Actual source code: ex5.c

  1: /*$Id: ex5.c,v 1.22 2001/04/10 19:37:12 bsmith Exp $*/

  3: /* Program usage:  ex3 [-help] [all PETSc options] */

  5: static char help[] ="Solves a simple time-dependent linear PDE (the heat equation).n
  6: Input parameters include:n
  7:   -m <points>, where <points> = number of grid pointsn
  8:   -time_dependent_rhs : Treat the problem as having a time-dependent right-hand siden
  9:   -debug              : Activate debugging printoutsn
 10:   -nox                : Deactivate x-window graphicsnn";

 12: /*
 13:    Concepts: TS^time-dependent linear problems
 14:    Concepts: TS^heat equation
 15:    Concepts: TS^diffusion equation
 16:    Processors: 1
 17: */

 19: /* ------------------------------------------------------------------------

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

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

 37:    We compare the approximate solution with the exact solution, given by
 38:        u_exact(x,t) = exp(-36*pi*pi*t) * cos(6*pi*x) +
 39:                       3*exp(-4*pi*pi*t) * cos(2*pi*x)

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

 47:     The parallel version of this code is ts/examples/tutorials/ex4.c

 49:   ------------------------------------------------------------------------- */

 51: /* 
 52:    Include "petscts.h" so that we can use TS solvers.  Note that this file
 53:    automatically includes:
 54:      petsc.h       - base PETSc routines   petscvec.h  - vectors
 55:      petscsys.h    - system routines       petscmat.h  - matrices
 56:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
 57:      petscviewer.h - viewers               petscpc.h   - preconditioners
 58:      petscsles.h   - linear solvers        petscsnes.h - nonlinear solvers
 59: */

 61: #include "petscts.h"

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

 76: /* 
 77:    User-defined routines
 78: */
 79: extern int InitialConditions(Vec,AppCtx*);
 80: extern int RHSMatrixHeat(TS,double,Mat*,Mat*,MatStructure*,void*);
 81: extern int Monitor(TS,int,double,Vec,void*);
 82: extern int ExactSolution(double,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:   double        time_total_max = 100.0; /* default max total time */
 91:   int           time_steps_max = 100;   /* default max timesteps */
 92:   PetscDraw     draw;                   /* drawing context */
 93:   int           ierr,steps,size,m;
 94:   PetscTruth    flg;
 95:   double        dt,ftime;

 97:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 98:      Initialize program and set problem parameters
 99:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
100: 
101:   PetscInitialize(&argc,&argv,(char*)0,help);
102:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
103:   if (size != 1) SETERRQ(1,"This is a uniprocessor example only!");

105:   m    = 60;
106:   PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
107:   PetscOptionsHasName(PETSC_NULL,"-debug",&appctx.debug);
108:   appctx.m        = m;
109:   appctx.h        = 1.0/(m-1.0);
110:   appctx.norm_2   = 0.0;
111:   appctx.norm_max = 0.0;
112:   PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processorn");

114:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
115:      Create vector data structures
116:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

118:   /* 
119:      Create vector data structures for approximate and exact solutions
120:   */
121:   VecCreateSeq(PETSC_COMM_SELF,m,&u);
122:   VecDuplicate(u,&appctx.solution);

124:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
125:      Set up displays to show graphs of the solution and error 
126:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

135:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
136:      Create timestepping solver context
137:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

139:   TSCreate(PETSC_COMM_SELF,TS_LINEAR,&ts);

141:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
142:      Set optional user-defined monitoring routine
143:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

145:   TSSetMonitor(ts,Monitor,&appctx,PETSC_NULL);

147:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

149:      Create matrix data structure; set matrix evaluation routine.
150:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

152:   MatCreate(PETSC_COMM_SELF,PETSC_DECIDE,PETSC_DECIDE,m,m,&A);
153:   MatSetFromOptions(A);

155:   PetscOptionsHasName(PETSC_NULL,"-time_dependent_rhs",&flg);
156:   if (flg) {
157:     /*
158:        For linear problems with a time-dependent f(u,t) in the equation 
159:        u_t = f(u,t), the user provides the discretized right-hand-side
160:        as a time-dependent matrix.
161:     */
162:     TSSetRHSMatrix(ts,A,A,RHSMatrixHeat,&appctx);
163:   } else {
164:     /*
165:        For linear problems with a time-independent f(u) in the equation 
166:        u_t = f(u), the user provides the discretized right-hand-side
167:        as a matrix only once, and then sets a null matrix evaluation
168:        routine.
169:     */
170:     MatStructure A_structure;
171:     RHSMatrixHeat(ts,0.0,&A,&A,&A_structure,&appctx);
172:     TSSetRHSMatrix(ts,A,A,PETSC_NULL,&appctx);
173:   }

175:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
176:      Set solution vector and initial timestep
177:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

