Actual source code: ex3.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: Processors: 1
16: */
18: /* ------------------------------------------------------------------------
20: This program solves the one-dimensional heat equation (also called the
21: diffusion equation),
22: u_t = u_xx,
23: on the domain 0 <= x <= 1, with the boundary conditions
24: u(t,0) = 0, u(t,1) = 0,
25: and the initial condition
26: u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
27: This is a linear, second-order, parabolic equation.
29: We discretize the right-hand side using finite differences with
30: uniform grid spacing h:
31: u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
32: We then demonstrate time evolution using the various TS methods by
33: running the program via
34: ex3 -ts_type <timestepping solver>
36: We compare the approximate solution with the exact solution, given by
37: u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
38: 3*exp(-4*pi*pi*t) * sin(2*pi*x)
40: Notes:
41: This code demonstrates the TS solver interface to two variants of
42: linear problems, u_t = f(u,t), namely
43: - time-dependent f: f(u,t) is a function of t
44: - time-independent f: f(u,t) is simply f(u)
46: The parallel version of this code is ts/examples/tutorials/ex4.c
48: ------------------------------------------------------------------------- */
50: /*
51: Include "petscts.h" so that we can use TS solvers. Note that this file
52: automatically includes:
53: petscsys.h - base PETSc routines petscvec.h - vectors
54: petscmat.h - matrices
55: petscis.h - index sets petscksp.h - Krylov subspace methods
56: petscviewer.h - viewers petscpc.h - preconditioners
57: petscksp.h - linear solvers petscsnes.h - nonlinear solvers
58: */
60: #include <petscts.h>
62: /*
63: User-defined application context - contains data needed by the
64: application-provided call-back routines.
65: */
66: typedef struct {
67: Vec solution; /* global exact solution vector */
68: PetscInt m; /* total number of grid points */
69: PetscReal h; /* mesh width h = 1/(m-1) */
70: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
71: PetscViewer viewer1,viewer2; /* viewers for the solution and error */
72: PetscReal norm_2,norm_max; /* error norms */
73: } AppCtx;
75: /*
76: User-defined routines
77: */
78: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
79: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat*,Mat*,MatStructure*,void*);
80: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
81: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);
82: extern PetscErrorCode MyBCRoutine(TS,PetscReal,Vec,void*);
86: int main(int argc,char **argv)
87: {
88: AppCtx appctx; /* user-defined application context */
89: TS ts; /* timestepping context */
90: Mat A; /* matrix data structure */
91: Vec u; /* approximate solution vector */
92: PetscReal time_total_max = 100.0; /* default max total time */
93: PetscInt time_steps_max = 100; /* default max timesteps */
94: PetscDraw draw; /* drawing context */
96: PetscInt steps,m;
97: PetscMPIInt size;
98: PetscReal dt;
99: PetscBool 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(PETSC_COMM_SELF,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 processor\n");
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: TSMonitorSet(ts,Monitor,&appctx,PETSC_NULL);
152: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
154: Create matrix data structure; set matrix evaluation routine.
155: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
157: MatCreate(PETSC_COMM_SELF,&A);
158: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
159: MatSetFromOptions(A);
160: MatSetUp(A);
162: flg = PETSC_FALSE;
163: PetscOptionsGetBool(PETSC_NULL,"-time_dependent_rhs",&flg,PETSC_NULL);
164: if (flg) {
165: /*
166: For linear problems with a time-dependent f(u,t) in the equation
167: u_t = f(u,t), the user provides the discretized right-hand-side
168: as a time-dependent matrix.
169: */
170: TSSetRHSFunction(ts,PETSC_NULL,TSComputeRHSFunctionLinear,&appctx);
171: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
172: } else {
173: /*
174: For linear problems with a time-independent f(u) in the equation
175: u_t = f(u), the user provides the discretized right-hand-side
176: as a matrix only once, and then sets a null matrix evaluation
177: routine.
178: */
179: MatStructure A_structure;
180: RHSMatrixHeat(ts,0.0,u,&A,&A,&A_structure,&appctx);
181: TSSetRHSFunction(ts,PETSC_NULL,TSComputeRHSFunctionLinear,&appctx);
182: TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
183: }
185: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
186: Set solution vector and initial timestep
187: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
189: dt = appctx.h*appctx.h/2.0;
190: TSSetInitialTimeStep(ts,0.0,dt);
192: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
193: Customize timestepping solver:
194: - Set the solution method to be the Backward Euler method.
195: - Set timestepping duration info
196: Then set runtime options, which can override these defaults.
197: For example,
198: -ts_max_steps <maxsteps> -ts_final_time <maxtime>
199: to override the defaults set by TSSetDuration().
200: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
202: TSSetDuration(ts,time_steps_max,time_total_max);
203: TSSetFromOptions(ts);
205: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
206: Solve the problem
207: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
209: /*
210: Evaluate initial conditions
211: */
212: InitialConditions(u,&appctx);
214: /*
215: Run the timestepping solver
216: */
217: TSSolve(ts,u,PETSC_NULL);
218: TSGetTimeStepNumber(ts,&steps);
220: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
221: View timestepping solver info
222: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
224: PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %G, avg. error (max norm) = %G\n",
225: appctx.norm_2/steps,appctx.norm_max/steps);
226: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
228: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
229: Free work space. All PETSc objects should be destroyed when they
230: are no longer needed.
