Actual source code: ex4.c

petsc-3.7.3 2016-07-24
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  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: /*
 10:    Concepts: TS^time-dependent linear problems
 11:    Concepts: TS^heat equation
 12:    Concepts: TS^diffusion equation
 13:    Processors: n
 14: */

 16: /* ------------------------------------------------------------------------

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

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

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

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

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

 46:   ------------------------------------------------------------------------- */

 48: /*
 49:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs) to manage
 50:    the parallel grid.  Include "petscts.h" so that we can use TS solvers.
 51:    Note that this file automatically includes:
 52:      petscsys.h       - base PETSc routines   petscvec.h  - vectors
 53:      petscmat.h  - matrices
 54:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
 55:      petscviewer.h - viewers               petscpc.h   - preconditioners
 56:      petscksp.h   - linear solvers        petscsnes.h - nonlinear solvers
 57: */

 59: #include <petscdm.h>
 60: #include <petscdmda.h>
 61: #include <petscts.h>
 62: #include <petscdraw.h>

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

 81: /*
 82:    User-defined routines
 83: */
 84: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
 85: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat,Mat,void*);
 86: extern PetscErrorCode RHSFunctionHeat(TS,PetscReal,Vec,Vec,void*);
 87: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
 88: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);

 92: int main(int argc,char **argv)
 93: {
 94:   AppCtx         appctx;                 /* user-defined application context */
 95:   TS             ts;                     /* timestepping context */
 96:   Mat            A;                      /* matrix data structure */
 97:   Vec            u;                      /* approximate solution vector */
 98:   PetscReal      time_total_max = 1.0;   /* default max total time */
 99:   PetscInt       time_steps_max = 100;   /* default max timesteps */
100:   PetscDraw      draw;                   /* drawing context */
102:   PetscInt       steps,m;
103:   PetscMPIInt    size;
104:   PetscReal      dt,ftime;
105:   PetscBool      flg;
106:   TSProblemType  tsproblem = TS_LINEAR;

108:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
109:      Initialize program and set problem parameters
110:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

115:   m               = 60;
116:   PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
117:   PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);
118:   appctx.m        = m;
119:   appctx.h        = 1.0/(m-1.0);
120:   appctx.norm_2   = 0.0;
121:   appctx.norm_max = 0.0;

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

126:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
127:      Create vector data structures
128:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
129:   /*
130:      Create distributed array (DMDA) to manage parallel grid and vectors
131:      and to set up the ghost point communication pattern.  There are M
132:      total grid values spread equally among all the processors.
133:   */

135:   DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,m,1,1,NULL,&appctx.da);

137:   /*
138:      Extract global and local vectors from DMDA; we use these to store the
139:      approximate solution.  Then duplicate these for remaining vectors that
140:      have the same types.
141:   */
142:   DMCreateGlobalVector(appctx.da,&u);
143:   DMCreateLocalVector(appctx.da,&appctx.u_local);

145:   /*
146:      Create local work vector for use in evaluating right-hand-side function;
147:      create global work vector for storing exact solution.
148:   */
149:   VecDuplicate(appctx.u_local,&appctx.localwork);
150:   VecDuplicate(u,&appctx.solution);

152:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
153:      Set up displays to show graphs of the solution and error
154:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

156:   PetscViewerDrawOpen(PETSC_COMM_WORLD,0,"",80,380,400,160,&appctx.viewer1);
157:   PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
158:   PetscDrawSetDoubleBuffer(draw);
159:   PetscViewerDrawOpen(PETSC_COMM_WORLD,0,"",80,0,400,160,&appctx.viewer2);
160:   PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
161:   PetscDrawSetDoubleBuffer(draw);

163:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
164:      Create timestepping solver context
165:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

167:   TSCreate(PETSC_COMM_WORLD,&ts);

169:   flg  = PETSC_FALSE;
170:   PetscOptionsGetBool(NULL,NULL,"-nonlinear",&flg,NULL);
171:   TSSetProblemType(ts,flg ? TS_NONLINEAR : TS_LINEAR);

173:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
174:      Set optional user-defined monitoring routine
175:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
176:   TSMonitorSet(ts,Monitor,&appctx,NULL);

178:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

180:      Create matrix data structure; set matrix evaluation routine.
181:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

183:   MatCreate(PETSC_COMM_WORLD,&A);
184:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
185:   MatSetFromOptions(A);
186:   MatSetUp(A);

188:   flg  = PETSC_FALSE;
189:   PetscOptionsGetBool(NULL,NULL,"-time_dependent_rhs",&flg,NULL);
190:   if (flg) {
191:     /*
192:        For linear problems with a time-dependent f(u,t) in the equation
193:        u_t = f(u,t), the user provides the discretized right-hand-side
194:        as a time-dependent matrix.
195:     */
196:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
197:     TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
198:   } else {
199:     /*
200:        For linear problems with a time-independent f(u) in the equation
201:        u_t = f(u), the user provides the discretized right-hand-side
202:        as a matrix only once, and then sets a null matrix evaluation
203:        routine.
204:     */
205:     RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
206:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
207:     TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
208:   }

210:   if (tsproblem == TS_NONLINEAR) {
211:     SNES snes;
212:     TSSetRHSFunction(ts,NULL,RHSFunctionHeat,&appctx);
213:     TSGetSNES(ts,&snes);
214:     SNESSetJacobian(snes,NULL,NULL,SNESComputeJacobianDefault,NULL);
215:   }

217:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
218:      Set solution vector and initial timestep
219:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

221:   dt   = appctx.h*appctx.h/2.0;
222:   TSSetInitialTimeStep(ts,0.0,dt);
223:   TSSetSolution(ts,u);

225:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
226:      Customize timestepping solver:
227:        - Set the solution method to be the Backward Euler method.
228:        - Set timestepping duration info
229:      Then set runtime options, which can override these defaults.
230:      For example,
231:           -ts_max_steps <maxsteps> -ts_final_time <maxtime>
232:      to override the defaults set by TSSetDuration().
233:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

235:   TSSetDuration(ts,time_steps_max,time_total_max);
236:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
237:   TSSetFromOptions(ts);

239:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
240:      Solve the problem
241:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

243:   /*
244:      Evaluate initial conditions
245:   */
246:   InitialConditions(u,&appctx);

248:   /*
249:      Run the timestepping solver
250:   */
251:   TSSolve(ts,u);
252:   TSGetSolveTime(ts,&ftime);
253:   TSGetTimeStepNumber(ts,&steps);

255:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
256:      View timestepping solver info
257:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
258:   PetscPrintf(PETSC_COMM_WORLD,"Total timesteps %D, Final time %g\n",steps,(double)ftime);
259:   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));

261:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
262:      Free work space.  All PETSc objects should be destroyed when they
263:      are no longer needed.
264:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

266:   TSDestroy(&ts);
267:   MatDestroy(&A);
268:   VecDestroy(&u);
269:   PetscViewerDestroy(&appctx.viewer1);
270:   PetscViewerDestroy(&appctx.viewer2);
271:   VecDestroy(&appctx.localwork);
272:   VecDestroy(&appctx.solution);
273:   VecDestroy(&appctx.u_local);
274:   DMDestroy(&appctx.da);

276:   /*
277:      Always call PetscFinalize() before exiting a program.  This routine
278:        - finalizes the PETSc libraries as well as MPI
279:        - provides summary and diagnostic information if certain runtime
280:          options are chosen (e.g., -log_summary).
281:   */
282:   PetscFinalize();
283:   return 0;
284: }
285: /* --------------------------------------------------------------------- */
288: /*
289:    InitialConditions - Computes the solution at the initial time.

