Actual source code: ex6.c
1: #ifdef PETSC_RCS_HEADER
2: static char vcid[] = "$Id: ex3.c,v 1.7 1999/07/21 14:35:38 curfman Exp curfman $";
3: #endif
5: /* Program usage: ex3 [-help] [all PETSc options] */
7: static char help[] ="Solves a simple time-dependent linear PDE (the heat equation).\n\
8: Input parameters include:\n\
9: -m <points>, where <points> = number of grid points\n\
10: -time_dependent_rhs : Treat the problem as having a time-dependent right-hand side\n\
11: -time_dependent_bc : Treat the problem as having time-dependent boundary conditions\n\
12: -debug : Activate debugging printouts\n\
13: -nox : Deactivate x-window graphics\n\n";
15: /*
16: Concepts: TS^time-dependent linear problems
17: Concepts: TS^heat equation
18: Concepts: TS^diffusion equation
19: Routines: TSCreate(); TSSetSolution(); TSSetRHSMatrix();
20: Routines: TSSetInitialTimeStep(); TSSetDuration(); TSSetMonitor();
21: Routines: TSSetFromOptions(); TSStep(); TSDestroy();
22: Routines: TSSetTimeStep(); TSGetTimeStep();
23: Processors: 1
24: */
26: /* ------------------------------------------------------------------------
28: This program solves the one-dimensional heat equation (also called the
29: diffusion equation),
30: u_t = u_xx,
31: on the domain 0 <= x <= 1, with the boundary conditions
32: u(t,0) = 0, u(t,1) = 0,
33: and the initial condition
34: u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
35: This is a linear, second-order, parabolic equation.
37: We discretize the right-hand side using finite differences with
38: uniform grid spacing h:
39: u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
40: We then demonstrate time evolution using the various TS methods by
41: running the program via
42: ex3 -ts_type <timestepping solver>
44: We compare the approximate solution with the exact solution, given by
45: u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
46: 3*exp(-4*pi*pi*t) * sin(2*pi*x)
48: Notes:
49: This code demonstrates the TS solver interface to two variants of
50: linear problems, u_t = f(u,t), namely
51: - time-dependent f: f(u,t) is a function of t
52: - time-independent f: f(u,t) is simply f(u)
54: The parallel version of this code is ts/examples/tutorials/ex4.c
56: ------------------------------------------------------------------------- */
58: /*
59: Include "ts.h" so that we can use TS solvers. Note that this file
60: automatically includes:
61: petsc.h - base PETSc routines vec.h - vectors
62: sys.h - system routines mat.h - matrices
63: is.h - index sets ksp.h - Krylov subspace methods
64: viewer.h - viewers pc.h - preconditioners
65: snes.h - nonlinear solvers
66: */
68: #include petscts.h
70: /*
71: User-defined application context - contains data needed by the
72: application-provided call-back routines.
73: */
74: typedef struct {
75: Vec solution; /* global exact solution vector */
76: int m; /* total number of grid points */
77: double h; /* mesh width h = 1/(m-1) */
78: int debug; /* flag (1 indicates activation of debugging printouts) */
79: PetscViewer viewer1, viewer2; /* viewers for the solution and error */
80: double norm_2, norm_max; /* error norms */
81: } AppCtx;
83: /*
84: User-defined routines
85: */
86: extern int InitialConditions(Vec,AppCtx*);
87: extern int RHSMatrixHeat(TS,PetscReal,Mat*,Mat*,MatStructure*,void*);
88: extern int Monitor(TS,int,PetscReal,Vec,void*);
89: extern int ExactSolution(PetscReal,Vec,AppCtx*);
90: extern int MyBCRoutine(TS,PetscReal,Vec,void*);
94: int main(int argc,char **argv)
95: {
96: AppCtx appctx; /* user-defined application context */
97: TS ts; /* timestepping context */
98: Mat A; /* matrix data structure */
99: Vec u; /* approximate solution vector */
100: double time_total_max = 100.0; /* default max total time */
101: int time_steps_max = 100; /* default max timesteps */
102: PetscDraw draw; /* drawing context */
103: PetscTruth flg;
104: int ierr, steps, size, m;
105: double dt;
106: PetscReal ftime;
108: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
109: Initialize program and set problem parameters
110: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
111:
112: PetscInitialize(&argc,&argv,(char*)0,help);
113: MPI_Comm_size(PETSC_COMM_WORLD,&size);
114: if (size != 1) SETERRQ(1,"This is a uniprocessor example only!");
116: m = 60;
117: PetscOptionsGetInt(PETSC_NULL,"-m",&m,&flg);
118: PetscOptionsHasName(PETSC_NULL,"-debug",&flg);
119: appctx.m = m;
120: appctx.h = 1.0/(m-1.0);
121: appctx.norm_2 = 0.0;
122: appctx.norm_max = 0.0;
123: appctx.debug = (int) flg;
124: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");
126: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
