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: /* ------------------------------------------------------------------------
11: This program solves the one-dimensional heat equation (also called the
12: diffusion equation),
13: u_t = u_xx,
14: on the domain 0 <= x <= 1, with the boundary conditions
15: u(t,0) = 0, u(t,1) = 0,
16: and the initial condition
17: u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
18: This is a linear, second-order, parabolic equation.
20: We discretize the right-hand side using finite differences with
21: uniform grid spacing h:
22: u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
23: We then demonstrate time evolution using the various TS methods by
24: running the program via
25: ex3 -ts_type <timestepping solver>
27: We compare the approximate solution with the exact solution, given by
28: u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
29: 3*exp(-4*pi*pi*t) * sin(2*pi*x)
31: Notes:
32: This code demonstrates the TS solver interface to two variants of
33: linear problems, u_t = f(u,t), namely
34: - time-dependent f: f(u,t) is a function of t
35: - time-independent f: f(u,t) is simply f(u)
37: The parallel version of this code is ts/tutorials/ex4.c
39: ------------------------------------------------------------------------- */
41: /*
42: Include "ts.h" so that we can use TS solvers. Note that this file
43: automatically includes:
44: petscsys.h - base PETSc routines vec.h - vectors
45: sys.h - system routines mat.h - matrices
46: is.h - index sets ksp.h - Krylov subspace methods
47: viewer.h - viewers pc.h - preconditioners
48: snes.h - nonlinear solvers
49: */
51: #include <petscts.h>
52: #include <petscdraw.h>
54: /*
55: User-defined application context - contains data needed by the
56: application-provided call-back routines.
57: */
58: typedef struct {
59: Vec solution; /* global exact solution vector */
60: PetscInt m; /* total number of grid points */
61: PetscReal h; /* mesh width h = 1/(m-1) */
62: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
63: PetscViewer viewer1, viewer2; /* viewers for the solution and error */
64: PetscReal norm_2, norm_max; /* error norms */
65: } AppCtx;
67: /*
68: User-defined routines
69: */
70: extern PetscErrorCode InitialConditions(Vec, AppCtx *);
71: extern PetscErrorCode RHSMatrixHeat(TS, PetscReal, Vec, Mat, Mat, void *);
72: extern PetscErrorCode Monitor(TS, PetscInt, PetscReal, Vec, void *);
73: extern PetscErrorCode ExactSolution(PetscReal, Vec, AppCtx *);
74: extern PetscErrorCode MyBCRoutine(TS, PetscReal, Vec, void *);
76: int main(int argc, char **argv)
77: {
78: AppCtx appctx; /* user-defined application context */
79: TS ts; /* timestepping context */
80: Mat A; /* matrix data structure */
81: Vec u; /* approximate solution vector */
82: PetscReal time_total_max = 100.0; /* default max total time */
83: PetscInt time_steps_max = 100; /* default max timesteps */
84: PetscDraw draw; /* drawing context */
85: PetscInt steps, m;
86: PetscMPIInt size;
87: PetscReal dt;
88: PetscReal ftime;
89: PetscBool flg;
90: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
91: Initialize program and set problem parameters
92: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
95: PetscInitialize(&argc, &argv, (char *)0, help);
96: MPI_Comm_size(PETSC_COMM_WORLD, &size);
99: m = 60;
100: PetscOptionsGetInt(NULL, NULL, "-m", &m, NULL);
101: PetscOptionsHasName(NULL, NULL, "-debug", &appctx.debug);
103: appctx.m = m;
104: appctx.h = 1.0 / (m - 1.0);
105: appctx.norm_2 = 0.0;
106: appctx.norm_max = 0.0;
108: PetscPrintf(PETSC_COMM_SELF, "Solving a linear TS problem on 1 processor\n");
110: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
111: Create vector data structures
112: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
114: /*
115: Create vector data structures for approximate and exact solutions
116: */
117: VecCreateSeq(PETSC_COMM_SELF, m, &u);
118: VecDuplicate(u, &appctx.solution);
120: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
121: Set up displays to show graphs of the solution and error
122: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
124: PetscViewerDrawOpen(PETSC_COMM_SELF, 0, "", 80, 380, 400, 160, &appctx.viewer1);
125: PetscViewerDrawGetDraw(appctx.viewer1, 0, &draw);
126: PetscDrawSetDoubleBuffer(draw);
127: PetscViewerDrawOpen(PETSC_COMM_SELF, 0, "", 80, 0, 400, 160, &appctx.viewer2);
128: PetscViewerDrawGetDraw(appctx.viewer2, 0, &draw);
129: PetscDrawSetDoubleBuffer(draw);
131: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
132: Create timestepping solver context
133: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
135: TSCreate(PETSC_COMM_SELF, &ts);
136: TSSetProblemType(ts, TS_LINEAR);
138: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
139: Set optional user-defined monitoring routine
140: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
142: TSMonitorSet(ts, Monitor, &appctx, NULL);
144: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
146: Create matrix data structure; set matrix evaluation routine.
