Actual source code: ex6.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: /*
10:    Concepts: TS^time-dependent linear problems
11:    Concepts: TS^heat equation
12:    Concepts: TS^diffusion equation
13:    Routines: TSCreate(); TSSetSolution(); TSSetRHSJacobian(), TSSetIJacobian();
14:    Routines: TSSetTimeStep(); TSSetMaxTime(); TSMonitorSet();
15:    Routines: TSSetFromOptions(); TSStep(); TSDestroy();
16:    Routines: TSSetTimeStep(); TSGetTimeStep();
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/tutorials/ex4.c

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

52: /*
53:    Include "ts.h" so that we can use TS solvers.  Note that this file
54:    automatically includes:
55:      petscsys.h  - base PETSc routines   vec.h  - vectors
56:      sys.h    - system routines       mat.h  - matrices
57:      is.h     - index sets            ksp.h  - Krylov subspace methods
58:      viewer.h - viewers               pc.h   - preconditioners
59:      snes.h - nonlinear solvers
60: */

62: #include <petscts.h>
63: #include <petscdraw.h>

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

78: /*
79:    User-defined routines
80: */
81: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
82: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat,Mat,void*);
83: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
84: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);
85: extern PetscErrorCode MyBCRoutine(TS,PetscReal,Vec,void*);

87: int main(int argc,char **argv)
88: {
89:   AppCtx         appctx;                 /* user-defined application context */
90:   TS             ts;                     /* timestepping context */
91:   Mat            A;                      /* matrix data structure */
92:   Vec            u;                      /* approximate solution vector */
93:   PetscReal      time_total_max = 100.0; /* default max total time */
94:   PetscInt       time_steps_max = 100;   /* default max timesteps */
95:   PetscDraw      draw;                   /* drawing context */
97:   PetscInt       steps, m;
98:   PetscMPIInt    size;
99:   PetscReal      dt;
100:   PetscReal      ftime;
101:   PetscBool      flg;
102:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
103:      Initialize program and set problem parameters
104:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

106:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
107:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
108:   if (size != 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_WRONG_MPI_SIZE,"This is a uniprocessor example only!");

110:   m    = 60;
111:   PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
112:   PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);

114:   appctx.m        = m;
115:   appctx.h        = 1.0/(m-1.0);
116:   appctx.norm_2   = 0.0;
117:   appctx.norm_max = 0.0;

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

121:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
122:      Create vector data structures
123:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

125:   /*
126:      Create vector data structures for approximate and exact solutions
127:   */
128:   VecCreateSeq(PETSC_COMM_SELF,m,&u);
129:   VecDuplicate(u,&appctx.solution);

131:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
132:      Set up displays to show graphs of the solution and error
133:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

142:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
143:      Create timestepping solver context
144:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

146:   TSCreate(PETSC_COMM_SELF,&ts);
147:   TSSetProblemType(ts,TS_LINEAR);

149:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
150:      Set optional user-defined monitoring routine
151:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

155:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

157:      Create matrix data structure; set matrix evaluation routine.
158:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

160:   MatCreate(PETSC_COMM_SELF,&A);
161:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
162:   MatSetFromOptions(A);
163:   MatSetUp(A);

165:   PetscOptionsHasName(NULL,NULL,"-time_dependent_rhs",&flg);
166:   if (flg) {
167:     /*
168:        For linear problems with a time-dependent f(u,t) in the equation
169:        u_t = f(u,t), the user provides the discretized right-hand-side
170:        as a time-dependent matrix.
171:     */
172:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
173:     TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
174:   } else {
175:     /*
176:        For linear problems with a time-independent f(u) in the equation
177:        u_t = f(u), the user provides the discretized right-hand-side
178:        as a matrix only once, and then sets a null matrix evaluation
179:        routine.
180:     */
181:     RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
182:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
183:     TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
184:   }

186:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
187:      Set solution vector and initial timestep
188:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