127: Create vector data structures
128: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
130: /*
131: Create vector data structures for approximate and exact solutions
132: */
133: VecCreateSeq(PETSC_COMM_SELF,m,&u);
134: VecDuplicate(u,&appctx.solution);
136: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
137: Set up displays to show graphs of the solution and error
138: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
140: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
141: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
142: PetscDrawSetDoubleBuffer(draw);
143: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
144: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
145: PetscDrawSetDoubleBuffer(draw);
147: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
148: Create timestepping solver context
149: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
151: TSCreate(PETSC_COMM_SELF,&ts);
152: TSSetProblemType(ts,TS_LINEAR);
154: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
155: Set optional user-defined monitoring routine
156: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
158: TSSetMonitor(ts,Monitor,&appctx,PETSC_NULL);
160: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
162: Create matrix data structure; set matrix evaluation routine.
163: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
165: MatCreate(PETSC_COMM_SELF,PETSC_DECIDE,PETSC_DECIDE,m,m,&A);
167: PetscOptionsHasName(PETSC_NULL,"-time_dependent_rhs",&flg);
168: if (flg) {
169: /*
170: For linear problems with a time-dependent f(u,t) in the equation
171: u_t = f(u,t), the user provides the discretized right-hand-side
172: as a time-dependent matrix.
173: */
174: TSSetRHSMatrix(ts,A,A,RHSMatrixHeat,&appctx);
175: } else {
176: /*
177: For linear problems with a time-independent f(u) in the equation
178: u_t = f(u), the user provides the discretized right-hand-side
179: as a matrix only once, and then sets a null matrix evaluation
180: routine.
181: */
182: MatStructure A_structure;
183: RHSMatrixHeat(ts,0.0,&A,&A,&A_structure,&appctx);
184: TSSetRHSMatrix(ts,A,A,PETSC_NULL,&appctx);
185: }
187: /* Treat the problem as having time-dependent boundary conditions */
188: PetscOptionsHasName(PETSC_NULL,"-time_dependent_bc",&flg);
189: if (flg) {
190: TSSetRHSBoundaryConditions(ts,MyBCRoutine,&appctx);
191: }
193: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
194: Set solution vector and initial timestep
195: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
197: dt = appctx.h*appctx.h/2.0;
198: TSSetInitialTimeStep(ts,0.0,dt);
199: TSSetSolution(ts,u);
201: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
202: Customize timestepping solver:
203: - Set the solution method to be the Backward Euler method.
204: - Set timestepping duration info
205: Then set runtime options, which can override these defaults.
206: For example,
207: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
208: to override the defaults set by TSSetDuration().
209: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
211: TSSetDuration(ts,time_steps_max,time_total_max);
212: TSSetFromOptions(ts);
214: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
215: Solve the problem
216: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
218: /*
219: Evaluate initial conditions
220: */
221: InitialConditions(u,&appctx);
223: /*
224: Run the timestepping solver
225: */
226: TSStep(ts,&steps,&ftime);
228: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
229: View timestepping solver info
230: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
232: PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %g, avg. error (max norm) = %g\n",
233: appctx.norm_2/steps,appctx.norm_max/steps);
234: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
236: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
237: Free work space. All PETSc objects should be destroyed when they
238: are no longer needed.
239: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
241: TSDestroy(ts);
242: MatDestroy(A);
243: VecDestroy(u);
244: PetscViewerDestroy(appctx.viewer1);
245: PetscViewerDestroy(appctx.viewer2);
246: VecDestroy(appctx.solution);
248: /*
249: Always call PetscFinalize() before exiting a program. This routine
250: - finalizes the PETSc libraries as well as MPI
251: - provides summary and diagnostic information if certain runtime
252: options are chosen (e.g., -log_summary).
253: */
254: PetscFinalize();
255: return 0;
256: }
257: /* --------------------------------------------------------------------- */
260: /*
261: InitialConditions - Computes the solution at the initial time.
