Actual source code: ex6.c
petsc-3.4.5 2014-06-29
2: static char help[] ="Solves a simple time-dependent linear PDE (the heat equation).\n\
3: Input parameters include:\n\
4: -m <points>, where <points> = number of grid points\n\
5: -time_dependent_rhs : Treat the problem as having a time-dependent right-hand side\n\
6: -debug : Activate debugging printouts\n\
7: -nox : Deactivate x-window graphics\n\n";
9: /*
10: Concepts: TS^time-dependent linear problems
11: Concepts: TS^heat equation
12: Concepts: TS^diffusion equation
13: Routines: TSCreate(); TSSetSolution(); TSSetRHSJacobian(), TSSetIJacobian();
14: Routines: TSSetInitialTimeStep(); TSSetDuration(); TSMonitorSet();
15: Routines: TSSetFromOptions(); TSStep(); TSDestroy();
16: Routines: TSSetTimeStep(); TSGetTimeStep();
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 "ts.h" so that we can use TS solvers. Note that this file
54: automatically includes:
55: petscsys.h - base PETSc routines vec.h - vectors
56: sys.h - system routines mat.h - matrices
57: is.h - index sets ksp.h - Krylov subspace methods
58: viewer.h - viewers pc.h - preconditioners
59: snes.h - nonlinear solvers
60: */
62: #include <petscts.h>
63: #include <petscdraw.h>
65: /*
66: User-defined application context - contains data needed by the
67: application-provided call-back routines.
68: */
69: typedef struct {
70: Vec solution; /* global exact solution vector */
71: PetscInt m; /* total number of grid points */
72: PetscReal h; /* mesh width h = 1/(m-1) */
73: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
74: PetscViewer viewer1, viewer2; /* viewers for the solution and error */
75: PetscReal norm_2, norm_max; /* error norms */
76: } AppCtx;
78: /*
79: User-defined routines
80: */
81: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
82: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat*,Mat*,MatStructure*,void*);
83: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
84: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);
85: extern PetscErrorCode MyBCRoutine(TS,PetscReal,Vec,void*);
89: int main(int argc,char **argv)
90: {
91: AppCtx appctx; /* user-defined application context */
92: TS ts; /* timestepping context */
93: Mat A; /* matrix data structure */
94: Vec u; /* approximate solution vector */
95: PetscReal time_total_max = 100.0; /* default max total time */
96: PetscInt time_steps_max = 100; /* default max timesteps */
97: PetscDraw draw; /* drawing context */
99: PetscInt steps, m;
100: PetscMPIInt size;
101: PetscReal dt;
102: PetscReal ftime;
103: PetscBool flg;
104: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
105: Initialize program and set problem parameters
106: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
108: PetscInitialize(&argc,&argv,(char*)0,help);
109: MPI_Comm_size(PETSC_COMM_WORLD,&size);
110: if (size != 1) SETERRQ(PETSC_COMM_SELF,1,"This is a uniprocessor example only!");
112: m = 60;
113: PetscOptionsGetInt(NULL,"-m",&m,NULL);
114: PetscOptionsHasName(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;
121: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");
123: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
124: Create vector data structures
125: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
127: /*
128: Create vector data structures for approximate and exact solutions
129: */
130: VecCreateSeq(PETSC_COMM_SELF,m,&u);
131: VecDuplicate(u,&appctx.solution);
133: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
134: Set up displays to show graphs of the solution and error
135: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
137: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
138: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
139: PetscDrawSetDoubleBuffer(draw);
140: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
141: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
142: PetscDrawSetDoubleBuffer(draw);
144: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
145: Create timestepping solver context
146: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
148: TSCreate(PETSC_COMM_SELF,&ts);
149: TSSetProblemType(ts,TS_LINEAR);
151: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
152: Set optional user-defined monitoring routine
153: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
155: TSMonitorSet(ts,Monitor,&appctx,NULL);
157: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
159: Create matrix data structure; set matrix evaluation routine.
