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: int main(int argc,char **argv)
85: {
86: AppCtx appctx; /* user-defined application context */
87: TS ts; /* timestepping context */
88: Mat A; /* matrix data structure */
89: Vec u; /* approximate solution vector */
90: PetscReal time_total_max = 100.0; /* default max total time */
91: int time_steps_max = 100; /* default max timesteps */
92: PetscDraw draw; /* drawing context */
93: int ierr,steps,size,m;
94: PetscTruth flg;
95: PetscReal dt,ftime;
97: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
98: Initialize program and set problem parameters
99: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
100:
101: PetscInitialize(&argc,&argv,(char*)0,help);
102: MPI_Comm_size(PETSC_COMM_WORLD,&size);
103: if (size != 1) SETERRQ(1,"This is a uniprocessor example only!");
105: m = 60;
106: PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
107: PetscOptionsHasName(PETSC_NULL,"-debug",&appctx.debug);
108: appctx.m = m;
109: appctx.h = 1.0/(m-1.0);
110: appctx.norm_2 = 0.0;
111: appctx.norm_max = 0.0;
112: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processorn");
114: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
115: Create vector data structures
116: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
118: /*
119: Create vector data structures for approximate and exact solutions
120: */
121: VecCreateSeq(PETSC_COMM_SELF,m,&u);
122: VecDuplicate(u,&appctx.solution);
124: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
125: Set up displays to show graphs of the solution and error
126: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
128: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
129: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
130: PetscDrawSetDoubleBuffer(draw);
131: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
132: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
133: PetscDrawSetDoubleBuffer(draw);
135: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
136: Create timestepping solver context
137: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
139: TSCreate(PETSC_COMM_SELF,&ts);
140: TSSetProblemType(ts,TS_LINEAR);
142: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
143: Set optional user-defined monitoring routine
144: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
146: TSSetMonitor(ts,Monitor,&appctx,PETSC_NULL);
148: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
150: Create matrix data structure; set matrix evaluation routine.
151: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
153: MatCreate(PETSC_COMM_SELF,PETSC_DECIDE,PETSC_DECIDE,m,m,&A);
154: MatSetFromOptions(A);
156: PetscOptionsHasName(PETSC_NULL,"-time_dependent_rhs",&flg);
157: if (flg) {
158: /*
159: For linear problems with a time-dependent f(u,t) in the equation
160: u_t = f(u,t), the user provides the discretized right-hand-side
161: as a time-dependent matrix.
162: */
163: TSSetRHSMatrix(ts,A,A,RHSMatrixHeat,&appctx);
164: } else {
165: /*
166: For linear problems with a time-independent f(u) in the equation
167: u_t = f(u), the user provides the discretized right-hand-side
168: as a matrix only once, and then sets a null matrix evaluation
169: routine.
170: */
171: MatStructure A_structure;
172: RHSMatrixHeat(ts,0.0,&A,&A,&A_structure,&appctx);
173: TSSetRHSMatrix(ts,A,A,PETSC_NULL,&appctx);
174: }
176: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
177: Set solution vector and initial timestep
178: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
180: dt = appctx.h*appctx.h/2.0;
181: TSSetInitialTimeStep(ts,0.0,dt);
182: TSSetSolution(ts,u);
184: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
185: Customize timestepping solver:
186: - Set the solution method to be the Backward Euler method.
187: - Set timestepping duration info
188: Then set runtime options, which can override these defaults.
189: For example,
190: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
191: to override the defaults set by TSSetDuration().
192: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
194: TSSetDuration(ts,time_steps_max,time_total_max);
195: TSSetFromOptions(ts);
197: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
198: Solve the problem
199: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
201: /*
202: Evaluate initial conditions
203: */
204: InitialConditions(u,&appctx);
206: /*
207: Run the timestepping solver
208: */
209: TSStep(ts,&steps,&ftime);
211: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
212: View timestepping solver info
213: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
215: PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %g, avg. error (max norm) = %gn",
216: appctx.norm_2/steps,appctx.norm_max/steps);
217: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
219: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
220: Free work space. All PETSc objects should be destroyed when they
221: are no longer needed.
