Actual source code: ex3.c
1: /*$Id: ex3.c,v 1.28 2001/08/10 03:34:17 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: -time_dependent_bc : Treat the problem as having time-dependent boundary conditionsn
10: -debug : Activate debugging printoutsn
11: -nox : Deactivate x-window graphicsnn";
13: /*
14: Concepts: TS^time-dependent linear problems
15: Concepts: TS^heat equation
16: Concepts: TS^diffusion equation
17: Processors: 1
18: */
20: /* ------------------------------------------------------------------------
22: This program solves the one-dimensional heat equation (also called the
23: diffusion equation),
24: u_t = u_xx,
25: on the domain 0 <= x <= 1, with the boundary conditions
26: u(t,0) = 0, u(t,1) = 0,
27: and the initial condition
28: u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
29: This is a linear, second-order, parabolic equation.
31: We discretize the right-hand side using finite differences with
32: uniform grid spacing h:
33: u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
34: We then demonstrate time evolution using the various TS methods by
35: running the program via
36: ex3 -ts_type <timestepping solver>
38: We compare the approximate solution with the exact solution, given by
39: u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
40: 3*exp(-4*pi*pi*t) * sin(2*pi*x)
42: Notes:
43: This code demonstrates the TS solver interface to two variants of
44: linear problems, u_t = f(u,t), namely
45: - time-dependent f: f(u,t) is a function of t
46: - time-independent f: f(u,t) is simply f(u)
48: The parallel version of this code is ts/examples/tutorials/ex4.c
50: ------------------------------------------------------------------------- */
52: /*
53: Include "petscts.h" so that we can use TS solvers. Note that this file
54: automatically includes:
55: petsc.h - base PETSc routines petscvec.h - vectors
56: petscsys.h - system routines petscmat.h - matrices
57: petscis.h - index sets petscksp.h - Krylov subspace methods
58: petscviewer.h - viewers petscpc.h - preconditioners
59: petscsles.h - linear solvers petscsnes.h - nonlinear solvers
60: */
62: #include petscts.h
64: /*
65: User-defined application context - contains data needed by the
66: application-provided call-back routines.
67: */
68: typedef struct {
69: Vec solution; /* global exact solution vector */
70: int m; /* total number of grid points */
71: PetscReal h; /* mesh width h = 1/(m-1) */
72: PetscTruth debug; /* flag (1 indicates activation of debugging printouts) */
73: PetscViewer viewer1,viewer2; /* viewers for the solution and error */
74: PetscReal norm_2,norm_max; /* error norms */
75: } AppCtx;
77: /*
78: User-defined routines
79: */
80: extern int InitialConditions(Vec,AppCtx*);
81: extern int RHSMatrixHeat(TS,PetscReal,Mat*,Mat*,MatStructure*,void*);
82: extern int Monitor(TS,int,PetscReal,Vec,void*);
83: extern int ExactSolution(PetscReal,Vec,AppCtx*);
84: extern int MyBCRoutine(TS,PetscReal,Vec,void*);
86: int main(int argc,char **argv)
87: {
88: AppCtx appctx; /* user-defined application context */
89: TS ts; /* timestepping context */
90: Mat A; /* matrix data structure */
91: Vec u; /* approximate solution vector */
92: PetscReal time_total_max = 100.0; /* default max total time */
93: int time_steps_max = 100; /* default max timesteps */
94: PetscDraw draw; /* drawing context */
95: int ierr,steps,size,m;
96: PetscReal dt,ftime;
97: PetscTruth flg;
99: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
100: Initialize program and set problem parameters
101: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
102:
103: PetscInitialize(&argc,&argv,(char*)0,help);
104: MPI_Comm_size(PETSC_COMM_WORLD,&size);
105: if (size != 1) SETERRQ(1,"This is a uniprocessor example only!");
107: m = 60;
108: PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
109: PetscOptionsHasName(PETSC_NULL,"-debug",&appctx.debug);
110: appctx.m = m;
111: appctx.h = 1.0/(m-1.0);
112: appctx.norm_2 = 0.0;
113: appctx.norm_max = 0.0;
114: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processorn");
116: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
117: Create vector data structures
118: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
120: /*
121: Create vector data structures for approximate and exact solutions
122: */
123: VecCreateSeq(PETSC_COMM_SELF,m,&u);
124: VecDuplicate(u,&appctx.solution);
126: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
127: Set up displays to show graphs of the solution and error
128: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
130: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
131: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
132: PetscDrawSetDoubleBuffer(draw);
133: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
134: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
135: PetscDrawSetDoubleBuffer(draw);
137: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
138: Create timestepping solver context
139: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
141: TSCreate(PETSC_COMM_SELF,&ts);
142: TSSetProblemType(ts,TS_LINEAR);
144: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
145: Set optional user-defined monitoring routine
146: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
148: TSSetMonitor(ts,Monitor,&appctx,PETSC_NULL);
150: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
152: Create matrix data structure; set matrix evaluation routine.
