Actual source code: ex3.c

petsc-3.4.3 2013-10-15
```  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: 1
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:        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 parallel version of this code is ts/examples/tutorials/ex4.c

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

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

58: #include <petscts.h>
59: #include <petscdraw.h>

61: /*
62:    User-defined application context - contains data needed by the
63:    application-provided call-back routines.
64: */
65: typedef struct {
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*,MatStructure*,void*);
79: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
80: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);

84: int main(int argc,char **argv)
85: {
86:   AppCtx         appctx;                 /* user-defined application context */
87:   TS             ts;                     /* timestepping context */
88:   Mat            A;                      /* matrix data structure */
89:   Vec            u;                      /* approximate solution vector */
90:   PetscReal      time_total_max = 100.0; /* default max total time */
91:   PetscInt       time_steps_max = 100;   /* default max timesteps */
92:   PetscDraw      draw;                   /* drawing context */
94:   PetscInt       steps,m;
95:   PetscMPIInt    size;
96:   PetscReal      dt;
97:   PetscBool      flg;

99:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
100:      Initialize program and set problem parameters
101:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

103:   PetscInitialize(&argc,&argv,(char*)0,help);
104:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
105:   if (size != 1) SETERRQ(PETSC_COMM_SELF,1,"This is a uniprocessor example only!");

107:   m    = 60;
108:   PetscOptionsGetInt(NULL,"-m",&m,NULL);
109:   PetscOptionsHasName(NULL,"-debug",&appctx.debug);

111:   appctx.m        = m;
112:   appctx.h        = 1.0/(m-1.0);
113:   appctx.norm_2   = 0.0;
114:   appctx.norm_max = 0.0;

116:   PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");

118:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
119:      Create vector data structures
120:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

128:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
129:      Set up displays to show graphs of the solution and error
130:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

139:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
140:      Create timestepping solver context
141:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

143:   TSCreate(PETSC_COMM_SELF,&ts);
144:   TSSetProblemType(ts,TS_LINEAR);

146:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
147:      Set optional user-defined monitoring routine
148:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

150:   TSMonitorSet(ts,Monitor,&appctx,NULL);

152:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

154:      Create matrix data structure; set matrix evaluation routine.
155:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

157:   MatCreate(PETSC_COMM_SELF,&A);
158:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
159:   MatSetFromOptions(A);
160:   MatSetUp(A);

162:   flg  = PETSC_FALSE;
163:   PetscOptionsGetBool(NULL,"-time_dependent_rhs",&flg,NULL);
164:   if (flg) {
165:     /*
166:        For linear problems with a time-dependent f(u,t) in the equation
167:        u_t = f(u,t), the user provides the discretized right-hand-side
168:        as a time-dependent matrix.
169:     */
170:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
171:     TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
172:   } else {
173:     /*
174:        For linear problems with a time-independent f(u) in the equation
175:        u_t = f(u), the user provides the discretized right-hand-side
176:        as a matrix only once, and then sets the special Jacobian evaluation
177:        routine TSComputeRHSJacobianConstant() which will NOT recompute the Jacobian.
178:     */
179:     MatStructure A_structure;
180:     RHSMatrixHeat(ts,0.0,u,&A,&A,&A_structure,&appctx);
181:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
182:     TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
183:   }

185:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
186:      Set solution vector and initial timestep
187:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

189:   dt   = appctx.h*appctx.h/2.0;
190:   TSSetInitialTimeStep(ts,0.0,dt);

192:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
193:      Customize timestepping solver:
194:        - Set the solution method to be the Backward Euler method.
195:        - Set timestepping duration info
196:      Then set runtime options, which can override these defaults.
197:      For example,
198:           -ts_max_steps <maxsteps> -ts_final_time <maxtime>
199:      to override the defaults set by TSSetDuration().
200:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

205:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
206:      Solve the problem
207:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

214:   /*
215:      Run the timestepping solver
216:   */
217:   TSSolve(ts,u);
218:   TSGetTimeStepNumber(ts,&steps);

220:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
221:      View timestepping solver info
222:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

227:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
228:      Free work space.  All PETSc objects should be destroyed when they
229:      are no longer needed.
230:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

239:   /*
240:      Always call PetscFinalize() before exiting a program.  This routine
241:        - finalizes the PETSc libraries as well as MPI
242:        - provides summary and diagnostic information if certain runtime
243:          options are chosen (e.g., -log_summary).
244:   */
245:   PetscFinalize();
246:   return 0;
247: }
248: /* --------------------------------------------------------------------- */
251: /*
252:    InitialConditions - Computes the solution at the initial time.

254:    Input Parameter:
255:    u - uninitialized solution vector (global)
256:    appctx - user-defined application context

258:    Output Parameter:
259:    u - vector with solution at initial time (global)
260: */
261: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
262: {
263:   PetscScalar    *u_localptr,h = appctx->h;
265:   PetscInt       i;

267:   /*
268:     Get a pointer to vector data.
269:     - For default PETSc vectors, VecGetArray() returns a pointer to
270:       the data array.  Otherwise, the routine is implementation dependent.
271:     - You MUST call VecRestoreArray() when you no longer need access to
272:       the array.
273:     - Note that the Fortran interface to VecGetArray() differs from the
274:       C version.  See the users manual for details.
275:   */
276:   VecGetArray(u,&u_localptr);

278:   /*
279:      We initialize the solution array by simply writing the solution
280:      directly into the array locations.  Alternatively, we could use
281:      VecSetValues() or VecSetValuesLocal().
282:   */
283:   for (i=0; i<appctx->m; i++) u_localptr[i] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);

285:   /*
286:      Restore vector
287:   */
288:   VecRestoreArray(u,&u_localptr);

290:   /*
291:      Print debugging information if desired
292:   */
293:   if (appctx->debug) {
294:     PetscPrintf(PETSC_COMM_WORLD,"Initial guess vector\n");
295:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
296:   }

298:   return 0;
299: }
300: /* --------------------------------------------------------------------- */
303: /*
304:    ExactSolution - Computes the exact solution at a given time.

306:    Input Parameters:
307:    t - current time
308:    solution - vector in which exact solution will be computed
309:    appctx - user-defined application context

311:    Output Parameter:
312:    solution - vector with the newly computed exact solution
313: */
314: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
315: {
316:   PetscScalar    *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
318:   PetscInt       i;

320:   /*
321:      Get a pointer to vector data.
322:   */
323:   VecGetArray(solution,&s_localptr);

325:   /*
326:      Simply write the solution directly into the array locations.
327:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
328:   */
329:   ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc);
330:   ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
331:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
332:   for (i=0; i<appctx->m; i++) s_localptr[i] = PetscSinScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i)*ex2;

334:   /*
335:      Restore vector
336:   */
337:   VecRestoreArray(solution,&s_localptr);
338:   return 0;
339: }
340: /* --------------------------------------------------------------------- */
343: /*
344:    Monitor - User-provided routine to monitor the solution computed at
345:    each timestep.  This example plots the solution and computes the
346:    error in two different norms.

