Actual source code: ex5.c

```  1: static char help[] = "Solves a simple time-dependent linear PDE (the heat equation).\n\
2: Input parameters include:\n\
3:   -m <points>, where <points> = number of grid points\n\
4:   -time_dependent_rhs : Treat the problem as having a time-dependent right-hand side\n\
5:   -debug              : Activate debugging printouts\n\
6:   -nox                : Deactivate x-window graphics\n\n";

8: /* ------------------------------------------------------------------------

10:    This program solves the one-dimensional heat equation (also called the
11:    diffusion equation),
12:        u_t = u_xx,
13:    on the domain 0 <= x <= 1, with the boundary conditions
14:        u(t,0) = 1, u(t,1) = 1,
15:    and the initial condition
16:        u(0,x) = cos(6*pi*x) + 3*cos(2*pi*x).
17:    This is a linear, second-order, parabolic equation.

19:    We discretize the right-hand side using finite differences with
20:    uniform grid spacing h:
21:        u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
22:    We then demonstrate time evolution using the various TS methods by
23:    running the program via
24:        ex3 -ts_type <timestepping solver>

26:    We compare the approximate solution with the exact solution, given by
27:        u_exact(x,t) = exp(-36*pi*pi*t) * cos(6*pi*x) +
28:                       3*exp(-4*pi*pi*t) * cos(2*pi*x)

30:    Notes:
31:    This code demonstrates the TS solver interface to two variants of
32:    linear problems, u_t = f(u,t), namely
33:      - time-dependent f:   f(u,t) is a function of t
34:      - time-independent f: f(u,t) is simply just f(u)

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

38:   ------------------------------------------------------------------------- */

40: /*
41:    Include "petscts.h" so that we can use TS solvers.  Note that this file
42:    automatically includes:
43:      petscsys.h       - base PETSc routines   petscvec.h  - vectors
44:      petscmat.h  - matrices
45:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
46:      petscviewer.h - viewers               petscpc.h   - preconditioners
47:      petscksp.h   - linear solvers        petscsnes.h - nonlinear solvers
48: */
49: #include <petscts.h>
50: #include <petscdraw.h>

52: /*
53:    User-defined application context - contains data needed by the
54:    application-provided call-back routines.
55: */
56: typedef struct {
57:   Vec         solution;         /* global exact solution vector */
58:   PetscInt    m;                /* total number of grid points */
59:   PetscReal   h;                /* mesh width h = 1/(m-1) */
60:   PetscBool   debug;            /* flag (1 indicates activation of debugging printouts) */
61:   PetscViewer viewer1, viewer2; /* viewers for the solution and error */
62:   PetscReal   norm_2, norm_max; /* error norms */
63: } AppCtx;

65: /*
66:    User-defined routines
67: */
68: extern PetscErrorCode InitialConditions(Vec, AppCtx *);
69: extern PetscErrorCode RHSMatrixHeat(TS, PetscReal, Vec, Mat, Mat, void *);
70: extern PetscErrorCode Monitor(TS, PetscInt, PetscReal, Vec, void *);
71: extern PetscErrorCode ExactSolution(PetscReal, Vec, AppCtx *);

73: int main(int argc, char **argv)
74: {
75:   AppCtx      appctx;                 /* user-defined application context */
76:   TS          ts;                     /* timestepping context */
77:   Mat         A;                      /* matrix data structure */
78:   Vec         u;                      /* approximate solution vector */
79:   PetscReal   time_total_max = 100.0; /* default max total time */
80:   PetscInt    time_steps_max = 100;   /* default max timesteps */
81:   PetscDraw   draw;                   /* drawing context */
82:   PetscInt    steps, m;
83:   PetscMPIInt size;
84:   PetscBool   flg;
85:   PetscReal   dt, ftime;

87:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
88:      Initialize program and set problem parameters
89:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