153: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
155: MatCreate(PETSC_COMM_SELF,PETSC_DECIDE,PETSC_DECIDE,m,m,&A);
156: MatSetFromOptions(A);
158: PetscOptionsHasName(PETSC_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: TSSetRHSMatrix(ts,A,A,RHSMatrixHeat,&appctx);
166: } else {
167: /*
168: For linear problems with a time-independent f(u) in the equation
169: u_t = f(u), the user provides the discretized right-hand-side
170: as a matrix only once, and then sets a null matrix evaluation
171: routine.
172: */
173: MatStructure A_structure;
174: RHSMatrixHeat(ts,0.0,&A,&A,&A_structure,&appctx);
175: TSSetRHSMatrix(ts,A,A,PETSC_NULL,&appctx);
176: }
178: /* Treat the problem as having time-dependent boundary conditions */
179: PetscOptionsHasName(PETSC_NULL,"-time_dependent_bc",&flg);
180: if (flg) {
181: TSSetRHSBoundaryConditions(ts,MyBCRoutine,&appctx);
182: }
184: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
185: Set solution vector and initial timestep
186: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
188: dt = appctx.h*appctx.h/2.0;
189: TSSetInitialTimeStep(ts,0.0,dt);
190: TSSetSolution(ts,u);
192: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
193: Customize timestepping solver:
194: - Set the solution method to be the Backward Euler method.
195: - Set timestepping duration info
196: Then set runtime options, which can override these defaults.
197: For example,
198: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
199: to override the defaults set by TSSetDuration().
200: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
202: TSSetDuration(ts,time_steps_max,time_total_max);
203: TSSetFromOptions(ts);
205: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
206: Solve the problem
207: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
209: /*
210: Evaluate initial conditions
211: */
212: InitialConditions(u,&appctx);
214: /*
215: Run the timestepping solver
216: */
217: TSStep(ts,&steps,&ftime);
219: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
220: View timestepping solver info
221: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
223: PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %g, avg. error (max norm) = %gn",
224: appctx.norm_2/steps,appctx.norm_max/steps);
225: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
227: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
228: Free work space. All PETSc objects should be destroyed when they
229: are no longer needed.
230: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
232: TSDestroy(ts);
233: MatDestroy(A);
234: VecDestroy(u);
235: PetscViewerDestroy(appctx.viewer1);
236: PetscViewerDestroy(appctx.viewer2);
237: VecDestroy(appctx.solution);
239: /*
240: Always call PetscFinalize() before exiting a program. This routine
241: - finalizes the PETSc libraries as well as MPI
242: - provides summary and diagnostic information if certain runtime
243: options are chosen (e.g., -log_summary).
244: */
245: PetscFinalize();
246: return 0;
247: }
248: /* --------------------------------------------------------------------- */
249: /*
250: InitialConditions - Computes the solution at the initial time.