190:   dt   = appctx.h*appctx.h/2.0;
191:   TSSetTimeStep(ts,dt);
192:   TSSetSolution(ts,u);

194:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
195:      Customize timestepping solver:
196:        - Set the solution method to be the Backward Euler method.
197:        - Set timestepping duration info
198:      Then set runtime options, which can override these defaults.
199:      For example,
200:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
201:      to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
202:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

204:   TSSetMaxSteps(ts,time_steps_max);
205:   TSSetMaxTime(ts,time_total_max);
206:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
207:   TSSetFromOptions(ts);

209:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
210:      Solve the problem
211:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

213:   /*
214:      Evaluate initial conditions
215:   */
216:   InitialConditions(u,&appctx);

218:   /*
219:      Run the timestepping solver
220:   */
221:   TSSolve(ts,u);
222:   TSGetSolveTime(ts,&ftime);
223:   TSGetStepNumber(ts,&steps);

225:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
226:      View timestepping solver info
227:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

229:   PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %g, avg. error (max norm) = %g\n",(double)(appctx.norm_2/steps),(double)(appctx.norm_max/steps));
230:   TSView(ts,PETSC_VIEWER_STDOUT_SELF);

232:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
233:      Free work space.  All PETSc objects should be destroyed when they
234:      are no longer needed.
235:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

237:   TSDestroy(&ts);
238:   MatDestroy(&A);
239:   VecDestroy(&u);
240:   PetscViewerDestroy(&appctx.viewer1);
241:   PetscViewerDestroy(&appctx.viewer2);
242:   VecDestroy(&appctx.solution);

244:   /*
245:      Always call PetscFinalize() before exiting a program.  This routine
246:        - finalizes the PETSc libraries as well as MPI
247:        - provides summary and diagnostic information if certain runtime
248:          options are chosen (e.g., -log_view).
249:   */
250:   PetscFinalize();
251:   return ierr;
252: }
253: /* --------------------------------------------------------------------- */
254: /*
255:    InitialConditions - Computes the solution at the initial time.

257:    Input Parameter:
258:    u - uninitialized solution vector (global)
259:    appctx - user-defined application context

261:    Output Parameter:
262:    u - vector with solution at initial time (global)
263: */
264: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
265: {
266:   PetscScalar    *u_localptr;
267:   PetscInt       i;

270:   /*
271:     Get a pointer to vector data.
272:     - For default PETSc vectors, VecGetArray() returns a pointer to
273:       the data array.  Otherwise, the routine is implementation dependent.
274:     - You MUST call VecRestoreArray() when you no longer need access to
275:       the array.
276:     - Note that the Fortran interface to VecGetArray() differs from the
277:       C version.  See the users manual for details.
278:   */
279:   VecGetArray(u,&u_localptr);

281:   /*
282:      We initialize the solution array by simply writing the solution
283:      directly into the array locations.  Alternatively, we could use
284:      VecSetValues() or VecSetValuesLocal().
285:   */
286:   for (i=0; i<appctx->m; i++) u_localptr[i] = PetscSinReal(PETSC_PI*i*6.*appctx->h) + 3.*PetscSinReal(PETSC_PI*i*2.*appctx->h);

288:   /*
289:      Restore vector
290:   */
291:   VecRestoreArray(u,&u_localptr);

293:   /*
294:      Print debugging information if desired
295:   */
296:   if (appctx->debug) {
297:      VecView(u,PETSC_VIEWER_STDOUT_SELF);
298:   }

300:   return 0;
301: }
302: /* --------------------------------------------------------------------- */
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;
317:   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 = PetscExpReal(-36.*PETSC_PI*PETSC_PI*t); ex2 = PetscExpReal(-4.*PETSC_PI*PETSC_PI*t);
330:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
331:   for (i=0; i<appctx->m; i++) s_localptr[i] = PetscSinReal(PetscRealPart(sc1)*(PetscReal)i)*ex1 + 3.*PetscSinReal(PetscRealPart(sc2)*(PetscReal)i)*ex2;

