Actual source code: ex2.c
petsc-3.7.5 2017-01-01
2: static char help[] ="Solves a time-dependent nonlinear PDE. Uses implicit\n\
3: timestepping. Runtime options include:\n\
4: -M <xg>, where <xg> = number of grid points\n\
5: -debug : Activate debugging printouts\n\
6: -nox : Deactivate x-window graphics\n\n";
8: /*
9: Concepts: TS^time-dependent nonlinear problems
10: Processors: n
11: */
13: /* ------------------------------------------------------------------------
15: This program solves the PDE
17: u * u_xx
18: u_t = ---------
19: 2*(t+1)^2
21: on the domain 0 <= x <= 1, with boundary conditions
22: u(t,0) = t + 1, u(t,1) = 2*t + 2,
23: and initial condition
24: u(0,x) = 1 + x*x.
26: The exact solution is:
27: u(t,x) = (1 + x*x) * (1 + t)
29: Note that since the solution is linear in time and quadratic in x,
30: the finite difference scheme actually computes the "exact" solution.
32: We use by default the backward Euler method.
34: ------------------------------------------------------------------------- */
36: /*
37: Include "petscts.h" to use the PETSc timestepping routines. Note that
38: this file automatically includes "petscsys.h" and other lower-level
39: PETSc include files.
41: Include the "petscdmda.h" to allow us to use the distributed array data
42: structures to manage the parallel grid.
43: */
44: #include <petscts.h>
45: #include <petscdm.h>
46: #include <petscdmda.h>
47: #include <petscdraw.h>
49: /*
50: User-defined application context - contains data needed by the
51: application-provided callback routines.
52: */
53: typedef struct {
54: MPI_Comm comm; /* communicator */
55: DM da; /* distributed array data structure */
56: Vec localwork; /* local ghosted work vector */
57: Vec u_local; /* local ghosted approximate solution vector */
58: Vec solution; /* global exact solution vector */
59: PetscInt m; /* total number of grid points */
60: PetscReal h; /* mesh width: h = 1/(m-1) */
61: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
62: } AppCtx;
64: /*
65: User-defined routines, provided below.
66: */
67: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
68: extern PetscErrorCode RHSFunction(TS,PetscReal,Vec,Vec,void*);
69: extern PetscErrorCode RHSJacobian(TS,PetscReal,Vec,Mat,Mat,void*);
70: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
71: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);
75: int main(int argc,char **argv)
76: {
77: AppCtx appctx; /* user-defined application context */
78: TS ts; /* timestepping context */
79: Mat A; /* Jacobian matrix data structure */
80: Vec u; /* approximate solution vector */
81: PetscInt time_steps_max = 100; /* default max timesteps */
83: PetscReal dt;
84: PetscReal time_total_max = 100.0; /* default max total time */
85: PetscBool mymonitor = PETSC_FALSE;
86: PetscReal bounds[] = {1.0, 3.3};
88: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
89: Initialize program and set problem parameters
90: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
92: PetscInitialize(&argc,&argv,(char*)0,help);
93: PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),1,bounds);
95: appctx.comm = PETSC_COMM_WORLD;
96: appctx.m = 60;
98: PetscOptionsGetInt(NULL,NULL,"-M",&appctx.m,NULL);
99: PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);
100: PetscOptionsHasName(NULL,NULL,"-mymonitor",&mymonitor);
102: appctx.h = 1.0/(appctx.m-1.0);
104: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
105: Create vector data structures
106: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
108: /*
109: Create distributed array (DMDA) to manage parallel grid and vectors
110: and to set up the ghost point communication pattern. There are M
111: total grid values spread equally among all the processors.
112: */
113: DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,appctx.m,1,1,NULL,&appctx.da);
115: /*
116: Extract global and local vectors from DMDA; we use these to store the
117: approximate solution. Then duplicate these for remaining vectors that
118: have the same types.
119: */
120: DMCreateGlobalVector(appctx.da,&u);
121: DMCreateLocalVector(appctx.da,&appctx.u_local);
123: /*
124: Create local work vector for use in evaluating right-hand-side function;
125: create global work vector for storing exact solution.