222: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
224: TSDestroy(ts);
225: MatDestroy(A);
226: VecDestroy(u);
227: PetscViewerDestroy(appctx.viewer1);
228: PetscViewerDestroy(appctx.viewer2);
229: VecDestroy(appctx.solution);
231: /*
232: Always call PetscFinalize() before exiting a program. This routine
233: - finalizes the PETSc libraries as well as MPI
234: - provides summary and diagnostic information if certain runtime
235: options are chosen (e.g., -log_summary).
236: */
237: PetscFinalize();
238: return 0;
239: }
240: /* --------------------------------------------------------------------- */
241: /*
242: InitialConditions - Computes the solution at the initial time.
244: Input Parameter:
245: u - uninitialized solution vector (global)
246: appctx - user-defined application context
248: Output Parameter:
249: u - vector with solution at initial time (global)
250: */
251: int InitialConditions(Vec u,AppCtx *appctx)
252: {
253: PetscScalar *u_localptr,h = appctx->h;
254: int i,ierr;
256: /*
257: Get a pointer to vector data.
258: - For default PETSc vectors, VecGetArray() returns a pointer to
259: the data array. Otherwise, the routine is implementation dependent.
260: - You MUST call VecRestoreArray() when you no longer need access to
261: the array.
262: - Note that the Fortran interface to VecGetArray() differs from the
263: C version. See the users manual for details.
264: */
265: VecGetArray(u,&u_localptr);
267: /*
268: We initialize the solution array by simply writing the solution
269: directly into the array locations. Alternatively, we could use
270: VecSetValues() or VecSetValuesLocal().
271: */
272: for (i=0; i<appctx->m; i++) {
273: u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);
274: }
276: /*
277: Restore vector
278: */
279: VecRestoreArray(u,&u_localptr);
281: /*
282: Print debugging information if desired
283: */
284: if (appctx->debug) {
285: printf("initial guess vectorn");
286: VecView(u,PETSC_VIEWER_STDOUT_SELF);
287: }
289: return 0;
290: }
291: /* --------------------------------------------------------------------- */
292: /*
293: ExactSolution - Computes the exact solution at a given time.
295: Input Parameters:
296: t - current time
297: solution - vector in which exact solution will be computed
298: appctx - user-defined application context
300: Output Parameter:
301: solution - vector with the newly computed exact solution
302: */
303: int ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
304: {
305: PetscScalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
306: int i,ierr;
308: /*
309: Get a pointer to vector data.
310: */
311: VecGetArray(solution,&s_localptr);
313: /*
314: Simply write the solution directly into the array locations.
315: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
316: */
317: ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc); ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
318: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
319: for (i=0; i<appctx->m; i++) {
320: s_localptr[i] = PetscCosScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscCosScalar(sc2*(PetscReal)i)*ex2;
321: }
323: /*
324: Restore vector
325: */
326: VecRestoreArray(solution,&s_localptr);
327: return 0;
328: }
329: /* --------------------------------------------------------------------- */
330: /*
331: Monitor - User-provided routine to monitor the solution computed at
332: each timestep. This example plots the solution and computes the
333: error in two different norms.
335: Input Parameters:
336: ts - the timestep context
337: step - the count of the current step (with 0 meaning the
338: initial condition)
339: time - the current time
340: u - the solution at this timestep
341: ctx - the user-provided context for this monitoring routine.
342: In this case we use the application context which contains
343: information about the problem size, workspace and the exact
344: solution.
