Actual source code: ex4.c

petsc-main 2021-04-20
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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/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*);

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 = 1.0;   /* 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,ftime;
103:   PetscBool      flg;
104:   TSProblemType  tsproblem = TS_LINEAR;

106:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
107:      Initialize program and set problem parameters
108:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

110:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
111:   appctx.comm = PETSC_COMM_WORLD;

113:   m               = 60;
114:   PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
115:   PetscOptionsHasName(NULL,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;

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

124:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
125:      Create vector data structures
126:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
127:   /*
128:      Create distributed array (DMDA) to manage parallel grid and vectors
129:      and to set up the ghost point communication pattern.  There are M
130:      total grid values spread equally among all the processors.
131:   */

133:   DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,m,1,1,NULL,&appctx.da);
134:   DMSetFromOptions(appctx.da);
135:   DMSetUp(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:   TSSetTimeStep(ts,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_max_time <maxtime>
232:      to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
233:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

235:   TSSetMaxSteps(ts,time_steps_max);
236:   TSSetMaxTime(ts,time_total_max);
237:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
238:   TSSetFromOptions(ts);

240:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
241:      Solve the problem
242:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

256:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
257:      View timestepping solver info
258:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
259:   PetscPrintf(PETSC_COMM_WORLD,"Total timesteps %D, Final time %g\n",steps,(double)ftime);
260:   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));

262:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
263:      Free work space.  All PETSc objects should be destroyed when they
264:      are no longer needed.
265:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

277:   /*
278:      Always call PetscFinalize() before exiting a program.  This routine
279:        - finalizes the PETSc libraries as well as MPI
280:        - provides summary and diagnostic information if certain runtime
281:          options are chosen (e.g., -log_view).
282:   */
283:   PetscFinalize();
284:   return ierr;
285: }
286: /* --------------------------------------------------------------------- */
287: /*
288:    InitialConditions - Computes the solution at the initial time.

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

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

303:   /*
304:      Determine starting point of each processor's range of
305:      grid values.
306:   */
307:   VecGetOwnershipRange(u,&mybase,&myend);

309:   /*
310:     Get a pointer to vector data.
311:     - For default PETSc vectors, VecGetArray() returns a pointer to
312:       the data array.  Otherwise, the routine is implementation dependent.
313:     - You MUST call VecRestoreArray() when you no longer need access to
314:       the array.
315:     - Note that the Fortran interface to VecGetArray() differs from the
316:       C version.  See the users manual for details.
317:   */
318:   VecGetArray(u,&u_localptr);

320:   /*
321:      We initialize the solution array by simply writing the solution
322:      directly into the array locations.  Alternatively, we could use
323:      VecSetValues() or VecSetValuesLocal().
324:   */
325:   for (i=mybase; i<myend; i++) u_localptr[i-mybase] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);

327:   /*
328:      Restore vector
329:   */
330:   VecRestoreArray(u,&u_localptr);

332:   /*
333:      Print debugging information if desired
334:   */
335:   if (appctx->debug) {
336:     PetscPrintf(appctx->comm,"initial guess vector\n");
337:     VecView(u,PETSC_VIEWER_STDOUT_WORLD);
338:   }

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

346:    Input Parameters:
347:    t - current time
348:    solution - vector in which exact solution will be computed
349:    appctx - user-defined application context

351:    Output Parameter:
352:    solution - vector with the newly computed exact solution
353: */
354: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
355: {
356:   PetscScalar    *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2;
357:   PetscInt       i,mybase,myend;

360:   /*
361:      Determine starting and ending points of each processor's
362:      range of grid values
363:   */
364:   VecGetOwnershipRange(solution,&mybase,&myend);

366:   /*
367:      Get a pointer to vector data.
368:   */
369:   VecGetArray(solution,&s_localptr);

371:   /*
372:      Simply write the solution directly into the array locations.
373:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
374:   */
375:   ex1 = PetscExpReal(-36.*PETSC_PI*PETSC_PI*t); ex2 = PetscExpReal(-4.*PETSC_PI*PETSC_PI*t);
376:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
377:   for (i=mybase; i<myend; i++) s_localptr[i-mybase] = PetscSinScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i)*ex2;

379:   /*
380:      Restore vector
381:   */
382:   VecRestoreArray(solution,&s_localptr);
383:   return 0;
384: }
385: /* --------------------------------------------------------------------- */
386: /*
387:    Monitor - User-provided routine to monitor the solution computed at
388:    each timestep.  This example plots the solution and computes the
389:    error in two different norms.

