Actual source code: ex6.c
1: static char help[] = "Solves a simple time-dependent linear PDE (the heat equation).\n\
2: Input parameters include:\n\
3: -m <points>, where <points> = number of grid points\n\
4: -time_dependent_rhs : Treat the problem as having a time-dependent right-hand side\n\
5: -debug : Activate debugging printouts\n\
6: -nox : Deactivate x-window graphics\n\n";
8: /* ------------------------------------------------------------------------
10: This program solves the one-dimensional heat equation (also called the
11: diffusion equation),
12: u_t = u_xx,
13: on the domain 0 <= x <= 1, with the boundary conditions
14: u(t,0) = 0, u(t,1) = 0,
15: and the initial condition
16: u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
17: This is a linear, second-order, parabolic equation.
19: We discretize the right-hand side using finite differences with
20: uniform grid spacing h:
21: u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
22: We then demonstrate time evolution using the various TS methods by
23: running the program via
24: ex3 -ts_type <timestepping solver>
26: We compare the approximate solution with the exact solution, given by
27: u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
28: 3*exp(-4*pi*pi*t) * sin(2*pi*x)
30: Notes:
31: This code demonstrates the TS solver interface to two variants of
32: linear problems, u_t = f(u,t), namely
33: - time-dependent f: f(u,t) is a function of t
34: - time-independent f: f(u,t) is simply f(u)
36: The parallel version of this code is ts/tutorials/ex4.c
38: ------------------------------------------------------------------------- */
40: /*
41: Include "petscts.h" so that we can use the TS ODE integrators. Note that this file
42: automatically includes:
43: petscsys.h - base PETSc routines
44: petscis.h - index sets
45: petscvec.h - vectors
46: petscmat.h - matrices
47: petscpc.h - preconditioners
48: petscksp.h - Krylov subspace methods
49: petscsnes.h - nonlinear solvers
50: */
52: #include <petscts.h>
53: #include <petscdraw.h>
55: /*
56: User-defined application context - contains data needed by the
57: application-provided call-back routines.
58: */
59: typedef struct {
60: Vec solution; /* global exact solution vector */
61: PetscInt m; /* total number of grid points */
62: PetscReal h; /* mesh width h = 1/(m-1) */
63: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
64: PetscViewer viewer1, viewer2; /* viewers for the solution and error */
65: PetscReal norm_2, norm_max; /* error norms */
66: } AppCtx;
68: /*
69: User-defined routines
70: */
71: extern PetscErrorCode InitialConditions(Vec, AppCtx *);
72: extern PetscErrorCode RHSMatrixHeat(TS, PetscReal, Vec, Mat, Mat, void *);
73: extern PetscErrorCode Monitor(TS, PetscInt, PetscReal, Vec, void *);
74: extern PetscErrorCode ExactSolution(PetscReal, Vec, AppCtx *);
75: extern PetscErrorCode MyBCRoutine(TS, PetscReal, Vec, void *);
77: int main(int argc, char **argv)
78: {
79: AppCtx appctx; /* user-defined application context */
80: TS ts; /* timestepping context */
81: Mat A; /* matrix data structure */
82: Vec u; /* approximate solution vector */
83: PetscReal time_total_max = 100.0; /* default max total time */
84: PetscInt time_steps_max = 100; /* default max timesteps */
85: PetscDraw draw; /* drawing context */
86: PetscInt steps, m;
87: PetscMPIInt size;
88: PetscReal dt;
89: PetscReal ftime;
90: PetscBool flg;
91: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
92: Initialize program and set problem parameters
93: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
95: PetscFunctionBeginUser;
96: PetscCall(PetscInitialize(&argc, &argv, NULL, help));
97: MPI_Comm_size(PETSC_COMM_WORLD, &size);
98: PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!");
100: m = 60;
101: PetscCall(PetscOptionsGetInt(NULL, NULL, "-m", &m, NULL));
102: PetscCall(PetscOptionsHasName(NULL, NULL, "-debug", &appctx.debug));
104: appctx.m = m;
105: appctx.h = 1.0 / (m - 1.0);
106: appctx.norm_2 = 0.0;
107: appctx.norm_max = 0.0;
109: PetscCall(PetscPrintf(PETSC_COMM_SELF, "Solving a linear TS problem on 1 processor\n"));
111: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
112: Create vector data structures
113: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
115: /*
116: Create vector data structures for approximate and exact solutions
117: */
118: PetscCall(VecCreateSeq(PETSC_COMM_SELF, m, &u));
119: PetscCall(VecDuplicate(u, &appctx.solution));
121: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
122: Set up displays to show graphs of the solution and error
123: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
125: PetscCall(PetscViewerDrawOpen(PETSC_COMM_SELF, NULL, "", 80, 380, 400, 160, &appctx.viewer1));
126: PetscCall(PetscViewerDrawGetDraw(appctx.viewer1, 0, &draw));
127: PetscCall(PetscDrawSetDoubleBuffer(draw));
128: PetscCall(PetscViewerDrawOpen(PETSC_COMM_SELF, NULL, "", 80, 0, 400, 160, &appctx.viewer2));
129: PetscCall(PetscViewerDrawGetDraw(appctx.viewer2, 0, &draw));
130: PetscCall(PetscDrawSetDoubleBuffer(draw));
132: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
133: Create timestepping solver context
134: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
136: PetscCall(TSCreate(PETSC_COMM_SELF, &ts));
137: PetscCall(TSSetProblemType(ts, TS_LINEAR));
139: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
140: Set optional user-defined monitoring routine
141: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
143: PetscCall(TSMonitorSet(ts, Monitor, &appctx, NULL));
145: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
147: Create matrix data structure; set matrix evaluation routine.
