Actual source code: ex5.c

petsc-dev 2014-02-02
Report Typos and Errors
  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) = 1, u(t,1) = 1,
 23:    and the initial condition
 24:        u(0,x) = cos(6*pi*x) + 3*cos(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) * cos(6*pi*x) +
 36:                       3*exp(-4*pi*pi*t) * cos(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 just 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: */
 57: #include <petscts.h>
 58: #include <petscdraw.h>

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

 73: /*
 74:    User-defined routines
 75: */
 76: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
 77: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat*,Mat*,MatStructure*,void*);
 78: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
 79: 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 = 100.0; /* default max total time */
 90:   PetscInt       time_steps_max = 100;   /* default max timesteps */
 91:   PetscDraw      draw;                   /* drawing context */
 93:   PetscInt       steps,m;
 94:   PetscMPIInt    size;
 95:   PetscBool      flg;
 96:   PetscReal      dt,ftime;

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

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

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

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

116:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
117:      Create vector data structures
118:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

120:   /*
121:      Create vector data structures for approximate and exact solutions
122:   */
123:   VecCreateSeq(PETSC_COMM_SELF,m,&u);
124:   VecDuplicate(u,&appctx.solution);

126:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
127:      Set up displays to show graphs of the solution and error
128:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

137:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
138:      Create timestepping solver context
139:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

141:   TSCreate(PETSC_COMM_SELF,&ts);
142:   TSSetProblemType(ts,TS_LINEAR);

144:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
145:      Set optional user-defined monitoring routine
146:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

150:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

152:      Create matrix data structure; set matrix evaluation routine.
153:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

160:   PetscOptionsHasName(NULL,"-time_dependent_rhs",&flg);
161:   if (flg) {
162:     /*
163:        For linear problems with a time-dependent f(u,t) in the equation
164:        u_t = f(u,t), the user provides the discretized right-hand-side
165:        as a time-dependent matrix.
166:     */
167:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
168:     TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
169:   } else {
170:     /*
171:        For linear problems with a time-independent f(u) in the equation
172:        u_t = f(u), the user provides the discretized right-hand-side
173:        as a matrix only once, and then sets a null matrix evaluation
174:        routine.
175:     */
176:     MatStructure A_structure;
177:     RHSMatrixHeat(ts,0.0,u,&A,&A,&A_structure,&appctx);
178:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
179:     TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
180:   }

182:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
183:      Set solution vector and initial timestep
184:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

186:   dt   = appctx.h*appctx.h/2.0;
187:   TSSetInitialTimeStep(ts,0.0,dt);
188:   TSSetSolution(ts,u);

190:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
191:      Customize timestepping solver:
192:        - Set the solution method to be the Backward Euler method.
193:        - Set timestepping duration info
194:      Then set runtime options, which can override these defaults.
195:      For example,
196:           -ts_max_steps <maxsteps> -ts_final_time <maxtime>
197:      to override the defaults set by TSSetDuration().
198:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

200:   TSSetDuration(ts,time_steps_max,time_total_max);
201:   TSSetFromOptions(ts);

203:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
204:      Solve the problem
205:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

207:   /*
208:      Evaluate initial conditions
209:   */
210:   InitialConditions(u,&appctx);

212:   /*
213:      Run the timestepping solver
214:   */
215:   TSSolve(ts,u);
216:   TSGetSolveTime(ts,&ftime);
217:   TSGetTimeStepNumber(ts,&steps);

219:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
220:      View timestepping solver info
221:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

223:   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));
224:   TSView(ts,PETSC_VIEWER_STDOUT_SELF);

226:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
227:      Free work space.  All PETSc objects should be destroyed when they
228:      are no longer needed.
229:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

238:   /*
239:      Always call PetscFinalize() before exiting a program.  This routine
240:        - finalizes the PETSc libraries as well as MPI
241:        - provides summary and diagnostic information if certain runtime
242:          options are chosen (e.g., -log_summary).
243:   */
244:   PetscFinalize();
245:   return 0;
246: }
247: /* --------------------------------------------------------------------- */
250: /*
251:    InitialConditions - Computes the solution at the initial time.

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

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

266:   /*
267:     Get a pointer to vector data.
268:     - For default PETSc vectors, VecGetArray() returns a pointer to
269:       the data array.  Otherwise, the routine is implementation dependent.
270:     - You MUST call VecRestoreArray() when you no longer need access to
271:       the array.
272:     - Note that the Fortran interface to VecGetArray() differs from the
273:       C version.  See the users manual for details.
274:   */
275:   VecGetArray(u,&u_localptr);

277:   /*
278:      We initialize the solution array by simply writing the solution
279:      directly into the array locations.  Alternatively, we could use
280:      VecSetValues() or VecSetValuesLocal().
281:   */
282:   for (i=0; i<appctx->m; i++) u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);

284:   /*
285:      Restore vector
286:   */
287:   VecRestoreArray(u,&u_localptr);

289:   /*
290:      Print debugging information if desired
291:   */
292:   if (appctx->debug) {
293:     printf("initial guess vector\n");
294:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
295:   }

297:   return 0;
298: }
299: /* --------------------------------------------------------------------- */
302: /*
303:    ExactSolution - Computes the exact solution at a given time.