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

408:   /*
409:      View a graph of the current iterate
410:   */
411:   VecView(u,appctx->viewer2);

413:   /*
414:      Compute the exact solution
415:   */
416:   ExactSolution(time,appctx->solution,appctx);

418:   /*
419:      Print debugging information if desired
420:   */
421:   if (appctx->debug) {
422:     PetscPrintf(appctx->comm,"Computed solution vector\n");
423:     VecView(u,PETSC_VIEWER_STDOUT_WORLD);
424:     PetscPrintf(appctx->comm,"Exact solution vector\n");
425:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
426:   }

428:   /*
429:      Compute the 2-norm and max-norm of the error
430:   */
431:   VecAXPY(appctx->solution,-1.0,u);
432:   VecNorm(appctx->solution,NORM_2,&norm_2);
433:   norm_2 = PetscSqrtReal(appctx->h)*norm_2;
434:   VecNorm(appctx->solution,NORM_MAX,&norm_max);
435:   if (norm_2   < 1e-14) norm_2   = 0;
436:   if (norm_max < 1e-14) norm_max = 0;

438:   /*
439:      PetscPrintf() causes only the first processor in this
440:      communicator to print the timestep information.
441:   */
442:   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);
443:   appctx->norm_2   += norm_2;
444:   appctx->norm_max += norm_max;

446:   /*
447:      View a graph of the error
448:   */
449:   VecView(appctx->solution,appctx->viewer1);

451:   /*
452:      Print debugging information if desired
453:   */
454:   if (appctx->debug) {
455:     PetscPrintf(appctx->comm,"Error vector\n");
456:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
457:   }

459:   return 0;
460: }

462: /* --------------------------------------------------------------------- */
463: /*
464:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
465:    matrix for the heat equation.

467:    Input Parameters:
468:    ts - the TS context
469:    t - current time
470:    global_in - global input vector
471:    dummy - optional user-defined context, as set by TSetRHSJacobian()

473:    Output Parameters:
474:    AA - Jacobian matrix
475:    BB - optionally different preconditioning matrix
476:    str - flag indicating matrix structure

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

499:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
500:      Compute entries for the locally owned part of the matrix
501:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

505:   /*
506:      Set matrix rows corresponding to boundary data
507:   */

509:   if (mstart == 0) {  /* first processor only */
510:     v[0] = 1.0;
511:     MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
512:     mstart++;
513:   }

515:   if (mend == appctx->m) { /* last processor only */
516:     mend--;
517:     v[0] = 1.0;
518:     MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
519:   }

521:   /*
522:      Set matrix rows corresponding to interior data.  We construct the
523:      matrix one row at a time.
524:   */
525:   v[0] = sone; v[1] = stwo; v[2] = sone;
526:   for (i=mstart; i<mend; i++) {
527:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
528:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
529:   }

531:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
532:      Complete the matrix assembly process and set some options
533:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
534:   /*
535:      Assemble matrix, using the 2-step process:
536:        MatAssemblyBegin(), MatAssemblyEnd()
537:      Computations can be done while messages are in transition
538:      by placing code between these two statements.
539:   */
540:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
541:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

543:   /*
544:      Set and option to indicate that we will never add a new nonzero location
545:      to the matrix. If we do, it will generate an error.
546:   */
547:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

549:   return 0;
550: }

552: PetscErrorCode RHSFunctionHeat(TS ts,PetscReal t,Vec globalin,Vec globalout,void *ctx)
553: {
555:   Mat            A;

558:   TSGetRHSJacobian(ts,&A,NULL,NULL,&ctx);
559:   RHSMatrixHeat(ts,t,globalin,A,NULL,ctx);
560:   /* MatView(A,PETSC_VIEWER_STDOUT_WORLD); */
561:   MatMult(A,globalin,globalout);
562:   return(0);
563: }

565: /*TEST

567:     test:
568:       args: -ts_view -nox

570:     test:
571:       suffix: 2
572:       args: -ts_view -nox
573:       nsize: 3

575:     test:
576:       suffix: 3
577:       args: -ts_view -nox -nonlinear

579:     test:
580:       suffix: 4
581:       args: -ts_view -nox -nonlinear
582:       nsize: 3
583:       timeoutfactor: 3

585:     test:
586:       suffix: sundials
587:       requires: sundials2
588:       args: -nox -ts_type sundials -ts_max_steps 5 -nonlinear
589:       nsize: 4

591: TEST*/

```