126: */
127: VecDuplicate(appctx.u_local,&appctx.localwork);
128: VecDuplicate(u,&appctx.solution);
130: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
131: Create timestepping solver context; set callback routine for
132: right-hand-side function evaluation.
133: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
135: TSCreate(PETSC_COMM_WORLD,&ts);
136: TSSetProblemType(ts,TS_NONLINEAR);
137: TSSetRHSFunction(ts,NULL,RHSFunction,&appctx);
139: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
140: Set optional user-defined monitoring routine
141: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
143: if (mymonitor) {
144: TSMonitorSet(ts,Monitor,&appctx,NULL);
145: }
147: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
148: For nonlinear problems, the user can provide a Jacobian evaluation
149: routine (or use a finite differencing approximation).
151: Create matrix data structure; set Jacobian evaluation routine.
152: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
154: MatCreate(PETSC_COMM_WORLD,&A);
155: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,appctx.m,appctx.m);
156: MatSetFromOptions(A);
157: MatSetUp(A);
158: TSSetRHSJacobian(ts,A,A,RHSJacobian,&appctx);
160: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
161: Set solution vector and initial timestep
162: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
164: dt = appctx.h/2.0;
165: TSSetInitialTimeStep(ts,0.0,dt);
167: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
168: Customize timestepping solver:
169: - Set the solution method to be the Backward Euler method.
170: - Set timestepping duration info
171: Then set runtime options, which can override these defaults.
172: For example,
173: -ts_max_steps <maxsteps> -ts_final_time <maxtime>
174: to override the defaults set by TSSetDuration().
175: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
177: TSSetType(ts,TSBEULER);
178: TSSetDuration(ts,time_steps_max,time_total_max);
179: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
180: TSSetFromOptions(ts);
181:
182: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
183: Solve the problem
184: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
186: /*
187: Evaluate initial conditions
188: */
189: InitialConditions(u,&appctx);
191: /*
192: Run the timestepping solver
193: */
194: TSSolve(ts,u);
196: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
197: Free work space. All PETSc objects should be destroyed when they
198: are no longer needed.
199: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
201: TSDestroy(&ts);
202: VecDestroy(&u);
203: MatDestroy(&A);
204: DMDestroy(&appctx.da);
205: VecDestroy(&appctx.localwork);
206: VecDestroy(&appctx.solution);
207: VecDestroy(&appctx.u_local);
209: /*
210: Always call PetscFinalize() before exiting a program. This routine
211: - finalizes the PETSc libraries as well as MPI
212: - provides summary and diagnostic information if certain runtime
213: options are chosen (e.g., -log_summary).
214: */
215: PetscFinalize();
216: return 0;
217: }
218: /* --------------------------------------------------------------------- */
221: /*
222: InitialConditions - Computes the solution at the initial time.
224: Input Parameters:
225: u - uninitialized solution vector (global)
226: appctx - user-defined application context
228: Output Parameter:
229: u - vector with solution at initial time (global)
230: */
231: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
232: {
233: PetscScalar *u_localptr,h = appctx->h,x;
234: PetscInt i,mybase,myend;
237: /*
238: Determine starting point of each processor's range of
239: grid values.
240: */
241: VecGetOwnershipRange(u,&mybase,&myend);
243: /*
244: Get a pointer to vector data.
245: - For default PETSc vectors, VecGetArray() returns a pointer to
246: the data array. Otherwise, the routine is implementation dependent.
247: - You MUST call VecRestoreArray() when you no longer need access to
248: the array.
249: - Note that the Fortran interface to VecGetArray() differs from the
250: C version. See the users manual for details.
251: */
252: VecGetArray(u,&u_localptr);
254: /*
255: We initialize the solution array by simply writing the solution
256: directly into the array locations. Alternatively, we could use
257: VecSetValues() or VecSetValuesLocal().
258: */
259: for (i=mybase; i<myend; i++) {
260: x = h*(PetscReal)i; /* current location in global grid */
261: u_localptr[i-mybase] = 1.0 + x*x;
262: }
264: /*
265: Restore vector
266: */
267: VecRestoreArray(u,&u_localptr);
269: /*
270: Print debugging information if desired
271: */
272: if (appctx->debug) {
273: PetscPrintf(appctx->comm,"initial guess vector\n");
274: VecView(u,PETSC_VIEWER_STDOUT_WORLD);
275: }
277: return 0;
278: }
279: /* --------------------------------------------------------------------- */
282: /*
283: ExactSolution - Computes the exact solution at a given time.