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

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

319:   /*
320:      Get a pointer to vector data.
321:   */
322:   VecGetArray(solution,&s_localptr);

324:   /*
325:      Simply write the solution directly into the array locations.
326:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
327:   */
328:   ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc); ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
329:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
330:   for (i=0; i<appctx->m; i++) s_localptr[i] = PetscCosScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscCosScalar(sc2*(PetscReal)i)*ex2;

332:   /*
333:      Restore vector
334:   */
335:   VecRestoreArray(solution,&s_localptr);
336:   return 0;
337: }
338: /* --------------------------------------------------------------------- */
341: /*
342:    Monitor - User-provided routine to monitor the solution computed at
343:    each timestep.  This example plots the solution and computes the
344:    error in two different norms.

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

363:   /*
364:      View a graph of the current iterate
365:   */
366:   VecView(u,appctx->viewer2);

368:   /*
369:      Compute the exact solution
370:   */
371:   ExactSolution(time,appctx->solution,appctx);

373:   /*
374:      Print debugging information if desired
375:   */
376:   if (appctx->debug) {
377:     printf("Computed solution vector\n");
378:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
379:     printf("Exact solution vector\n");
380:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
381:   }

383:   /*
384:      Compute the 2-norm and max-norm of the error
385:   */
386:   VecAXPY(appctx->solution,-1.0,u);
387:   VecNorm(appctx->solution,NORM_2,&norm_2);
388:   norm_2 = PetscSqrtReal(appctx->h)*norm_2;
389:   VecNorm(appctx->solution,NORM_MAX,&norm_max);

391:   PetscPrintf(PETSC_COMM_WORLD,"Timestep %D: time = %g, 2-norm error = %g, max norm error = %g\n",step,(double)time,(double)norm_2,(double)norm_max);
392:   appctx->norm_2   += norm_2;
393:   appctx->norm_max += norm_max;

395:   /*
396:      View a graph of the error
397:   */
398:   VecView(appctx->solution,appctx->viewer1);

400:   /*
401:      Print debugging information if desired
402:   */
403:   if (appctx->debug) {
404:     printf("Error vector\n");
405:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
406:   }

408:   return 0;
409: }
410: /* --------------------------------------------------------------------- */
413: /*
414:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
415:    matrix for the heat equation.

417:    Input Parameters:
418:    ts - the TS context
419:    t - current time
420:    global_in - global input vector
421:    dummy - optional user-defined context, as set by TSetRHSJacobian()

423:    Output Parameters:
424:    AA - Jacobian matrix
425:    BB - optionally different preconditioning matrix
426:    str - flag indicating matrix structure

428:   Notes:
429:   Recall that MatSetValues() uses 0-based row and column numbers
430:   in Fortran as well as in C.
431: */
432: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
433: {
434:   Mat            A       = *AA;                /* Jacobian matrix */
435:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined application context */
436:   PetscInt       mstart  = 0;
437:   PetscInt       mend    = appctx->m;
439:   PetscInt       i,idx[3];
440:   PetscScalar    v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;

442:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
443:      Compute entries for the locally owned part of the matrix
444:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
445:   /*
446:      Set matrix rows corresponding to boundary data
447:   */

449:   mstart = 0;
450:   v[0]   = 1.0;
451:   MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
452:   mstart++;

454:   mend--;
455:   v[0] = 1.0;
456:   MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);

458:   /*
459:      Set matrix rows corresponding to interior data.  We construct the
460:      matrix one row at a time.
461:   */
462:   v[0] = sone; v[1] = stwo; v[2] = sone;
463:   for (i=mstart; i<mend; i++) {
464:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
465:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
466:   }

468:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
469:      Complete the matrix assembly process and set some options
470:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
471:   /*
472:      Assemble matrix, using the 2-step process:
473:        MatAssemblyBegin(), MatAssemblyEnd()
474:      Computations can be done while messages are in transition
475:      by placing code between these two statements.
476:   */
477:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
478:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

480:   /*
481:      Set flag to indicate that the Jacobian matrix retains an identical
482:      nonzero structure throughout all timestepping iterations (although the
483:      values of the entries change). Thus, we can save some work in setting
484:      up the preconditioner (e.g., no need to redo symbolic factorization for
485:      ILU/ICC preconditioners).
486:       - If the nonzero structure of the matrix is different during
487:         successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
488:         must be used instead.  If you are unsure whether the matrix
489:         structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
490:       - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc
491:         believes your assertion and does not check the structure
492:         of the matrix.  If you erroneously claim that the structure
493:         is the same when it actually is not, the new preconditioner
494:         will not function correctly.  Thus, use this optimization
495:         feature with caution!
496:   */
497:   *str = SAME_NONZERO_PATTERN;

499:   /*
500:      Set and option to indicate that we will never add a new nonzero location
501:      to the matrix. If we do, it will generate an error.
502:   */
503:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

505:   return 0;
506: }