Actual source code: ex6.c
petsc-3.3-p7 2013-05-11
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: -debug : Activate debugging printouts\n\
9: -nox : Deactivate x-window graphics\n\n";
11: /*
12: Concepts: TS^time-dependent linear problems
13: Concepts: TS^heat equation
14: Concepts: TS^diffusion equation
15: Routines: TSCreate(); TSSetSolution(); TSSetRHSJacobian(), TSSetIJacobian();
16: Routines: TSSetInitialTimeStep(); TSSetDuration(); TSMonitorSet();
17: Routines: TSSetFromOptions(); TSStep(); TSDestroy();
18: Routines: TSSetTimeStep(); TSGetTimeStep();
19: Processors: 1
20: */
22: /* ------------------------------------------------------------------------
24: This program solves the one-dimensional heat equation (also called the
25: diffusion equation),
26: u_t = u_xx,
27: on the domain 0 <= x <= 1, with the boundary conditions
28: u(t,0) = 0, u(t,1) = 0,
29: and the initial condition
30: u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
31: This is a linear, second-order, parabolic equation.
33: We discretize the right-hand side using finite differences with
34: uniform grid spacing h:
35: u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
36: We then demonstrate time evolution using the various TS methods by
37: running the program via
38: ex3 -ts_type <timestepping solver>
40: We compare the approximate solution with the exact solution, given by
41: u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
42: 3*exp(-4*pi*pi*t) * sin(2*pi*x)
44: Notes:
45: This code demonstrates the TS solver interface to two variants of
46: linear problems, u_t = f(u,t), namely
47: - time-dependent f: f(u,t) is a function of t
48: - time-independent f: f(u,t) is simply f(u)
50: The parallel version of this code is ts/examples/tutorials/ex4.c
52: ------------------------------------------------------------------------- */
54: /*
55: Include "ts.h" so that we can use TS solvers. Note that this file
56: automatically includes:
57: petscsys.h - base PETSc routines vec.h - vectors
58: sys.h - system routines mat.h - matrices
59: is.h - index sets ksp.h - Krylov subspace methods
60: viewer.h - viewers pc.h - preconditioners
61: snes.h - nonlinear solvers
62: */
64: #include <petscts.h>
66: /*
67: User-defined application context - contains data needed by the
68: application-provided call-back routines.
69: */
70: typedef struct {
71: Vec solution; /* global exact solution vector */
72: PetscInt m; /* total number of grid points */
73: PetscReal h; /* mesh width h = 1/(m-1) */
74: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
75: PetscViewer viewer1, viewer2; /* viewers for the solution and error */
76: PetscReal norm_2, norm_max; /* error norms */
77: } AppCtx;
79: /*
80: User-defined routines
81: */
82: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
83: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat*,Mat*,MatStructure*,void*);
84: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
85: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);
86: extern PetscErrorCode MyBCRoutine(TS,PetscReal,Vec,void*);
90: int main(int argc,char **argv)
91: {
92: AppCtx appctx; /* user-defined application context */
93: TS ts; /* timestepping context */
94: Mat A; /* matrix data structure */
95: Vec u; /* approximate solution vector */
96: PetscReal time_total_max = 100.0; /* default max total time */
97: PetscInt time_steps_max = 100; /* default max timesteps */
98: PetscDraw draw; /* drawing context */
100: PetscInt steps, m;
101: PetscMPIInt size;
102: PetscReal dt;
103: PetscReal ftime;
104: PetscBool flg;
105: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
106: Initialize program and set problem parameters
107: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
108:
109: PetscInitialize(&argc,&argv,(char*)0,help);
110: MPI_Comm_size(PETSC_COMM_WORLD,&size);
111: if (size != 1) SETERRQ(PETSC_COMM_SELF,1,"This is a uniprocessor example only!");
113: m = 60;
114: PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
115: PetscOptionsHasName(PETSC_NULL,"-debug",&appctx.debug);
116: appctx.m = m;
117: appctx.h = 1.0/(m-1.0);
118: appctx.norm_2 = 0.0;
119: appctx.norm_max = 0.0;
120: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");
122: PetscOptionsGetInt(PETSC_NULL,"-time_steps_max",&time_steps_max,PETSC_NULL);
124: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
125: Create vector data structures
126: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
128: /*
129: Create vector data structures for approximate and exact solutions
130: */
131: VecCreateSeq(PETSC_COMM_SELF,m,&u);
132: VecDuplicate(u,&appctx.solution);
134: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
135: Set up displays to show graphs of the solution and error
136: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
138: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
139: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
140: PetscDrawSetDoubleBuffer(draw);
141: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
142: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
143: PetscDrawSetDoubleBuffer(draw);
145: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
146: Create timestepping solver context
147: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
149: TSCreate(PETSC_COMM_SELF,&ts);
150: TSSetProblemType(ts,TS_LINEAR);
152: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
153: Set optional user-defined monitoring routine
154: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
156: TSMonitorSet(ts,Monitor,&appctx,PETSC_NULL);
158: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
160: Create matrix data structure; set matrix evaluation routine.
