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: #undef __FUNCT__
88: int main(int argc,char **argv)
89: {
90: AppCtx appctx; /* user-defined application context */
91: TS ts; /* timestepping context */
92: Mat A; /* matrix data structure */
93: Vec u; /* approximate solution vector */
94: PetscReal time_total_max = 100.0; /* default max total time */
95: int time_steps_max = 100; /* default max timesteps */
96: PetscDraw draw; /* drawing context */
97: int ierr,steps,size,m;
98: PetscReal dt,ftime;
99: PetscTruth flg;
101: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
102: Initialize program and set problem parameters
103: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
104:
105: PetscInitialize(&argc,&argv,(char*)0,help);
106: MPI_Comm_size(PETSC_COMM_WORLD,&size);
107: if (size != 1) SETERRQ(1,"This is a uniprocessor example only!");
109: m = 60;
110: PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
111: PetscOptionsHasName(PETSC_NULL,"-debug",&appctx.debug);
112: appctx.m = m;
113: appctx.h = 1.0/(m-1.0);
114: appctx.norm_2 = 0.0;
115: appctx.norm_max = 0.0;
116: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processorn");
118: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
119: Create vector data structures
120: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
122: /*
123: Create vector data structures for approximate and exact solutions
124: */
125: VecCreateSeq(PETSC_COMM_SELF,m,&u);
126: VecDuplicate(u,&appctx.solution);
128: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
129: Set up displays to show graphs of the solution and error
130: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
132: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
133: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
134: PetscDrawSetDoubleBuffer(draw);
135: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
136: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
137: PetscDrawSetDoubleBuffer(draw);
139: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
140: Create timestepping solver context
141: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
143: TSCreate(PETSC_COMM_SELF,&ts);
144: TSSetProblemType(ts,TS_LINEAR);
146: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
147: Set optional user-defined monitoring routine
148: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
150: TSSetMonitor(ts,Monitor,&appctx,PETSC_NULL);
152: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
154: Create matrix data structure; set matrix evaluation routine.
155: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
157: MatCreate(PETSC_COMM_SELF,PETSC_DECIDE,PETSC_DECIDE,m,m,&A);
158: MatSetFromOptions(A);
160: PetscOptionsHasName(PETSC_NULL,"-time_dependent_rhs",&flg);
161: if (flg) {
162: /*
163: For linear problems with a time-dependent f(u,t) in the equation
164: u_t = f(u,t), the user provides the discretized right-hand-side
165: as a time-dependent matrix.
166: */
167: TSSetRHSMatrix(ts,A,A,RHSMatrixHeat,&appctx);
168: } else {
169: /*
170: For linear problems with a time-independent f(u) in the equation
171: u_t = f(u), the user provides the discretized right-hand-side
172: as a matrix only once, and then sets a null matrix evaluation
173: routine.
174: */
175: MatStructure A_structure;
176: RHSMatrixHeat(ts,0.0,&A,&A,&A_structure,&appctx);
177: TSSetRHSMatrix(ts,A,A,PETSC_NULL,&appctx);
178: }
180: /* Treat the problem as having time-dependent boundary conditions */
181: PetscOptionsHasName(PETSC_NULL,"-time_dependent_bc",&flg);
182: if (flg) {
183: TSSetRHSBoundaryConditions(ts,MyBCRoutine,&appctx);
184: }
186: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
187: Set solution vector and initial timestep
188: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
190: dt = appctx.h*appctx.h/2.0;
191: TSSetInitialTimeStep(ts,0.0,dt);
192: TSSetSolution(ts,u);
194: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
195: Customize timestepping solver:
196: - Set the solution method to be the Backward Euler method.
197: - Set timestepping duration info
198: Then set runtime options, which can override these defaults.
199: For example,
200: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
201: to override the defaults set by TSSetDuration().
202: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
204: TSSetDuration(ts,time_steps_max,time_total_max);
205: TSSetFromOptions(ts);
207: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
208: Solve the problem
209: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
211: /*
212: Evaluate initial conditions
213: */
214: InitialConditions(u,&appctx);
216: /*
217: Run the timestepping solver
218: */
219: TSStep(ts,&steps,&ftime);
221: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
222: View timestepping solver info
223: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
225: PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %g, avg. error (max norm) = %gn",
226: appctx.norm_2/steps,appctx.norm_max/steps);
227: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
229: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
230: Free work space. All PETSc objects should be destroyed when they
231: are no longer needed.