91:   PetscFunctionBeginUser;
92:   PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));
93:   PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
94:   PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!");

96:   m = 60;
97:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-m", &m, NULL));
98:   PetscCall(PetscOptionsHasName(NULL, NULL, "-debug", &appctx.debug));
99:   appctx.m        = m;
100:   appctx.h        = 1.0 / (m - 1.0);
101:   appctx.norm_2   = 0.0;
102:   appctx.norm_max = 0.0;

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

106:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
107:      Create vector data structures
108:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

110:   /*
111:      Create vector data structures for approximate and exact solutions
112:   */
113:   PetscCall(VecCreateSeq(PETSC_COMM_SELF, m, &u));
114:   PetscCall(VecDuplicate(u, &appctx.solution));

116:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
117:      Set up displays to show graphs of the solution and error
118:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

120:   PetscCall(PetscViewerDrawOpen(PETSC_COMM_SELF, 0, "", 80, 380, 400, 160, &appctx.viewer1));
121:   PetscCall(PetscViewerDrawGetDraw(appctx.viewer1, 0, &draw));
122:   PetscCall(PetscDrawSetDoubleBuffer(draw));
123:   PetscCall(PetscViewerDrawOpen(PETSC_COMM_SELF, 0, "", 80, 0, 400, 160, &appctx.viewer2));
124:   PetscCall(PetscViewerDrawGetDraw(appctx.viewer2, 0, &draw));
125:   PetscCall(PetscDrawSetDoubleBuffer(draw));

127:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
128:      Create timestepping solver context
129:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

131:   PetscCall(TSCreate(PETSC_COMM_SELF, &ts));
132:   PetscCall(TSSetProblemType(ts, TS_LINEAR));

134:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
135:      Set optional user-defined monitoring routine
136:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

138:   PetscCall(TSMonitorSet(ts, Monitor, &appctx, NULL));

140:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

142:      Create matrix data structure; set matrix evaluation routine.
143:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

145:   PetscCall(MatCreate(PETSC_COMM_SELF, &A));
146:   PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, m, m));
147:   PetscCall(MatSetFromOptions(A));
148:   PetscCall(MatSetUp(A));

150:   PetscCall(PetscOptionsHasName(NULL, NULL, "-time_dependent_rhs", &flg));
151:   if (flg) {
152:     /*
153:        For linear problems with a time-dependent f(u,t) in the equation
154:        u_t = f(u,t), the user provides the discretized right-hand side
155:        as a time-dependent matrix.
156:     */
157:     PetscCall(TSSetRHSFunction(ts, NULL, TSComputeRHSFunctionLinear, &appctx));
158:     PetscCall(TSSetRHSJacobian(ts, A, A, RHSMatrixHeat, &appctx));
159:   } else {
160:     /*
161:        For linear problems with a time-independent f(u) in the equation
162:        u_t = f(u), the user provides the discretized right-hand side
163:        as a matrix only once, and then sets a null matrix evaluation
164:        routine.
165:     */
166:     PetscCall(RHSMatrixHeat(ts, 0.0, u, A, A, &appctx));
167:     PetscCall(TSSetRHSFunction(ts, NULL, TSComputeRHSFunctionLinear, &appctx));
168:     PetscCall(TSSetRHSJacobian(ts, A, A, TSComputeRHSJacobianConstant, &appctx));
169:   }

171:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
172:      Set solution vector and initial timestep
173:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

175:   dt = appctx.h * appctx.h / 2.0;
176:   PetscCall(TSSetTimeStep(ts, dt));
177:   PetscCall(TSSetSolution(ts, u));

179:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
180:      Customize timestepping solver:
181:        - Set the solution method to be the Backward Euler method.
182:        - Set timestepping duration info
183:      Then set runtime options, which can override these defaults.