161: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
163: MatCreate(PETSC_COMM_SELF,&A);
164: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
165: MatSetFromOptions(A);
166: MatSetUp(A);
168: PetscOptionsHasName(PETSC_NULL,"-time_dependent_rhs",&flg);
169: if (flg) {
170: /*
171: For linear problems with a time-dependent f(u,t) in the equation
172: u_t = f(u,t), the user provides the discretized right-hand-side
173: as a time-dependent matrix.
174: */
175: TSSetRHSFunction(ts,PETSC_NULL,TSComputeRHSFunctionLinear,&appctx);
176: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
177: } else {
178: /*
179: For linear problems with a time-independent f(u) in the equation
180: u_t = f(u), the user provides the discretized right-hand-side
181: as a matrix only once, and then sets a null matrix evaluation
182: routine.
183: */
184: MatStructure A_structure;
185: RHSMatrixHeat(ts,0.0,u,&A,&A,&A_structure,&appctx);
186: TSSetRHSFunction(ts,PETSC_NULL,TSComputeRHSFunctionLinear,&appctx);
187: TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
188: }
190: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
191: Set solution vector and initial timestep
192: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
194: dt = appctx.h*appctx.h/2.0;
195: TSSetInitialTimeStep(ts,0.0,dt);
196: TSSetSolution(ts,u);
198: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
199: Customize timestepping solver:
200: - Set the solution method to be the Backward Euler method.
201: - Set timestepping duration info
202: Then set runtime options, which can override these defaults.
203: For example,
204: -ts_max_steps <maxsteps> -ts_final_time <maxtime>
205: to override the defaults set by TSSetDuration().
206: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
208: TSSetDuration(ts,time_steps_max,time_total_max);
209: TSSetFromOptions(ts);
211: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
212: Solve the problem
213: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
215: /*
216: Evaluate initial conditions
217: */
218: InitialConditions(u,&appctx);
220: /*
221: Run the timestepping solver
222: */
223: TSSolve(ts,u,&ftime);
224: TSGetTimeStepNumber(ts,&steps);
226: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
227: View timestepping solver info
228: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
230: PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %G, avg. error (max norm) = %G\n",
231: appctx.norm_2/steps,appctx.norm_max/steps);
232: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
234: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
235: Free work space. All PETSc objects should be destroyed when they
236: are no longer needed.
237: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
239: TSDestroy(&ts);
240: MatDestroy(&A);
241: VecDestroy(&u);
242: PetscViewerDestroy(&appctx.viewer1);
243: PetscViewerDestroy(&appctx.viewer2);
244: VecDestroy(&appctx.solution);
246: /*
247: Always call PetscFinalize() before exiting a program. This routine
248: - finalizes the PETSc libraries as well as MPI
249: - provides summary and diagnostic information if certain runtime
250: options are chosen (e.g., -log_summary).
251: */
252: PetscFinalize();
253: return 0;
254: }
255: /* --------------------------------------------------------------------- */
258: /*
259: InitialConditions - Computes the solution at the initial time.
261: Input Parameter:
262: u - uninitialized solution vector (global)
263: appctx - user-defined application context
265: Output Parameter:
266: u - vector with solution at initial time (global)
267: */
268: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
269: {
270: PetscScalar *u_localptr;
271: PetscInt i;
274: /*
275: Get a pointer to vector data.
276: - For default PETSc vectors, VecGetArray() returns a pointer to
277: the data array. Otherwise, the routine is implementation dependent.
278: - You MUST call VecRestoreArray() when you no longer need access to
279: the array.
280: - Note that the Fortran interface to VecGetArray() differs from the
281: C version. See the users manual for details.
282: */
283: VecGetArray(u,&u_localptr);
285: /*
286: We initialize the solution array by simply writing the solution
287: directly into the array locations. Alternatively, we could use
288: VecSetValues() or VecSetValuesLocal().
289: */
290: for (i=0; i<appctx->m; i++) {
291: u_localptr[i] = sin(PETSC_PI*i*6.*appctx->h) + 3.*sin(PETSC_PI*i*2.*appctx->h);
292: }
294: /*
295: Restore vector
296: */
297: VecRestoreArray(u,&u_localptr);
299: /*
300: Print debugging information if desired
301: */
302: if (appctx->debug) {
303: printf("initial guess vector\n");
304: VecView(u,PETSC_VIEWER_STDOUT_SELF);
305: }
307: return 0;
308: }
309: /* --------------------------------------------------------------------- */
312: /*
313: ExactSolution - Computes the exact solution at a given time.