160: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
162: MatCreate(PETSC_COMM_SELF,&A);
163: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
164: MatSetFromOptions(A);
165: MatSetUp(A);
167: PetscOptionsHasName(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: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
175: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
176: } else {
177: /*
178: For linear problems with a time-independent f(u) in the equation
179: u_t = f(u), the user provides the discretized right-hand-side
180: as a matrix only once, and then sets a null matrix evaluation
181: routine.
182: */
183: MatStructure A_structure;
184: RHSMatrixHeat(ts,0.0,u,&A,&A,&A_structure,&appctx);
185: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
186: TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
187: }
189: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
190: Set solution vector and initial timestep
191: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
193: dt = appctx.h*appctx.h/2.0;
194: TSSetInitialTimeStep(ts,0.0,dt);
195: TSSetSolution(ts,u);
197: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
198: Customize timestepping solver:
199: - Set the solution method to be the Backward Euler method.
200: - Set timestepping duration info
201: Then set runtime options, which can override these defaults.
202: For example,
203: -ts_max_steps <maxsteps> -ts_final_time <maxtime>
204: to override the defaults set by TSSetDuration().
205: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
207: TSSetDuration(ts,time_steps_max,time_total_max);
208: TSSetFromOptions(ts);
210: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
211: Solve the problem
212: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
214: /*
215: Evaluate initial conditions
216: */
217: InitialConditions(u,&appctx);
219: /*
220: Run the timestepping solver
221: */
222: TSSolve(ts,u);
223: TSGetSolveTime(ts,&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++) u_localptr[i] = sin(PETSC_PI*i*6.*appctx->h) + 3.*sin(PETSC_PI*i*2.*appctx->h);
292: /*
293: Restore vector
294: */
295: VecRestoreArray(u,&u_localptr);
297: /*
298: Print debugging information if desired
299: */
300: if (appctx->debug) {
301: printf("initial guess vector\n");
302: VecView(u,PETSC_VIEWER_STDOUT_SELF);
303: }
305: return 0;
306: }
307: /* --------------------------------------------------------------------- */
310: /*
311: ExactSolution - Computes the exact solution at a given time.
313: Input Parameters:
314: t - current time
315: solution - vector in which exact solution will be computed
316: appctx - user-defined application context
318: Output Parameter:
319: solution - vector with the newly computed exact solution
320: */
321: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
322: {
323: PetscScalar *s_localptr, h = appctx->h, ex1, ex2, sc1, sc2;
324: PetscInt i;
327: /*
328: Get a pointer to vector data.
329: */
330: VecGetArray(solution,&s_localptr);
332: /*
333: Simply write the solution directly into the array locations.
334: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
335: */
336: ex1 = exp(-36.*PETSC_PI*PETSC_PI*t); ex2 = exp(-4.*PETSC_PI*PETSC_PI*t);
337: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
338: for (i=0; i<appctx->m; i++) s_localptr[i] = sin(PetscRealPart(sc1)*(PetscReal)i)*ex1 + 3.*sin(PetscRealPart(sc2)*(PetscReal)i)*ex2;
340: /*
341: Restore vector
342: */
343: VecRestoreArray(solution,&s_localptr);
344: return 0;
345: }
346: /* --------------------------------------------------------------------- */
349: /*
350: Monitor - User-provided routine to monitor the solution computed at
351: each timestep. This example plots the solution and computes the
352: error in two different norms.
354: This example also demonstrates changing the timestep via TSSetTimeStep().
356: Input Parameters:
357: ts - the timestep context
358: step - the count of the current step (with 0 meaning the
359: initial condition)
360: crtime - the current time
361: u - the solution at this timestep
362: ctx - the user-provided context for this monitoring routine.
363: In this case we use the application context which contains
364: information about the problem size, workspace and the exact
365: solution.