285: Input Parameters:
286: t - current time
287: solution - vector in which exact solution will be computed
288: appctx - user-defined application context
290: Output Parameter:
291: solution - vector with the newly computed exact solution
292: */
293: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
294: {
295: PetscScalar *s_localptr,h = appctx->h,x;
296: PetscInt i,mybase,myend;
299: /*
300: Determine starting and ending points of each processor's
301: range of grid values
302: */
303: VecGetOwnershipRange(solution,&mybase,&myend);
305: /*
306: Get a pointer to vector data.
307: */
308: VecGetArray(solution,&s_localptr);
310: /*
311: Simply write the solution directly into the array locations.
312: Alternatively, we could use VecSetValues() or VecSetValuesLocal().
313: */
314: for (i=mybase; i<myend; i++) {
315: x = h*(PetscReal)i;
316: s_localptr[i-mybase] = (t + 1.0)*(1.0 + x*x);
317: }
319: /*
320: Restore vector
321: */
322: VecRestoreArray(solution,&s_localptr);
323: return 0;
324: }
325: /* --------------------------------------------------------------------- */
328: /*
329: Monitor - User-provided routine to monitor the solution computed at
330: each timestep. This example plots the solution and computes the
331: error in two different norms.
333: Input Parameters:
334: ts - the timestep context
335: step - the count of the current step (with 0 meaning the
336: initial condition)
337: time - the current time
338: u - the solution at this timestep
339: ctx - the user-provided context for this monitoring routine.
340: In this case we use the application context which contains
341: information about the problem size, workspace and the exact
342: solution.
343: */
344: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
345: {
346: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
348: PetscReal en2,en2s,enmax;
349: PetscDraw draw;
351: /*
352: We use the default X windows viewer
353: PETSC_VIEWER_DRAW_(appctx->comm)
354: that is associated with the current communicator. This saves
355: the effort of calling PetscViewerDrawOpen() to create the window.
356: Note that if we wished to plot several items in separate windows we
357: would create each viewer with PetscViewerDrawOpen() and store them in
358: the application context, appctx.
360: PetscReal buffering makes graphics look better.
361: */
362: PetscViewerDrawGetDraw(PETSC_VIEWER_DRAW_(appctx->comm),0,&draw);
363: PetscDrawSetDoubleBuffer(draw);
364: VecView(u,PETSC_VIEWER_DRAW_(appctx->comm));
366: /*
367: Compute the exact solution at this timestep
368: */
369: ExactSolution(time,appctx->solution,appctx);
371: /*
372: Print debugging information if desired
373: */
374: if (appctx->debug) {
375: PetscPrintf(appctx->comm,"Computed solution vector\n");
376: VecView(u,PETSC_VIEWER_STDOUT_WORLD);
377: PetscPrintf(appctx->comm,"Exact solution vector\n");
378: VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
379: }
381: /*
382: Compute the 2-norm and max-norm of the error
383: */
384: VecAXPY(appctx->solution,-1.0,u);
385: VecNorm(appctx->solution,NORM_2,&en2);
386: en2s = PetscSqrtReal(appctx->h)*en2; /* scale the 2-norm by the grid spacing */
387: VecNorm(appctx->solution,NORM_MAX,&enmax);
389: /*
390: PetscPrintf() causes only the first processor in this
391: communicator to print the timestep information.
392: */
393: PetscPrintf(appctx->comm,"Timestep %D: time = %g 2-norm error = %g max norm error = %g\n",step,(double)time,(double)en2s,(double)enmax);
395: /*
396: Print debugging information if desired
397: */
398: if (appctx->debug) {
399: PetscPrintf(appctx->comm,"Error vector\n");
400: VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
401: }
402: return 0;
403: }
404: /* --------------------------------------------------------------------- */
407: /*
408: RHSFunction - User-provided routine that evalues the right-hand-side
409: function of the ODE. This routine is set in the main program by
410: calling TSSetRHSFunction(). We compute:
411: global_out = F(global_in)
413: Input Parameters:
414: ts - timesteping context
415: t - current time
416: global_in - vector containing the current iterate
417: ctx - (optional) user-provided context for function evaluation.