148: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
150: PetscCall(MatCreate(PETSC_COMM_SELF, &A));
151: PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, m, m));
152: PetscCall(MatSetFromOptions(A));
153: PetscCall(MatSetUp(A));
155: PetscCall(PetscOptionsHasName(NULL, NULL, "-time_dependent_rhs", &flg));
156: if (flg) {
157: /*
158: For linear problems with a time-dependent f(u,t) in the equation
159: u_t = f(u,t), the user provides the discretized right-hand-side
160: as a time-dependent matrix.
161: */
162: PetscCall(TSSetRHSFunction(ts, NULL, TSComputeRHSFunctionLinear, &appctx));
163: PetscCall(TSSetRHSJacobian(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: PetscCall(RHSMatrixHeat(ts, 0.0, u, A, A, &appctx));
172: PetscCall(TSSetRHSFunction(ts, NULL, TSComputeRHSFunctionLinear, &appctx));
173: PetscCall(TSSetRHSJacobian(ts, A, A, TSComputeRHSJacobianConstant, &appctx));
174: }
176: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
177: Set solution vector and initial timestep
178: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
180: dt = appctx.h * appctx.h / 2.0;
181: PetscCall(TSSetTimeStep(ts, dt));
182: PetscCall(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 TSSetMaxSteps()/TSSetMaxTime().
192: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
194: PetscCall(TSSetMaxSteps(ts, time_steps_max));
195: PetscCall(TSSetMaxTime(ts, time_total_max));
196: PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
197: PetscCall(TSSetFromOptions(ts));
199: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
200: Solve the problem
201: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
203: /*
204: Evaluate initial conditions
205: */
206: PetscCall(InitialConditions(u, &appctx));
208: /*
209: Run the timestepping solver
210: */
211: PetscCall(TSSolve(ts, u));
212: PetscCall(TSGetSolveTime(ts, &ftime));
213: PetscCall(TSGetStepNumber(ts, &steps));
215: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
216: View timestepping solver info
217: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
219: PetscCall(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)));
220: PetscCall(TSView(ts, PETSC_VIEWER_STDOUT_SELF));
222: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
223: Free work space. All PETSc objects should be destroyed when they
224: are no longer needed.
225: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
227: PetscCall(TSDestroy(&ts));
228: PetscCall(MatDestroy(&A));
229: PetscCall(VecDestroy(&u));
230: PetscCall(PetscViewerDestroy(&appctx.viewer1));
231: PetscCall(PetscViewerDestroy(&appctx.viewer2));
232: PetscCall(VecDestroy(&appctx.solution));
234: /*
235: Always call PetscFinalize() before exiting a program. This routine
236: - finalizes the PETSc libraries as well as MPI
237: - provides summary and diagnostic information if certain runtime
238: options are chosen (e.g., -log_view).
239: */
240: PetscCall(PetscFinalize());
241: return 0;
242: }
243: /* --------------------------------------------------------------------- */
244: /*
245: InitialConditions - Computes the solution at the initial time.
