Actual source code: ex6.c
petsc-3.7.5 2017-01-01
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: TSSetInitialTimeStep(); TSSetDuration(); 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/examples/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*);
89: int main(int argc,char **argv)
90: {
91: AppCtx appctx; /* user-defined application context */
92: TS ts; /* timestepping context */
93: Mat A; /* matrix data structure */
94: Vec u; /* approximate solution vector */
95: PetscReal time_total_max = 100.0; /* default max total time */
96: PetscInt time_steps_max = 100; /* default max timesteps */
97: PetscDraw draw; /* drawing context */
99: PetscInt steps, m;
100: PetscMPIInt size;
101: PetscReal dt;
102: PetscReal ftime;
103: PetscBool flg;
104: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
105: Initialize program and set problem parameters
106: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
108: PetscInitialize(&argc,&argv,(char*)0,help);
109: MPI_Comm_size(PETSC_COMM_WORLD,&size);
110: if (size != 1) SETERRQ(PETSC_COMM_SELF,1,"This is a uniprocessor example only!");
112: m = 60;
113: PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
114: PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);
116: appctx.m = m;
117: appctx.h = 1.0/(m-1.0);
118: appctx.norm_2 = 0.0;
119: appctx.norm_max = 0.0;
121: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");
123: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
124: Create vector data structures
125: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
127: /*
128: Create vector data structures for approximate and exact solutions
129: */
130: VecCreateSeq(PETSC_COMM_SELF,m,&u);
131: VecDuplicate(u,&appctx.solution);
133: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
134: Set up displays to show graphs of the solution and error
135: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
137: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
138: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
139: PetscDrawSetDoubleBuffer(draw);
140: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
141: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
142: PetscDrawSetDoubleBuffer(draw);
144: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
145: Create timestepping solver context
146: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
148: TSCreate(PETSC_COMM_SELF,&ts);
149: TSSetProblemType(ts,TS_LINEAR);
151: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
152: Set optional user-defined monitoring routine
153: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
155: TSMonitorSet(ts,Monitor,&appctx,NULL);
157: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
159: Create matrix data structure; set matrix evaluation routine.
160: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
162: MatCreate(PETSC_COMM_SELF,&A);
163: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
164: MatSetFromOptions(A);
165: MatSetUp(A);
167: PetscOptionsHasName(NULL,NULL,"-time_dependent_rhs",&flg);
168: if (flg) {
169: /*
170: For linear problems with a time-dependent f(u,t) in the equation
171: u_t = f(u,t), the user provides the discretized right-hand-side
172: as a time-dependent matrix.
173: */
174: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
175: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
176: } else {
177: /*
178: For linear problems with a time-independent f(u) in the equation
179: u_t = f(u), the user provides the discretized right-hand-side
180: as a matrix only once, and then sets a null matrix evaluation
181: routine.
182: */
183: RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
184: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
185: TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
186: }
188: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
189: Set solution vector and initial timestep
190: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
192: dt = appctx.h*appctx.h/2.0;
193: TSSetInitialTimeStep(ts,0.0,dt);
194: TSSetSolution(ts,u);
196: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
197: Customize timestepping solver:
198: - Set the solution method to be the Backward Euler method.
199: - Set timestepping duration info
200: Then set runtime options, which can override these defaults.
201: For example,
202: -ts_max_steps <maxsteps> -ts_final_time <maxtime>
203: to override the defaults set by TSSetDuration().
204: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
206: TSSetDuration(ts,time_steps_max,time_total_max);
207: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
208: TSSetFromOptions(ts);
210: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
211: Solve the problem
212: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
214: /*
215: Evaluate initial conditions
216: */
217: InitialConditions(u,&appctx);
219: /*
220: Run the timestepping solver
221: */
222: TSSolve(ts,u);
223: TSGetSolveTime(ts,&ftime);
224: TSGetTimeStepNumber(ts,&steps);
226: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
227: View timestepping solver info
228: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
230: 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));
231: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
233: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
234: Free work space. All PETSc objects should be destroyed when they
235: are no longer needed.