263: Input Parameter:
264: u - uninitialized solution vector (global)
265: appctx - user-defined application context
267: Output Parameter:
268: u - vector with solution at initial time (global)
269: */
270: int InitialConditions(Vec u,AppCtx *appctx)
271: {
272: PetscScalar *u_localptr;
273: /*PetscScalar h = appctx->h;*/
274: int i, ierr;
276: /*
277: Get a pointer to vector data.
278: - For default PETSc vectors, VecGetArray() returns a pointer to
279: the data array. Otherwise, the routine is implementation dependent.
280: - You MUST call VecRestoreArray() when you no longer need access to
281: the array.
282: - Note that the Fortran interface to VecGetArray() differs from the
283: C version. See the users manual for details.
284: */
285: VecGetArray(u,&u_localptr);
287: /*
288: We initialize the solution array by simply writing the solution
289: directly into the array locations. Alternatively, we could use
290: VecSetValues() or VecSetValuesLocal().
291: */
292: for (i=0; i<appctx->m; i++) {
293: u_localptr[i] = 0.0;
294: /* u_localptr[i] = sin(PETSC_PI*i*6.*h) + 3.*sin(PETSC_PI*i*2.*h); */
295: }
297: /*
298: Restore vector
299: */
300: VecRestoreArray(u,&u_localptr);
302: /*
303: Print debugging information if desired
304: */
305: if (appctx->debug) {
306: printf("initial guess vector\n");
307: VecView(u,PETSC_VIEWER_STDOUT_SELF);
308: }
310: return 0;
311: }
312: /* --------------------------------------------------------------------- */
315: /*
316: ExactSolution - Computes the exact solution at a given time.
318: Input Parameters:
319: t - current time
320: solution - vector in which exact solution will be computed
321: appctx - user-defined application context
323: Output Parameter:
324: solution - vector with the newly computed exact solution
325: */
326: int ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
327: {
328: PetscScalar *s_localptr, h = appctx->h, ex1, ex2, sc1, sc2;
329: int i, ierr;
331: /*
332: Get a pointer to vector data.
333: */
334: VecGetArray(solution,&s_localptr);
336: /*
337: Simply write the solution directly into the array locations.
338: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
339: */
340: ex1 = exp(-36.*PETSC_PI*PETSC_PI*t); ex2 = exp(-4.*PETSC_PI*PETSC_PI*t);
341: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
342: for (i=0; i<appctx->m; i++) {
343: s_localptr[i] = sin(PetscRealPart(sc1)*(double)i)*ex1 + 3.*sin(PetscRealPart(sc2)*(double)i)*ex2;
344: }
346: /*
347: Restore vector
348: */
349: VecRestoreArray(solution,&s_localptr);
350: return 0;
351: }
352: /* --------------------------------------------------------------------- */
355: /*
356: Monitor - User-provided routine to monitor the solution computed at
357: each timestep. This example plots the solution and computes the
358: error in two different norms.
360: This example also demonstrates changing the timestep via TSSetTimeStep().
362: Input Parameters:
363: ts - the timestep context
364: step - the count of the current step (with 0 meaning the
365: initial condition)
366: ctime - the current time
367: u - the solution at this timestep
368: ctx - the user-provided context for this monitoring routine.
369: In this case we use the application context which contains
370: information about the problem size, workspace and the exact
371: solution.