147: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
149: MatCreate(PETSC_COMM_SELF, &A);
150: MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, m, m);
151: MatSetFromOptions(A);
152: MatSetUp(A);
154: PetscOptionsHasName(NULL, NULL, "-time_dependent_rhs", &flg);
155: if (flg) {
156: /*
157: For linear problems with a time-dependent f(u,t) in the equation
158: u_t = f(u,t), the user provides the discretized right-hand-side
159: as a time-dependent matrix.
160: */
161: TSSetRHSFunction(ts, NULL, TSComputeRHSFunctionLinear, &appctx);
162: TSSetRHSJacobian(ts, A, A, RHSMatrixHeat, &appctx);
163: } else {
164: /*
165: For linear problems with a time-independent f(u) in the equation
166: u_t = f(u), the user provides the discretized right-hand-side
167: as a matrix only once, and then sets a null matrix evaluation
168: routine.
169: */
170: RHSMatrixHeat(ts, 0.0, u, A, A, &appctx);
171: TSSetRHSFunction(ts, NULL, TSComputeRHSFunctionLinear, &appctx);
172: TSSetRHSJacobian(ts, A, A, TSComputeRHSJacobianConstant, &appctx);
173: }
175: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
176: Set solution vector and initial timestep
177: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
179: dt = appctx.h * appctx.h / 2.0;
180: TSSetTimeStep(ts, dt);
181: TSSetSolution(ts, u);
183: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
184: Customize timestepping solver:
185: - Set the solution method to be the Backward Euler method.
186: - Set timestepping duration info
187: Then set runtime options, which can override these defaults.
188: For example,
189: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
190: to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
191: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
193: TSSetMaxSteps(ts, time_steps_max);
194: TSSetMaxTime(ts, time_total_max);
195: TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER);
196: TSSetFromOptions(ts);
198: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
199: Solve the problem
200: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
202: /*
203: Evaluate initial conditions
204: */
205: InitialConditions(u, &appctx);
207: /*
208: Run the timestepping solver
209: */
210: TSSolve(ts, u);
211: TSGetSolveTime(ts, &ftime);
212: TSGetStepNumber(ts, &steps);
214: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
215: View timestepping solver info
216: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
218: 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));
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_view).
238: */
239: PetscFinalize();
240: return 0;
241: }
242: /* --------------------------------------------------------------------- */
243: /*
244: InitialConditions - Computes the solution at the initial time.
246: Input Parameter:
247: u - uninitialized solution vector (global)
248: appctx - user-defined application context
250: Output Parameter:
251: u - vector with solution at initial time (global)
252: */
253: PetscErrorCode InitialConditions(Vec u, AppCtx *appctx)
254: {
255: PetscScalar *u_localptr;
256: PetscInt i;
258: /*
259: Get a pointer to vector data.
260: - For default PETSc vectors, VecGetArray() returns a pointer to
261: the data array. Otherwise, the routine is implementation dependent.
262: - You MUST call VecRestoreArray() when you no longer need access to
263: the array.
264: - Note that the Fortran interface to VecGetArray() differs from the
265: C version. See the users manual for details.