236: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
238: TSDestroy(&ts);
239: MatDestroy(&A);
240: VecDestroy(&u);
241: PetscViewerDestroy(&appctx.viewer1);
242: PetscViewerDestroy(&appctx.viewer2);
243: VecDestroy(&appctx.solution);
245: /*
246: Always call PetscFinalize() before exiting a program. This routine
247: - finalizes the PETSc libraries as well as MPI
248: - provides summary and diagnostic information if certain runtime
249: options are chosen (e.g., -log_summary).
250: */
251: PetscFinalize();
252: return 0;
253: }
254: /* --------------------------------------------------------------------- */
257: /*
258: InitialConditions - Computes the solution at the initial time.
260: Input Parameter:
261: u - uninitialized solution vector (global)
262: appctx - user-defined application context
264: Output Parameter:
265: u - vector with solution at initial time (global)
266: */
267: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
268: {
269: PetscScalar *u_localptr;
270: PetscInt i;
273: /*
274: Get a pointer to vector data.
275: - For default PETSc vectors, VecGetArray() returns a pointer to
276: the data array. Otherwise, the routine is implementation dependent.
277: - You MUST call VecRestoreArray() when you no longer need access to
278: the array.
279: - Note that the Fortran interface to VecGetArray() differs from the
280: C version. See the users manual for details.
281: */
282: VecGetArray(u,&u_localptr);
284: /*
285: We initialize the solution array by simply writing the solution
286: directly into the array locations. Alternatively, we could use
287: VecSetValues() or VecSetValuesLocal().
288: */
289: 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);
291: /*
292: Restore vector
293: */
294: VecRestoreArray(u,&u_localptr);
296: /*
297: Print debugging information if desired
298: */
299: if (appctx->debug) {
300: VecView(u,PETSC_VIEWER_STDOUT_SELF);
301: }
303: return 0;
304: }
305: /* --------------------------------------------------------------------- */
308: /*
309: ExactSolution - Computes the exact solution at a given time.
311: Input Parameters:
312: t - current time
313: solution - vector in which exact solution will be computed
314: appctx - user-defined application context
316: Output Parameter:
317: solution - vector with the newly computed exact solution
318: */
319: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
320: {
321: PetscScalar *s_localptr, h = appctx->h, ex1, ex2, sc1, sc2;
322: PetscInt i;
325: /*
326: Get a pointer to vector data.
327: */
328: VecGetArray(solution,&s_localptr);
330: /*
331: Simply write the solution directly into the array locations.
332: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
333: */
334: ex1 = PetscExpReal(-36.*PETSC_PI*PETSC_PI*t); ex2 = PetscExpReal(-4.*PETSC_PI*PETSC_PI*t);
335: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
336: for (i=0; i<appctx->m; i++) s_localptr[i] = PetscSinReal(PetscRealPart(sc1)*(PetscReal)i)*ex1 + 3.*PetscSinReal(PetscRealPart(sc2)*(PetscReal)i)*ex2;
338: /*
339: Restore vector
340: */
341: VecRestoreArray(solution,&s_localptr);
342: return 0;
343: }
344: /* --------------------------------------------------------------------- */
347: /*
348: Monitor - User-provided routine to monitor the solution computed at
349: each timestep. This example plots the solution and computes the
350: error in two different norms.
352: This example also demonstrates changing the timestep via TSSetTimeStep().
354: Input Parameters:
355: ts - the timestep context
356: step - the count of the current step (with 0 meaning the
357: initial condition)
358: crtime - the current time
359: u - the solution at this timestep
360: ctx - the user-provided context for this monitoring routine.
361: In this case we use the application context which contains
362: information about the problem size, workspace and the exact
363: solution.