247: Input Parameter:
248: u - uninitialized solution vector (global)
249: appctx - user-defined application context
251: Output Parameter:
252: u - vector with solution at initial time (global)
253: */
254: PetscErrorCode InitialConditions(Vec u, AppCtx *appctx)
255: {
256: PetscScalar *u_localptr;
257: PetscInt i;
259: PetscFunctionBeginUser;
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: PetscCall(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++) u_localptr[i] = PetscSinReal(PETSC_PI * i * 6. * appctx->h) + 3. * PetscSinReal(PETSC_PI * i * 2. * appctx->h);
278: /*
279: Restore vector
280: */
281: PetscCall(VecRestoreArray(u, &u_localptr));
283: /*
284: Print debugging information if desired
285: */
286: if (appctx->debug) PetscCall(VecView(u, PETSC_VIEWER_STDOUT_SELF));
287: PetscFunctionReturn(PETSC_SUCCESS);
288: }
289: /* --------------------------------------------------------------------- */
290: /*
291: ExactSolution - Computes the exact solution at a given time.
293: Input Parameters:
294: t - current time
295: solution - vector in which exact solution will be computed
296: appctx - user-defined application context
298: Output Parameter:
299: solution - vector with the newly computed exact solution
300: */
301: PetscErrorCode ExactSolution(PetscReal t, Vec solution, AppCtx *appctx)
302: {
303: PetscScalar *s_localptr, h = appctx->h, ex1, ex2, sc1, sc2;
304: PetscInt i;
306: PetscFunctionBeginUser;
307: /*
308: Get a pointer to vector data.
309: */
310: PetscCall(VecGetArray(solution, &s_localptr));
312: /*
313: Simply write the solution directly into the array locations.
314: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
315: */
316: ex1 = PetscExpReal(-36. * PETSC_PI * PETSC_PI * t);
317: ex2 = PetscExpReal(-4. * PETSC_PI * PETSC_PI * t);
318: sc1 = PETSC_PI * 6. * h;
319: sc2 = PETSC_PI * 2. * h;
320: for (i = 0; i < appctx->m; i++) s_localptr[i] = PetscSinReal(PetscRealPart(sc1) * (PetscReal)i) * ex1 + 3. * PetscSinReal(PetscRealPart(sc2) * (PetscReal)i) * ex2;
322: /*
323: Restore vector
324: */
325: PetscCall(VecRestoreArray(solution, &s_localptr));
326: PetscFunctionReturn(PETSC_SUCCESS);
327: }
328: /* --------------------------------------------------------------------- */
329: /*
330: Monitor - User-provided routine to monitor the solution computed at
331: each timestep. This example plots the solution and computes the
332: error in two different norms.
334: This example also demonstrates changing the timestep via TSSetTimeStep().
336: Input Parameters:
337: ts - the timestep context
338: step - the count of the current step (with 0 meaning the
339: initial condition)
340: crtime - the current time
341: u - the solution at this timestep
342: ctx - the user-provided context for this monitoring routine.
343: In this case we use the application context which contains
344: information about the problem size, workspace and the exact
345: solution.
346: */
347: PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal crtime, Vec u, PetscCtx ctx)
348: {
349: AppCtx *appctx = (AppCtx *)ctx; /* user-defined application context */
350: PetscReal norm_2, norm_max, dt, dttol;
351: PetscBool flg;
353: PetscFunctionBeginUser;
354: /*
355: View a graph of the current iterate
356: */
357: PetscCall(VecView(u, appctx->viewer2));
359: /*
360: Compute the exact solution
361: */
362: PetscCall(ExactSolution(crtime, appctx->solution, appctx));
364: /*
365: Print debugging information if desired
366: */
367: if (appctx->debug) {
368: PetscCall(PetscPrintf(PETSC_COMM_SELF, "Computed solution vector\n"));
369: PetscCall(VecView(u, PETSC_VIEWER_STDOUT_SELF));
370: PetscCall(PetscPrintf(PETSC_COMM_SELF, "Exact solution vector\n"));
371: PetscCall(VecView(appctx->solution, PETSC_VIEWER_STDOUT_SELF));
372: }
374: /*
375: Compute the 2-norm and max-norm of the error
376: */
377: PetscCall(VecAXPY(appctx->solution, -1.0, u));
378: PetscCall(VecNorm(appctx->solution, NORM_2, &norm_2));
379: norm_2 = PetscSqrtReal(appctx->h) * norm_2;
380: PetscCall(VecNorm(appctx->solution, NORM_MAX, &norm_max));
382: PetscCall(TSGetTimeStep(ts, &dt));
383: if (norm_2 > 1.e-2) PetscCall(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));
384: appctx->norm_2 += norm_2;
385: appctx->norm_max += norm_max;
387: dttol = .0001;
388: PetscCall(PetscOptionsGetReal(NULL, NULL, "-dttol", &dttol, &flg));
389: if (dt < dttol) {
390: dt *= .999;
391: PetscCall(TSSetTimeStep(ts, dt));
392: }
394: /*
395: View a graph of the error
396: */
397: PetscCall(VecView(appctx->solution, appctx->viewer1));
399: /*
400: Print debugging information if desired
401: */
402: if (appctx->debug) {
403: PetscCall(PetscPrintf(PETSC_COMM_SELF, "Error vector\n"));
404: PetscCall(VecView(appctx->solution, PETSC_VIEWER_STDOUT_SELF));
405: }
406: PetscFunctionReturn(PETSC_SUCCESS);
407: }
408: /* --------------------------------------------------------------------- */
409: /*
410: RHSMatrixHeat - User-provided routine to compute the right-hand-side
411: matrix for the heat equation.
