Actual source code: ex3.c

  1: /*$Id: ex3.c,v 1.24 2001/04/10 19:37:12 bsmith Exp $*/

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

  5: static char help[] ="Solves a simple time-dependent linear PDE (the heat equation).n
  6: Input parameters include:n
  7:   -m <points>, where <points> = number of grid pointsn
  8:   -time_dependent_rhs : Treat the problem as having a time-dependent right-hand siden
  9:   -time_dependent_bc  : Treat the problem as having time-dependent boundary conditionsn
 10:   -debug              : Activate debugging printoutsn
 11:   -nox                : Deactivate x-window graphicsnn";

 13: /*
 14:    Concepts: TS^time-dependent linear problems
 15:    Concepts: TS^heat equation
 16:    Concepts: TS^diffusion equation
 17:    Processors: 1
 18: */

 20: /* ------------------------------------------------------------------------

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

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

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

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

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

 50:   ------------------------------------------------------------------------- */

 52: /* 
 53:    Include "petscts.h" so that we can use TS solvers.  Note that this file
 54:    automatically includes:
 55:      petsc.h       - base PETSc routines   petscvec.h  - vectors
 56:      petscsys.h    - system routines       petscmat.h  - matrices
 57:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
 58:      petscviewer.h - viewers               petscpc.h   - preconditioners
 59:      petscsles.h   - linear solvers        petscsnes.h - nonlinear solvers
 60: */

 62:  #include petscts.h

 64: /* 
 65:    User-defined application context - contains data needed by the 
 66:    application-provided call-back routines.
 67: */
 68: typedef struct {
 69:   Vec        solution;          /* global exact solution vector */
 70:   int        m;                 /* total number of grid points */
 71:   double     h;                 /* mesh width h = 1/(m-1) */
 72:   PetscTruth debug;             /* flag (1 indicates activation of debugging printouts) */
 73:   PetscViewer     viewer1,viewer2;  /* viewers for the solution and error */
 74:   double     norm_2,norm_max;  /* error norms */
 75: } AppCtx;

 77: /* 
 78:    User-defined routines
 79: */
 80: extern int InitialConditions(Vec,AppCtx*);
 81: extern int RHSMatrixHeat(TS,double,Mat*,Mat*,MatStructure*,void*);
 82: extern int Monitor(TS,int,double,Vec,void*);
 83: extern int ExactSolution(double,Vec,AppCtx*);
 84: extern int MyBCRoutine(TS,double,Vec,void*);

 86: int main(int argc,char **argv)
 87: {
 88:   AppCtx        appctx;                 /* user-defined application context */
 89:   TS            ts;                     /* timestepping context */
 90:   Mat           A;                      /* matrix data structure */
 91:   Vec           u;                      /* approximate solution vector */
 92:   double        time_total_max = 100.0; /* default max total time */
 93:   int           time_steps_max = 100;   /* default max timesteps */
 94:   PetscDraw     draw;                   /* drawing context */
 95:   int           ierr,steps,size,m;
 96:   double        dt,ftime;
 97:   PetscTruth    flg;

 99:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
100:      Initialize program and set problem parameters
101:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
102: 
103:   PetscInitialize(&argc,&argv,(char*)0,help);
104:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
105:   if (size != 1) SETERRQ(1,"This is a uniprocessor example only!");

107:   m    = 60;
108:   PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
109:   PetscOptionsHasName(PETSC_NULL,"-debug",&appctx.debug);
110:   appctx.m        = m;
111:   appctx.h        = 1.0/(m-1.0);
112:   appctx.norm_2   = 0.0;
113:   appctx.norm_max = 0.0;
114:   PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processorn");

116:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
117:      Create vector data structures
118:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

120:   /* 
121:      Create vector data structures for approximate and exact solutions
122:   */
123:   VecCreateSeq(PETSC_COMM_SELF,m,&u);
124:   VecDuplicate(u,&appctx.solution);

126:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
127:      Set up displays to show graphs of the solution and error 
128:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

130:   PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
131:   PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
132:   PetscDrawSetDoubleBuffer(draw);
133:   PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
134:   PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
135:   PetscDrawSetDoubleBuffer(draw);

137:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
138:      Create timestepping solver context
139:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

141:   TSCreate(PETSC_COMM_SELF,TS_LINEAR,&ts);

143:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
144:      Set optional user-defined monitoring routine
145:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

147:   TSSetMonitor(ts,Monitor,&appctx,PETSC_NULL);

149:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

151:      Create matrix data structure; set matrix evaluation routine.
152:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

154:   MatCreate(PETSC_COMM_SELF,PETSC_DECIDE,PETSC_DECIDE,m,m,&A);
155:   MatSetFromOptions(A);

157:   PetscOptionsHasName(PETSC_NULL,"-time_dependent_rhs",&flg);
158:   if (flg) {
159:     /*
160:        For linear problems with a time-dependent f(u,t) in the equation 
161:        u_t = f(u,t), the user provides the discretized right-hand-side
162:        as a time-dependent matrix.
163:     */
164:     TSSetRHSMatrix(ts,A,A,RHSMatrixHeat,&appctx);
165:   } else {
166:     /*
167:        For linear problems with a time-independent f(u) in the equation 
168:        u_t = f(u), the user provides the discretized right-hand-side
169:        as a matrix only once, and then sets a null matrix evaluation
170:        routine.
171:     */
172:     MatStructure A_structure;
173:     RHSMatrixHeat(ts,0.0,&A,&A,&A_structure,&appctx);
174:     TSSetRHSMatrix(ts,A,A,PETSC_NULL,&appctx);
175:   }

177:   /* Treat the problem as having time-dependent boundary conditions */
178:   PetscOptionsHasName(PETSC_NULL,"-time_dependent_bc",&flg);
179:   if (flg) {
180:     TSSetRHSBoundaryConditions(ts,MyBCRoutine,&appctx);
181:   }

183:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
184:      Set solution vector and initial timestep
185:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

187:   dt = appctx.h*appctx.h/2.0;
188:   TSSetInitialTimeStep(ts,0.0,dt);
189:   TSSetSolution(ts,u);

191:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
192:      Customize timestepping solver:  
193:        - Set the solution method to be the Backward Euler method.
194:        - Set timestepping duration info 
195:      Then set runtime options, which can override these defaults.
196:      For example,
197:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
198:      to override the defaults set by TSSetDuration().
199:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

201:   TSSetDuration(ts,time_steps_max,time_total_max);
202:   TSSetFromOptions(ts);

204:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
205:      Solve the problem
206:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

208:   /*
209:      Evaluate initial conditions
210:   */
211:   InitialConditions(u,&appctx);

213:   /*
214:      Run the timestepping solver
215:   */
216:   TSStep(ts,&steps,&ftime);

218:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
219:      View timestepping solver info
220:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

222:   PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %g, avg. error (max norm) = %gn",
223:               appctx.norm_2/steps,appctx.norm_max/steps);
224:   TSView(ts,PETSC_VIEWER_STDOUT_SELF);

226:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
227:      Free work space.  All PETSc objects should be destroyed when they
228:      are no longer needed.
229:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

231:   TSDestroy(ts);
232:   MatDestroy(A);
233:   VecDestroy(u);
234:   PetscViewerDestroy(appctx.viewer1);
235:   PetscViewerDestroy(appctx.viewer2);
236:   VecDestroy(appctx.solution);

238:   /*
239:      Always call PetscFinalize() before exiting a program.  This routine
240:        - finalizes the PETSc libraries as well as MPI
241:        - provides summary and diagnostic information if certain runtime
242:          options are chosen (e.g., -log_summary). 
243:   */
244:   PetscFinalize();
245:   return 0;
246: }
247: /* --------------------------------------------------------------------- */
248: /*
249:    InitialConditions - Computes the solution at the initial time. 

