Actual source code: ex4.c


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

  9: /* ------------------------------------------------------------------------

 11:    This program solves the one-dimensional heat equation (also called the
 12:    diffusion equation),
 13:        u_t = u_xx,
 14:    on the domain 0 <= x <= 1, with the boundary conditions
 15:        u(t,0) = 0, u(t,1) = 0,
 16:    and the initial condition
 17:        u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
 18:    This is a linear, second-order, parabolic equation.

 20:    We discretize the right-hand side using finite differences with
 21:    uniform grid spacing h:
 22:        u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
 23:    We then demonstrate time evolution using the various TS methods by
 24:    running the program via
 25:        mpiexec -n <procs> ex3 -ts_type <timestepping solver>

 27:    We compare the approximate solution with the exact solution, given by
 28:        u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
 29:                       3*exp(-4*pi*pi*t) * sin(2*pi*x)

 31:    Notes:
 32:    This code demonstrates the TS solver interface to two variants of
 33:    linear problems, u_t = f(u,t), namely
 34:      - time-dependent f:   f(u,t) is a function of t
 35:      - time-independent f: f(u,t) is simply f(u)

 37:     The uniprocessor version of this code is ts/tutorials/ex3.c

 39:   ------------------------------------------------------------------------- */

 41: /*
 42:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs) to manage
 43:    the parallel grid.  Include "petscts.h" so that we can use TS solvers.
 44:    Note that this file automatically includes:
 45:      petscsys.h       - base PETSc routines   petscvec.h  - vectors
 46:      petscmat.h  - matrices
 47:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
 48:      petscviewer.h - viewers               petscpc.h   - preconditioners
 49:      petscksp.h   - linear solvers        petscsnes.h - nonlinear solvers
 50: */

 52: #include <petscdm.h>
 53: #include <petscdmda.h>
 54: #include <petscts.h>
 55: #include <petscdraw.h>

 57: /*
 58:    User-defined application context - contains data needed by the
 59:    application-provided call-back routines.
 60: */
 61: typedef struct {
 62:   MPI_Comm    comm;              /* communicator */
 63:   DM          da;                /* distributed array data structure */
 64:   Vec         localwork;         /* local ghosted work vector */
 65:   Vec         u_local;           /* local ghosted approximate solution vector */
 66:   Vec         solution;          /* global exact solution vector */
 67:   PetscInt    m;                 /* total number of grid points */
 68:   PetscReal   h;                 /* mesh width h = 1/(m-1) */
 69:   PetscBool   debug;             /* flag (1 indicates activation of debugging printouts) */
 70:   PetscViewer viewer1,viewer2;  /* viewers for the solution and error */
 71:   PetscReal   norm_2,norm_max;  /* error norms */
 72: } AppCtx;

 74: /*
 75:    User-defined routines
 76: */
 77: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
 78: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat,Mat,void*);
 79: extern PetscErrorCode RHSFunctionHeat(TS,PetscReal,Vec,Vec,void*);
 80: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
 81: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);

 83: int main(int argc,char **argv)
 84: {
 85:   AppCtx         appctx;                 /* user-defined application context */
 86:   TS             ts;                     /* timestepping context */
 87:   Mat            A;                      /* matrix data structure */
 88:   Vec            u;                      /* approximate solution vector */
 89:   PetscReal      time_total_max = 1.0;   /* default max total time */
 90:   PetscInt       time_steps_max = 100;   /* default max timesteps */
 91:   PetscDraw      draw;                   /* drawing context */
 92:   PetscInt       steps,m;
 93:   PetscMPIInt    size;
 94:   PetscReal      dt,ftime;
 95:   PetscBool      flg;
 96:   TSProblemType  tsproblem = TS_LINEAR;

 98:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 99:      Initialize program and set problem parameters
100:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

102:   PetscInitialize(&argc,&argv,(char*)0,help);
103:   appctx.comm = PETSC_COMM_WORLD;

105:   m               = 60;
106:   PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
107:   PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);
108:   appctx.m        = m;
109:   appctx.h        = 1.0/(m-1.0);
110:   appctx.norm_2   = 0.0;
111:   appctx.norm_max = 0.0;

113:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
114:   PetscPrintf(PETSC_COMM_WORLD,"Solving a linear TS problem, number of processors = %d\n",size);

116:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
117:      Create vector data structures
118:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
119:   /*
120:      Create distributed array (DMDA) to manage parallel grid and vectors
121:      and to set up the ghost point communication pattern.  There are M
122:      total grid values spread equally among all the processors.
123:   */

125:   DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,m,1,1,NULL,&appctx.da);
126:   DMSetFromOptions(appctx.da);
127:   DMSetUp(appctx.da);

