Actual source code: ex5.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) = 1, u(t,1) = 1,
16: and the initial condition
17: u(0,x) = cos(6*pi*x) + 3*cos(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) * cos(6*pi*x) +
29: 3*exp(-4*pi*pi*t) * cos(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 just f(u)
37: The parallel version of this code is ts/tutorials/ex4.c
39: ------------------------------------------------------------------------- */
41: /*
42: Include "petscts.h" so that we can use TS solvers. Note that this file
43: automatically includes:
44: petscsys.h - base PETSc routines petscvec.h - vectors
45: petscmat.h - matrices
46: petscis.h - index sets petscksp.h - Krylov subspace methods
47: petscviewer.h - viewers petscpc.h - preconditioners
48: petscksp.h - linear solvers petscsnes.h - nonlinear solvers
49: */
50: #include <petscts.h>
51: #include <petscdraw.h>
53: /*
54: User-defined application context - contains data needed by the
55: application-provided call-back routines.
56: */
57: typedef struct {
58: Vec solution; /* global exact solution vector */
59: PetscInt m; /* total number of grid points */
60: PetscReal h; /* mesh width h = 1/(m-1) */
61: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
62: PetscViewer viewer1, viewer2; /* viewers for the solution and error */
63: PetscReal norm_2, norm_max; /* error norms */
64: } AppCtx;
66: /*
67: User-defined routines
68: */
69: extern PetscErrorCode InitialConditions(Vec, AppCtx *);
70: extern PetscErrorCode RHSMatrixHeat(TS, PetscReal, Vec, Mat, Mat, void *);
71: extern PetscErrorCode Monitor(TS, PetscInt, PetscReal, Vec, void *);
72: extern PetscErrorCode ExactSolution(PetscReal, Vec, AppCtx *);
74: int main(int argc, char **argv)
75: {
76: AppCtx appctx; /* user-defined application context */
77: TS ts; /* timestepping context */
78: Mat A; /* matrix data structure */
79: Vec u; /* approximate solution vector */
80: PetscReal time_total_max = 100.0; /* default max total time */
81: PetscInt time_steps_max = 100; /* default max timesteps */
82: PetscDraw draw; /* drawing context */
83: PetscInt steps, m;
84: PetscMPIInt size;
85: PetscBool flg;
86: PetscReal dt, ftime;
88: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
89: Initialize program and set problem parameters
90: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
93: PetscInitialize(&argc, &argv, (char *)0, help);
94: MPI_Comm_size(PETSC_COMM_WORLD, &size);
97: m = 60;
98: PetscOptionsGetInt(NULL, NULL, "-m", &m, NULL);
99: PetscOptionsHasName(NULL, NULL, "-debug", &appctx.debug);
100: appctx.m = m;
101: appctx.h = 1.0 / (m - 1.0);
102: appctx.norm_2 = 0.0;
103: appctx.norm_max = 0.0;
105: PetscPrintf(PETSC_COMM_SELF, "Solving a linear TS problem on 1 processor\n");
107: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
108: Create vector data structures
109: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
111: /*
112: Create vector data structures for approximate and exact solutions
113: */
114: VecCreateSeq(PETSC_COMM_SELF, m, &u);
115: VecDuplicate(u, &appctx.solution);
117: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
118: Set up displays to show graphs of the solution and error
119: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
121: PetscViewerDrawOpen(PETSC_COMM_SELF, 0, "", 80, 380, 400, 160, &appctx.viewer1);
122: PetscViewerDrawGetDraw(appctx.viewer1, 0, &draw);
123: PetscDrawSetDoubleBuffer(draw);
124: PetscViewerDrawOpen(PETSC_COMM_SELF, 0, "", 80, 0, 400, 160, &appctx.viewer2);
125: PetscViewerDrawGetDraw(appctx.viewer2, 0, &draw);
126: PetscDrawSetDoubleBuffer(draw);
128: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
129: Create timestepping solver context
130: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
132: TSCreate(PETSC_COMM_SELF, &ts);
133: TSSetProblemType(ts, TS_LINEAR);
135: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
136: Set optional user-defined monitoring routine
137: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
139: TSMonitorSet(ts, Monitor, &appctx, NULL);
141: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
143: Create matrix data structure; set matrix evaluation routine.
144: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
146: MatCreate(PETSC_COMM_SELF, &A);
147: MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, m, m);
148: MatSetFromOptions(A);
149: MatSetUp(A);
151: PetscOptionsHasName(NULL, NULL, "-time_dependent_rhs", &flg);
152: if (flg) {
153: /*
154: For linear problems with a time-dependent f(u,t) in the equation
155: u_t = f(u,t), the user provides the discretized right-hand-side
156: as a time-dependent matrix.