266: */
267: VecGetArray(u, &u_localptr);
269: /*
270: We initialize the solution array by simply writing the solution
271: directly into the array locations. Alternatively, we could use
272: VecSetValues() or VecSetValuesLocal().
273: */
274: 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);
276: /*
277: Restore vector
278: */
279: VecRestoreArray(u, &u_localptr);
281: /*
282: Print debugging information if desired
283: */
284: if (appctx->debug) VecView(u, PETSC_VIEWER_STDOUT_SELF);
286: return 0;
287: }
288: /* --------------------------------------------------------------------- */
289: /*
290: ExactSolution - Computes the exact solution at a given time.
292: Input Parameters:
293: t - current time
294: solution - vector in which exact solution will be computed
295: appctx - user-defined application context
297: Output Parameter:
298: solution - vector with the newly computed exact solution
299: */
300: PetscErrorCode ExactSolution(PetscReal t, Vec solution, AppCtx *appctx)
301: {
302: PetscScalar *s_localptr, h = appctx->h, ex1, ex2, sc1, sc2;
303: PetscInt i;
305: /*
306: Get a pointer to vector data.
307: */
308: VecGetArray(solution, &s_localptr);
310: /*
311: Simply write the solution directly into the array locations.
312: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
313: */
314: ex1 = PetscExpReal(-36. * PETSC_PI * PETSC_PI * t);
315: ex2 = PetscExpReal(-4. * PETSC_PI * PETSC_PI * t);
316: sc1 = PETSC_PI * 6. * h;
317: sc2 = PETSC_PI * 2. * h;
318: for (i = 0; i < appctx->m; i++) s_localptr[i] = PetscSinReal(PetscRealPart(sc1) * (PetscReal)i) * ex1 + 3. * PetscSinReal(PetscRealPart(sc2) * (PetscReal)i) * ex2;
320: /*
321: Restore vector
322: */
323: VecRestoreArray(solution, &s_localptr);
324: return 0;
325: }
326: /* --------------------------------------------------------------------- */
327: /*
328: Monitor - User-provided routine to monitor the solution computed at
329: each timestep. This example plots the solution and computes the
330: error in two different norms.
332: This example also demonstrates changing the timestep via TSSetTimeStep().
334: Input Parameters:
335: ts - the timestep context
336: step - the count of the current step (with 0 meaning the
337: initial condition)
338: crtime - the current time
339: u - the solution at this timestep
340: ctx - the user-provided context for this monitoring routine.
341: In this case we use the application context which contains
342: information about the problem size, workspace and the exact
343: solution.
344: */
345: PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal crtime, Vec u, void *ctx)
346: {
347: AppCtx *appctx = (AppCtx *)ctx; /* user-defined application context */
348: PetscReal norm_2, norm_max, dt, dttol;
349: PetscBool flg;
351: /*
352: View a graph of the current iterate
353: */
354: VecView(u, appctx->viewer2);
356: /*
357: Compute the exact solution
358: */
359: ExactSolution(crtime, appctx->solution, appctx);
361: /*
362: Print debugging information if desired
363: */
364: if (appctx->debug) {
365: PetscPrintf(PETSC_COMM_SELF, "Computed solution vector\n");
366: VecView(u, PETSC_VIEWER_STDOUT_SELF);
367: PetscPrintf(PETSC_COMM_SELF, "Exact solution vector\n");
368: VecView(appctx->solution, PETSC_VIEWER_STDOUT_SELF);
369: }
371: /*
372: Compute the 2-norm and max-norm of the error
373: */
374: VecAXPY(appctx->solution, -1.0, u);
375: VecNorm(appctx->solution, NORM_2, &norm_2);
376: norm_2 = PetscSqrtReal(appctx->h) * norm_2;
377: VecNorm(appctx->solution, NORM_MAX, &norm_max);
379: TSGetTimeStep(ts, &dt);
380: if (norm_2 > 1.e-2) PetscPrintf(PETSC_COMM_SELF, "Timestep %" PetscInt_FMT ": 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);
381: appctx->norm_2 += norm_2;
382: appctx->norm_max += norm_max;
384: dttol = .0001;
385: PetscOptionsGetReal(NULL, NULL, "-dttol", &dttol, &flg);
386: if (dt < dttol) {
387: dt *= .999;
388: TSSetTimeStep(ts, dt);
389: }
391: /*
392: View a graph of the error
393: */
394: VecView(appctx->solution, appctx->viewer1);
396: /*
397: Print debugging information if desired
398: */
399: if (appctx->debug) {
400: PetscPrintf(PETSC_COMM_SELF, "Error vector\n");
401: VecView(appctx->solution, PETSC_VIEWER_STDOUT_SELF);
402: }
404: return 0;
405: }
406: /* --------------------------------------------------------------------- */
407: /*
408: RHSMatrixHeat - User-provided routine to compute the right-hand-side
409: matrix for the heat equation.
