Actual source code: ex4.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: Processors: n
14: */
16: /* ------------------------------------------------------------------------
18: This program solves the one-dimensional heat equation (also called the
19: diffusion equation),
20: u_t = u_xx,
21: on the domain 0 <= x <= 1, with the boundary conditions
22: u(t,0) = 0, u(t,1) = 0,
23: and the initial condition
24: u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
25: This is a linear, second-order, parabolic equation.
27: We discretize the right-hand side using finite differences with
28: uniform grid spacing h:
29: u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
30: We then demonstrate time evolution using the various TS methods by
31: running the program via
32: mpiexec -n <procs> ex3 -ts_type <timestepping solver>
34: We compare the approximate solution with the exact solution, given by
35: u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
36: 3*exp(-4*pi*pi*t) * sin(2*pi*x)
38: Notes:
39: This code demonstrates the TS solver interface to two variants of
40: linear problems, u_t = f(u,t), namely
41: - time-dependent f: f(u,t) is a function of t
42: - time-independent f: f(u,t) is simply f(u)
44: The uniprocessor version of this code is ts/examples/tutorials/ex3.c
46: ------------------------------------------------------------------------- */
48: /*
49: Include "petscdmda.h" so that we can use distributed arrays (DMDAs) to manage
50: the parallel grid. Include "petscts.h" so that we can use TS solvers.
51: Note that this file automatically includes:
52: petscsys.h - base PETSc routines petscvec.h - vectors
53: petscmat.h - matrices
54: petscis.h - index sets petscksp.h - Krylov subspace methods
55: petscviewer.h - viewers petscpc.h - preconditioners
56: petscksp.h - linear solvers petscsnes.h - nonlinear solvers
57: */
59: #include <petscdmda.h>
60: #include <petscts.h>
61: #include <petscdraw.h>
63: /*
64: User-defined application context - contains data needed by the
65: application-provided call-back routines.
66: */
67: typedef struct {
68: MPI_Comm comm; /* communicator */
69: DM da; /* distributed array data structure */
70: Vec localwork; /* local ghosted work vector */
71: Vec u_local; /* local ghosted approximate solution vector */
72: Vec solution; /* global exact solution vector */
73: PetscInt m; /* total number of grid points */
74: PetscReal h; /* mesh width h = 1/(m-1) */
75: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
76: PetscViewer viewer1,viewer2; /* viewers for the solution and error */
77: PetscReal norm_2,norm_max; /* error norms */
78: } AppCtx;
80: /*
81: User-defined routines
82: */
83: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
84: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat*,Mat*,MatStructure*,void*);
85: extern PetscErrorCode RHSFunctionHeat(TS,PetscReal,Vec,Vec,void*);
86: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
87: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);
91: int main(int argc,char **argv)
92: {
93: AppCtx appctx; /* user-defined application context */
94: TS ts; /* timestepping context */
95: Mat A; /* matrix data structure */
96: Vec u; /* approximate solution vector */
97: PetscReal time_total_max = 1.0; /* default max total time */
98: PetscInt time_steps_max = 100; /* default max timesteps */
99: PetscDraw draw; /* drawing context */
101: PetscInt steps,m;
102: PetscMPIInt size;
103: PetscReal dt,ftime;
104: PetscBool flg;
105: TSProblemType tsproblem = TS_LINEAR;
107: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
108: Initialize program and set problem parameters
109: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
111: PetscInitialize(&argc,&argv,(char*)0,help);
112: appctx.comm = PETSC_COMM_WORLD;
114: m = 60;
115: PetscOptionsGetInt(NULL,"-m",&m,NULL);
116: PetscOptionsHasName(NULL,"-debug",&appctx.debug);
117: appctx.m = m;
118: appctx.h = 1.0/(m-1.0);
119: appctx.norm_2 = 0.0;
120: appctx.norm_max = 0.0;
122: MPI_Comm_size(PETSC_COMM_WORLD,&size);
123: PetscPrintf(PETSC_COMM_WORLD,"Solving a linear TS problem, number of processors = %d\n",size);
125: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
126: Create vector data structures
127: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
128: /*
129: Create distributed array (DMDA) to manage parallel grid and vectors
130: and to set up the ghost point communication pattern. There are M
131: total grid values spread equally among all the processors.
132: */
134: DMDACreate1d(PETSC_COMM_WORLD,DMDA_BOUNDARY_NONE,m,1,1,NULL,&appctx.da);
136: /*
137: Extract global and local vectors from DMDA; we use these to store the
138: approximate solution. Then duplicate these for remaining vectors that
139: have the same types.
