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: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
92: PetscInitialize(&argc,&argv,(char*)0,help);
93: MPI_Comm_size(PETSC_COMM_WORLD,&size);
96: m = 60;
97: PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
98: PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);
99: appctx.m = m;
100: appctx.h = 1.0/(m-1.0);
101: appctx.norm_2 = 0.0;
102: appctx.norm_max = 0.0;
104: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");
106: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
107: Create vector data structures
108: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
110: /*
111: Create vector data structures for approximate and exact solutions
112: */
113: VecCreateSeq(PETSC_COMM_SELF,m,&u);
114: VecDuplicate(u,&appctx.solution);
116: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
117: Set up displays to show graphs of the solution and error
118: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
120: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
121: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
122: PetscDrawSetDoubleBuffer(draw);
123: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
124: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
125: PetscDrawSetDoubleBuffer(draw);
127: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
128: Create timestepping solver context
129: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
131: TSCreate(PETSC_COMM_SELF,&ts);
132: TSSetProblemType(ts,TS_LINEAR);
134: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
135: Set optional user-defined monitoring routine
136: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
138: TSMonitorSet(ts,Monitor,&appctx,NULL);
140: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
142: Create matrix data structure; set matrix evaluation routine.
143: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
145: MatCreate(PETSC_COMM_SELF,&A);
146: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
147: MatSetFromOptions(A);
148: MatSetUp(A);
150: PetscOptionsHasName(NULL,NULL,"-time_dependent_rhs",&flg);
151: if (flg) {
152: /*
153: For linear problems with a time-dependent f(u,t) in the equation
154: u_t = f(u,t), the user provides the discretized right-hand-side
155: as a time-dependent matrix.
156: */
157: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
158: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
159: } else {
160: /*
161: For linear problems with a time-independent f(u) in the equation
162: u_t = f(u), the user provides the discretized right-hand-side
163: as a matrix only once, and then sets a null matrix evaluation
164: routine.
165: */
166: RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
167: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
168: TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
169: }
171: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
172: Set solution vector and initial timestep
173: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
175: dt = appctx.h*appctx.h/2.0;
176: TSSetTimeStep(ts,dt);
177: TSSetSolution(ts,u);
179: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
180: Customize timestepping solver:
181: - Set the solution method to be the Backward Euler method.
182: - Set timestepping duration info
183: Then set runtime options, which can override these defaults.
184: For example,
185: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
186: to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
187: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
189: TSSetMaxSteps(ts,time_steps_max);
190: TSSetMaxTime(ts,time_total_max);
191: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
192: TSSetFromOptions(ts);
194: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
195: Solve the problem
196: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
198: /*
199: Evaluate initial conditions
200: */
201: InitialConditions(u,&appctx);
203: /*
204: Run the timestepping solver
205: */
206: TSSolve(ts,u);
207: TSGetSolveTime(ts,&ftime);
208: TSGetStepNumber(ts,&steps);
210: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
211: View timestepping solver info
212: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
214: 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));
215: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
217: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
218: Free work space. All PETSc objects should be destroyed when they
219: are no longer needed.
220: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
222: TSDestroy(&ts);
223: MatDestroy(&A);
224: VecDestroy(&u);
225: PetscViewerDestroy(&appctx.viewer1);
226: PetscViewerDestroy(&appctx.viewer2);
227: VecDestroy(&appctx.solution);
229: /*
230: Always call PetscFinalize() before exiting a program. This routine
231: - finalizes the PETSc libraries as well as MPI
232: - provides summary and diagnostic information if certain runtime
233: options are chosen (e.g., -log_view).
234: */
235: PetscFinalize();
236: return 0;
237: }
238: /* --------------------------------------------------------------------- */
239: /*
240: InitialConditions - Computes the solution at the initial time.
242: Input Parameter:
243: u - uninitialized solution vector (global)
244: appctx - user-defined application context
246: Output Parameter:
247: u - vector with solution at initial time (global)
248: */
249: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
250: {
251: PetscScalar *u_localptr,h = appctx->h;
252: PetscInt i;
254: /*
255: Get a pointer to vector data.