232: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
234: TSDestroy(ts);
235: MatDestroy(A);
236: VecDestroy(u);
237: PetscViewerDestroy(appctx.viewer1);
238: PetscViewerDestroy(appctx.viewer2);
239: VecDestroy(appctx.solution);
241: /*
242: Always call PetscFinalize() before exiting a program. This routine
243: - finalizes the PETSc libraries as well as MPI
244: - provides summary and diagnostic information if certain runtime
245: options are chosen (e.g., -log_summary).
246: */
247: PetscFinalize();
248: return 0;
249: }
250: /* --------------------------------------------------------------------- */
251: #undef __FUNCT__
253: /*
254: InitialConditions - Computes the solution at the initial time.
256: Input Parameter:
257: u - uninitialized solution vector (global)
258: appctx - user-defined application context
260: Output Parameter:
261: u - vector with solution at initial time (global)
262: */
263: int InitialConditions(Vec u,AppCtx *appctx)
264: {
265: PetscScalar *u_localptr,h = appctx->h;
266: int i,ierr;
268: /*
269: Get a pointer to vector data.
270: - For default PETSc vectors, VecGetArray() returns a pointer to
271: the data array. Otherwise, the routine is implementation dependent.
272: - You MUST call VecRestoreArray() when you no longer need access to
273: the array.
274: - Note that the Fortran interface to VecGetArray() differs from the
275: C version. See the users manual for details.
276: */
277: VecGetArray(u,&u_localptr);
279: /*
280: We initialize the solution array by simply writing the solution
281: directly into the array locations. Alternatively, we could use
282: VecSetValues() or VecSetValuesLocal().
283: */
284: for (i=0; i<appctx->m; i++) {
285: u_localptr[i] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);
286: }
288: /*
289: Restore vector
290: */
291: VecRestoreArray(u,&u_localptr);
293: /*
294: Print debugging information if desired
295: */
296: if (appctx->debug) {
297: printf("initial guess vectorn");
298: VecView(u,PETSC_VIEWER_STDOUT_SELF);
299: }
301: return 0;
302: }
303: /* --------------------------------------------------------------------- */
304: #undef __FUNCT__
306: /*
307: ExactSolution - Computes the exact solution at a given time.
309: Input Parameters:
310: t - current time
311: solution - vector in which exact solution will be computed
312: appctx - user-defined application context
314: Output Parameter:
315: solution - vector with the newly computed exact solution
316: */
317: int ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
318: {
319: PetscScalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
320: int i,ierr;
322: /*
323: Get a pointer to vector data.
324: */
325: VecGetArray(solution,&s_localptr);
327: /*
328: Simply write the solution directly into the array locations.
329: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
330: */
331: ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc);
332: ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
333: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
334: for (i=0; i<appctx->m; i++) {
335: s_localptr[i] = PetscSinScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i)*ex2;
336: }
338: /*
339: Restore vector
340: */
341: VecRestoreArray(solution,&s_localptr);
342: return 0;
343: }
344: /* --------------------------------------------------------------------- */
345: #undef __FUNCT__
347: /*
348: Monitor - User-provided routine to monitor the solution computed at
349: each timestep. This example plots the solution and computes the
350: error in two different norms.
352: This example also demonstrates changing the timestep via TSSetTimeStep().
354: Input Parameters:
355: ts - the timestep context
356: step - the count of the current step (with 0 meaning the
357: initial condition)
358: time - the current time
359: u - the solution at this timestep
360: ctx - the user-provided context for this monitoring routine.
361: In this case we use the application context which contains
362: information about the problem size, workspace and the exact
363: solution.