140: */
141: DMCreateGlobalVector(appctx.da,&u);
142: DMCreateLocalVector(appctx.da,&appctx.u_local);
144: /*
145: Create local work vector for use in evaluating right-hand-side function;
146: create global work vector for storing exact solution.
147: */
148: VecDuplicate(appctx.u_local,&appctx.localwork);
149: VecDuplicate(u,&appctx.solution);
151: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
152: Set up displays to show graphs of the solution and error
153: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
155: PetscViewerDrawOpen(PETSC_COMM_WORLD,0,"",80,380,400,160,&appctx.viewer1);
156: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
157: PetscDrawSetDoubleBuffer(draw);
158: PetscViewerDrawOpen(PETSC_COMM_WORLD,0,"",80,0,400,160,&appctx.viewer2);
159: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
160: PetscDrawSetDoubleBuffer(draw);
162: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
163: Create timestepping solver context
164: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
166: TSCreate(PETSC_COMM_WORLD,&ts);
168: flg = PETSC_FALSE;
169: PetscOptionsGetBool(NULL,"-nonlinear",&flg,NULL);
170: TSSetProblemType(ts,flg ? TS_NONLINEAR : TS_LINEAR);
172: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
173: Set optional user-defined monitoring routine
174: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
175: TSMonitorSet(ts,Monitor,&appctx,NULL);
177: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
179: Create matrix data structure; set matrix evaluation routine.
180: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
182: MatCreate(PETSC_COMM_WORLD,&A);
183: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
184: MatSetFromOptions(A);
185: MatSetUp(A);
187: flg = PETSC_FALSE;
188: PetscOptionsGetBool(NULL,"-time_dependent_rhs",&flg,NULL);
189: if (flg) {
190: /*
191: For linear problems with a time-dependent f(u,t) in the equation
192: u_t = f(u,t), the user provides the discretized right-hand-side
193: as a time-dependent matrix.
194: */
195: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
196: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
197: } else {
198: /*
199: For linear problems with a time-independent f(u) in the equation
200: u_t = f(u), the user provides the discretized right-hand-side
201: as a matrix only once, and then sets a null matrix evaluation
202: routine.
203: */
204: MatStructure A_structure;
205: RHSMatrixHeat(ts,0.0,u,&A,&A,&A_structure,&appctx);
206: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
207: TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
208: }
210: if (tsproblem == TS_NONLINEAR) {
211: SNES snes;
212: TSSetRHSFunction(ts,NULL,RHSFunctionHeat,&appctx);
213: TSGetSNES(ts,&snes);
214: SNESSetJacobian(snes,NULL,NULL,SNESComputeJacobianDefault,NULL);
215: }
217: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
218: Set solution vector and initial timestep
219: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
221: dt = appctx.h*appctx.h/2.0;
222: TSSetInitialTimeStep(ts,0.0,dt);
223: TSSetSolution(ts,u);
225: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
226: Customize timestepping solver:
227: - Set the solution method to be the Backward Euler method.
228: - Set timestepping duration info
229: Then set runtime options, which can override these defaults.
230: For example,
231: -ts_max_steps <maxsteps> -ts_final_time <maxtime>
232: to override the defaults set by TSSetDuration().
233: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
235: TSSetDuration(ts,time_steps_max,time_total_max);
236: TSSetFromOptions(ts);
238: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
239: Solve the problem
240: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
242: /*
243: Evaluate initial conditions
244: */
245: InitialConditions(u,&appctx);
247: /*
248: Run the timestepping solver
249: */
250: TSSolve(ts,u);
251: TSGetSolveTime(ts,&ftime);
252: TSGetTimeStepNumber(ts,&steps);
254: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
255: View timestepping solver info
256: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
257: PetscPrintf(PETSC_COMM_WORLD,"Total timesteps %D, Final time %G\n",steps,ftime);
258: PetscPrintf(PETSC_COMM_WORLD,"Avg. error (2 norm) = %G Avg. error (max norm) = %G\n",appctx.norm_2/steps,appctx.norm_max/steps);
260: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
261: Free work space. All PETSc objects should be destroyed when they
262: are no longer needed.
263: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
265: TSDestroy(&ts);
266: MatDestroy(&A);
267: VecDestroy(&u);
268: PetscViewerDestroy(&appctx.viewer1);
269: PetscViewerDestroy(&appctx.viewer2);
270: VecDestroy(&appctx.localwork);
271: VecDestroy(&appctx.solution);
272: VecDestroy(&appctx.u_local);
273: DMDestroy(&appctx.da);
275: /*
276: Always call PetscFinalize() before exiting a program. This routine
277: - finalizes the PETSc libraries as well as MPI
278: - provides summary and diagnostic information if certain runtime
279: options are chosen (e.g., -log_summary).