291:    Input Parameter:
292:    u - uninitialized solution vector (global)
293:    appctx - user-defined application context

295:    Output Parameter:
296:    u - vector with solution at initial time (global)
297: */
298: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
299: {
300:   PetscScalar    *u_localptr,h = appctx->h;
301:   PetscInt       i,mybase,myend;

304:   /*
305:      Determine starting point of each processor's range of
306:      grid values.
307:   */
308:   VecGetOwnershipRange(u,&mybase,&myend);

310:   /*
311:     Get a pointer to vector data.
312:     - For default PETSc vectors, VecGetArray() returns a pointer to
313:       the data array.  Otherwise, the routine is implementation dependent.
314:     - You MUST call VecRestoreArray() when you no longer need access to
315:       the array.
316:     - Note that the Fortran interface to VecGetArray() differs from the
317:       C version.  See the users manual for details.
318:   */
319:   VecGetArray(u,&u_localptr);

321:   /*
322:      We initialize the solution array by simply writing the solution
323:      directly into the array locations.  Alternatively, we could use
324:      VecSetValues() or VecSetValuesLocal().
325:   */
326:   for (i=mybase; i<myend; i++) u_localptr[i-mybase] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);

328:   /*
329:      Restore vector
330:   */
331:   VecRestoreArray(u,&u_localptr);

333:   /*
334:      Print debugging information if desired
335:   */
336:   if (appctx->debug) {
337:     PetscPrintf(appctx->comm,"initial guess vector\n");
338:     VecView(u,PETSC_VIEWER_STDOUT_WORLD);
339:   }

341:   return 0;
342: }
343: /* --------------------------------------------------------------------- */
346: /*
347:    ExactSolution - Computes the exact solution at a given time.

349:    Input Parameters:
350:    t - current time
351:    solution - vector in which exact solution will be computed
352:    appctx - user-defined application context

354:    Output Parameter:
355:    solution - vector with the newly computed exact solution
356: */
357: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
358: {
359:   PetscScalar    *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2;
360:   PetscInt       i,mybase,myend;

363:   /*
364:      Determine starting and ending points of each processor's
365:      range of grid values
366:   */
367:   VecGetOwnershipRange(solution,&mybase,&myend);

369:   /*
370:      Get a pointer to vector data.
371:   */
372:   VecGetArray(solution,&s_localptr);

374:   /*
375:      Simply write the solution directly into the array locations.
376:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
377:   */
378:   ex1 = PetscExpReal(-36.*PETSC_PI*PETSC_PI*t); ex2 = PetscExpReal(-4.*PETSC_PI*PETSC_PI*t);
379:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
380:   for (i=mybase; i<myend; i++) s_localptr[i-mybase] = PetscSinScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i)*ex2;

382:   /*
383:      Restore vector
384:   */
385:   VecRestoreArray(solution,&s_localptr);
386:   return 0;
387: }
388: /* --------------------------------------------------------------------- */
391: /*
392:    Monitor - User-provided routine to monitor the solution computed at
393:    each timestep.  This example plots the solution and computes the
394:    error in two different norms.

396:    Input Parameters:
397:    ts     - the timestep context
398:    step   - the count of the current step (with 0 meaning the
399:              initial condition)
400:    time   - the current time
401:    u      - the solution at this timestep
402:    ctx    - the user-provided context for this monitoring routine.
403:             In this case we use the application context which contains
404:             information about the problem size, workspace and the exact
405:             solution.
406: */
407: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
408: {
409:   AppCtx         *appctx = (AppCtx*) ctx;   /* user-defined application context */
411:   PetscReal      norm_2,norm_max;

413:   /*
414:      View a graph of the current iterate
415:   */
416:   VecView(u,appctx->viewer2);

418:   /*
419:      Compute the exact solution
420:   */
421:   ExactSolution(time,appctx->solution,appctx);

423:   /*
424:      Print debugging information if desired
425:   */
426:   if (appctx->debug) {
427:     PetscPrintf(appctx->comm,"Computed solution vector\n");
428:     VecView(u,PETSC_VIEWER_STDOUT_WORLD);
429:     PetscPrintf(appctx->comm,"Exact solution vector\n");
430:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
431:   }