231: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
233: TSDestroy(&ts);
234: MatDestroy(&A);
235: VecDestroy(&u);
236: PetscViewerDestroy(&appctx.viewer1);
237: PetscViewerDestroy(&appctx.viewer2);
238: VecDestroy(&appctx.solution);
240: /*
241: Always call PetscFinalize() before exiting a program. This routine
242: - finalizes the PETSc libraries as well as MPI
243: - provides summary and diagnostic information if certain runtime
244: options are chosen (e.g., -log_summary).
245: */
246: PetscFinalize();
247: return 0;
248: }
249: /* --------------------------------------------------------------------- */
252: /*
253: InitialConditions - Computes the solution at the initial time.
255: Input Parameter:
256: u - uninitialized solution vector (global)
257: appctx - user-defined application context
259: Output Parameter:
260: u - vector with solution at initial time (global)
261: */
262: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
263: {
264: PetscScalar *u_localptr,h = appctx->h;
266: PetscInt i;
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 vector\n");
298: VecView(u,PETSC_VIEWER_STDOUT_SELF);
299: }
301: return 0;
302: }
303: /* --------------------------------------------------------------------- */
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: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
318: {
319: PetscScalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
321: PetscInt i;
323: /*
324: Get a pointer to vector data.
325: */
326: VecGetArray(solution,&s_localptr);
328: /*
329: Simply write the solution directly into the array locations.
330: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
331: */
332: ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc);
333: ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
334: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
335: for (i=0; i<appctx->m; i++) {
336: s_localptr[i] = PetscSinScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i)*ex2;
337: }
339: /*
340: Restore vector
341: */
342: VecRestoreArray(solution,&s_localptr);
343: return 0;
344: }
345: /* --------------------------------------------------------------------- */
348: /*
349: Monitor - User-provided routine to monitor the solution computed at
350: each timestep. This example plots the solution and computes the
351: error in two different norms.
353: This example also demonstrates changing the timestep via TSSetTimeStep().
355: Input Parameters:
356: ts - the timestep context
357: step - the count of the current step (with 0 meaning the
358: initial condition)
359: time - the current time
360: u - the solution at this timestep
361: ctx - the user-provided context for this monitoring routine.
362: In this case we use the application context which contains
363: information about the problem size, workspace and the exact
364: solution.
365: */
366: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
367: {
368: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
370: PetscReal norm_2,norm_max,dt,dttol;
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 vector\n");
386: VecView(u,PETSC_VIEWER_STDOUT_SELF);
387: printf("Exact solution vector\n");
388: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
389: }
391: /*
392: Compute the 2-norm and max-norm of the error
393: */
394: VecAXPY(appctx->solution,-1.0,u);
395: VecNorm(appctx->solution,NORM_2,&norm_2);
396: norm_2 = PetscSqrtReal(appctx->h)*norm_2;
397: VecNorm(appctx->solution,NORM_MAX,&norm_max);
399: TSGetTimeStep(ts,&dt);
400: PetscPrintf(PETSC_COMM_WORLD,"Timestep %3D: step size = %-11g, time = %-11g, 2-norm error = %-11g, max norm error = %-11g\n",
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 vector\n");
422: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
423: }
425: return 0;
426: }
427: /* --------------------------------------------------------------------- */
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: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,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: PetscInt mstart = 0;
454: PetscInt mend = appctx->m;
456: PetscInt i,idx[3];
457: PetscScalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;
459: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
460: Compute entries for the locally owned part of the matrix
461: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
462: /*
463: Set matrix rows corresponding to boundary data
464: */
466: mstart = 0;
467: v[0] = 1.0;
468: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
469: mstart++;
471: mend--;
472: v[0] = 1.0;
473: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
475: /*
476: Set matrix rows corresponding to interior data. We construct the
477: matrix one row at a time.
478: */
479: v[0] = sone; v[1] = stwo; v[2] = sone;
480: for (i=mstart; i<mend; i++) {
481: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
482: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
483: }
485: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
486: Complete the matrix assembly process and set some options
487: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
488: /*
489: Assemble matrix, using the 2-step process:
490: MatAssemblyBegin(), MatAssemblyEnd()
491: Computations can be done while messages are in transition
492: by placing code between these two statements.
493: */
494: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
495: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
497: /*
498: Set flag to indicate that the Jacobian matrix retains an identical
499: nonzero structure throughout all timestepping iterations (although the
500: values of the entries change). Thus, we can save some work in setting
501: up the preconditioner (e.g., no need to redo symbolic factorization for
502: ILU/ICC preconditioners).
503: - If the nonzero structure of the matrix is different during
504: successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
505: must be used instead. If you are unsure whether the matrix
506: structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
507: - Caution: If you specify SAME_NONZERO_PATTERN, PETSc
508: believes your assertion and does not check the structure
509: of the matrix. If you erroneously claim that the structure
510: is the same when it actually is not, the new preconditioner
511: will not function correctly. Thus, use this optimization
512: feature with caution!
513: */
514: *str = SAME_NONZERO_PATTERN;
516: /*
517: Set and option to indicate that we will never add a new nonzero location
518: to the matrix. If we do, it will generate an error.
519: */
520: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
522: return 0;
523: }
524: /* --------------------------------------------------------------------- */
527: /*
528: Input Parameters:
529: ts - the TS context
530: t - current time
531: f - function
532: ctx - optional user-defined context, as set by TSetBCFunction()
533: */
534: PetscErrorCode MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
535: {
536: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
537: PetscErrorCode ierr,m = appctx->m;
538: PetscScalar *fa;
540: VecGetArray(f,&fa);
541: fa[0] = 0.0;
542: fa[m-1] = 0.0;
543: VecRestoreArray(f,&fa);
544: printf("t=%g\n",t);
545:
546: return 0;
547: }