418: In this case we use the appctx defined above.
420: Output Parameter:
421: global_out - vector containing the newly evaluated function
422: */
423: PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec global_in,Vec global_out,void *ctx)
424: {
425: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
426: DM da = appctx->da; /* distributed array */
427: Vec local_in = appctx->u_local; /* local ghosted input vector */
428: Vec localwork = appctx->localwork; /* local ghosted work vector */
429: PetscErrorCode ierr;
430: PetscInt i,localsize;
431: PetscMPIInt rank,size;
432: PetscScalar *copyptr,sc;
433: const PetscScalar *localptr;
435: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
436: Get ready for local function computations
437: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
438: /*
439: Scatter ghost points to local vector, using the 2-step process
440: DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
441: By placing code between these two statements, computations can be
442: done while messages are in transition.
443: */
444: DMGlobalToLocalBegin(da,global_in,INSERT_VALUES,local_in);
445: DMGlobalToLocalEnd(da,global_in,INSERT_VALUES,local_in);
447: /*
448: Access directly the values in our local INPUT work array
449: */
450: VecGetArrayRead(local_in,&localptr);
452: /*
453: Access directly the values in our local OUTPUT work array
454: */
455: VecGetArray(localwork,©ptr);
457: sc = 1.0/(appctx->h*appctx->h*2.0*(1.0+t)*(1.0+t));
459: /*
460: Evaluate our function on the nodes owned by this processor
461: */
462: VecGetLocalSize(local_in,&localsize);
464: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
465: Compute entries for the locally owned part
466: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
468: /*
469: Handle boundary conditions: This is done by using the boundary condition
470: u(t,boundary) = g(t,boundary)
471: for some function g. Now take the derivative with respect to t to obtain
472: u_{t}(t,boundary) = g_{t}(t,boundary)
474: In our case, u(t,0) = t + 1, so that u_{t}(t,0) = 1
475: and u(t,1) = 2t+ 2, so that u_{t}(t,1) = 2
476: */
477: MPI_Comm_rank(appctx->comm,&rank);
478: MPI_Comm_size(appctx->comm,&size);
479: if (!rank) copyptr[0] = 1.0;
480: if (rank == size-1) copyptr[localsize-1] = 2.0;
482: /*
483: Handle the interior nodes where the PDE is replace by finite
484: difference operators.
485: */
486: for (i=1; i<localsize-1; i++) copyptr[i] = localptr[i] * sc * (localptr[i+1] + localptr[i-1] - 2.0*localptr[i]);
488: /*
489: Restore vectors
490: */
491: VecRestoreArrayRead(local_in,&localptr);
492: VecRestoreArray(localwork,©ptr);
494: /*
495: Insert values from the local OUTPUT vector into the global
496: output vector
497: */
498: DMLocalToGlobalBegin(da,localwork,INSERT_VALUES,global_out);
499: DMLocalToGlobalEnd(da,localwork,INSERT_VALUES,global_out);
501: /* Print debugging information if desired */
502: if (appctx->debug) {
503: PetscPrintf(appctx->comm,"RHS function vector\n");
504: VecView(global_out,PETSC_VIEWER_STDOUT_WORLD);
505: }
507: return 0;
508: }
509: /* --------------------------------------------------------------------- */
512: /*
513: RHSJacobian - User-provided routine to compute the Jacobian of
514: the nonlinear right-hand-side function of the ODE.
516: Input Parameters:
517: ts - the TS context
518: t - current time
519: global_in - global input vector
520: dummy - optional user-defined context, as set by TSetRHSJacobian()
522: Output Parameters:
523: AA - Jacobian matrix
524: BB - optionally different preconditioning matrix
525: str - flag indicating matrix structure
527: Notes:
528: RHSJacobian computes entries for the locally owned part of the Jacobian.