433:   /*
434:      Compute the 2-norm and max-norm of the error
435:   */
436:   VecAXPY(appctx->solution,-1.0,u);
437:   VecNorm(appctx->solution,NORM_2,&norm_2);
438:   norm_2 = PetscSqrtReal(appctx->h)*norm_2;
439:   VecNorm(appctx->solution,NORM_MAX,&norm_max);
440:   if (norm_2   < 1e-14) norm_2   = 0;
441:   if (norm_max < 1e-14) norm_max = 0;

443:   /*
444:      PetscPrintf() causes only the first processor in this
445:      communicator to print the timestep information.
446:   */
447:   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);
448:   appctx->norm_2   += norm_2;
449:   appctx->norm_max += norm_max;

451:   /*
452:      View a graph of the error
453:   */
454:   VecView(appctx->solution,appctx->viewer1);

456:   /*
457:      Print debugging information if desired
458:   */
459:   if (appctx->debug) {
460:     PetscPrintf(appctx->comm,"Error vector\n");
461:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
462:   }

464:   return 0;
465: }

467: /* --------------------------------------------------------------------- */
470: /*
471:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
472:    matrix for the heat equation.

474:    Input Parameters:
475:    ts - the TS context
476:    t - current time
477:    global_in - global input vector
478:    dummy - optional user-defined context, as set by TSetRHSJacobian()

480:    Output Parameters:
481:    AA - Jacobian matrix
482:    BB - optionally different preconditioning matrix
483:    str - flag indicating matrix structure

485:   Notes:
486:   RHSMatrixHeat computes entries for the locally owned part of the system.
487:    - Currently, all PETSc parallel matrix formats are partitioned by
488:      contiguous chunks of rows across the processors.
489:    - Each processor needs to insert only elements that it owns
490:      locally (but any non-local elements will be sent to the
491:      appropriate processor during matrix assembly).
492:    - Always specify global row and columns of matrix entries when
493:      using MatSetValues(); we could alternatively use MatSetValuesLocal().
494:    - Here, we set all entries for a particular row at once.
495:    - Note that MatSetValues() uses 0-based row and column numbers
496:      in Fortran as well as in C.
497: */
498: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
499: {
500:   Mat            A       = AA;              /* Jacobian matrix */
501:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined application context */
503:   PetscInt       i,mstart,mend,idx[3];
504:   PetscScalar    v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;

506:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
507:      Compute entries for the locally owned part of the matrix
508:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

512:   /*
513:      Set matrix rows corresponding to boundary data
514:   */

516:   if (mstart == 0) {  /* first processor only */
517:     v[0] = 1.0;
518:     MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
519:     mstart++;
520:   }

522:   if (mend == appctx->m) { /* last processor only */
523:     mend--;
524:     v[0] = 1.0;
525:     MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
526:   }

528:   /*
529:      Set matrix rows corresponding to interior data.  We construct the
530:      matrix one row at a time.
531:   */
532:   v[0] = sone; v[1] = stwo; v[2] = sone;
533:   for (i=mstart; i<mend; i++) {
534:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
535:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
536:   }

538:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
539:      Complete the matrix assembly process and set some options
540:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
541:   /*
542:      Assemble matrix, using the 2-step process:
543:        MatAssemblyBegin(), MatAssemblyEnd()
544:      Computations can be done while messages are in transition
545:      by placing code between these two statements.
546:   */
547:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
548:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

550:   /*
551:      Set and option to indicate that we will never add a new nonzero location
552:      to the matrix. If we do, it will generate an error.
553:   */
554:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

556:   return 0;
557: }

561: PetscErrorCode RHSFunctionHeat(TS ts,PetscReal t,Vec globalin,Vec globalout,void *ctx)
562: {
564:   Mat            A;

567:   TSGetRHSJacobian(ts,&A,NULL,NULL,&ctx);
568:   RHSMatrixHeat(ts,t,globalin,A,NULL,ctx);
569:   /* MatView(A,PETSC_VIEWER_STDOUT_WORLD); */
570:   MatMult(A,globalin,globalout);
571:   return(0);
572: }