Actual source code: ex3.c
2: /* Program usage: ex3 [-help] [all PETSc options] */
4: static char help[] ="Solves a simple time-dependent linear PDE (the heat equation).\n\
5: Input parameters include:\n\
6: -m <points>, where <points> = number of grid points\n\
7: -time_dependent_rhs : Treat the problem as having a time-dependent right-hand side\n\
8: -time_dependent_bc : Treat the problem as having time-dependent boundary conditions\n\
9: -debug : Activate debugging printouts\n\
10: -nox : Deactivate x-window graphics\n\n";
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) = 0, u(t,1) = 0,
26: and the initial condition
27: u(0,x) = sin(6*pi*x) + 3*sin(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) * sin(6*pi*x) +
39: 3*exp(-4*pi*pi*t) * sin(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 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: petscksp.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: PetscInt m; /* total number of grid points */
70: PetscReal 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: PetscReal norm_2,norm_max; /* error norms */
74: } AppCtx;
76: /*
77: User-defined routines
78: */
87: int main(int argc,char **argv)
88: {
89: AppCtx appctx; /* user-defined application context */
90: TS ts; /* timestepping context */
91: Mat A; /* matrix data structure */
92: Vec u; /* approximate solution vector */
93: PetscReal time_total_max = 100.0; /* default max total time */
94: PetscInt time_steps_max = 100; /* default max timesteps */
95: PetscDraw draw; /* drawing context */
97: PetscInt steps,m;
98: PetscMPIInt size;
99: PetscReal dt,ftime;
100: PetscTruth flg;
102: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
103: Initialize program and set problem parameters
104: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
105:
106: PetscInitialize(&argc,&argv,(char*)0,help);
107: MPI_Comm_size(PETSC_COMM_WORLD,&size);
108: if (size != 1) SETERRQ(1,"This is a uniprocessor example only!");
110: m = 60;
111: PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
112: PetscOptionsHasName(PETSC_NULL,"-debug",&appctx.debug);
113: appctx.m = m;
114: appctx.h = 1.0/(m-1.0);
115: appctx.norm_2 = 0.0;
116: appctx.norm_max = 0.0;
117: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");
119: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
120: Create vector data structures
121: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
123: /*
124: Create vector data structures for approximate and exact solutions
125: */
126: VecCreateSeq(PETSC_COMM_SELF,m,&u);
127: VecDuplicate(u,&appctx.solution);
129: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
130: Set up displays to show graphs of the solution and error
131: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
133: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
134: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
135: PetscDrawSetDoubleBuffer(draw);
136: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
137: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
138: PetscDrawSetDoubleBuffer(draw);
140: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
141: Create timestepping solver context
142: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
144: TSCreate(PETSC_COMM_SELF,&ts);
145: TSSetProblemType(ts,TS_LINEAR);
147: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
148: Set optional user-defined monitoring routine
149: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
151: TSSetMonitor(ts,Monitor,&appctx,PETSC_NULL);
153: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
155: Create matrix data structure; set matrix evaluation routine.
156: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
158: MatCreate(PETSC_COMM_SELF,&A);
159: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
160: MatSetFromOptions(A);
162: PetscOptionsHasName(PETSC_NULL,"-time_dependent_rhs",&flg);
163: if (flg) {
164: /*
165: For linear problems with a time-dependent f(u,t) in the equation
166: u_t = f(u,t), the user provides the discretized right-hand-side
167: as a time-dependent matrix.
168: */
169: TSSetRHSMatrix(ts,A,A,RHSMatrixHeat,&appctx);
170: } else {
171: /*
172: For linear problems with a time-independent f(u) in the equation
173: u_t = f(u), the user provides the discretized right-hand-side
174: as a matrix only once, and then sets a null matrix evaluation
175: routine.
176: */
177: MatStructure A_structure;
178: RHSMatrixHeat(ts,0.0,&A,&A,&A_structure,&appctx);
179: TSSetRHSMatrix(ts,A,A,PETSC_NULL,&appctx);
180: }
182: /* Treat the problem as having time-dependent boundary conditions */
183: PetscOptionsHasName(PETSC_NULL,"-time_dependent_bc",&flg);
184: if (flg) {
185: TSSetRHSBoundaryConditions(ts,MyBCRoutine,&appctx);
186: }
188: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
189: Set solution vector and initial timestep
190: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
192: dt = appctx.h*appctx.h/2.0;
193: TSSetInitialTimeStep(ts,0.0,dt);
194: TSSetSolution(ts,u);
196: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
197: Customize timestepping solver:
198: - Set the solution method to be the Backward Euler method.
