Actual source code: ex6.c
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: Routines: TSCreate(); TSSetSolution(); TSSetRHSJacobian(), TSSetIJacobian();
14: Routines: TSSetTimeStep(); TSSetMaxTime(); TSMonitorSet();
15: Routines: TSSetFromOptions(); TSStep(); TSDestroy();
16: Routines: TSSetTimeStep(); TSGetTimeStep();
17: Processors: 1
18: */
20: /* ------------------------------------------------------------------------
22: This program solves the one-dimensional heat equation (also called the
23: diffusion equation),
24: u_t = u_xx,
25: on the domain 0 <= x <= 1, with the boundary conditions
26: u(t,0) = 0, u(t,1) = 0,
27: and the initial condition
28: u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
29: This is a linear, second-order, parabolic equation.
31: We discretize the right-hand side using finite differences with
32: uniform grid spacing h:
33: u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
34: We then demonstrate time evolution using the various TS methods by
35: running the program via
36: ex3 -ts_type <timestepping solver>
38: We compare the approximate solution with the exact solution, given by
39: u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
40: 3*exp(-4*pi*pi*t) * sin(2*pi*x)
42: Notes:
43: This code demonstrates the TS solver interface to two variants of
44: linear problems, u_t = f(u,t), namely
45: - time-dependent f: f(u,t) is a function of t
46: - time-independent f: f(u,t) is simply f(u)
48: The parallel version of this code is ts/tutorials/ex4.c
50: ------------------------------------------------------------------------- */
52: /*
53: Include "ts.h" so that we can use TS solvers. Note that this file
54: automatically includes:
55: petscsys.h - base PETSc routines vec.h - vectors
56: sys.h - system routines mat.h - matrices
57: is.h - index sets ksp.h - Krylov subspace methods
58: viewer.h - viewers pc.h - preconditioners
59: snes.h - nonlinear solvers
60: */
62: #include <petscts.h>
63: #include <petscdraw.h>
65: /*
66: User-defined application context - contains data needed by the
67: application-provided call-back routines.
68: */
69: typedef struct {
70: Vec solution; /* global exact solution vector */
71: PetscInt m; /* total number of grid points */
72: PetscReal h; /* mesh width h = 1/(m-1) */
73: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
74: PetscViewer viewer1, viewer2; /* viewers for the solution and error */
75: PetscReal norm_2, norm_max; /* error norms */
76: } AppCtx;
78: /*
79: User-defined routines
80: */
81: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
82: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat,Mat,void*);
83: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
84: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);
85: extern PetscErrorCode MyBCRoutine(TS,PetscReal,Vec,void*);
87: int main(int argc,char **argv)
88: {
89: AppCtx appctx; /* user-defined application context */
90: TS ts; /* timestepping context */
91: Mat A; /* matrix data structure */
92: Vec u; /* approximate solution vector */
93: PetscReal time_total_max = 100.0; /* default max total time */
94: PetscInt time_steps_max = 100; /* default max timesteps */
95: PetscDraw draw; /* drawing context */
96: PetscInt steps, m;
97: PetscMPIInt size;
98: PetscReal dt;
99: PetscReal ftime;
100: PetscBool flg;
101: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
102: Initialize program and set problem parameters
103: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
105: PetscInitialize(&argc,&argv,(char*)0,help);
106: MPI_Comm_size(PETSC_COMM_WORLD,&size);
109: m = 60;
110: PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
111: PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);
113: appctx.m = m;
114: appctx.h = 1.0/(m-1.0);
115: appctx.norm_2 = 0.0;
116: appctx.norm_max = 0.0;
118: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");
120: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
121: Create vector data structures
122: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
124: /*
125: Create vector data structures for approximate and exact solutions
126: */
127: VecCreateSeq(PETSC_COMM_SELF,m,&u);
128: VecDuplicate(u,&appctx.solution);
130: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
131: Set up displays to show graphs of the solution and error
132: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
134: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
135: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
136: PetscDrawSetDoubleBuffer(draw);
137: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
138: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
139: PetscDrawSetDoubleBuffer(draw);
141: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
142: Create timestepping solver context
143: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
145: TSCreate(PETSC_COMM_SELF,&ts);
146: TSSetProblemType(ts,TS_LINEAR);
148: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
149: Set optional user-defined monitoring routine
150: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
152: TSMonitorSet(ts,Monitor,&appctx,NULL);
154: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
156: Create matrix data structure; set matrix evaluation routine.
157: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
159: MatCreate(PETSC_COMM_SELF,&A);
160: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
161: MatSetFromOptions(A);
162: MatSetUp(A);
164: PetscOptionsHasName(NULL,NULL,"-time_dependent_rhs",&flg);
165: if (flg) {
166: /*
167: For linear problems with a time-dependent f(u,t) in the equation
168: u_t = f(u,t), the user provides the discretized right-hand-side
169: as a time-dependent matrix.