364: */
365: int Monitor(TS ts,int step,PetscReal time,Vec u,void *ctx)
366: {
367: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
368: int ierr;
369: PetscReal norm_2,norm_max,dt,dttol;
370: PetscScalar mone = -1.0;
371: /*
372: View a graph of the current iterate
373: */
374: VecView(u,appctx->viewer2);
376: /*
377: Compute the exact solution
378: */
379: ExactSolution(time,appctx->solution,appctx);
381: /*
382: Print debugging information if desired
383: */
384: if (appctx->debug) {
385: printf("Computed solution vectorn");
386: VecView(u,PETSC_VIEWER_STDOUT_SELF);
387: printf("Exact solution vectorn");
388: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
389: }
391: /*
392: Compute the 2-norm and max-norm of the error
393: */
394: VecAXPY(&mone,u,appctx->solution);
395: VecNorm(appctx->solution,NORM_2,&norm_2);
396: norm_2 = sqrt(appctx->h)*norm_2;
397: VecNorm(appctx->solution,NORM_MAX,&norm_max);
399: TSGetTimeStep(ts,&dt);
400: printf("Timestep %3d: step size = %-11g, time = %-11g, 2-norm error = %-11g, max norm error = %-11gn",
401: step,dt,time,norm_2,norm_max);
402: appctx->norm_2 += norm_2;
403: appctx->norm_max += norm_max;
405: dttol = .0001;
406: PetscOptionsGetReal(PETSC_NULL,"-dttol",&dttol,PETSC_NULL);
407: if (dt < dttol) {
408: dt *= .999;
409: TSSetTimeStep(ts,dt);
410: }
412: /*
413: View a graph of the error
414: */
415: VecView(appctx->solution,appctx->viewer1);
417: /*
418: Print debugging information if desired
419: */
420: if (appctx->debug) {
421: printf("Error vectorn");
422: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
423: }
425: return 0;
426: }
427: /* --------------------------------------------------------------------- */
428: #undef __FUNCT__
430: /*
431: RHSMatrixHeat - User-provided routine to compute the right-hand-side
432: matrix for the heat equation.
434: Input Parameters:
435: ts - the TS context
436: t - current time
437: global_in - global input vector
438: dummy - optional user-defined context, as set by TSetRHSJacobian()
440: Output Parameters:
441: AA - Jacobian matrix
442: BB - optionally different preconditioning matrix
443: str - flag indicating matrix structure
445: Notes:
446: Recall that MatSetValues() uses 0-based row and column numbers
447: in Fortran as well as in C.
448: */
449: int RHSMatrixHeat(TS ts,PetscReal t,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
450: {
451: Mat A = *AA; /* Jacobian matrix */
452: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
453: int mstart = 0;
454: int mend = appctx->m;
455: int ierr,i,idx[3];
456: PetscScalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;
458: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
459: Compute entries for the locally owned part of the matrix
460: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
461: /*
462: Set matrix rows corresponding to boundary data
463: */
465: mstart = 0;
466: v[0] = 1.0;
467: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
468: mstart++;
470: mend--;
471: v[0] = 1.0;
472: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
474: /*
475: Set matrix rows corresponding to interior data. We construct the
476: matrix one row at a time.
477: */
478: v[0] = sone; v[1] = stwo; v[2] = sone;
479: for (i=mstart; i<mend; i++) {
480: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
481: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
482: }
484: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
485: Complete the matrix assembly process and set some options
486: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
487: /*
488: Assemble matrix, using the 2-step process:
489: MatAssemblyBegin(), MatAssemblyEnd()
490: Computations can be done while messages are in transition
491: by placing code between these two statements.
492: */
493: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
494: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
496: /*
497: Set flag to indicate that the Jacobian matrix retains an identical
498: nonzero structure throughout all timestepping iterations (although the
499: values of the entries change). Thus, we can save some work in setting
500: up the preconditioner (e.g., no need to redo symbolic factorization for
501: ILU/ICC preconditioners).
502: - If the nonzero structure of the matrix is different during
503: successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
504: must be used instead. If you are unsure whether the matrix
505: structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
506: - Caution: If you specify SAME_NONZERO_PATTERN, PETSc
507: believes your assertion and does not check the structure
508: of the matrix. If you erroneously claim that the structure
509: is the same when it actually is not, the new preconditioner
510: will not function correctly. Thus, use this optimization
511: feature with caution!
512: */
513: *str = SAME_NONZERO_PATTERN;
515: /*
516: Set and option to indicate that we will never add a new nonzero location
517: to the matrix. If we do, it will generate an error.
518: */
519: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR);
521: return 0;
522: }
523: /* --------------------------------------------------------------------- */
524: #undef __FUNCT__
526: /*
527: Input Parameters:
528: ts - the TS context
529: t - current time
530: f - function
531: ctx - optional user-defined context, as set by TSetBCFunction()
532: */
533: int MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
534: {
535: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
536: int ierr,m = appctx->m;
537: PetscScalar *fa;
539: VecGetArray(f,&fa);
540: fa[0] = 0.0;
541: fa[m-1] = 0.0;
542: VecRestoreArray(f,&fa);
543: printf("t=%gn",t);
544:
545: return 0;
546: }