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

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

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

365:   /*
366:      View a graph of the current iterate
367:   */
368:   VecView(u,appctx->viewer2);

370:   /*
371:      Compute the exact solution
372:   */
373:   ExactSolution(crtime,appctx->solution,appctx);

375:   /*
376:      Print debugging information if desired
377:   */
378:   if (appctx->debug) {
379:     PetscPrintf(PETSC_COMM_SELF,"Computed solution vector\n");
380:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
381:     PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");
382:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
383:   }

385:   /*
386:      Compute the 2-norm and max-norm of the error
387:   */
388:   VecAXPY(appctx->solution,-1.0,u);
389:   VecNorm(appctx->solution,NORM_2,&norm_2);
390:   norm_2 = PetscSqrtReal(appctx->h)*norm_2;
391:   VecNorm(appctx->solution,NORM_MAX,&norm_max);

393:   TSGetTimeStep(ts,&dt);
394:   if (norm_2 > 1.e-2) {
395:     PetscPrintf(PETSC_COMM_SELF,"Timestep %D: step size = %g, time = %g, 2-norm error = %g, max norm error = %g\n",step,(double)dt,(double)crtime,(double)norm_2,(double)norm_max);
396:   }
397:   appctx->norm_2   += norm_2;
398:   appctx->norm_max += norm_max;

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

407:   /*
408:      View a graph of the error
409:   */
410:   VecView(appctx->solution,appctx->viewer1);

412:   /*
413:      Print debugging information if desired
414:   */
415:   if (appctx->debug) {
416:     PetscPrintf(PETSC_COMM_SELF,"Error vector\n");
417:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
418:   }

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

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

433:    Output Parameters:
434:    AA - Jacobian matrix
435:    BB - optionally different preconditioning matrix
436:    str - flag indicating matrix structure

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

452:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
453:      Compute entries for the locally owned part of the matrix
454:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
455:   /*
456:      Set matrix rows corresponding to boundary data
457:   */

459:   mstart = 0;
460:   v[0]   = 1.0;
461:   MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
462:   mstart++;

464:   mend--;
465:   v[0] = 1.0;
466:   MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);

468:   /*
469:      Set matrix rows corresponding to interior data.  We construct the
470:      matrix one row at a time.
471:   */
472:   v[0] = sone; v[1] = stwo; v[2] = sone;
473:   for (i=mstart; i<mend; i++) {
474:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
475:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
476:   }

478:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
479:      Complete the matrix assembly process and set some options
480:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
481:   /*
482:      Assemble matrix, using the 2-step process:
483:        MatAssemblyBegin(), MatAssemblyEnd()
484:      Computations can be done while messages are in transition
485:      by placing code between these two statements.
486:   */
487:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
488:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

490:   /*
491:      Set and option to indicate that we will never add a new nonzero location
492:      to the matrix. If we do, it will generate an error.
493:   */
494:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

496:   return 0;
497: }
498: /* --------------------------------------------------------------------- */
499: /*
500:    Input Parameters:
501:    ts - the TS context
502:    t - current time
503:    f - function
504:    ctx - optional user-defined context, as set by TSetBCFunction()
505:  */
506: PetscErrorCode MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
507: {
508:   AppCtx         *appctx = (AppCtx*) ctx;      /* user-defined application context */
510:   PetscInt       m = appctx->m;
511:   PetscScalar    *fa;

513:   VecGetArray(f,&fa);
514:   fa[0]   = 0.0;
515:   fa[m-1] = 1.0;
516:   VecRestoreArray(f,&fa);
517:   PetscPrintf(PETSC_COMM_SELF,"t=%g\n",(double)t);

519:   return 0;
520: }

522: /*TEST

524:     test:
525:       args: -nox -ts_max_steps 4

527: TEST*/
```