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

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

367:   /*
368:      View a graph of the current iterate
369:   */
370:   VecView(u,appctx->viewer2);

372:   /*
373:      Compute the exact solution
374:   */
375:   ExactSolution(time,appctx->solution,appctx);

377:   /*
378:      Print debugging information if desired
379:   */
380:   if (appctx->debug) {
381:     PetscPrintf(PETSC_COMM_SELF,"Computed solution vector\n");
382:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
383:     PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");
384:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
385:   }

387:   /*
388:      Compute the 2-norm and max-norm of the error
389:   */
390:   VecAXPY(appctx->solution,-1.0,u);
391:   VecNorm(appctx->solution,NORM_2,&norm_2);
392:   norm_2 = PetscSqrtReal(appctx->h)*norm_2;
393:   VecNorm(appctx->solution,NORM_MAX,&norm_max);

395:   TSGetTimeStep(ts,&dt);
396:   PetscPrintf(PETSC_COMM_WORLD,"Timestep %3D: step size = %-11g, time = %-11g, 2-norm error = %-11g, max norm error = %-11g\n",step,dt,time,norm_2,norm_max);

398:   appctx->norm_2   += norm_2;
399:   appctx->norm_max += norm_max;

401:   dttol = .0001;
402:   PetscOptionsGetReal(NULL,"-dttol",&dttol,NULL);
403:   if (dt < dttol) {
404:     dt  *= .999;
405:     TSSetTimeStep(ts,dt);
406:   }

408:   /*
409:      View a graph of the error
410:   */
411:   VecView(appctx->solution,appctx->viewer1);

413:   /*
414:      Print debugging information if desired
415:   */
416:   if (appctx->debug) {
417:     PetscPrintf(PETSC_COMM_SELF,"Error vector\n");
418:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
419:   }

421:   return 0;
422: }
423: /* --------------------------------------------------------------------- */
426: /*
427:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
428:    matrix for the heat equation.

430:    Input Parameters:
431:    ts - the TS context
432:    t - current time
433:    global_in - global input vector
434:    dummy - optional user-defined context, as set by TSetRHSJacobian()

436:    Output Parameters:
437:    AA - Jacobian matrix
438:    BB - optionally different preconditioning matrix
439:    str - flag indicating matrix structure

441:    Notes:
442:    Recall that MatSetValues() uses 0-based row and column numbers
443:    in Fortran as well as in C.
444: */
445: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
446: {
447:   Mat            A       = *AA;                /* Jacobian matrix */
448:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined application context */
449:   PetscInt       mstart  = 0;
450:   PetscInt       mend    = appctx->m;
452:   PetscInt       i,idx[3];
453:   PetscScalar    v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;

455:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
456:      Compute entries for the locally owned part of the matrix
457:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
458:   /*
459:      Set matrix rows corresponding to boundary data
460:   */

462:   mstart = 0;
463:   v[0]   = 1.0;
464:   MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
465:   mstart++;

467:   mend--;
468:   v[0] = 1.0;
469:   MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);

471:   /*
472:      Set matrix rows corresponding to interior data.  We construct the
473:      matrix one row at a time.
474:   */
475:   v[0] = sone; v[1] = stwo; v[2] = sone;
476:   for (i=mstart; i<mend; i++) {
477:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
478:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
479:   }

481:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
482:      Complete the matrix assembly process and set some options
483:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
484:   /*
485:      Assemble matrix, using the 2-step process:
486:        MatAssemblyBegin(), MatAssemblyEnd()
487:      Computations can be done while messages are in transition
488:      by placing code between these two statements.
489:   */
490:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
491:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

493:   /*
494:      Set flag to indicate that the Jacobian matrix retains an identical
495:      nonzero structure throughout all timestepping iterations (although the
496:      values of the entries change). Thus, we can save some work in setting
497:      up the preconditioner (e.g., no need to redo symbolic factorization for
498:      ILU/ICC preconditioners).
499:       - If the nonzero structure of the matrix is different during
500:         successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
501:         must be used instead.  If you are unsure whether the matrix
502:         structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
503:       - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc
504:         believes your assertion and does not check the structure
505:         of the matrix.  If you erroneously claim that the structure
506:         is the same when it actually is not, the new preconditioner
507:         will not function correctly.  Thus, use this optimization
508:         feature with caution!
509:   */
510:   *str = SAME_NONZERO_PATTERN;

512:   /*
513:      Set and option to indicate that we will never add a new nonzero location
514:      to the matrix. If we do, it will generate an error.
515:   */
516:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

518:   return 0;
519: }
```