170: */
171: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
172: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
173: } else {
174: /*
175: For linear problems with a time-independent f(u) in the equation
176: u_t = f(u), the user provides the discretized right-hand-side
177: as a matrix only once, and then sets a null matrix evaluation
178: routine.
179: */
180: RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
181: TSSetRHSFunction(ts,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: TSSetTimeStep(ts,dt);
191: TSSetSolution(ts,u);
193: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
194: Customize timestepping solver:
195: - Set the solution method to be the Backward Euler method.
196: - Set timestepping duration info
197: Then set runtime options, which can override these defaults.
198: For example,
199: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
200: to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
201: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
203: TSSetMaxSteps(ts,time_steps_max);
204: TSSetMaxTime(ts,time_total_max);
205: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
206: TSSetFromOptions(ts);
208: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
209: Solve the problem
210: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
212: /*
213: Evaluate initial conditions
214: */
215: InitialConditions(u,&appctx);
217: /*
218: Run the timestepping solver
219: */
220: TSSolve(ts,u);
221: TSGetSolveTime(ts,&ftime);
222: TSGetStepNumber(ts,&steps);
224: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
225: View timestepping solver info
226: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
228: PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %g, avg. error (max norm) = %g\n",(double)(appctx.norm_2/steps),(double)(appctx.norm_max/steps));
229: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
231: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
232: Free work space. All PETSc objects should be destroyed when they
233: are no longer needed.
234: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
236: TSDestroy(&ts);
237: MatDestroy(&A);
238: VecDestroy(&u);
239: PetscViewerDestroy(&appctx.viewer1);
240: PetscViewerDestroy(&appctx.viewer2);
241: VecDestroy(&appctx.solution);
243: /*
244: Always call PetscFinalize() before exiting a program. This routine
245: - finalizes the PETSc libraries as well as MPI
246: - provides summary and diagnostic information if certain runtime
247: options are chosen (e.g., -log_view).
248: */
249: PetscFinalize();
250: return 0;
251: }
252: /* --------------------------------------------------------------------- */
253: /*
254: InitialConditions - Computes the solution at the initial time.
256: Input Parameter:
257: u - uninitialized solution vector (global)
258: appctx - user-defined application context
260: Output Parameter:
261: u - vector with solution at initial time (global)
262: */
263: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
264: {
265: PetscScalar *u_localptr;
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++) u_localptr[i] = PetscSinReal(PETSC_PI*i*6.*appctx->h) + 3.*PetscSinReal(PETSC_PI*i*2.*appctx->h);
286: /*
287: Restore vector
288: */
289: VecRestoreArray(u,&u_localptr);
291: /*
292: Print debugging information if desired
293: */
294: if (appctx->debug) {
295: VecView(u,PETSC_VIEWER_STDOUT_SELF);
296: }
298: return 0;
299: }
300: /* --------------------------------------------------------------------- */
301: /*
302: ExactSolution - Computes the exact solution at a given time.
304: Input Parameters:
305: t - current time
306: solution - vector in which exact solution will be computed
307: appctx - user-defined application context
309: Output Parameter:
310: solution - vector with the newly computed exact solution
311: */
312: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
313: {
314: PetscScalar *s_localptr, h = appctx->h, ex1, ex2, sc1, sc2;
315: PetscInt i;
317: /*
318: Get a pointer to vector data.
319: */
320: VecGetArray(solution,&s_localptr);
322: /*
323: Simply write the solution directly into the array locations.
324: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
325: */
326: ex1 = PetscExpReal(-36.*PETSC_PI*PETSC_PI*t); ex2 = PetscExpReal(-4.*PETSC_PI*PETSC_PI*t);
327: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
328: for (i=0; i<appctx->m; i++) s_localptr[i] = PetscSinReal(PetscRealPart(sc1)*(PetscReal)i)*ex1 + 3.*PetscSinReal(PetscRealPart(sc2)*(PetscReal)i)*ex2;
330: /*
331: Restore vector
332: */
333: VecRestoreArray(solution,&s_localptr);
334: return 0;
335: }
336: /* --------------------------------------------------------------------- */
337: /*
338: Monitor - User-provided routine to monitor the solution computed at
339: each timestep. This example plots the solution and computes the
340: error in two different norms.
342: This example also demonstrates changing the timestep via TSSetTimeStep().
344: Input Parameters:
345: ts - the timestep context
346: step - the count of the current step (with 0 meaning the
347: initial condition)
348: crtime - the current time
349: u - the solution at this timestep
350: ctx - the user-provided context for this monitoring routine.