129:   /*
130:      Extract global and local vectors from DMDA; we use these to store the
131:      approximate solution.  Then duplicate these for remaining vectors that
132:      have the same types.
133:   */
134:   DMCreateGlobalVector(appctx.da,&u);
135:   DMCreateLocalVector(appctx.da,&appctx.u_local);

137:   /*
138:      Create local work vector for use in evaluating right-hand-side function;
139:      create global work vector for storing exact solution.
140:   */
141:   VecDuplicate(appctx.u_local,&appctx.localwork);
142:   VecDuplicate(u,&appctx.solution);

144:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
145:      Set up displays to show graphs of the solution and error
146:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

148:   PetscViewerDrawOpen(PETSC_COMM_WORLD,0,"",80,380,400,160,&appctx.viewer1);
149:   PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
150:   PetscDrawSetDoubleBuffer(draw);
151:   PetscViewerDrawOpen(PETSC_COMM_WORLD,0,"",80,0,400,160,&appctx.viewer2);
152:   PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
153:   PetscDrawSetDoubleBuffer(draw);

155:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
156:      Create timestepping solver context
157:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

159:   TSCreate(PETSC_COMM_WORLD,&ts);

161:   flg  = PETSC_FALSE;
162:   PetscOptionsGetBool(NULL,NULL,"-nonlinear",&flg,NULL);
163:   TSSetProblemType(ts,flg ? TS_NONLINEAR : TS_LINEAR);

165:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
166:      Set optional user-defined monitoring routine
167:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
168:   TSMonitorSet(ts,Monitor,&appctx,NULL);

170:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

172:      Create matrix data structure; set matrix evaluation routine.
173:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

175:   MatCreate(PETSC_COMM_WORLD,&A);
176:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
177:   MatSetFromOptions(A);
178:   MatSetUp(A);

180:   flg  = PETSC_FALSE;
181:   PetscOptionsGetBool(NULL,NULL,"-time_dependent_rhs",&flg,NULL);
182:   if (flg) {
183:     /*
184:        For linear problems with a time-dependent f(u,t) in the equation
185:        u_t = f(u,t), the user provides the discretized right-hand-side
186:        as a time-dependent matrix.
187:     */
188:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
189:     TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
190:   } else {
191:     /*
192:        For linear problems with a time-independent f(u) in the equation
193:        u_t = f(u), the user provides the discretized right-hand-side
194:        as a matrix only once, and then sets a null matrix evaluation
195:        routine.
196:     */
197:     RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
198:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
199:     TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
200:   }

202:   if (tsproblem == TS_NONLINEAR) {
203:     SNES snes;
204:     TSSetRHSFunction(ts,NULL,RHSFunctionHeat,&appctx);
205:     TSGetSNES(ts,&snes);
206:     SNESSetJacobian(snes,NULL,NULL,SNESComputeJacobianDefault,NULL);
207:   }

209:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
210:      Set solution vector and initial timestep
211:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

213:   dt   = appctx.h*appctx.h/2.0;
214:   TSSetTimeStep(ts,dt);
215:   TSSetSolution(ts,u);

217:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
218:      Customize timestepping solver:
219:        - Set the solution method to be the Backward Euler method.
220:        - Set timestepping duration info
221:      Then set runtime options, which can override these defaults.
222:      For example,
223:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
224:      to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
225:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

227:   TSSetMaxSteps(ts,time_steps_max);
228:   TSSetMaxTime(ts,time_total_max);
229:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
230:   TSSetFromOptions(ts);

232:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
233:      Solve the problem
234:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

236:   /*
237:      Evaluate initial conditions
238:   */
239:   InitialConditions(u,&appctx);

241:   /*
242:      Run the timestepping solver
243:   */
244:   TSSolve(ts,u);
245:   TSGetSolveTime(ts,&ftime);
246:   TSGetStepNumber(ts,&steps);

248:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
249:      View timestepping solver info
250:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
251:   PetscPrintf(PETSC_COMM_WORLD,"Total timesteps %D, Final time %g\n",steps,(double)ftime);
252:   PetscPrintf(PETSC_COMM_WORLD,"Avg. error (2 norm) = %g Avg. error (max norm) = %g\n",(double)(appctx.norm_2/steps),(double)(appctx.norm_max/steps));

254:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
255:      Free work space.  All PETSc objects should be destroyed when they
256:      are no longer needed.
257:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

259:   TSDestroy(&ts);
260:   MatDestroy(&A);
261:   VecDestroy(&u);
262:   PetscViewerDestroy(&appctx.viewer1);
263:   PetscViewerDestroy(&appctx.viewer2);
264:   VecDestroy(&appctx.localwork);
265:   VecDestroy(&appctx.solution);
266:   VecDestroy(&appctx.u_local);
267:   DMDestroy(&appctx.da);

269:   /*
270:      Always call PetscFinalize() before exiting a program.  This routine
271:        - finalizes the PETSc libraries as well as MPI
272:        - provides summary and diagnostic information if certain runtime
273:          options are chosen (e.g., -log_view).
274:   */
275:   PetscFinalize();
276:   return 0;
277: }
278: /* --------------------------------------------------------------------- */
279: /*
280:    InitialConditions - Computes the solution at the initial time.