252: Input Parameter:
253: u - uninitialized solution vector (global)
254: appctx - user-defined application context
256: Output Parameter:
257: u - vector with solution at initial time (global)
258: */
259: int InitialConditions(Vec u,AppCtx *appctx)
260: {
261: PetscScalar *u_localptr,h = appctx->h;
262: int i,ierr;
264: /*
265: Get a pointer to vector data.
266: - For default PETSc vectors, VecGetArray() returns a pointer to
267: the data array. Otherwise, the routine is implementation dependent.
268: - You MUST call VecRestoreArray() when you no longer need access to
269: the array.
270: - Note that the Fortran interface to VecGetArray() differs from the
271: C version. See the users manual for details.
272: */
273: VecGetArray(u,&u_localptr);
275: /*
276: We initialize the solution array by simply writing the solution
277: directly into the array locations. Alternatively, we could use
278: VecSetValues() or VecSetValuesLocal().
279: */
280: for (i=0; i<appctx->m; i++) {
281: u_localptr[i] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);
282: }
284: /*
285: Restore vector
286: */
287: VecRestoreArray(u,&u_localptr);
289: /*
290: Print debugging information if desired
291: */
292: if (appctx->debug) {
293: printf("initial guess vectorn");
294: VecView(u,PETSC_VIEWER_STDOUT_SELF);
295: }
297: return 0;
298: }
299: /* --------------------------------------------------------------------- */
300: /*
301: ExactSolution - Computes the exact solution at a given time.
303: Input Parameters:
304: t - current time
305: solution - vector in which exact solution will be computed
306: appctx - user-defined application context
308: Output Parameter:
309: solution - vector with the newly computed exact solution
310: */
311: int ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
312: {
313: PetscScalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
314: int i,ierr;
316: /*
317: Get a pointer to vector data.
318: */
319: VecGetArray(solution,&s_localptr);
321: /*
322: Simply write the solution directly into the array locations.
323: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
324: */
325: ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc);
326: ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
327: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
328: for (i=0; i<appctx->m; i++) {
329: s_localptr[i] = PetscSinScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i)*ex2;
330: }
332: /*
333: Restore vector
334: */
335: VecRestoreArray(solution,&s_localptr);
336: return 0;
337: }
338: /* --------------------------------------------------------------------- */
339: /*
340: Monitor - User-provided routine to monitor the solution computed at
341: each timestep. This example plots the solution and computes the
342: error in two different norms.
344: This example also demonstrates changing the timestep via TSSetTimeStep().
346: Input Parameters:
347: ts - the timestep context
348: step - the count of the current step (with 0 meaning the
349: initial condition)
350: time - the current time
351: u - the solution at this timestep
352: ctx - the user-provided context for this monitoring routine.
353: In this case we use the application context which contains
354: information about the problem size, workspace and the exact
355: solution.
356: */
357: int Monitor(TS ts,int step,PetscReal time,Vec u,void *ctx)
358: {
359: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
360: int ierr;
361: PetscReal norm_2,norm_max,dt,dttol;
362: PetscScalar mone = -1.0;
363: /*
364: View a graph of the current iterate
365: */
366: VecView(u,appctx->viewer2);
368: /*
369: Compute the exact solution
370: */
371: ExactSolution(time,appctx->solution,appctx);
373: /*
374: Print debugging information if desired
375: */
376: if (appctx->debug) {
377: printf("Computed solution vectorn");
378: VecView(u,PETSC_VIEWER_STDOUT_SELF);
379: printf("Exact solution vectorn");
380: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
381: }
383: /*
384: Compute the 2-norm and max-norm of the error
385: */
386: VecAXPY(&mone,u,appctx->solution);
387: VecNorm(appctx->solution,NORM_2,&norm_2);
388: norm_2 = sqrt(appctx->h)*norm_2;
389: VecNorm(appctx->solution,NORM_MAX,&norm_max);
391: TSGetTimeStep(ts,&dt);
392: printf("Timestep %3d: step size = %-11g, time = %-11g, 2-norm error = %-11g, max norm error = %-11gn",
393: step,dt,time,norm_2,norm_max);
394: appctx->norm_2 += norm_2;
395: appctx->norm_max += norm_max;
397: dttol = .0001;
398: PetscOptionsGetReal(PETSC_NULL,"-dttol",&dttol,PETSC_NULL);
399: if (dt < dttol) {
400: dt *= .999;
401: TSSetTimeStep(ts,dt);
402: }
404: /*
405: View a graph of the error
406: */
407: VecView(appctx->solution,appctx->viewer1);
409: /*
410: Print debugging information if desired
411: */
412: if (appctx->debug) {
413: printf("Error vectorn");
414: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
415: }
417: return 0;
418: }
419: /* --------------------------------------------------------------------- */
420: /*
421: RHSMatrixHeat - User-provided routine to compute the right-hand-side
422: matrix for the heat equation.