529: - Currently, all PETSc parallel matrix formats are partitioned by
530: contiguous chunks of rows across the processors.
531: - Each processor needs to insert only elements that it owns
532: locally (but any non-local elements will be sent to the
533: appropriate processor during matrix assembly).
534: - Always specify global row and columns of matrix entries when
535: using MatSetValues().
536: - Here, we set all entries for a particular row at once.
537: - Note that MatSetValues() uses 0-based row and column numbers
538: in Fortran as well as in C.
539: */
540: PetscErrorCode RHSJacobian(TS ts,PetscReal t,Vec global_in,Mat AA,Mat BB,void *ctx)
541: {
542: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
543: Vec local_in = appctx->u_local; /* local ghosted input vector */
544: DM da = appctx->da; /* distributed array */
545: PetscScalar v[3],sc;
546: const PetscScalar *localptr;
547: PetscErrorCode ierr;
548: PetscInt i,mstart,mend,mstarts,mends,idx[3],is;
550: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
551: Get ready for local Jacobian computations
552: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
553: /*
554: Scatter ghost points to local vector, using the 2-step process
555: DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
556: By placing code between these two statements, computations can be
557: done while messages are in transition.
558: */
559: DMGlobalToLocalBegin(da,global_in,INSERT_VALUES,local_in);
560: DMGlobalToLocalEnd(da,global_in,INSERT_VALUES,local_in);
562: /*
563: Get pointer to vector data
564: */
565: VecGetArrayRead(local_in,&localptr);
567: /*
568: Get starting and ending locally owned rows of the matrix
569: */
570: MatGetOwnershipRange(BB,&mstarts,&mends);
571: mstart = mstarts; mend = mends;
573: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
574: Compute entries for the locally owned part of the Jacobian.
575: - Currently, all PETSc parallel matrix formats are partitioned by
576: contiguous chunks of rows across the processors.
577: - Each processor needs to insert only elements that it owns
578: locally (but any non-local elements will be sent to the
579: appropriate processor during matrix assembly).
580: - Here, we set all entries for a particular row at once.
581: - We can set matrix entries either using either
582: MatSetValuesLocal() or MatSetValues().
583: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
585: /*
586: Set matrix rows corresponding to boundary data
587: */
588: if (mstart == 0) {
589: v[0] = 0.0;
590: MatSetValues(BB,1,&mstart,1,&mstart,v,INSERT_VALUES);
591: mstart++;
592: }
593: if (mend == appctx->m) {
594: mend--;
595: v[0] = 0.0;
596: MatSetValues(BB,1,&mend,1,&mend,v,INSERT_VALUES);
597: }
599: /*
600: Set matrix rows corresponding to interior data. We construct the
601: matrix one row at a time.
602: */
603: sc = 1.0/(appctx->h*appctx->h*2.0*(1.0+t)*(1.0+t));
604: for (i=mstart; i<mend; i++) {
605: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
606: is = i - mstart + 1;
607: v[0] = sc*localptr[is];
608: v[1] = sc*(localptr[is+1] + localptr[is-1] - 4.0*localptr[is]);
609: v[2] = sc*localptr[is];
610: MatSetValues(BB,1,&i,3,idx,v,INSERT_VALUES);
611: }
613: /*
614: Restore vector
615: */
616: VecRestoreArrayRead(local_in,&localptr);
618: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
619: Complete the matrix assembly process and set some options
620: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
621: /*
622: Assemble matrix, using the 2-step process:
623: MatAssemblyBegin(), MatAssemblyEnd()
624: Computations can be done while messages are in transition
625: by placing code between these two statements.
626: */
627: MatAssemblyBegin(BB,MAT_FINAL_ASSEMBLY);
628: MatAssemblyEnd(BB,MAT_FINAL_ASSEMBLY);
629: if (BB != AA) {
630: MatAssemblyBegin(AA,MAT_FINAL_ASSEMBLY);
631: MatAssemblyEnd(AA,MAT_FINAL_ASSEMBLY);
632: }
634: /*
635: Set and option to indicate that we will never add a new nonzero location
636: to the matrix. If we do, it will generate an error.
637: */
638: MatSetOption(BB,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
640: return 0;
641: }