282:    Input Parameter:
283:    u - uninitialized solution vector (global)
284:    appctx - user-defined application context

286:    Output Parameter:
287:    u - vector with solution at initial time (global)
288: */
289: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
290: {
291:   PetscScalar    *u_localptr,h = appctx->h;
292:   PetscInt       i,mybase,myend;

294:   /*
295:      Determine starting point of each processor's range of
296:      grid values.
297:   */
298:   VecGetOwnershipRange(u,&mybase,&myend);

300:   /*
301:     Get a pointer to vector data.
302:     - For default PETSc vectors, VecGetArray() returns a pointer to
303:       the data array.  Otherwise, the routine is implementation dependent.
304:     - You MUST call VecRestoreArray() when you no longer need access to
305:       the array.
306:     - Note that the Fortran interface to VecGetArray() differs from the
307:       C version.  See the users manual for details.
308:   */
309:   VecGetArray(u,&u_localptr);

311:   /*
312:      We initialize the solution array by simply writing the solution
313:      directly into the array locations.  Alternatively, we could use
314:      VecSetValues() or VecSetValuesLocal().
315:   */
316:   for (i=mybase; i<myend; i++) u_localptr[i-mybase] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);

318:   /*
319:      Restore vector
320:   */
321:   VecRestoreArray(u,&u_localptr);

323:   /*
324:      Print debugging information if desired
325:   */
326:   if (appctx->debug) {
327:     PetscPrintf(appctx->comm,"initial guess vector\n");
328:     VecView(u,PETSC_VIEWER_STDOUT_WORLD);
329:   }

331:   return 0;
332: }
333: /* --------------------------------------------------------------------- */
334: /*
335:    ExactSolution - Computes the exact solution at a given time.

337:    Input Parameters:
338:    t - current time
339:    solution - vector in which exact solution will be computed
340:    appctx - user-defined application context

342:    Output Parameter:
343:    solution - vector with the newly computed exact solution
344: */
345: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
346: {
347:   PetscScalar    *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2;
348:   PetscInt       i,mybase,myend;

350:   /*
351:      Determine starting and ending points of each processor's
352:      range of grid values
353:   */
354:   VecGetOwnershipRange(solution,&mybase,&myend);

356:   /*
357:      Get a pointer to vector data.
358:   */
359:   VecGetArray(solution,&s_localptr);

361:   /*
362:      Simply write the solution directly into the array locations.
363:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
364:   */
365:   ex1 = PetscExpReal(-36.*PETSC_PI*PETSC_PI*t); ex2 = PetscExpReal(-4.*PETSC_PI*PETSC_PI*t);
366:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
367:   for (i=mybase; i<myend; i++) s_localptr[i-mybase] = PetscSinScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i)*ex2;

369:   /*
370:      Restore vector
371:   */
372:   VecRestoreArray(solution,&s_localptr);
373:   return 0;
374: }
375: /* --------------------------------------------------------------------- */
376: /*
377:    Monitor - User-provided routine to monitor the solution computed at
378:    each timestep.  This example plots the solution and computes the
379:    error in two different norms.

381:    Input Parameters:
382:    ts     - the timestep context
383:    step   - the count of the current step (with 0 meaning the
384:              initial condition)
385:    time   - the current time
386:    u      - the solution at this timestep
387:    ctx    - the user-provided context for this monitoring routine.
388:             In this case we use the application context which contains
389:             information about the problem size, workspace and the exact
390:             solution.
391: */
392: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
393: {
394:   AppCtx         *appctx = (AppCtx*) ctx;   /* user-defined application context */
395:   PetscReal      norm_2,norm_max;

397:   /*
398:      View a graph of the current iterate
399:   */
400:   VecView(u,appctx->viewer2);

402:   /*
403:      Compute the exact solution
404:   */
405:   ExactSolution(time,appctx->solution,appctx);

407:   /*
408:      Print debugging information if desired
409:   */
410:   if (appctx->debug) {
411:     PetscPrintf(appctx->comm,"Computed solution vector\n");
412:     VecView(u,PETSC_VIEWER_STDOUT_WORLD);
413:     PetscPrintf(appctx->comm,"Exact solution vector\n");
414:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
415:   }

417:   /*
418:      Compute the 2-norm and max-norm of the error
419:   */
420:   VecAXPY(appctx->solution,-1.0,u);
421:   VecNorm(appctx->solution,NORM_2,&norm_2);
422:   norm_2 = PetscSqrtReal(appctx->h)*norm_2;
423:   VecNorm(appctx->solution,NORM_MAX,&norm_max);
424:   if (norm_2   < 1e-14) norm_2   = 0;
425:   if (norm_max < 1e-14) norm_max = 0;