184:      For example,
185:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
186:      to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
187:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

189:   PetscCall(TSSetMaxSteps(ts, time_steps_max));
190:   PetscCall(TSSetMaxTime(ts, time_total_max));
191:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
192:   PetscCall(TSSetFromOptions(ts));

194:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
195:      Solve the problem
196:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

198:   /*
199:      Evaluate initial conditions
200:   */
201:   PetscCall(InitialConditions(u, &appctx));

203:   /*
204:      Run the timestepping solver
205:   */
206:   PetscCall(TSSolve(ts, u));
207:   PetscCall(TSGetSolveTime(ts, &ftime));
208:   PetscCall(TSGetStepNumber(ts, &steps));

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

214:   PetscCall(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)));
215:   PetscCall(TSView(ts, PETSC_VIEWER_STDOUT_SELF));

217:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
218:      Free work space.  All PETSc objects should be destroyed when they
219:      are no longer needed.
220:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

229:   /*
230:      Always call PetscFinalize() before exiting a program.  This routine
231:        - finalizes the PETSc libraries as well as MPI
232:        - provides summary and diagnostic information if certain runtime
233:          options are chosen (e.g., -log_view).
234:   */
235:   PetscCall(PetscFinalize());
236:   return 0;
237: }
238: /* --------------------------------------------------------------------- */
239: /*
240:    InitialConditions - Computes the solution at the initial time.

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

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

254:   PetscFunctionBeginUser;
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:   PetscCall(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++) u_localptr[i] = PetscCosScalar(PETSC_PI * i * 6. * h) + 3. * PetscCosScalar(PETSC_PI * i * 2. * h);

273:   /*
274:      Restore vector
275:   */
276:   PetscCall(VecRestoreArray(u, &u_localptr));

278:   /*
279:      Print debugging information if desired
280:   */
281:   if (appctx->debug) {
282:     printf("initial guess vector\n");
283:     PetscCall(VecView(u, PETSC_VIEWER_STDOUT_SELF));
284:   }
285:   PetscFunctionReturn(PETSC_SUCCESS);
286: }
287: /* --------------------------------------------------------------------- */
288: /*
289:    ExactSolution - Computes the exact solution at a given time.

291:    Input Parameters:
292:    t - current time
293:    solution - vector in which exact solution will be computed
294:    appctx - user-defined application context

296:    Output Parameter:
297:    solution - vector with the newly computed exact solution
298: */
299: PetscErrorCode ExactSolution(PetscReal t, Vec solution, AppCtx *appctx)
300: {
301:   PetscScalar *s_localptr, h = appctx->h, ex1, ex2, sc1, sc2, tc = t;
302:   PetscInt     i;

304:   PetscFunctionBeginUser;
305:   /*
306:      Get a pointer to vector data.
307:   */
308:   PetscCall(VecGetArray(solution, &s_localptr));

310:   /*
311:      Simply write the solution directly into the array locations.
312:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
313:   */
314:   ex1 = PetscExpScalar(-36. * PETSC_PI * PETSC_PI * tc);
315:   ex2 = PetscExpScalar(-4. * PETSC_PI * PETSC_PI * tc);
316:   sc1 = PETSC_PI * 6. * h;
317:   sc2 = PETSC_PI * 2. * h;
318:   for (i = 0; i < appctx->m; i++) s_localptr[i] = PetscCosScalar(sc1 * (PetscReal)i) * ex1 + 3. * PetscCosScalar(sc2 * (PetscReal)i) * ex2;

320:   /*
321:      Restore vector
322:   */
323:   PetscCall(VecRestoreArray(solution, &s_localptr));
324:   PetscFunctionReturn(PETSC_SUCCESS);
325: }
326: /* --------------------------------------------------------------------- */
327: /*
328:    Monitor - User-provided routine to monitor the solution computed at
329:    each timestep.  This example plots the solution and computes the
330:    error in two different norms.