351: In this case we use the application context which contains
352: information about the problem size, workspace and the exact
353: solution.
354: */
355: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal crtime,Vec u,void *ctx)
356: {
357: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
358: PetscReal norm_2, norm_max, dt, dttol;
359: PetscBool flg;
361: /*
362: View a graph of the current iterate
363: */
364: VecView(u,appctx->viewer2);
366: /*
367: Compute the exact solution
368: */
369: ExactSolution(crtime,appctx->solution,appctx);
371: /*
372: Print debugging information if desired
373: */
374: if (appctx->debug) {
375: PetscPrintf(PETSC_COMM_SELF,"Computed solution vector\n");
376: VecView(u,PETSC_VIEWER_STDOUT_SELF);
377: PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");
378: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
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,&norm_2);
386: norm_2 = PetscSqrtReal(appctx->h)*norm_2;
387: VecNorm(appctx->solution,NORM_MAX,&norm_max);
389: TSGetTimeStep(ts,&dt);
390: if (norm_2 > 1.e-2) {
391: PetscPrintf(PETSC_COMM_SELF,"Timestep %D: step size = %g, time = %g, 2-norm error = %g, max norm error = %g\n",step,(double)dt,(double)crtime,(double)norm_2,(double)norm_max);
392: }
393: appctx->norm_2 += norm_2;
394: appctx->norm_max += norm_max;
396: dttol = .0001;
397: PetscOptionsGetReal(NULL,NULL,"-dttol",&dttol,&flg);
398: if (dt < dttol) {
399: dt *= .999;
400: TSSetTimeStep(ts,dt);
401: }
403: /*
404: View a graph of the error
405: */
406: VecView(appctx->solution,appctx->viewer1);
408: /*
409: Print debugging information if desired
410: */
411: if (appctx->debug) {
412: PetscPrintf(PETSC_COMM_SELF,"Error vector\n");
413: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
414: }
416: return 0;
417: }
418: /* --------------------------------------------------------------------- */
419: /*
420: RHSMatrixHeat - User-provided routine to compute the right-hand-side
421: matrix for the heat equation.
423: Input Parameters:
424: ts - the TS context
425: t - current time
426: global_in - global input vector
427: dummy - optional user-defined context, as set by TSetRHSJacobian()
429: Output Parameters:
430: AA - Jacobian matrix
431: BB - optionally different preconditioning matrix
432: str - flag indicating matrix structure
434: Notes:
435: Recall that MatSetValues() uses 0-based row and column numbers
436: in Fortran as well as in C.
437: */
438: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
439: {
440: Mat A = AA; /* Jacobian matrix */
441: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
442: PetscInt mstart = 0;
443: PetscInt mend = appctx->m;
444: PetscInt i, idx[3];
445: PetscScalar v[3], stwo = -2./(appctx->h*appctx->h), sone = -.5*stwo;
447: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
448: Compute entries for the locally owned part of the matrix
449: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
450: /*
451: Set matrix rows corresponding to boundary data
452: */
454: mstart = 0;
455: v[0] = 1.0;
456: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
457: mstart++;
459: mend--;
460: v[0] = 1.0;
461: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
463: /*
464: Set matrix rows corresponding to interior data. We construct the
465: matrix one row at a time.
466: */
467: v[0] = sone; v[1] = stwo; v[2] = sone;
468: for (i=mstart; i<mend; i++) {
469: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
470: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
471: }
473: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
474: Complete the matrix assembly process and set some options
475: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
476: /*
477: Assemble matrix, using the 2-step process:
478: MatAssemblyBegin(), MatAssemblyEnd()
479: Computations can be done while messages are in transition
480: by placing code between these two statements.
481: */
482: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
483: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
485: /*
486: Set and option to indicate that we will never add a new nonzero location
487: to the matrix. If we do, it will generate an error.
488: */
489: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
491: return 0;
492: }
493: /* --------------------------------------------------------------------- */
494: /*
495: Input Parameters:
496: ts - the TS context
497: t - current time
498: f - function
499: ctx - optional user-defined context, as set by TSetBCFunction()
500: */
501: PetscErrorCode MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
502: {
503: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
504: PetscInt m = appctx->m;
505: PetscScalar *fa;
507: VecGetArray(f,&fa);
508: fa[0] = 0.0;
509: fa[m-1] = 1.0;
510: VecRestoreArray(f,&fa);
511: PetscPrintf(PETSC_COMM_SELF,"t=%g\n",(double)t);
513: return 0;
514: }
516: /*TEST
518: test:
519: args: -nox -ts_max_steps 4
521: TEST*/