427:   /*
428:      PetscPrintf() causes only the first processor in this
429:      communicator to print the timestep information.
430:   */
431:   PetscPrintf(appctx->comm,"Timestep %D: time = %g 2-norm error = %g max norm error = %g\n",step,(double)time,(double)norm_2,(double)norm_max);
432:   appctx->norm_2   += norm_2;
433:   appctx->norm_max += norm_max;

435:   /*
436:      View a graph of the error
437:   */
438:   VecView(appctx->solution,appctx->viewer1);

440:   /*
441:      Print debugging information if desired
442:   */
443:   if (appctx->debug) {
444:     PetscPrintf(appctx->comm,"Error vector\n");
445:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
446:   }

448:   return 0;
449: }

451: /* --------------------------------------------------------------------- */
452: /*
453:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
454:    matrix for the heat equation.

456:    Input Parameters:
457:    ts - the TS context
458:    t - current time
459:    global_in - global input vector
460:    dummy - optional user-defined context, as set by TSetRHSJacobian()

462:    Output Parameters:
463:    AA - Jacobian matrix
464:    BB - optionally different preconditioning matrix
465:    str - flag indicating matrix structure

467:   Notes:
468:   RHSMatrixHeat computes entries for the locally owned part of the system.
469:    - Currently, all PETSc parallel matrix formats are partitioned by
470:      contiguous chunks of rows across the processors.
471:    - Each processor needs to insert only elements that it owns
472:      locally (but any non-local elements will be sent to the
473:      appropriate processor during matrix assembly).
474:    - Always specify global row and columns of matrix entries when
475:      using MatSetValues(); we could alternatively use MatSetValuesLocal().
476:    - Here, we set all entries for a particular row at once.
477:    - Note that MatSetValues() uses 0-based row and column numbers
478:      in Fortran as well as in C.
479: */
480: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
481: {
482:   Mat            A       = AA;              /* Jacobian matrix */
483:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined application context */
484:   PetscInt       i,mstart,mend,idx[3];
485:   PetscScalar    v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;

487:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
488:      Compute entries for the locally owned part of the matrix
489:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

491:   MatGetOwnershipRange(A,&mstart,&mend);

493:   /*
494:      Set matrix rows corresponding to boundary data
495:   */

497:   if (mstart == 0) {  /* first processor only */
498:     v[0] = 1.0;
499:     MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
500:     mstart++;
501:   }

503:   if (mend == appctx->m) { /* last processor only */
504:     mend--;
505:     v[0] = 1.0;
506:     MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
507:   }

509:   /*
510:      Set matrix rows corresponding to interior data.  We construct the
511:      matrix one row at a time.
512:   */
513:   v[0] = sone; v[1] = stwo; v[2] = sone;
514:   for (i=mstart; i<mend; i++) {
515:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
516:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
517:   }

519:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
520:      Complete the matrix assembly process and set some options
521:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
522:   /*
523:      Assemble matrix, using the 2-step process:
524:        MatAssemblyBegin(), MatAssemblyEnd()
525:      Computations can be done while messages are in transition
526:      by placing code between these two statements.
527:   */
528:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
529:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

531:   /*
532:      Set and option to indicate that we will never add a new nonzero location
533:      to the matrix. If we do, it will generate an error.
534:   */
535:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

537:   return 0;
538: }

540: PetscErrorCode RHSFunctionHeat(TS ts,PetscReal t,Vec globalin,Vec globalout,void *ctx)
541: {
542:   Mat            A;

545:   TSGetRHSJacobian(ts,&A,NULL,NULL,&ctx);
546:   RHSMatrixHeat(ts,t,globalin,A,NULL,ctx);
547:   /* MatView(A,PETSC_VIEWER_STDOUT_WORLD); */
548:   MatMult(A,globalin,globalout);
549:   return 0;
550: }

552: /*TEST

554:     test:
555:       args: -ts_view -nox

557:     test:
558:       suffix: 2
559:       args: -ts_view -nox
560:       nsize: 3

562:     test:
563:       suffix: 3
564:       args: -ts_view -nox -nonlinear

566:     test:
567:       suffix: 4
568:       args: -ts_view -nox -nonlinear
569:       nsize: 3
570:       timeoutfactor: 3

572:     test:
573:       suffix: sundials
574:       requires: sundials2
575:       args: -nox -ts_type sundials -ts_max_steps 5 -nonlinear
576:       nsize: 4

578:     test:
579:       suffix: sundials_dense
580:       requires: sundials2
581:       args: -nox -ts_type sundials -ts_sundials_use_dense -ts_max_steps 5 -nonlinear
582:       nsize: 1

584: TEST*/