366: */
367: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal crtime,Vec u,void *ctx)
368: {
369: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
371: PetscReal norm_2, norm_max, dt, dttol;
372: PetscBool flg;
374: /*
375: View a graph of the current iterate
376: */
377: VecView(u,appctx->viewer2);
379: /*
380: Compute the exact solution
381: */
382: ExactSolution(crtime,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 = PetscSqrtReal(appctx->h)*norm_2;
400: VecNorm(appctx->solution,NORM_MAX,&norm_max);
402: TSGetTimeStep(ts,&dt);
403: if (norm_2 > 1.e-2) {
404: printf("Timestep %d: step size = %G, time = %G, 2-norm error = %G, max norm error = %G\n",
405: (int)step,dt,crtime,norm_2,norm_max);
406: }
407: appctx->norm_2 += norm_2;
408: appctx->norm_max += norm_max;
410: dttol = .0001;
411: PetscOptionsGetReal(NULL,"-dttol",&dttol,&flg);
412: if (dt < dttol) {
413: dt *= .999;
414: TSSetTimeStep(ts,dt);
415: }
417: /*
418: View a graph of the error
419: */
420: VecView(appctx->solution,appctx->viewer1);
422: /*
423: Print debugging information if desired
424: */
425: if (appctx->debug) {
426: printf("Error vector\n");
427: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
428: }
430: return 0;
431: }
432: /* --------------------------------------------------------------------- */
435: /*
436: RHSMatrixHeat - User-provided routine to compute the right-hand-side
437: matrix for the heat equation.
439: Input Parameters:
440: ts - the TS context
441: t - current time
442: global_in - global input vector
443: dummy - optional user-defined context, as set by TSetRHSJacobian()
445: Output Parameters:
446: AA - Jacobian matrix
447: BB - optionally different preconditioning matrix
448: str - flag indicating matrix structure
450: Notes:
451: Recall that MatSetValues() uses 0-based row and column numbers
452: in Fortran as well as in C.
453: */
454: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
455: {
456: Mat A = *AA; /* Jacobian matrix */
457: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
458: PetscInt mstart = 0;
459: PetscInt mend = appctx->m;
461: PetscInt i, idx[3];
462: PetscScalar v[3], stwo = -2./(appctx->h*appctx->h), sone = -.5*stwo;
464: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
465: Compute entries for the locally owned part of the matrix
466: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
467: /*
468: Set matrix rows corresponding to boundary data
469: */
471: mstart = 0;
472: v[0] = 1.0;
473: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
474: mstart++;
476: mend--;
477: v[0] = 1.0;
478: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
480: /*
481: Set matrix rows corresponding to interior data. We construct the
482: matrix one row at a time.
483: */
484: v[0] = sone; v[1] = stwo; v[2] = sone;
485: for (i=mstart; i<mend; i++) {
486: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
487: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
488: }
490: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
491: Complete the matrix assembly process and set some options
492: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
493: /*
494: Assemble matrix, using the 2-step process:
495: MatAssemblyBegin(), MatAssemblyEnd()
496: Computations can be done while messages are in transition
497: by placing code between these two statements.
498: */
499: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
500: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
502: /*
503: Set flag to indicate that the Jacobian matrix retains an identical
504: nonzero structure throughout all timestepping iterations (although the
505: values of the entries change). Thus, we can save some work in setting
506: up the preconditioner (e.g., no need to redo symbolic factorization for
507: ILU/ICC preconditioners).
508: - If the nonzero structure of the matrix is different during
509: successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
510: must be used instead. If you are unsure whether the matrix
511: structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
512: - Caution: If you specify SAME_NONZERO_PATTERN, PETSc
513: believes your assertion and does not check the structure
514: of the matrix. If you erroneously claim that the structure
515: is the same when it actually is not, the new preconditioner
516: will not function correctly. Thus, use this optimization
517: feature with caution!
518: */
519: *str = SAME_NONZERO_PATTERN;
521: /*
522: Set and option to indicate that we will never add a new nonzero location
523: to the matrix. If we do, it will generate an error.
524: */
525: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
527: return 0;
528: }
529: /* --------------------------------------------------------------------- */
532: /*
533: Input Parameters:
534: ts - the TS context
535: t - current time
536: f - function
537: ctx - optional user-defined context, as set by TSetBCFunction()
538: */
539: PetscErrorCode MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
540: {
541: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
543: PetscInt m = appctx->m;
544: PetscScalar *fa;
546: VecGetArray(f,&fa);
547: fa[0] = 0.0;
548: fa[m-1] = 1.0;
549: VecRestoreArray(f,&fa);
550: printf("t=%g\n",t);
552: return 0;
553: }