179:   dt = appctx.h*appctx.h/2.0;
180:   TSSetInitialTimeStep(ts,0.0,dt);
181:   TSSetSolution(ts,u);

183:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
184:      Customize timestepping solver:  
185:        - Set the solution method to be the Backward Euler method.
186:        - Set timestepping duration info 
187:      Then set runtime options, which can override these defaults.
188:      For example,
189:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
190:      to override the defaults set by TSSetDuration().
191:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

193:   TSSetDuration(ts,time_steps_max,time_total_max);
194:   TSSetFromOptions(ts);

196:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
197:      Solve the problem
198:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

200:   /*
201:      Evaluate initial conditions
202:   */
203:   InitialConditions(u,&appctx);

205:   /*
206:      Run the timestepping solver
207:   */
208:   TSStep(ts,&steps,&ftime);

210:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
211:      View timestepping solver info
212:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

214:   PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %g, avg. error (max norm) = %gn",
215:               appctx.norm_2/steps,appctx.norm_max/steps);
216:   TSView(ts,PETSC_VIEWER_STDOUT_SELF);

218:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
219:      Free work space.  All PETSc objects should be destroyed when they
220:      are no longer needed.
221:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

223:   TSDestroy(ts);
224:   MatDestroy(A);
225:   VecDestroy(u);
226:   PetscViewerDestroy(appctx.viewer1);
227:   PetscViewerDestroy(appctx.viewer2);
228:   VecDestroy(appctx.solution);

230:   /*
231:      Always call PetscFinalize() before exiting a program.  This routine
232:        - finalizes the PETSc libraries as well as MPI
233:        - provides summary and diagnostic information if certain runtime
234:          options are chosen (e.g., -log_summary). 
235:   */
236:   PetscFinalize();
237:   return 0;
238: }
239: /* --------------------------------------------------------------------- */
240: /*
241:    InitialConditions - Computes the solution at the initial time. 

243:    Input Parameter:
244:    u - uninitialized solution vector (global)
245:    appctx - user-defined application context

247:    Output Parameter:
248:    u - vector with solution at initial time (global)
249: */
250: int InitialConditions(Vec u,AppCtx *appctx)
251: {
252:   Scalar *u_localptr,h = appctx->h;
253:   int    i,ierr;

255:   /* 
256:     Get a pointer to vector data.
257:     - For default PETSc vectors, VecGetArray() returns a pointer to
258:       the data array.  Otherwise, the routine is implementation dependent.
259:     - You MUST call VecRestoreArray() when you no longer need access to
260:       the array.
261:     - Note that the Fortran interface to VecGetArray() differs from the
262:       C version.  See the users manual for details.
263:   */
264:   VecGetArray(u,&u_localptr);

266:   /* 
267:      We initialize the solution array by simply writing the solution
268:      directly into the array locations.  Alternatively, we could use
269:      VecSetValues() or VecSetValuesLocal().
270:   */
271:   for (i=0; i<appctx->m; i++) {
272:     u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);
273:   }

275:   /* 
276:      Restore vector
277:   */
278:   VecRestoreArray(u,&u_localptr);

280:   /* 
281:      Print debugging information if desired
282:   */
283:   if (appctx->debug) {
284:      printf("initial guess vectorn");
285:      VecView(u,PETSC_VIEWER_STDOUT_SELF);
286:   }

288:   return 0;
289: }
290: /* --------------------------------------------------------------------- */
291: /*
292:    ExactSolution - Computes the exact solution at a given time.

294:    Input Parameters:
295:    t - current time
296:    solution - vector in which exact solution will be computed
297:    appctx - user-defined application context

299:    Output Parameter:
300:    solution - vector with the newly computed exact solution
301: */
302: int ExactSolution(double t,Vec solution,AppCtx *appctx)
303: {
304:   Scalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2;
305:   int    i,ierr;

307:   /*
308:      Get a pointer to vector data.
309:   */
310:   VecGetArray(solution,&s_localptr);

312:   /* 
313:      Simply write the solution directly into the array locations.
314:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
315:   */
316:   ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*t); ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*t);
317:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
318:   for (i=0; i<appctx->m; i++) {
319:     s_localptr[i] = PetscCosScalar(sc1*(double)i)*ex1 + 3.*PetscCosScalar(sc2*(double)i)*ex2;
320:   }

322:   /* 
323:      Restore vector
324:   */
325:   VecRestoreArray(solution,&s_localptr);
326:   return 0;
327: }
328: /* --------------------------------------------------------------------- */
329: /*
330:    Monitor - User-provided routine to monitor the solution computed at 
331:    each timestep.  This example plots the solution and computes the
332:    error in two different norms.