345: */
346: int Monitor(TS ts,int step,PetscReal time,Vec u,void *ctx)
347: {
348: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
349: int ierr;
350: PetscReal norm_2,norm_max;
351: PetscScalar mone = -1.0;
353: /*
354: View a graph of the current iterate
355: */
356: VecView(u,appctx->viewer2);
358: /*
359: Compute the exact solution
360: */
361: ExactSolution(time,appctx->solution,appctx);
363: /*
364: Print debugging information if desired
365: */
366: if (appctx->debug) {
367: printf("Computed solution vectorn");
368: VecView(u,PETSC_VIEWER_STDOUT_SELF);
369: printf("Exact solution vectorn");
370: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
371: }
373: /*
374: Compute the 2-norm and max-norm of the error
375: */
376: VecAXPY(&mone,u,appctx->solution);
377: VecNorm(appctx->solution,NORM_2,&norm_2);
378: norm_2 = sqrt(appctx->h)*norm_2;
379: VecNorm(appctx->solution,NORM_MAX,&norm_max);
381: printf("Timestep %d: time = %g, 2-norm error = %g, max norm error = %gn",
382: step,time,norm_2,norm_max);
383: appctx->norm_2 += norm_2;
384: appctx->norm_max += norm_max;
386: /*
387: View a graph of the error
388: */
389: VecView(appctx->solution,appctx->viewer1);
391: /*
392: Print debugging information if desired
393: */
394: if (appctx->debug) {
395: printf("Error vectorn");
396: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
397: }
399: return 0;
400: }
401: /* --------------------------------------------------------------------- */
402: /*
403: RHSMatrixHeat - User-provided routine to compute the right-hand-side
404: matrix for the heat equation.
406: Input Parameters:
407: ts - the TS context
408: t - current time
409: global_in - global input vector
410: dummy - optional user-defined context, as set by TSetRHSJacobian()
412: Output Parameters:
413: AA - Jacobian matrix
414: BB - optionally different preconditioning matrix
415: str - flag indicating matrix structure
417: Notes:
418: Recall that MatSetValues() uses 0-based row and column numbers
419: in Fortran as well as in C.
420: */
421: int RHSMatrixHeat(TS ts,PetscReal t,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
422: {
423: Mat A = *AA; /* Jacobian matrix */
424: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
425: int mstart = 0;
426: int mend = appctx->m;
427: int ierr,i,idx[3];
428: PetscScalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;
430: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
431: Compute entries for the locally owned part of the matrix
432: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
433: /*
434: Set matrix rows corresponding to boundary data
435: */
437: mstart = 0;
438: v[0] = 1.0;
439: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
440: mstart++;
442: mend--;
443: v[0] = 1.0;
444: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
446: /*
447: Set matrix rows corresponding to interior data. We construct the
448: matrix one row at a time.
449: */
450: v[0] = sone; v[1] = stwo; v[2] = sone;
451: for (i=mstart; i<mend; i++) {
452: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
453: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
454: }
456: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
457: Complete the matrix assembly process and set some options
458: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
459: /*
460: Assemble matrix, using the 2-step process:
461: MatAssemblyBegin(), MatAssemblyEnd()
462: Computations can be done while messages are in transition
463: by placing code between these two statements.
464: */
465: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
466: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
468: /*
469: Set flag to indicate that the Jacobian matrix retains an identical
470: nonzero structure throughout all timestepping iterations (although the
471: values of the entries change). Thus, we can save some work in setting
472: up the preconditioner (e.g., no need to redo symbolic factorization for
473: ILU/ICC preconditioners).
474: - If the nonzero structure of the matrix is different during
475: successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
476: must be used instead. If you are unsure whether the matrix
477: structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
478: - Caution: If you specify SAME_NONZERO_PATTERN, PETSc
479: believes your assertion and does not check the structure
480: of the matrix. If you erroneously claim that the structure
481: is the same when it actually is not, the new preconditioner
482: will not function correctly. Thus, use this optimization
483: feature with caution!
484: */
485: *str = SAME_NONZERO_PATTERN;
487: /*
488: Set and option to indicate that we will never add a new nonzero location
489: to the matrix. If we do, it will generate an error.
490: */
491: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR);
493: return 0;
494: }