Actual source code: ex5.c
1: /*$Id: ex5.c,v 1.25 2001/08/10 03:34:17 bsmith Exp $*/
3: /* Program usage: ex3 [-help] [all PETSc options] */
5: static char help[] ="Solves a simple time-dependent linear PDE (the heat equation).n
6: Input parameters include:n
7: -m <points>, where <points> = number of grid pointsn
8: -time_dependent_rhs : Treat the problem as having a time-dependent right-hand siden
9: -debug : Activate debugging printoutsn
10: -nox : Deactivate x-window graphicsnn";
12: /*
13: Concepts: TS^time-dependent linear problems
14: Concepts: TS^heat equation
15: Concepts: TS^diffusion equation
16: Processors: 1
17: */
19: /* ------------------------------------------------------------------------
21: This program solves the one-dimensional heat equation (also called the
22: diffusion equation),
23: u_t = u_xx,
24: on the domain 0 <= x <= 1, with the boundary conditions
25: u(t,0) = 1, u(t,1) = 1,
26: and the initial condition
27: u(0,x) = cos(6*pi*x) + 3*cos(2*pi*x).
28: This is a linear, second-order, parabolic equation.
30: We discretize the right-hand side using finite differences with
31: uniform grid spacing h:
32: u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
33: We then demonstrate time evolution using the various TS methods by
34: running the program via
35: ex3 -ts_type <timestepping solver>
37: We compare the approximate solution with the exact solution, given by
38: u_exact(x,t) = exp(-36*pi*pi*t) * cos(6*pi*x) +
39: 3*exp(-4*pi*pi*t) * cos(2*pi*x)
41: Notes:
42: This code demonstrates the TS solver interface to two variants of
43: linear problems, u_t = f(u,t), namely
44: - time-dependent f: f(u,t) is a function of t
45: - time-independent f: f(u,t) is simply just f(u)
47: The parallel version of this code is ts/examples/tutorials/ex4.c
49: ------------------------------------------------------------------------- */
51: /*
52: Include "petscts.h" so that we can use TS solvers. Note that this file
53: automatically includes:
54: petsc.h - base PETSc routines petscvec.h - vectors
55: petscsys.h - system routines petscmat.h - matrices
56: petscis.h - index sets petscksp.h - Krylov subspace methods
57: petscviewer.h - viewers petscpc.h - preconditioners
58: petscsles.h - linear solvers petscsnes.h - nonlinear solvers
59: */
61: #include petscts.h
63: /*
64: User-defined application context - contains data needed by the
65: application-provided call-back routines.
66: */
67: typedef struct {
68: Vec solution; /* global exact solution vector */
69: int m; /* total number of grid points */
70: PetscReal h; /* mesh width h = 1/(m-1) */
71: PetscTruth debug; /* flag (1 indicates activation of debugging printouts) */
72: PetscViewer viewer1,viewer2; /* viewers for the solution and error */
73: PetscReal norm_2,norm_max; /* error norms */
74: } AppCtx;
76: /*
77: User-defined routines
78: */
79: extern int InitialConditions(Vec,AppCtx*);
80: extern int RHSMatrixHeat(TS,PetscReal,Mat*,Mat*,MatStructure*,void*);
81: extern int Monitor(TS,int,PetscReal,Vec,void*);
82: extern int ExactSolution(PetscReal,Vec,AppCtx*);
84: #undef __FUNCT__
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: int time_steps_max = 100; /* default max timesteps */
94: PetscDraw draw; /* drawing context */
95: int ierr,steps,size,m;
96: PetscTruth flg;
97: PetscReal dt,ftime;
99: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
100: Initialize program and set problem parameters
101: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
102:
103: PetscInitialize(&argc,&argv,(char*)0,help);
104: MPI_Comm_size(PETSC_COMM_WORLD,&size);
105: if (size != 1) SETERRQ(1,"This is a uniprocessor example only!");
107: m = 60;
108: PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
109: PetscOptionsHasName(PETSC_NULL,"-debug",&appctx.debug);
110: appctx.m = m;
111: appctx.h = 1.0/(m-1.0);
112: appctx.norm_2 = 0.0;
113: appctx.norm_max = 0.0;
114: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processorn");
116: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
117: Create vector data structures
118: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
120: /*
121: Create vector data structures for approximate and exact solutions
122: */
123: VecCreateSeq(PETSC_COMM_SELF,m,&u);
124: VecDuplicate(u,&appctx.solution);
126: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
127: Set up displays to show graphs of the solution and error
128: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
130: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
131: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
132: PetscDrawSetDoubleBuffer(draw);
133: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
134: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
135: PetscDrawSetDoubleBuffer(draw);
137: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
138: Create timestepping solver context
139: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
141: TSCreate(PETSC_COMM_SELF,&ts);
142: TSSetProblemType(ts,TS_LINEAR);
144: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
145: Set optional user-defined monitoring routine
146: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
148: TSSetMonitor(ts,Monitor,&appctx,PETSC_NULL);
150: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
152: Create matrix data structure; set matrix evaluation routine.
