Actual source code: ex5.c

petsc-master 2020-01-21
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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: 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,void*);
78: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
79: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);

81: int main(int argc,char **argv)
82: {
83:   AppCtx         appctx;                 /* user-defined application context */
84:   TS             ts;                     /* timestepping context */
85:   Mat            A;                      /* matrix data structure */
86:   Vec            u;                      /* approximate solution vector */
87:   PetscReal      time_total_max = 100.0; /* default max total time */
88:   PetscInt       time_steps_max = 100;   /* default max timesteps */
89:   PetscDraw      draw;                   /* drawing context */
91:   PetscInt       steps,m;
92:   PetscMPIInt    size;
93:   PetscBool      flg;
94:   PetscReal      dt,ftime;

96:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
97:      Initialize program and set problem parameters
98:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

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

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);
140:   TSSetProblemType(ts,TS_LINEAR);

142:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
143:      Set optional user-defined monitoring routine
144:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

148:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

150:      Create matrix data structure; set matrix evaluation routine.
151:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

179:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
180:      Set solution vector and initial timestep
181:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

183:   dt   = appctx.h*appctx.h/2.0;
184:   TSSetTimeStep(ts,dt);
185:   TSSetSolution(ts,u);

187:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
188:      Customize timestepping solver:
189:        - Set the solution method to be the Backward Euler method.
190:        - Set timestepping duration info
191:      Then set runtime options, which can override these defaults.
192:      For example,
193:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
194:      to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
195:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

197:   TSSetMaxSteps(ts,time_steps_max);
198:   TSSetMaxTime(ts,time_total_max);
199:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
200:   TSSetFromOptions(ts);

202:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
203:      Solve the problem
204:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

206:   /*
207:      Evaluate initial conditions
208:   */
209:   InitialConditions(u,&appctx);

211:   /*
212:      Run the timestepping solver
213:   */
214:   TSSolve(ts,u);
215:   TSGetSolveTime(ts,&ftime);
216:   TSGetStepNumber(ts,&steps);

218:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
219:      View timestepping solver info
220:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

222:   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));
223:   TSView(ts,PETSC_VIEWER_STDOUT_SELF);

225:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
226:      Free work space.  All PETSc objects should be destroyed when they
227:      are no longer needed.
228:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

237:   /*
238:      Always call PetscFinalize() before exiting a program.  This routine
239:        - finalizes the PETSc libraries as well as MPI
240:        - provides summary and diagnostic information if certain runtime
241:          options are chosen (e.g., -log_view).
242:   */
243:   PetscFinalize();
244:   return ierr;
245: }
246: /* --------------------------------------------------------------------- */
247: /*
248:    InitialConditions - Computes the solution at the initial time.

250:    Input Parameter:
251:    u - uninitialized solution vector (global)
252:    appctx - user-defined application context

254:    Output Parameter:
255:    u - vector with solution at initial time (global)
256: */
257: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
258: {
259:   PetscScalar    *u_localptr,h = appctx->h;
260:   PetscInt       i;

263:   /*
264:     Get a pointer to vector data.
265:     - For default PETSc vectors, VecGetArray() returns a pointer to
266:       the data array.  Otherwise, the routine is implementation dependent.
267:     - You MUST call VecRestoreArray() when you no longer need access to
268:       the array.
269:     - Note that the Fortran interface to VecGetArray() differs from the
270:       C version.  See the users manual for details.
271:   */
272:   VecGetArray(u,&u_localptr);

274:   /*
275:      We initialize the solution array by simply writing the solution
276:      directly into the array locations.  Alternatively, we could use
277:      VecSetValues() or VecSetValuesLocal().
278:   */
279:   for (i=0; i<appctx->m; i++) u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);

281:   /*
282:      Restore vector
283:   */
284:   VecRestoreArray(u,&u_localptr);

286:   /*
287:      Print debugging information if desired
288:   */
289:   if (appctx->debug) {
290:     printf("initial guess vector\n");
291:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
292:   }

294:   return 0;
295: }
296: /* --------------------------------------------------------------------- */
297: /*
298:    ExactSolution - Computes the exact solution at a given time.