334:    Input Parameters:
335:    ts     - the timestep context
336:    step   - the count of the current step (with 0 meaning the
337:              initial condition)
338:    time   - the current time
339:    u      - the solution at this timestep
340:    ctx    - the user-provided context for this monitoring routine.
341:             In this case we use the application context which contains 
342:             information about the problem size, workspace and the exact 
343:             solution.
344: */
345: int Monitor(TS ts,int step,double time,Vec u,void *ctx)
346: {
347:   AppCtx   *appctx = (AppCtx*) ctx;   /* user-defined application context */
348:   int      ierr;
349:   double   norm_2,norm_max;
350:   Scalar   mone = -1.0;

352:   /* 
353:      View a graph of the current iterate
354:   */
355:   VecView(u,appctx->viewer2);

357:   /* 
358:      Compute the exact solution
359:   */
360:   ExactSolution(time,appctx->solution,appctx);

362:   /*
363:      Print debugging information if desired
364:   */
365:   if (appctx->debug) {
366:      printf("Computed solution vectorn");
367:      VecView(u,PETSC_VIEWER_STDOUT_SELF);
368:      printf("Exact solution vectorn");
369:      VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
370:   }

372:   /*
373:      Compute the 2-norm and max-norm of the error
374:   */
375:   VecAXPY(&mone,u,appctx->solution);
376:   VecNorm(appctx->solution,NORM_2,&norm_2);
377:   norm_2 = sqrt(appctx->h)*norm_2;
378:   VecNorm(appctx->solution,NORM_MAX,&norm_max);

380:   printf("Timestep %d: time = %g, 2-norm error = %g, max norm error = %gn",
381:          step,time,norm_2,norm_max);
382:   appctx->norm_2   += norm_2;
383:   appctx->norm_max += norm_max;

385:   /* 
386:      View a graph of the error
387:   */
388:   VecView(appctx->solution,appctx->viewer1);

390:   /*
391:      Print debugging information if desired
392:   */
393:   if (appctx->debug) {
394:      printf("Error vectorn");
395:      VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
396:   }

398:   return 0;
399: }
400: /* --------------------------------------------------------------------- */
401: /*
402:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
403:    matrix for the heat equation.

405:    Input Parameters:
406:    ts - the TS context
407:    t - current time
408:    global_in - global input vector
409:    dummy - optional user-defined context, as set by TSetRHSJacobian()

411:    Output Parameters:
412:    AA - Jacobian matrix
413:    BB - optionally different preconditioning matrix
414:    str - flag indicating matrix structure

416:   Notes:
417:   Recall that MatSetValues() uses 0-based row and column numbers
418:   in Fortran as well as in C.
419: */
420: int RHSMatrixHeat(TS ts,double t,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
421: {
422:   Mat    A = *AA;                      /* Jacobian matrix */
423:   AppCtx *appctx = (AppCtx*)ctx;     /* user-defined application context */
424:   int    mstart = 0;
425:   int    mend = appctx->m;
426:   int    ierr,i,idx[3];
427:   Scalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;

429:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
430:      Compute entries for the locally owned part of the matrix
431:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
432:   /* 
433:      Set matrix rows corresponding to boundary data
434:   */

436:   mstart = 0;
437:   v[0] = 1.0;
438:   MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
439:   mstart++;

441:   mend--;
442:   v[0] = 1.0;
443:   MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);

445:   /*
446:      Set matrix rows corresponding to interior data.  We construct the 
447:      matrix one row at a time.
448:   */
449:   v[0] = sone; v[1] = stwo; v[2] = sone;
450:   for (i=mstart; i<mend; i++) {
451:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
452:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
453:   }

455:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
456:      Complete the matrix assembly process and set some options
457:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
458:   /*
459:      Assemble matrix, using the 2-step process:
460:        MatAssemblyBegin(), MatAssemblyEnd()
461:      Computations can be done while messages are in transition
462:      by placing code between these two statements.
463:   */
464:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
465:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

467:   /*
468:      Set flag to indicate that the Jacobian matrix retains an identical
469:      nonzero structure throughout all timestepping iterations (although the
470:      values of the entries change). Thus, we can save some work in setting
471:      up the preconditioner (e.g., no need to redo symbolic factorization for
472:      ILU/ICC preconditioners).
473:       - If the nonzero structure of the matrix is different during
474:         successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
475:         must be used instead.  If you are unsure whether the matrix
476:         structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
477:       - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc
478:         believes your assertion and does not check the structure
479:         of the matrix.  If you erroneously claim that the structure
480:         is the same when it actually is not, the new preconditioner
481:         will not function correctly.  Thus, use this optimization
482:         feature with caution!
483:   */
484:   *str = SAME_NONZERO_PATTERN;

486:   /*
487:      Set and option to indicate that we will never add a new nonzero location 
488:      to the matrix. If we do, it will generate an error.
489:   */
490:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR);

492:   return 0;
493: }