251:    Input Parameter:
252:    u - uninitialized solution vector (global)
253:    appctx - user-defined application context

255:    Output Parameter:
256:    u - vector with solution at initial time (global)
257: */
258: int InitialConditions(Vec u,AppCtx *appctx)
259: {
260:   Scalar *u_localptr,h = appctx->h;
261:   int    i,ierr;

263:   /* 
264:     Get a pointer to vector data.
265:     - For default PETSc vectors, VecGetArray() returns a pointer to
266:       the data array.  Otherwise, the routine is implementation dependent.
267:     - You MUST call VecRestoreArray() when you no longer need access to
268:       the array.
269:     - Note that the Fortran interface to VecGetArray() differs from the
270:       C version.  See the users manual for details.
271:   */
272:   VecGetArray(u,&u_localptr);

274:   /* 
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().
278:   */
279:   for (i=0; i<appctx->m; i++) {
280:     u_localptr[i] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);
281:   }

283:   /* 
284:      Restore vector
285:   */
286:   VecRestoreArray(u,&u_localptr);

288:   /* 
289:      Print debugging information if desired
290:   */
291:   if (appctx->debug) {
292:      printf("initial guess vectorn");
293:      VecView(u,PETSC_VIEWER_STDOUT_SELF);
294:   }

296:   return 0;
297: }
298: /* --------------------------------------------------------------------- */
299: /*
300:    ExactSolution - Computes the exact solution at a given time.

302:    Input Parameters:
303:    t - current time
304:    solution - vector in which exact solution will be computed
305:    appctx - user-defined application context

307:    Output Parameter:
308:    solution - vector with the newly computed exact solution
309: */
310: int ExactSolution(double t,Vec solution,AppCtx *appctx)
311: {
312:   Scalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2;
313:   int    i,ierr;

315:   /*
316:      Get a pointer to vector data.
317:   */
318:   VecGetArray(solution,&s_localptr);

320:   /* 
321:      Simply write the solution directly into the array locations.
322:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
323:   */
324:   ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*t);
325:   ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*t);
326:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
327:   for (i=0; i<appctx->m; i++) {
328:     s_localptr[i] = PetscSinScalar(sc1*(double)i)*ex1 + 3.*PetscSinScalar(sc2*(double)i)*ex2;
329:   }

331:   /* 
332:      Restore vector
333:   */
334:   VecRestoreArray(solution,&s_localptr);
335:   return 0;
336: }
337: /* --------------------------------------------------------------------- */
338: /*
339:    Monitor - User-provided routine to monitor the solution computed at 
340:    each timestep.  This example plots the solution and computes the
341:    error in two different norms.

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

345:    Input Parameters:
346:    ts     - the timestep context
347:    step   - the count of the current step (with 0 meaning the
348:              initial condition)
349:    time   - the current time
350:    u      - the solution at this timestep
351:    ctx    - the user-provided context for this monitoring routine.
352:             In this case we use the application context which contains 
353:             information about the problem size, workspace and the exact 
354:             solution.
355: */
356: int Monitor(TS ts,int step,double time,Vec u,void *ctx)
357: {
358:   AppCtx     *appctx = (AppCtx*) ctx;   /* user-defined application context */
359:   int        ierr;
360:   double     norm_2,norm_max,dt,dttol;
361:   Scalar     mone = -1.0;
362:   /* 
363:      View a graph of the current iterate
364:   */
365:   VecView(u,appctx->viewer2);

367:   /* 
368:      Compute the exact solution
369:   */
370:   ExactSolution(time,appctx->solution,appctx);

372:   /*
373:      Print debugging information if desired
374:   */
375:   if (appctx->debug) {
376:      printf("Computed solution vectorn");
377:      VecView(u,PETSC_VIEWER_STDOUT_SELF);
378:      printf("Exact solution vectorn");
379:      VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
380:   }

382:   /*
383:      Compute the 2-norm and max-norm of the error
384:   */
385:   VecAXPY(&mone,u,appctx->solution);
386:   VecNorm(appctx->solution,NORM_2,&norm_2);
387:   norm_2 = sqrt(appctx->h)*norm_2;
388:   VecNorm(appctx->solution,NORM_MAX,&norm_max);

390:   TSGetTimeStep(ts,&dt);
391:   printf("Timestep %3d: step size = %-11g, time = %-11g, 2-norm error = %-11g, max norm error = %-11gn",
392:          step,dt,time,norm_2,norm_max);
393:   appctx->norm_2   += norm_2;
394:   appctx->norm_max += norm_max;