315: Input Parameters:
316: t - current time
317: solution - vector in which exact solution will be computed
318: appctx - user-defined application context
320: Output Parameter:
321: solution - vector with the newly computed exact solution
322: */
323: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
324: {
325: PetscScalar *s_localptr, h = appctx->h, ex1, ex2, sc1, sc2;
326: PetscInt i;
329: /*
330: Get a pointer to vector data.
331: */
332: VecGetArray(solution,&s_localptr);
334: /*
335: Simply write the solution directly into the array locations.
336: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
337: */
338: ex1 = exp(-36.*PETSC_PI*PETSC_PI*t); ex2 = exp(-4.*PETSC_PI*PETSC_PI*t);
339: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
340: for (i=0; i<appctx->m; i++) {
341: s_localptr[i] = sin(PetscRealPart(sc1)*(PetscReal)i)*ex1 + 3.*sin(PetscRealPart(sc2)*(PetscReal)i)*ex2;
342: }
344: /*
345: Restore vector
346: */
347: VecRestoreArray(solution,&s_localptr);
348: return 0;
349: }
350: /* --------------------------------------------------------------------- */
353: /*
354: Monitor - User-provided routine to monitor the solution computed at
355: each timestep. This example plots the solution and computes the
356: error in two different norms.
358: This example also demonstrates changing the timestep via TSSetTimeStep().
360: Input Parameters:
361: ts - the timestep context
362: step - the count of the current step (with 0 meaning the
363: initial condition)
364: crtime - the current time
365: u - the solution at this timestep
366: ctx - the user-provided context for this monitoring routine.
367: In this case we use the application context which contains
368: information about the problem size, workspace and the exact
369: solution.
370: */
371: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal crtime,Vec u,void *ctx)
372: {
373: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
375: PetscReal norm_2, norm_max, dt, dttol;
376: PetscBool flg;
378: /*
379: View a graph of the current iterate
380: */
381: VecView(u,appctx->viewer2);
383: /*
384: Compute the exact solution
385: */
386: ExactSolution(crtime,appctx->solution,appctx);
388: /*
389: Print debugging information if desired
390: */
391: if (appctx->debug) {
392: printf("Computed solution vector\n");
393: VecView(u,PETSC_VIEWER_STDOUT_SELF);
394: printf("Exact solution vector\n");
395: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
396: }
398: /*
399: Compute the 2-norm and max-norm of the error
400: */
401: VecAXPY(appctx->solution,-1.0,u);
402: VecNorm(appctx->solution,NORM_2,&norm_2);
403: norm_2 = PetscSqrtReal(appctx->h)*norm_2;
404: VecNorm(appctx->solution,NORM_MAX,&norm_max);
406: TSGetTimeStep(ts,&dt);
407: if (norm_2 > 1.e-2){
408: printf("Timestep %d: step size = %G, time = %G, 2-norm error = %G, max norm error = %G\n",
409: (int)step,dt,crtime,norm_2,norm_max);
410: }
411: appctx->norm_2 += norm_2;
412: appctx->norm_max += norm_max;
414: dttol = .0001;
415: PetscOptionsGetReal(PETSC_NULL,"-dttol",&dttol,&flg);
416: if (dt < dttol) {
417: dt *= .999;
418: TSSetTimeStep(ts,dt);
419: }
421: /*
422: View a graph of the error
423: */
424: VecView(appctx->solution,appctx->viewer1);
426: /*
427: Print debugging information if desired
428: */
429: if (appctx->debug) {
430: printf("Error vector\n");
431: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
432: }
434: return 0;
435: }
436: /* --------------------------------------------------------------------- */
439: /*
440: RHSMatrixHeat - User-provided routine to compute the right-hand-side
441: matrix for the heat equation.
443: Input Parameters:
444: ts - the TS context
445: t - current time
446: global_in - global input vector
447: dummy - optional user-defined context, as set by TSetRHSJacobian()
449: Output Parameters:
450: AA - Jacobian matrix
451: BB - optionally different preconditioning matrix
452: str - flag indicating matrix structure
454: Notes:
455: Recall that MatSetValues() uses 0-based row and column numbers
456: in Fortran as well as in C.
457: */
458: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
459: {
460: Mat A = *AA; /* Jacobian matrix */
461: AppCtx *appctx = (AppCtx *) ctx; /* user-defined application context */
462: PetscInt mstart = 0;
463: PetscInt mend = appctx->m;
465: PetscInt 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,PETSC_TRUE);
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: PetscErrorCode MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
544: {
545: AppCtx *appctx = (AppCtx *) ctx; /* user-defined application context */
547: PetscInt m = appctx->m;
548: PetscScalar *fa;
550: VecGetArray(f,&fa);
551: fa[0] = 0.0;
552: fa[m-1] = 1.0;
553: VecRestoreArray(f,&fa);
554: printf("t=%g\n",t);
555:
556: return 0;
557: }