280: */
281: PetscFinalize();
282: return 0;
283: }
284: /* --------------------------------------------------------------------- */
287: /*
288: InitialConditions - Computes the solution at the initial time.
290: Input Parameter:
291: u - uninitialized solution vector (global)
292: appctx - user-defined application context
294: Output Parameter:
295: u - vector with solution at initial time (global)
296: */
297: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
298: {
299: PetscScalar *u_localptr,h = appctx->h;
300: PetscInt i,mybase,myend;
303: /*
304: Determine starting point of each processor's range of
305: grid values.
306: */
307: VecGetOwnershipRange(u,&mybase,&myend);
309: /*
310: Get a pointer to vector data.
311: - For default PETSc vectors, VecGetArray() returns a pointer to
312: the data array. Otherwise, the routine is implementation dependent.
313: - You MUST call VecRestoreArray() when you no longer need access to
314: the array.
315: - Note that the Fortran interface to VecGetArray() differs from the
316: C version. See the users manual for details.
317: */
318: VecGetArray(u,&u_localptr);
320: /*
321: We initialize the solution array by simply writing the solution
322: directly into the array locations. Alternatively, we could use
323: VecSetValues() or VecSetValuesLocal().
324: */
325: for (i=mybase; i<myend; i++) u_localptr[i-mybase] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);
327: /*
328: Restore vector
329: */
330: VecRestoreArray(u,&u_localptr);
332: /*
333: Print debugging information if desired
334: */
335: if (appctx->debug) {
336: PetscPrintf(appctx->comm,"initial guess vector\n");
337: VecView(u,PETSC_VIEWER_STDOUT_WORLD);
338: }
340: return 0;
341: }
342: /* --------------------------------------------------------------------- */
345: /*
346: ExactSolution - Computes the exact solution at a given time.
348: Input Parameters:
349: t - current time
350: solution - vector in which exact solution will be computed
351: appctx - user-defined application context
353: Output Parameter:
354: solution - vector with the newly computed exact solution
355: */
356: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
357: {
358: PetscScalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2;
359: PetscInt i,mybase,myend;
362: /*
363: Determine starting and ending points of each processor's
364: range of grid values
365: */
366: VecGetOwnershipRange(solution,&mybase,&myend);
368: /*
369: Get a pointer to vector data.
370: */
371: VecGetArray(solution,&s_localptr);
373: /*
374: Simply write the solution directly into the array locations.
375: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
376: */
377: ex1 = exp(-36.*PETSC_PI*PETSC_PI*t); ex2 = exp(-4.*PETSC_PI*PETSC_PI*t);
378: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
379: for (i=mybase; i<myend; i++) s_localptr[i-mybase] = PetscSinScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i)*ex2;
381: /*
382: Restore vector
383: */
384: VecRestoreArray(solution,&s_localptr);
385: return 0;
386: }
387: /* --------------------------------------------------------------------- */
390: /*
391: Monitor - User-provided routine to monitor the solution computed at
392: each timestep. This example plots the solution and computes the
393: error in two different norms.
395: Input Parameters:
396: ts - the timestep context
397: step - the count of the current step (with 0 meaning the
398: initial condition)
399: time - the current time
400: u - the solution at this timestep
401: ctx - the user-provided context for this monitoring routine.
402: In this case we use the application context which contains
403: information about the problem size, workspace and the exact
404: solution.
405: */
406: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
407: {
408: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
410: PetscReal norm_2,norm_max;
412: /*
413: View a graph of the current iterate
414: */
415: VecView(u,appctx->viewer2);
417: /*
418: Compute the exact solution
419: */
420: ExactSolution(time,appctx->solution,appctx);
422: /*
423: Print debugging information if desired
424: */
425: if (appctx->debug) {
426: PetscPrintf(appctx->comm,"Computed solution vector\n");
427: VecView(u,PETSC_VIEWER_STDOUT_WORLD);
428: PetscPrintf(appctx->comm,"Exact solution vector\n");
429: VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
430: }
432: /*
433: Compute the 2-norm and max-norm of the error
434: */
435: VecAXPY(appctx->solution,-1.0,u);
436: VecNorm(appctx->solution,NORM_2,&norm_2);
437: norm_2 = PetscSqrtReal(appctx->h)*norm_2;
438: VecNorm(appctx->solution,NORM_MAX,&norm_max);
440: /*
441: PetscPrintf() causes only the first processor in this
442: communicator to print the timestep information.