364: */
365: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal crtime,Vec u,void *ctx)
366: {
367: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
369: PetscReal norm_2, norm_max, dt, dttol;
370: PetscBool flg;
372: /*
373: View a graph of the current iterate
374: */
375: VecView(u,appctx->viewer2);
377: /*
378: Compute the exact solution
379: */
380: ExactSolution(crtime,appctx->solution,appctx);
382: /*
383: Print debugging information if desired
384: */
385: if (appctx->debug) {
386: PetscPrintf(PETSC_COMM_SELF,"Computed solution vector\n");
387: VecView(u,PETSC_VIEWER_STDOUT_SELF);
388: PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");
389: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
390: }
392: /*
393: Compute the 2-norm and max-norm of the error
394: */
395: VecAXPY(appctx->solution,-1.0,u);
396: VecNorm(appctx->solution,NORM_2,&norm_2);
397: norm_2 = PetscSqrtReal(appctx->h)*norm_2;
398: VecNorm(appctx->solution,NORM_MAX,&norm_max);
400: TSGetTimeStep(ts,&dt);
401: if (norm_2 > 1.e-2) {
402: 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);
403: }
404: appctx->norm_2 += norm_2;
405: appctx->norm_max += norm_max;
407: dttol = .0001;
408: PetscOptionsGetReal(NULL,NULL,"-dttol",&dttol,&flg);
409: if (dt < dttol) {
410: dt *= .999;
411: TSSetTimeStep(ts,dt);
412: }
414: /*
415: View a graph of the error
416: */
417: VecView(appctx->solution,appctx->viewer1);
419: /*
420: Print debugging information if desired
421: */
422: if (appctx->debug) {
423: PetscPrintf(PETSC_COMM_SELF,"Error vector\n");
424: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
425: }
427: return 0;
428: }
429: /* --------------------------------------------------------------------- */
432: /*
433: RHSMatrixHeat - User-provided routine to compute the right-hand-side
434: matrix for the heat equation.
436: Input Parameters:
437: ts - the TS context
438: t - current time
439: global_in - global input vector
440: dummy - optional user-defined context, as set by TSetRHSJacobian()
442: Output Parameters:
443: AA - Jacobian matrix
444: BB - optionally different preconditioning matrix
445: str - flag indicating matrix structure
447: Notes:
448: Recall that MatSetValues() uses 0-based row and column numbers
449: in Fortran as well as in C.
450: */
451: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
452: {
453: Mat A = AA; /* Jacobian matrix */
454: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
455: PetscInt mstart = 0;
456: PetscInt mend = appctx->m;
458: PetscInt i, idx[3];
459: PetscScalar v[3], stwo = -2./(appctx->h*appctx->h), sone = -.5*stwo;
461: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
462: Compute entries for the locally owned part of the matrix
463: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
464: /*
465: Set matrix rows corresponding to boundary data
466: */
468: mstart = 0;
469: v[0] = 1.0;
470: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
471: mstart++;
473: mend--;
474: v[0] = 1.0;
475: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
477: /*
478: Set matrix rows corresponding to interior data. We construct the
479: matrix one row at a time.
480: */
481: v[0] = sone; v[1] = stwo; v[2] = sone;
482: for (i=mstart; i<mend; i++) {
483: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
484: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
485: }
487: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
488: Complete the matrix assembly process and set some options
489: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
490: /*
491: Assemble matrix, using the 2-step process:
492: MatAssemblyBegin(), MatAssemblyEnd()
493: Computations can be done while messages are in transition
494: by placing code between these two statements.
495: */
496: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
497: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
499: /*
500: Set and option to indicate that we will never add a new nonzero location
501: to the matrix. If we do, it will generate an error.
502: */
503: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
505: return 0;
506: }
507: /* --------------------------------------------------------------------- */
510: /*
511: Input Parameters:
512: ts - the TS context
513: t - current time
514: f - function
515: ctx - optional user-defined context, as set by TSetBCFunction()
516: */
517: PetscErrorCode MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
518: {
519: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
521: PetscInt m = appctx->m;
522: PetscScalar *fa;
524: VecGetArray(f,&fa);
525: fa[0] = 0.0;
526: fa[m-1] = 1.0;
527: VecRestoreArray(f,&fa);
528: PetscPrintf(PETSC_COMM_SELF,"t=%g\n",(double)t);
530: return 0;
531: }