396:   dttol = .0001;
397:   PetscOptionsGetDouble(PETSC_NULL,"-dttol",&dttol,PETSC_NULL);
398:   if (dt < dttol) {
399:     dt *= .999;
400:     TSSetTimeStep(ts,dt);
401:   }

403:   /* 
404:      View a graph of the error
405:   */
406:   VecView(appctx->solution,appctx->viewer1);

408:   /*
409:      Print debugging information if desired
410:   */
411:   if (appctx->debug) {
412:      printf("Error vectorn");
413:      VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
414:   }

416:   return 0;
417: }
418: /* --------------------------------------------------------------------- */
419: /*
420:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
421:    matrix for the heat equation.

423:    Input Parameters:
424:    ts - the TS context
425:    t - current time
426:    global_in - global input vector
427:    dummy - optional user-defined context, as set by TSetRHSJacobian()

429:    Output Parameters:
430:    AA - Jacobian matrix
431:    BB - optionally different preconditioning matrix
432:    str - flag indicating matrix structure

434:    Notes:
435:    Recall that MatSetValues() uses 0-based row and column numbers
436:    in Fortran as well as in C.
437: */
438: int RHSMatrixHeat(TS ts,double t,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
439: {
440:   Mat    A = *AA;                      /* Jacobian matrix */
441:   AppCtx *appctx = (AppCtx*)ctx;     /* user-defined application context */
442:   int    mstart = 0;
443:   int    mend = appctx->m;
444:   int    ierr,i,idx[3];
445:   Scalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;

447:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
448:      Compute entries for the locally owned part of the matrix
449:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
450:   /* 
451:      Set matrix rows corresponding to boundary data
452:   */

454:   mstart = 0;
455:   v[0] = 1.0;
456:   MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
457:   mstart++;

459:   mend--;
460:   v[0] = 1.0;
461:   MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);

463:   /*
464:      Set matrix rows corresponding to interior data.  We construct the 
465:      matrix one row at a time.
466:   */
467:   v[0] = sone; v[1] = stwo; v[2] = sone;
468:   for (i=mstart; i<mend; i++) {
469:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
470:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
471:   }

473:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
474:      Complete the matrix assembly process and set some options
475:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
476:   /*
477:      Assemble matrix, using the 2-step process:
478:        MatAssemblyBegin(), MatAssemblyEnd()
479:      Computations can be done while messages are in transition
480:      by placing code between these two statements.
481:   */
482:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
483:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

485:   /*
486:      Set flag to indicate that the Jacobian matrix retains an identical
487:      nonzero structure throughout all timestepping iterations (although the
488:      values of the entries change). Thus, we can save some work in setting
489:      up the preconditioner (e.g., no need to redo symbolic factorization for
490:      ILU/ICC preconditioners).
491:       - If the nonzero structure of the matrix is different during
492:         successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
493:         must be used instead.  If you are unsure whether the matrix
494:         structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
495:       - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc
496:         believes your assertion and does not check the structure
497:         of the matrix.  If you erroneously claim that the structure
498:         is the same when it actually is not, the new preconditioner
499:         will not function correctly.  Thus, use this optimization
500:         feature with caution!
501:   */
502:   *str = SAME_NONZERO_PATTERN;

504:   /*
505:      Set and option to indicate that we will never add a new nonzero location 
506:      to the matrix. If we do, it will generate an error.
507:   */
508:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR);

510:   return 0;
511: }
512: /* --------------------------------------------------------------------- */
513: /*
514:    Input Parameters:
515:    ts - the TS context
516:    t - current time
517:    f - function
518:    ctx - optional user-defined context, as set by TSetBCFunction()
519:  */
520: int MyBCRoutine(TS ts,double t,Vec f,void *ctx)
521: {
522:   AppCtx *appctx = (AppCtx*)ctx;     /* user-defined application context */
523:   int    ierr,m = appctx->m;
524:   Scalar *fa;

526:   VecGetArray(f,&fa);
527:   fa[0] = 0.0;
528:   fa[m-1] = 0.0;
529:   VecRestoreArray(f,&fa);
530:   printf("t=%gn",t);
531: 
532:   return 0;
533: }