153: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
155: MatCreate(PETSC_COMM_SELF,PETSC_DECIDE,PETSC_DECIDE,m,m,&A);
156: MatSetFromOptions(A);
158: PetscOptionsHasName(PETSC_NULL,"-time_dependent_rhs",&flg);
159: if (flg) {
160: /*
161: For linear problems with a time-dependent f(u,t) in the equation
162: u_t = f(u,t), the user provides the discretized right-hand-side
163: as a time-dependent matrix.
164: */
165: TSSetRHSMatrix(ts,A,A,RHSMatrixHeat,&appctx);
166: } else {
167: /*
168: For linear problems with a time-independent f(u) in the equation
169: u_t = f(u), the user provides the discretized right-hand-side
170: as a matrix only once, and then sets a null matrix evaluation
171: routine.
172: */
173: MatStructure A_structure;
174: RHSMatrixHeat(ts,0.0,&A,&A,&A_structure,&appctx);
175: TSSetRHSMatrix(ts,A,A,PETSC_NULL,&appctx);
176: }
178: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
179: Set solution vector and initial timestep
180: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
182: dt = appctx.h*appctx.h/2.0;
183: TSSetInitialTimeStep(ts,0.0,dt);
184: TSSetSolution(ts,u);
186: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
187: Customize timestepping solver:
188: - Set the solution method to be the Backward Euler method.
189: - Set timestepping duration info
190: Then set runtime options, which can override these defaults.
191: For example,
192: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
193: to override the defaults set by TSSetDuration().
194: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
196: TSSetDuration(ts,time_steps_max,time_total_max);
197: TSSetFromOptions(ts);
199: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
200: Solve the problem
201: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
203: /*
204: Evaluate initial conditions
205: */
206: InitialConditions(u,&appctx);
208: /*
209: Run the timestepping solver
210: */
211: TSStep(ts,&steps,&ftime);
213: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
214: View timestepping solver info
215: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
217: PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %g, avg. error (max norm) = %gn",
218: appctx.norm_2/steps,appctx.norm_max/steps);
219: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
221: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
222: Free work space. All PETSc objects should be destroyed when they
223: are no longer needed.
224: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
226: TSDestroy(ts);
227: MatDestroy(A);
228: VecDestroy(u);
229: PetscViewerDestroy(appctx.viewer1);
230: PetscViewerDestroy(appctx.viewer2);
231: VecDestroy(appctx.solution);
233: /*
234: Always call PetscFinalize() before exiting a program. This routine
235: - finalizes the PETSc libraries as well as MPI
236: - provides summary and diagnostic information if certain runtime
237: options are chosen (e.g., -log_summary).
238: */
239: PetscFinalize();
240: return 0;
241: }
242: /* --------------------------------------------------------------------- */
243: #undef __FUNCT__
245: /*
246: InitialConditions - Computes the solution at the initial time.
248: Input Parameter:
249: u - uninitialized solution vector (global)
250: appctx - user-defined application context
252: Output Parameter:
253: u - vector with solution at initial time (global)
254: */
255: int InitialConditions(Vec u,AppCtx *appctx)
256: {
257: PetscScalar *u_localptr,h = appctx->h;
258: int i,ierr;
260: /*
261: Get a pointer to vector data.
262: - For default PETSc vectors, VecGetArray() returns a pointer to
263: the data array. Otherwise, the routine is implementation dependent.
264: - You MUST call VecRestoreArray() when you no longer need access to
265: the array.
266: - Note that the Fortran interface to VecGetArray() differs from the
267: C version. See the users manual for details.
268: */
269: VecGetArray(u,&u_localptr);
271: /*
272: We initialize the solution array by simply writing the solution
273: directly into the array locations. Alternatively, we could use
274: VecSetValues() or VecSetValuesLocal().
275: */
276: for (i=0; i<appctx->m; i++) {
277: u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);
278: }
280: /*
281: Restore vector
282: */
283: VecRestoreArray(u,&u_localptr);
285: /*
286: Print debugging information if desired
287: */
288: if (appctx->debug) {
289: printf("initial guess vectorn");
290: VecView(u,PETSC_VIEWER_STDOUT_SELF);
291: }
293: return 0;
294: }
295: /* --------------------------------------------------------------------- */
296: #undef __FUNCT__
298: /*
299: ExactSolution - Computes the exact solution at a given time.
301: Input Parameters:
302: t - current time
303: solution - vector in which exact solution will be computed
304: appctx - user-defined application context
306: Output Parameter:
307: solution - vector with the newly computed exact solution
308: */
309: int ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
310: {
311: PetscScalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
312: int i,ierr;
314: /*
315: Get a pointer to vector data.
316: */
317: VecGetArray(solution,&s_localptr);
319: /*
320: Simply write the solution directly into the array locations.