300:    Input Parameters:
301:    t - current time
302:    solution - vector in which exact solution will be computed
303:    appctx - user-defined application context

305:    Output Parameter:
306:    solution - vector with the newly computed exact solution
307: */
308: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
309: {
310:   PetscScalar    *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
311:   PetscInt       i;

314:   /*
315:      Get a pointer to vector data.
316:   */
317:   VecGetArray(solution,&s_localptr);

319:   /*
320:      Simply write the solution directly into the array locations.
321:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
322:   */
323:   ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc); ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
324:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
325:   for (i=0; i<appctx->m; i++) s_localptr[i] = PetscCosScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscCosScalar(sc2*(PetscReal)i)*ex2;

327:   /*
328:      Restore vector
329:   */
330:   VecRestoreArray(solution,&s_localptr);
331:   return 0;
332: }
333: /* --------------------------------------------------------------------- */
334: /*
335:    Monitor - User-provided routine to monitor the solution computed at
336:    each timestep.  This example plots the solution and computes the
337:    error in two different norms.

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

356:   /*
357:      View a graph of the current iterate
358:   */
359:   VecView(u,appctx->viewer2);

361:   /*
362:      Compute the exact solution
363:   */
364:   ExactSolution(time,appctx->solution,appctx);

366:   /*
367:      Print debugging information if desired
368:   */
369:   if (appctx->debug) {
370:     printf("Computed solution vector\n");
371:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
372:     printf("Exact solution vector\n");
373:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
374:   }

376:   /*
377:      Compute the 2-norm and max-norm of the error
378:   */
379:   VecAXPY(appctx->solution,-1.0,u);
380:   VecNorm(appctx->solution,NORM_2,&norm_2);
381:   norm_2 = PetscSqrtReal(appctx->h)*norm_2;
382:   VecNorm(appctx->solution,NORM_MAX,&norm_max);
383:   if (norm_2   < 1e-14) norm_2   = 0;
384:   if (norm_max < 1e-14) norm_max = 0;

386:   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);
387:   appctx->norm_2   += norm_2;
388:   appctx->norm_max += norm_max;

390:   /*
391:      View a graph of the error
392:   */
393:   VecView(appctx->solution,appctx->viewer1);

395:   /*
396:      Print debugging information if desired
397:   */
398:   if (appctx->debug) {
399:     printf("Error vector\n");
400:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
401:   }

403:   return 0;
404: }
405: /* --------------------------------------------------------------------- */
406: /*
407:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
408:    matrix for the heat equation.

410:    Input Parameters:
411:    ts - the TS context
412:    t - current time
413:    global_in - global input vector
414:    dummy - optional user-defined context, as set by TSetRHSJacobian()

416:    Output Parameters:
417:    AA - Jacobian matrix
418:    BB - optionally different preconditioning matrix
419:    str - flag indicating matrix structure

421:   Notes:
422:   Recall that MatSetValues() uses 0-based row and column numbers
423:   in Fortran as well as in C.
424: */
425: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
426: {
427:   Mat            A       = AA;                /* Jacobian matrix */
428:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined application context */
429:   PetscInt       mstart  = 0;
430:   PetscInt       mend    = appctx->m;
432:   PetscInt       i,idx[3];
433:   PetscScalar    v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;

435:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
436:      Compute entries for the locally owned part of the matrix
437:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
438:   /*
439:      Set matrix rows corresponding to boundary data
440:   */

442:   mstart = 0;
443:   v[0]   = 1.0;
444:   MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
445:   mstart++;

447:   mend--;
448:   v[0] = 1.0;
449:   MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);

451:   /*
452:      Set matrix rows corresponding to interior data.  We construct the
453:      matrix one row at a time.
454:   */
455:   v[0] = sone; v[1] = stwo; v[2] = sone;
456:   for (i=mstart; i<mend; i++) {
457:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
458:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
459:   }

461:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
462:      Complete the matrix assembly process and set some options
463:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
464:   /*
465:      Assemble matrix, using the 2-step process:
466:        MatAssemblyBegin(), MatAssemblyEnd()
467:      Computations can be done while messages are in transition
468:      by placing code between these two statements.
469:   */
470:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
471:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

473:   /*
474:      Set and option to indicate that we will never add a new nonzero location
475:      to the matrix. If we do, it will generate an error.
476:   */
477:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

479:   return 0;
480: }

482: /*TEST

484:     test:
485:       requires: x

487:     test:
488:       suffix: nox
489:       args: -nox

491: TEST*/

```