199: - Set timestepping duration info
200: Then set runtime options, which can override these defaults.
201: For example,
202: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
203: to override the defaults set by TSSetDuration().
204: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
206: TSSetDuration(ts,time_steps_max,time_total_max);
207: TSSetFromOptions(ts);
209: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
210: Solve the problem
211: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
213: /*
214: Evaluate initial conditions
215: */
216: InitialConditions(u,&appctx);
218: /*
219: Run the timestepping solver
220: */
221: TSStep(ts,&steps,&ftime);
223: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
224: View timestepping solver info
225: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
227: PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %G, avg. error (max norm) = %G\n",
228: appctx.norm_2/steps,appctx.norm_max/steps);
229: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
231: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
232: Free work space. All PETSc objects should be destroyed when they
233: are no longer needed.
234: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
236: TSDestroy(ts);
237: MatDestroy(A);
238: VecDestroy(u);
239: PetscViewerDestroy(appctx.viewer1);
240: PetscViewerDestroy(appctx.viewer2);
241: VecDestroy(appctx.solution);
243: /*
244: Always call PetscFinalize() before exiting a program. This routine
245: - finalizes the PETSc libraries as well as MPI
246: - provides summary and diagnostic information if certain runtime
247: options are chosen (e.g., -log_summary).
248: */
249: PetscFinalize();
250: return 0;
251: }
252: /* --------------------------------------------------------------------- */
255: /*
256: InitialConditions - Computes the solution at the initial time.
258: Input Parameter:
259: u - uninitialized solution vector (global)
260: appctx - user-defined application context
262: Output Parameter:
263: u - vector with solution at initial time (global)
264: */
265: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
266: {
267: PetscScalar *u_localptr,h = appctx->h;
269: PetscInt i;
271: /*
272: Get a pointer to vector data.
273: - For default PETSc vectors, VecGetArray() returns a pointer to
274: the data array. Otherwise, the routine is implementation dependent.
275: - You MUST call VecRestoreArray() when you no longer need access to
276: the array.
277: - Note that the Fortran interface to VecGetArray() differs from the
278: C version. See the users manual for details.
279: */
280: VecGetArray(u,&u_localptr);
282: /*
283: We initialize the solution array by simply writing the solution
284: directly into the array locations. Alternatively, we could use
285: VecSetValues() or VecSetValuesLocal().
286: */
287: for (i=0; i<appctx->m; i++) {
288: u_localptr[i] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);
289: }
291: /*
292: Restore vector
293: */
294: VecRestoreArray(u,&u_localptr);
296: /*
297: Print debugging information if desired
298: */
299: if (appctx->debug) {
300: printf("initial guess vector\n");
301: VecView(u,PETSC_VIEWER_STDOUT_SELF);
302: }
304: return 0;
305: }
306: /* --------------------------------------------------------------------- */
309: /*
310: ExactSolution - Computes the exact solution at a given time.
312: Input Parameters:
313: t - current time
314: solution - vector in which exact solution will be computed
315: appctx - user-defined application context
317: Output Parameter:
318: solution - vector with the newly computed exact solution
319: */
320: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
321: {
322: PetscScalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
324: PetscInt i;
326: /*
327: Get a pointer to vector data.
328: */
329: VecGetArray(solution,&s_localptr);
331: /*
332: Simply write the solution directly into the array locations.
333: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
334: */
335: ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc);
336: ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
337: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
338: for (i=0; i<appctx->m; i++) {
339: s_localptr[i] = PetscSinScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i)*ex2;
340: }
342: /*
343: Restore vector
344: */
345: VecRestoreArray(solution,&s_localptr);
346: return 0;
347: }
348: /* --------------------------------------------------------------------- */
351: /*
352: Monitor - User-provided routine to monitor the solution computed at
353: each timestep. This example plots the solution and computes the
354: error in two different norms.
356: This example also demonstrates changing the timestep via TSSetTimeStep().
358: Input Parameters:
359: ts - the timestep context
360: step - the count of the current step (with 0 meaning the
361: initial condition)
362: time - the current time
363: u - the solution at this timestep
364: ctx - the user-provided context for this monitoring routine.
365: In this case we use the application context which contains
366: information about the problem size, workspace and the exact
367: solution.