321: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
322: */
323: ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc); ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
324: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
325: for (i=0; i<appctx->m; i++) {
326: s_localptr[i] = PetscCosScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscCosScalar(sc2*(PetscReal)i)*ex2;
327: }
329: /*
330: Restore vector
331: */
332: VecRestoreArray(solution,&s_localptr);
333: return 0;
334: }
335: /* --------------------------------------------------------------------- */
336: #undef __FUNCT__
338: /*
339: Monitor - User-provided routine to monitor the solution computed at
340: each timestep. This example plots the solution and computes the
341: error in two different norms.
343: Input Parameters:
344: ts - the timestep context
345: step - the count of the current step (with 0 meaning the
346: initial condition)
347: time - the current time
348: u - the solution at this timestep
349: ctx - the user-provided context for this monitoring routine.
350: In this case we use the application context which contains
351: information about the problem size, workspace and the exact
352: solution.
353: */
354: int Monitor(TS ts,int step,PetscReal time,Vec u,void *ctx)
355: {
356: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
357: int ierr;
358: PetscReal norm_2,norm_max;
359: PetscScalar mone = -1.0;
361: /*
362: View a graph of the current iterate
363: */
364: VecView(u,appctx->viewer2);
366: /*
367: Compute the exact solution
368: */
369: ExactSolution(time,appctx->solution,appctx);
371: /*
372: Print debugging information if desired
373: */
374: if (appctx->debug) {
375: printf("Computed solution vectorn");
376: VecView(u,PETSC_VIEWER_STDOUT_SELF);
377: printf("Exact solution vectorn");
378: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
379: }
381: /*
382: Compute the 2-norm and max-norm of the error
383: */
384: VecAXPY(&mone,u,appctx->solution);
385: VecNorm(appctx->solution,NORM_2,&norm_2);
386: norm_2 = sqrt(appctx->h)*norm_2;
387: VecNorm(appctx->solution,NORM_MAX,&norm_max);
389: printf("Timestep %d: time = %g, 2-norm error = %g, max norm error = %gn",
390: step,time,norm_2,norm_max);
391: appctx->norm_2 += norm_2;
392: appctx->norm_max += norm_max;
394: /*
395: View a graph of the error
396: */
397: VecView(appctx->solution,appctx->viewer1);
399: /*
400: Print debugging information if desired
401: */
402: if (appctx->debug) {
403: printf("Error vectorn");
404: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
405: }
407: return 0;
408: }
409: /* --------------------------------------------------------------------- */
410: #undef __FUNCT__
412: /*
413: RHSMatrixHeat - User-provided routine to compute the right-hand-side
414: matrix for the heat equation.
416: Input Parameters:
417: ts - the TS context
418: t - current time
419: global_in - global input vector
420: dummy - optional user-defined context, as set by TSetRHSJacobian()
422: Output Parameters:
423: AA - Jacobian matrix
424: BB - optionally different preconditioning matrix
425: str - flag indicating matrix structure
427: Notes:
428: Recall that MatSetValues() uses 0-based row and column numbers
429: in Fortran as well as in C.
430: */
431: int RHSMatrixHeat(TS ts,PetscReal t,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
432: {
433: Mat A = *AA; /* Jacobian matrix */
434: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
435: int mstart = 0;
436: int mend = appctx->m;
437: int ierr,i,idx[3];
438: PetscScalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;
440: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
441: Compute entries for the locally owned part of the matrix
442: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
443: /*
444: Set matrix rows corresponding to boundary data
445: */
447: mstart = 0;
448: v[0] = 1.0;
449: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
450: mstart++;
452: mend--;
453: v[0] = 1.0;
454: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
456: /*
457: Set matrix rows corresponding to interior data. We construct the
458: matrix one row at a time.
459: */
460: v[0] = sone; v[1] = stwo; v[2] = sone;
461: for (i=mstart; i<mend; i++) {
462: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
463: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
464: }
466: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
467: Complete the matrix assembly process and set some options
468: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
469: /*
470: Assemble matrix, using the 2-step process:
471: MatAssemblyBegin(), MatAssemblyEnd()
472: Computations can be done while messages are in transition
473: by placing code between these two statements.
474: */
475: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
476: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
478: /*
479: Set flag to indicate that the Jacobian matrix retains an identical
480: nonzero structure throughout all timestepping iterations (although the
481: values of the entries change). Thus, we can save some work in setting
482: up the preconditioner (e.g., no need to redo symbolic factorization for
483: ILU/ICC preconditioners).
484: - If the nonzero structure of the matrix is different during
485: successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
486: must be used instead. If you are unsure whether the matrix
487: structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
488: - Caution: If you specify SAME_NONZERO_PATTERN, PETSc
489: believes your assertion and does not check the structure
490: of the matrix. If you erroneously claim that the structure
491: is the same when it actually is not, the new preconditioner
492: will not function correctly. Thus, use this optimization
493: feature with caution!
494: */
495: *str = SAME_NONZERO_PATTERN;
497: /*
498: Set and option to indicate that we will never add a new nonzero location
499: to the matrix. If we do, it will generate an error.
500: */
501: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR);
503: return 0;
504: }