413: Input Parameters:
414: ts - the TS context
415: t - current time
416: global_in - global input vector
417: dummy - optional user-defined context, as set by TSetRHSJacobian()
419: Output Parameters:
420: AA - Jacobian matrix
421: BB - optionally different matrix used to construct the preconditioner
423: Notes:
424: Recall that MatSetValues() uses 0-based row and column numbers
425: in Fortran as well as in C.
426: */
427: PetscErrorCode RHSMatrixHeat(TS ts, PetscReal t, Vec X, Mat AA, Mat BB, PetscCtx ctx)
428: {
429: Mat A = AA; /* Jacobian matrix */
430: AppCtx *appctx = (AppCtx *)ctx; /* user-defined application context */
431: PetscInt mstart = 0;
432: PetscInt mend = appctx->m;
433: PetscInt i, idx[3];
434: PetscScalar v[3], stwo = -2. / (appctx->h * appctx->h), sone = -.5 * stwo;
436: PetscFunctionBeginUser;
437: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
438: Compute entries for the locally owned part of the matrix
439: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
440: /*
441: Set matrix rows corresponding to boundary data
442: */
444: mstart = 0;
445: v[0] = 1.0;
446: PetscCall(MatSetValues(A, 1, &mstart, 1, &mstart, v, INSERT_VALUES));
447: mstart++;
449: mend--;
450: v[0] = 1.0;
451: PetscCall(MatSetValues(A, 1, &mend, 1, &mend, v, INSERT_VALUES));
453: /*
454: Set matrix rows corresponding to interior data. We construct the
455: matrix one row at a time.
456: */
457: v[0] = sone;
458: v[1] = stwo;
459: v[2] = sone;
460: for (i = mstart; i < mend; i++) {
461: idx[0] = i - 1;
462: idx[1] = i;
463: idx[2] = i + 1;
464: PetscCall(MatSetValues(A, 1, &i, 3, idx, v, INSERT_VALUES));
465: }
467: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
468: Complete the matrix assembly process and set some options
469: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
470: /*
471: Assemble matrix, using the 2-step process:
472: MatAssemblyBegin(), MatAssemblyEnd()
473: Computations can be done while messages are in transition
474: by placing code between these two statements.
475: */
476: PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
477: PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
479: /*
480: Set and option to indicate that we will never add a new nonzero location
481: to the matrix. If we do, it will generate an error.
482: */
483: PetscCall(MatSetOption(A, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
484: PetscFunctionReturn(PETSC_SUCCESS);
485: }
486: /* --------------------------------------------------------------------- */
487: /*
488: Input Parameters:
489: ts - the TS context
490: t - current time
491: f - function
492: ctx - optional user-defined context, as set by TSetBCFunction()
493: */
494: PetscErrorCode MyBCRoutine(TS ts, PetscReal t, Vec f, PetscCtx ctx)
495: {
496: AppCtx *appctx = (AppCtx *)ctx; /* user-defined application context */
497: PetscInt m = appctx->m;
498: PetscScalar *fa;
500: PetscFunctionBeginUser;
501: PetscCall(VecGetArray(f, &fa));
502: fa[0] = 0.0;
503: fa[m - 1] = 1.0;
504: PetscCall(VecRestoreArray(f, &fa));
505: PetscCall(PetscPrintf(PETSC_COMM_SELF, "t=%g\n", (double)t));
506: PetscFunctionReturn(PETSC_SUCCESS);
507: }
509: /*TEST
511: test:
512: args: -nox -ts_max_steps 4
514: TEST*/