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

348:   PetscFunctionBeginUser;
349:   /*
350:      View a graph of the current iterate
351:   */
352:   PetscCall(VecView(u, appctx->viewer2));

354:   /*
355:      Compute the exact solution
356:   */
357:   PetscCall(ExactSolution(time, appctx->solution, appctx));

359:   /*
360:      Print debugging information if desired
361:   */
362:   if (appctx->debug) {
363:     printf("Computed solution vector\n");
364:     PetscCall(VecView(u, PETSC_VIEWER_STDOUT_SELF));
365:     printf("Exact solution vector\n");
366:     PetscCall(VecView(appctx->solution, PETSC_VIEWER_STDOUT_SELF));
367:   }

369:   /*
370:      Compute the 2-norm and max-norm of the error
371:   */
372:   PetscCall(VecAXPY(appctx->solution, -1.0, u));
373:   PetscCall(VecNorm(appctx->solution, NORM_2, &norm_2));
374:   norm_2 = PetscSqrtReal(appctx->h) * norm_2;
375:   PetscCall(VecNorm(appctx->solution, NORM_MAX, &norm_max));
376:   if (norm_2 < 1e-14) norm_2 = 0;
377:   if (norm_max < 1e-14) norm_max = 0;

379:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Timestep %" PetscInt_FMT ": time = %g, 2-norm error = %g, max norm error = %g\n", step, (double)time, (double)norm_2, (double)norm_max));
380:   appctx->norm_2 += norm_2;
381:   appctx->norm_max += norm_max;

383:   /*
384:      View a graph of the error
385:   */
386:   PetscCall(VecView(appctx->solution, appctx->viewer1));

388:   /*
389:      Print debugging information if desired
390:   */
391:   if (appctx->debug) {
392:     printf("Error vector\n");
393:     PetscCall(VecView(appctx->solution, PETSC_VIEWER_STDOUT_SELF));
394:   }
395:   PetscFunctionReturn(PETSC_SUCCESS);
396: }
397: /* --------------------------------------------------------------------- */
398: /*
399:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
400:    matrix for the heat equation.

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

408:    Output Parameters:
409:    AA - Jacobian matrix
410:    BB - optionally different preconditioning matrix
411:    str - flag indicating matrix structure

413:   Notes:
414:   Recall that MatSetValues() uses 0-based row and column numbers
415:   in Fortran as well as in C.
416: */
417: PetscErrorCode RHSMatrixHeat(TS ts, PetscReal t, Vec X, Mat AA, Mat BB, void *ctx)
418: {
419:   Mat         A      = AA;            /* Jacobian matrix */
420:   AppCtx     *appctx = (AppCtx *)ctx; /* user-defined application context */
421:   PetscInt    mstart = 0;
422:   PetscInt    mend   = appctx->m;
423:   PetscInt    i, idx[3];
424:   PetscScalar v[3], stwo = -2. / (appctx->h * appctx->h), sone = -.5 * stwo;

426:   PetscFunctionBeginUser;
427:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
428:      Compute entries for the locally owned part of the matrix
429:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
430:   /*
431:      Set matrix rows corresponding to boundary data
432:   */

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

439:   mend--;
440:   v[0] = 1.0;
441:   PetscCall(MatSetValues(A, 1, &mend, 1, &mend, v, INSERT_VALUES));

443:   /*
444:      Set matrix rows corresponding to interior data.  We construct the
445:      matrix one row at a time.
446:   */
447:   v[0] = sone;
448:   v[1] = stwo;
449:   v[2] = sone;
450:   for (i = mstart; i < mend; i++) {
451:     idx[0] = i - 1;
452:     idx[1] = i;
453:     idx[2] = i + 1;
454:     PetscCall(MatSetValues(A, 1, &i, 3, idx, v, INSERT_VALUES));
455:   }

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

469:   /*
470:      Set and option to indicate that we will never add a new nonzero location
471:      to the matrix. If we do, it will generate an error.
472:   */
473:   PetscCall(MatSetOption(A, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
474:   PetscFunctionReturn(PETSC_SUCCESS);
475: }

477: /*TEST

479:     test:
480:       requires: x

482:     test:
483:       suffix: nox
484:       args: -nox

486: TEST*/
```