157: */
158: TSSetRHSFunction(ts, NULL, TSComputeRHSFunctionLinear, &appctx);
159: TSSetRHSJacobian(ts, A, A, RHSMatrixHeat, &appctx);
160: } else {
161: /*
162: For linear problems with a time-independent f(u) in the equation
163: u_t = f(u), the user provides the discretized right-hand-side
164: as a matrix only once, and then sets a null matrix evaluation
165: routine.
166: */
167: RHSMatrixHeat(ts, 0.0, u, A, A, &appctx);
168: TSSetRHSFunction(ts, NULL, TSComputeRHSFunctionLinear, &appctx);
169: TSSetRHSJacobian(ts, A, A, TSComputeRHSJacobianConstant, &appctx);
170: }
172: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
173: Set solution vector and initial timestep
174: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
176: dt = appctx.h * appctx.h / 2.0;
177: TSSetTimeStep(ts, dt);
178: TSSetSolution(ts, u);
180: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
181: Customize timestepping solver:
182: - Set the solution method to be the Backward Euler method.
183: - Set timestepping duration info
184: Then set runtime options, which can override these defaults.
185: For example,
186: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
187: to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
188: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
190: TSSetMaxSteps(ts, time_steps_max);
191: TSSetMaxTime(ts, time_total_max);
192: TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER);
193: TSSetFromOptions(ts);
195: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
196: Solve the problem
197: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
199: /*
200: Evaluate initial conditions
201: */
202: InitialConditions(u, &appctx);
204: /*
205: Run the timestepping solver
206: */
207: TSSolve(ts, u);
208: TSGetSolveTime(ts, &ftime);
209: TSGetStepNumber(ts, &steps);
211: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
212: View timestepping solver info
213: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
215: 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));
216: TSView(ts, PETSC_VIEWER_STDOUT_SELF);
218: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
219: Free work space. All PETSc objects should be destroyed when they
220: are no longer needed.
221: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
223: TSDestroy(&ts);
224: MatDestroy(&A);
225: VecDestroy(&u);
226: PetscViewerDestroy(&appctx.viewer1);
227: PetscViewerDestroy(&appctx.viewer2);
228: VecDestroy(&appctx.solution);
230: /*
231: Always call PetscFinalize() before exiting a program. This routine
232: - finalizes the PETSc libraries as well as MPI
233: - provides summary and diagnostic information if certain runtime
234: options are chosen (e.g., -log_view).
235: */
236: PetscFinalize();
237: return 0;
238: }
239: /* --------------------------------------------------------------------- */
240: /*
241: InitialConditions - Computes the solution at the initial time.
243: Input Parameter:
244: u - uninitialized solution vector (global)
245: appctx - user-defined application context
247: Output Parameter:
248: u - vector with solution at initial time (global)
249: */
250: PetscErrorCode InitialConditions(Vec u, AppCtx *appctx)
251: {
252: PetscScalar *u_localptr, h = appctx->h;
253: PetscInt i;
255: /*
256: Get a pointer to vector data.
257: - For default PETSc vectors, VecGetArray() returns a pointer to
258: the data array. Otherwise, the routine is implementation dependent.
259: - You MUST call VecRestoreArray() when you no longer need access to
260: the array.
261: - Note that the Fortran interface to VecGetArray() differs from the
262: C version. See the users manual for details.
263: */
264: VecGetArray(u, &u_localptr);
266: /*
267: We initialize the solution array by simply writing the solution
268: directly into the array locations. Alternatively, we could use
269: VecSetValues() or VecSetValuesLocal().
270: */
271: for (i = 0; i < appctx->m; i++) u_localptr[i] = PetscCosScalar(PETSC_PI * i * 6. * h) + 3. * PetscCosScalar(PETSC_PI * i * 2. * h);
273: /*
274: Restore vector
275: */
276: VecRestoreArray(u, &u_localptr);
278: /*
279: Print debugging information if desired
280: */
281: if (appctx->debug) {
282: printf("initial guess vector\n");
283: VecView(u, PETSC_VIEWER_STDOUT_SELF);
284: }
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, tc = t;
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 = PetscExpScalar(-36. * PETSC_PI * PETSC_PI * tc);
315: ex2 = PetscExpScalar(-4. * PETSC_PI * PETSC_PI * tc);
316: sc1 = PETSC_PI * 6. * h;
317: sc2 = PETSC_PI * 2. * h;
318: for (i = 0; i < appctx->m; i++) s_localptr[i] = PetscCosScalar(sc1 * (PetscReal)i) * ex1 + 3. * PetscCosScalar(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: Input Parameters:
333: ts - the timestep context
334: step - the count of the current step (with 0 meaning the
335: initial condition)
336: time - the current time
337: u - the solution at this timestep
338: ctx - the user-provided context for this monitoring routine.