443: */
444: PetscPrintf(appctx->comm,"Timestep %D: time = %G 2-norm error = %G max norm error = %G\n",
445: step,time,norm_2,norm_max);
446: appctx->norm_2 += norm_2;
447: appctx->norm_max += norm_max;
449: /*
450: View a graph of the error
451: */
452: VecView(appctx->solution,appctx->viewer1);
454: /*
455: Print debugging information if desired
456: */
457: if (appctx->debug) {
458: PetscPrintf(appctx->comm,"Error vector\n");
459: VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
460: }
462: return 0;
463: }
465: /* --------------------------------------------------------------------- */
468: /*
469: RHSMatrixHeat - User-provided routine to compute the right-hand-side
470: matrix for the heat equation.
472: Input Parameters:
473: ts - the TS context
474: t - current time
475: global_in - global input vector
476: dummy - optional user-defined context, as set by TSetRHSJacobian()
478: Output Parameters:
479: AA - Jacobian matrix
480: BB - optionally different preconditioning matrix
481: str - flag indicating matrix structure
483: Notes:
484: RHSMatrixHeat computes entries for the locally owned part of the system.
485: - Currently, all PETSc parallel matrix formats are partitioned by
486: contiguous chunks of rows across the processors.
487: - Each processor needs to insert only elements that it owns
488: locally (but any non-local elements will be sent to the
489: appropriate processor during matrix assembly).
490: - Always specify global row and columns of matrix entries when
491: using MatSetValues(); we could alternatively use MatSetValuesLocal().
492: - Here, we set all entries for a particular row at once.
493: - Note that MatSetValues() uses 0-based row and column numbers
494: in Fortran as well as in C.
495: */
496: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
497: {
498: Mat A = *AA; /* Jacobian matrix */
499: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
501: PetscInt i,mstart,mend,idx[3];
502: PetscScalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;
504: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
505: Compute entries for the locally owned part of the matrix
506: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
508: MatGetOwnershipRange(A,&mstart,&mend);
510: /*
511: Set matrix rows corresponding to boundary data
512: */
514: if (mstart == 0) { /* first processor only */
515: v[0] = 1.0;
516: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
517: mstart++;
518: }
520: if (mend == appctx->m) { /* last processor only */
521: mend--;
522: v[0] = 1.0;
523: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
524: }
526: /*
527: Set matrix rows corresponding to interior data. We construct the
528: matrix one row at a time.
529: */
530: v[0] = sone; v[1] = stwo; v[2] = sone;
531: for (i=mstart; i<mend; i++) {
532: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
533: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
534: }
536: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
537: Complete the matrix assembly process and set some options
538: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
539: /*
540: Assemble matrix, using the 2-step process:
541: MatAssemblyBegin(), MatAssemblyEnd()
542: Computations can be done while messages are in transition
543: by placing code between these two statements.
544: */
545: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
546: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
548: /*
549: Set flag to indicate that the Jacobian matrix retains an identical
550: nonzero structure throughout all timestepping iterations (although the
551: values of the entries change). Thus, we can save some work in setting
552: up the preconditioner (e.g., no need to redo symbolic factorization for
553: ILU/ICC preconditioners).
554: - If the nonzero structure of the matrix is different during
555: successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
556: must be used instead. If you are unsure whether the matrix
557: structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
558: - Caution: If you specify SAME_NONZERO_PATTERN, PETSc
559: believes your assertion and does not check the structure
560: of the matrix. If you erroneously claim that the structure
561: is the same when it actually is not, the new preconditioner
562: will not function correctly. Thus, use this optimization
563: feature with caution!
564: */
565: *str = SAME_NONZERO_PATTERN;
567: /*
568: Set and option to indicate that we will never add a new nonzero location
569: to the matrix. If we do, it will generate an error.
570: */
571: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
573: return 0;
574: }
578: PetscErrorCode RHSFunctionHeat(TS ts,PetscReal t,Vec globalin,Vec globalout,void *ctx)
579: {
581: Mat A;
582: MatStructure A_structure;
585: TSGetRHSJacobian(ts,&A,NULL,NULL,&ctx);
586: RHSMatrixHeat(ts,t,globalin,&A,NULL,&A_structure,ctx);
587: /* MatView(A,PETSC_VIEWER_STDOUT_WORLD); */
588: MatMult(A,globalin,globalout);
589: return(0);
590: }