411: Input Parameters:
412: ts - the TS context
413: t - current time
414: global_in - global input vector
415: dummy - optional user-defined context, as set by TSetRHSJacobian()
417: Output Parameters:
418: AA - Jacobian matrix
419: BB - optionally different preconditioning matrix
420: str - flag indicating matrix structure
422: Notes:
423: Recall that MatSetValues() uses 0-based row and column numbers
424: in Fortran as well as in C.
425: */
426: PetscErrorCode RHSMatrixHeat(TS ts, PetscReal t, Vec X, Mat AA, Mat BB, void *ctx)
427: {
428: Mat A = AA; /* Jacobian matrix */
429: AppCtx *appctx = (AppCtx *)ctx; /* user-defined application context */
430: PetscInt mstart = 0;
431: PetscInt mend = appctx->m;
432: PetscInt i, idx[3];
433: PetscScalar v[3], stwo = -2. / (appctx->h * appctx->h), sone = -.5 * stwo;
435: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
436: Compute entries for the locally owned part of the matrix
437: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
438: /*
439: Set matrix rows corresponding to boundary data
440: */
442: mstart = 0;
443: v[0] = 1.0;
444: MatSetValues(A, 1, &mstart, 1, &mstart, v, INSERT_VALUES);
445: mstart++;
447: mend--;
448: v[0] = 1.0;
449: MatSetValues(A, 1, &mend, 1, &mend, v, INSERT_VALUES);
451: /*
452: Set matrix rows corresponding to interior data. We construct the
453: matrix one row at a time.
454: */
455: v[0] = sone;
456: v[1] = stwo;
457: v[2] = sone;
458: for (i = mstart; i < mend; i++) {
459: idx[0] = i - 1;
460: idx[1] = i;
461: idx[2] = i + 1;
462: MatSetValues(A, 1, &i, 3, idx, v, INSERT_VALUES);
463: }
465: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
466: Complete the matrix assembly process and set some options
467: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
468: /*
469: Assemble matrix, using the 2-step process:
470: MatAssemblyBegin(), MatAssemblyEnd()
471: Computations can be done while messages are in transition
472: by placing code between these two statements.
473: */
474: MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY);
475: MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY);
477: /*
478: Set and option to indicate that we will never add a new nonzero location
479: to the matrix. If we do, it will generate an error.
480: */
481: MatSetOption(A, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE);
483: return 0;
484: }
485: /* --------------------------------------------------------------------- */
486: /*
487: Input Parameters:
488: ts - the TS context
489: t - current time
490: f - function
491: ctx - optional user-defined context, as set by TSetBCFunction()
492: */
493: PetscErrorCode MyBCRoutine(TS ts, PetscReal t, Vec f, void *ctx)
494: {
495: AppCtx *appctx = (AppCtx *)ctx; /* user-defined application context */
496: PetscInt m = appctx->m;
497: PetscScalar *fa;
499: VecGetArray(f, &fa);
500: fa[0] = 0.0;
501: fa[m - 1] = 1.0;
502: VecRestoreArray(f, &fa);
503: PetscPrintf(PETSC_COMM_SELF, "t=%g\n", (double)t);
505: return 0;
506: }
508: /*TEST
510: test:
511: args: -nox -ts_max_steps 4
513: TEST*/