424: Input Parameters:
425: ts - the TS context
426: t - current time
427: global_in - global input vector
428: dummy - optional user-defined context, as set by TSetRHSJacobian()
430: Output Parameters:
431: AA - Jacobian matrix
432: BB - optionally different preconditioning matrix
433: str - flag indicating matrix structure
435: Notes:
436: Recall that MatSetValues() uses 0-based row and column numbers
437: in Fortran as well as in C.
438: */
439: int RHSMatrixHeat(TS ts,PetscReal t,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
440: {
441: Mat A = *AA; /* Jacobian matrix */
442: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
443: int mstart = 0;
444: int mend = appctx->m;
445: int ierr,i,idx[3];
446: PetscScalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;
448: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
449: Compute entries for the locally owned part of the matrix
450: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
451: /*
452: Set matrix rows corresponding to boundary data
453: */
455: mstart = 0;
456: v[0] = 1.0;
457: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
458: mstart++;
460: mend--;
461: v[0] = 1.0;
462: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
464: /*
465: Set matrix rows corresponding to interior data. We construct the
466: matrix one row at a time.
467: */
468: v[0] = sone; v[1] = stwo; v[2] = sone;
469: for (i=mstart; i<mend; i++) {
470: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
471: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
472: }
474: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
475: Complete the matrix assembly process and set some options
476: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
477: /*
478: Assemble matrix, using the 2-step process:
479: MatAssemblyBegin(), MatAssemblyEnd()
480: Computations can be done while messages are in transition
481: by placing code between these two statements.
482: */
483: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
484: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
486: /*
487: Set flag to indicate that the Jacobian matrix retains an identical
488: nonzero structure throughout all timestepping iterations (although the
489: values of the entries change). Thus, we can save some work in setting
490: up the preconditioner (e.g., no need to redo symbolic factorization for
491: ILU/ICC preconditioners).
492: - If the nonzero structure of the matrix is different during
493: successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
494: must be used instead. If you are unsure whether the matrix
495: structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
496: - Caution: If you specify SAME_NONZERO_PATTERN, PETSc
497: believes your assertion and does not check the structure
498: of the matrix. If you erroneously claim that the structure
499: is the same when it actually is not, the new preconditioner
500: will not function correctly. Thus, use this optimization
501: feature with caution!
502: */
503: *str = SAME_NONZERO_PATTERN;
505: /*
506: Set and option to indicate that we will never add a new nonzero location
507: to the matrix. If we do, it will generate an error.
508: */
509: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR);
511: return 0;
512: }
513: /* --------------------------------------------------------------------- */
514: /*
515: Input Parameters:
516: ts - the TS context
517: t - current time
518: f - function
519: ctx - optional user-defined context, as set by TSetBCFunction()
520: */
521: int MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
522: {
523: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
524: int ierr,m = appctx->m;
525: PetscScalar *fa;
527: VecGetArray(f,&fa);
528: fa[0] = 0.0;
529: fa[m-1] = 0.0;
530: VecRestoreArray(f,&fa);
531: printf("t=%gn",t);
532:
533: return 0;
534: }