339: In this case we use the application context which contains
340: information about the problem size, workspace and the exact
341: solution.
342: */
343: PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal time, Vec u, void *ctx)
344: {
345: AppCtx *appctx = (AppCtx *)ctx; /* user-defined application context */
346: PetscReal norm_2, norm_max;
348: /*
349: View a graph of the current iterate
350: */
351: VecView(u, appctx->viewer2);
353: /*
354: Compute the exact solution
355: */
356: ExactSolution(time, appctx->solution, appctx);
358: /*
359: Print debugging information if desired
360: */
361: if (appctx->debug) {
362: printf("Computed solution vector\n");
363: VecView(u, PETSC_VIEWER_STDOUT_SELF);
364: printf("Exact solution vector\n");
365: VecView(appctx->solution, PETSC_VIEWER_STDOUT_SELF);
366: }
368: /*
369: Compute the 2-norm and max-norm of the error
370: */
371: VecAXPY(appctx->solution, -1.0, u);
372: VecNorm(appctx->solution, NORM_2, &norm_2);
373: norm_2 = PetscSqrtReal(appctx->h) * norm_2;
374: VecNorm(appctx->solution, NORM_MAX, &norm_max);
375: if (norm_2 < 1e-14) norm_2 = 0;
376: if (norm_max < 1e-14) norm_max = 0;
378: PetscPrintf(PETSC_COMM_WORLD, "Timestep %" PetscInt_FMT ": time = %g, 2-norm error = %g, max norm error = %g\n", step, (double)time, (double)norm_2, (double)norm_max);
379: appctx->norm_2 += norm_2;
380: appctx->norm_max += norm_max;
382: /*
383: View a graph of the error
384: */
385: VecView(appctx->solution, appctx->viewer1);
387: /*
388: Print debugging information if desired
389: */
390: if (appctx->debug) {
391: printf("Error vector\n");
392: VecView(appctx->solution, PETSC_VIEWER_STDOUT_SELF);
393: }
395: return 0;
396: }
397: /* --------------------------------------------------------------------- */
398: /*
399: RHSMatrixHeat - User-provided routine to compute the right-hand-side
400: matrix for the heat equation.
402: Input Parameters:
403: ts - the TS context
404: t - current time
405: global_in - global input vector
406: dummy - optional user-defined context, as set by TSetRHSJacobian()
408: Output Parameters:
409: AA - Jacobian matrix
410: BB - optionally different preconditioning matrix
411: str - flag indicating matrix structure
413: Notes:
414: Recall that MatSetValues() uses 0-based row and column numbers
415: in Fortran as well as in C.
416: */
417: PetscErrorCode RHSMatrixHeat(TS ts, PetscReal t, Vec X, Mat AA, Mat BB, void *ctx)
418: {
419: Mat A = AA; /* Jacobian matrix */
420: AppCtx *appctx = (AppCtx *)ctx; /* user-defined application context */
421: PetscInt mstart = 0;
422: PetscInt mend = appctx->m;
423: PetscInt i, idx[3];
424: PetscScalar v[3], stwo = -2. / (appctx->h * appctx->h), sone = -.5 * stwo;
426: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
427: Compute entries for the locally owned part of the matrix
428: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
429: /*
430: Set matrix rows corresponding to boundary data
431: */
433: mstart = 0;
434: v[0] = 1.0;
435: MatSetValues(A, 1, &mstart, 1, &mstart, v, INSERT_VALUES);
436: mstart++;
438: mend--;
439: v[0] = 1.0;
440: MatSetValues(A, 1, &mend, 1, &mend, v, INSERT_VALUES);
442: /*
443: Set matrix rows corresponding to interior data. We construct the
444: matrix one row at a time.
445: */
446: v[0] = sone;
447: v[1] = stwo;
448: v[2] = sone;
449: for (i = mstart; i < mend; i++) {
450: idx[0] = i - 1;
451: idx[1] = i;
452: 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 and option to indicate that we will never add a new nonzero location
470: to the matrix. If we do, it will generate an error.
471: */
472: MatSetOption(A, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE);
474: return 0;
475: }
477: /*TEST
479: test:
480: requires: x
482: test:
483: suffix: nox
484: args: -nox
486: TEST*/