368: */
369: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
370: {
371: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
373: PetscReal norm_2,norm_max,dt,dttol;
374: /*
375: View a graph of the current iterate
376: */
377: VecView(u,appctx->viewer2);
379: /*
380: Compute the exact solution
381: */
382: ExactSolution(time,appctx->solution,appctx);
384: /*
385: Print debugging information if desired
386: */
387: if (appctx->debug) {
388: printf("Computed solution vector\n");
389: VecView(u,PETSC_VIEWER_STDOUT_SELF);
390: printf("Exact solution vector\n");
391: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
392: }
394: /*
395: Compute the 2-norm and max-norm of the error
396: */
397: VecAXPY(appctx->solution,-1.0,u);
398: VecNorm(appctx->solution,NORM_2,&norm_2);
399: norm_2 = sqrt(appctx->h)*norm_2;
400: VecNorm(appctx->solution,NORM_MAX,&norm_max);
402: TSGetTimeStep(ts,&dt);
403: PetscPrintf(PETSC_COMM_WORLD,"Timestep %3D: step size = %-11g, time = %-11g, 2-norm error = %-11g, max norm error = %-11g\n",
404: step,dt,time,norm_2,norm_max);
405: appctx->norm_2 += norm_2;
406: appctx->norm_max += norm_max;
408: dttol = .0001;
409: PetscOptionsGetReal(PETSC_NULL,"-dttol",&dttol,PETSC_NULL);
410: if (dt < dttol) {
411: dt *= .999;
412: TSSetTimeStep(ts,dt);
413: }
415: /*
416: View a graph of the error
417: */
418: VecView(appctx->solution,appctx->viewer1);
420: /*
421: Print debugging information if desired
422: */
423: if (appctx->debug) {
424: printf("Error vector\n");
425: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
426: }
428: return 0;
429: }
430: /* --------------------------------------------------------------------- */
433: /*
434: RHSMatrixHeat - User-provided routine to compute the right-hand-side
435: matrix for the heat equation.
437: Input Parameters:
438: ts - the TS context
439: t - current time
440: global_in - global input vector
441: dummy - optional user-defined context, as set by TSetRHSJacobian()
443: Output Parameters:
444: AA - Jacobian matrix
445: BB - optionally different preconditioning matrix
446: str - flag indicating matrix structure
448: Notes:
449: Recall that MatSetValues() uses 0-based row and column numbers
450: in Fortran as well as in C.
451: */
452: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
453: {
454: Mat A = *AA; /* Jacobian matrix */
455: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
456: PetscInt mstart = 0;
457: PetscInt mend = appctx->m;
459: PetscInt i,idx[3];
460: PetscScalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;
462: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
463: Compute entries for the locally owned part of the matrix
464: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
465: /*
466: Set matrix rows corresponding to boundary data
467: */
469: mstart = 0;
470: v[0] = 1.0;
471: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
472: mstart++;
474: mend--;
475: v[0] = 1.0;
476: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
478: /*
479: Set matrix rows corresponding to interior data. We construct the
480: matrix one row at a time.
481: */
482: v[0] = sone; v[1] = stwo; v[2] = sone;
483: for (i=mstart; i<mend; i++) {
484: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
485: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
486: }
488: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
489: Complete the matrix assembly process and set some options
490: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
491: /*
492: Assemble matrix, using the 2-step process:
493: MatAssemblyBegin(), MatAssemblyEnd()
494: Computations can be done while messages are in transition
495: by placing code between these two statements.
496: */
497: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
498: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
500: /*
501: Set flag to indicate that the Jacobian matrix retains an identical
502: nonzero structure throughout all timestepping iterations (although the
503: values of the entries change). Thus, we can save some work in setting
504: up the preconditioner (e.g., no need to redo symbolic factorization for
505: ILU/ICC preconditioners).
506: - If the nonzero structure of the matrix is different during
507: successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
508: must be used instead. If you are unsure whether the matrix
509: structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
510: - Caution: If you specify SAME_NONZERO_PATTERN, PETSc
511: believes your assertion and does not check the structure
512: of the matrix. If you erroneously claim that the structure
513: is the same when it actually is not, the new preconditioner
514: will not function correctly. Thus, use this optimization
515: feature with caution!
516: */
517: *str = SAME_NONZERO_PATTERN;
519: /*
520: Set and option to indicate that we will never add a new nonzero location
521: to the matrix. If we do, it will generate an error.
522: */
523: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR);
525: return 0;
526: }
527: /* --------------------------------------------------------------------- */
530: /*
531: Input Parameters:
532: ts - the TS context
533: t - current time
534: f - function
535: ctx - optional user-defined context, as set by TSetBCFunction()
536: */
537: PetscErrorCode MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
538: {
539: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
540: PetscErrorCode ierr,m = appctx->m;
541: PetscScalar *fa;
543: VecGetArray(f,&fa);
544: fa[0] = 0.0;
545: fa[m-1] = 0.0;
546: VecRestoreArray(f,&fa);
547: printf("t=%g\n",t);
548:
549: return 0;
550: }