372: */
373: int Monitor(TS ts,int step,PetscReal ctime,Vec u,void *ctx)
374: {
375: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
376: int ierr;
377: PetscReal norm_2, norm_max, dt, dttol;
378: PetscTruth flg;
379: PetscScalar mone = -1.0;
381: /*
382: View a graph of the current iterate
383: */
384: VecView(u,appctx->viewer2);
386: /*
387: Compute the exact solution
388: */
389: ExactSolution(ctime,appctx->solution,appctx);
391: /*
392: Print debugging information if desired
393: */
394: if (appctx->debug) {
395: printf("Computed solution vector\n");
396: VecView(u,PETSC_VIEWER_STDOUT_SELF);
397: printf("Exact solution vector\n");
398: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
399: }
401: /*
402: Compute the 2-norm and max-norm of the error
403: */
404: VecAXPY(&mone,u,appctx->solution);
405: VecNorm(appctx->solution,NORM_2,&norm_2);
406: norm_2 = sqrt(appctx->h)*norm_2;
407: VecNorm(appctx->solution,NORM_MAX,&norm_max);
409: TSGetTimeStep(ts,&dt);
410: printf("Timestep %d: step size = %g, time = %g, 2-norm error = %g, max norm error = %g\n",
411: step,dt,ctime,norm_2,norm_max);
412: appctx->norm_2 += norm_2;
413: appctx->norm_max += norm_max;
415: dttol = .0001;
416: PetscOptionsGetReal(PETSC_NULL,"-dttol",&dttol,&flg);
417: if (dt < dttol) {
418: dt *= .999;
419: TSSetTimeStep(ts,dt);
420: }
422: /*
423: View a graph of the error
424: */
425: VecView(appctx->solution,appctx->viewer1);
427: /*
428: Print debugging information if desired
429: */
430: if (appctx->debug) {
431: printf("Error vector\n");
432: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
433: }
435: return 0;
436: }
437: /* --------------------------------------------------------------------- */
440: /*
441: RHSMatrixHeat - User-provided routine to compute the right-hand-side
442: matrix for the heat equation.
444: Input Parameters:
445: ts - the TS context
446: t - current time
447: global_in - global input vector
448: dummy - optional user-defined context, as set by TSetRHSJacobian()
450: Output Parameters:
451: AA - Jacobian matrix
452: BB - optionally different preconditioning matrix
453: str - flag indicating matrix structure
455: Notes:
456: Recall that MatSetValues() uses 0-based row and column numbers
457: in Fortran as well as in C.
458: */
459: int RHSMatrixHeat(TS ts,PetscReal t,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
460: {
461: Mat A = *AA; /* Jacobian matrix */
462: AppCtx *appctx = (AppCtx *) ctx; /* user-defined application context */
463: int mstart = 0;
464: int mend = appctx->m;
465: int ierr, i, idx[3];
466: PetscScalar v[3], stwo = -2./(appctx->h*appctx->h), sone = -.5*stwo;
468: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
469: Compute entries for the locally owned part of the matrix
470: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
471: /*
472: Set matrix rows corresponding to boundary data
473: */
475: mstart = 0;
476: v[0] = 1.0;
477: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
478: mstart++;
480: mend--;
481: v[0] = 1.0;
482: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
484: /*
485: Set matrix rows corresponding to interior data. We construct the
486: matrix one row at a time.
487: */
488: v[0] = sone; v[1] = stwo; v[2] = sone;
489: for ( i=mstart; i<mend; i++ ) {
490: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
491: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
492: }
494: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
495: Complete the matrix assembly process and set some options
496: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
497: /*
498: Assemble matrix, using the 2-step process:
499: MatAssemblyBegin(), MatAssemblyEnd()
500: Computations can be done while messages are in transition
501: by placing code between these two statements.
502: */
503: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
504: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
506: /*
507: Set flag to indicate that the Jacobian matrix retains an identical
508: nonzero structure throughout all timestepping iterations (although the
509: values of the entries change). Thus, we can save some work in setting
510: up the preconditioner (e.g., no need to redo symbolic factorization for
511: ILU/ICC preconditioners).
512: - If the nonzero structure of the matrix is different during
513: successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
514: must be used instead. If you are unsure whether the matrix
515: structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
516: - Caution: If you specify SAME_NONZERO_PATTERN, PETSc
517: believes your assertion and does not check the structure
518: of the matrix. If you erroneously claim that the structure
519: is the same when it actually is not, the new preconditioner
520: will not function correctly. Thus, use this optimization
521: feature with caution!
522: */
523: *str = SAME_NONZERO_PATTERN;
525: /*
526: Set and option to indicate that we will never add a new nonzero location
527: to the matrix. If we do, it will generate an error.
528: */
529: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR);
531: return 0;
532: }
533: /* --------------------------------------------------------------------- */
536: /*
537: Input Parameters:
538: ts - the TS context
539: t - current time
540: f - function
541: ctx - optional user-defined context, as set by TSetBCFunction()
542: */
543: int MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
544: {
545: AppCtx *appctx = (AppCtx *) ctx; /* user-defined application context */
546: int ierr, m = appctx->m;
547: PetscScalar *fa;
549: VecGetArray(f,&fa);
550: fa[0] = 0.0;
551: fa[m-1] = 1.0;
552: VecRestoreArray(f,&fa);
553: printf("t=%g\n",t);
554:
555: return 0;
556: }