(f,&fa);
548:   printf("t=%g\n",t);
549:
550:   return 0;
551: }
```