256: - For default PETSc vectors, VecGetArray() returns a pointer to
257: the data array. Otherwise, the routine is implementation dependent.
258: - You MUST call VecRestoreArray() when you no longer need access to
259: the array.
260: - Note that the Fortran interface to VecGetArray() differs from the
261: C version. See the users manual for details.
262: */
263: VecGetArray(u,&u_localptr);
265: /*
266: We initialize the solution array by simply writing the solution
267: directly into the array locations. Alternatively, we could use
268: VecSetValues() or VecSetValuesLocal().
269: */
270: for (i=0; i<appctx->m; i++) u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);
272: /*
273: Restore vector
274: */
275: VecRestoreArray(u,&u_localptr);
277: /*
278: Print debugging information if desired
279: */
280: if (appctx->debug) {
281: printf("initial guess vector\n");
282: VecView(u,PETSC_VIEWER_STDOUT_SELF);
283: }
285: return 0;
286: }
287: /* --------------------------------------------------------------------- */
288: /*
289: ExactSolution - Computes the exact solution at a given time.
291: Input Parameters:
292: t - current time
293: solution - vector in which exact solution will be computed
294: appctx - user-defined application context
296: Output Parameter:
297: solution - vector with the newly computed exact solution
298: */
299: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
300: {
301: PetscScalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
302: PetscInt i;
304: /*
305: Get a pointer to vector data.
306: */
307: VecGetArray(solution,&s_localptr);
309: /*
310: Simply write the solution directly into the array locations.
311: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
312: */
313: ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc); ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
314: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
315: for (i=0; i<appctx->m; i++) s_localptr[i] = PetscCosScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscCosScalar(sc2*(PetscReal)i)*ex2;
317: /*
318: Restore vector
319: */
320: VecRestoreArray(solution,&s_localptr);
321: return 0;
322: }
323: /* --------------------------------------------------------------------- */
324: /*
325: Monitor - User-provided routine to monitor the solution computed at
326: each timestep. This example plots the solution and computes the
327: error in two different norms.
329: Input Parameters:
330: ts - the timestep context
331: step - the count of the current step (with 0 meaning the
332: initial condition)
333: time - the current time
334: u - the solution at this timestep
335: ctx - the user-provided context for this monitoring routine.
336: In this case we use the application context which contains
337: information about the problem size, workspace and the exact
338: solution.
339: */
340: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
341: {
342: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
343: PetscReal norm_2,norm_max;
345: /*
346: View a graph of the current iterate
347: */
348: VecView(u,appctx->viewer2);
350: /*
351: Compute the exact solution
352: */
353: ExactSolution(time,appctx->solution,appctx);
355: /*
356: Print debugging information if desired
357: */
358: if (appctx->debug) {
359: printf("Computed solution vector\n");
360: VecView(u,PETSC_VIEWER_STDOUT_SELF);
361: printf("Exact solution vector\n");
362: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
363: }
365: /*
366: Compute the 2-norm and max-norm of the error
367: */
368: VecAXPY(appctx->solution,-1.0,u);
369: VecNorm(appctx->solution,NORM_2,&norm_2);
370: norm_2 = PetscSqrtReal(appctx->h)*norm_2;
371: VecNorm(appctx->solution,NORM_MAX,&norm_max);
372: if (norm_2 < 1e-14) norm_2 = 0;
373: if (norm_max < 1e-14) norm_max = 0;
375: PetscPrintf(PETSC_COMM_WORLD,"Timestep %D: time = %g, 2-norm error = %g, max norm error = %g\n",step,(double)time,(double)norm_2,(double)norm_max);
376: appctx->norm_2 += norm_2;
377: appctx->norm_max += norm_max;
379: /*
380: View a graph of the error
381: */
382: VecView(appctx->solution,appctx->viewer1);
384: /*
385: Print debugging information if desired
386: */
387: if (appctx->debug) {
388: printf("Error vector\n");
389: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
390: }
392: return 0;
393: }
394: /* --------------------------------------------------------------------- */
395: /*
396: RHSMatrixHeat - User-provided routine to compute the right-hand-side
397: matrix for the heat equation.
399: Input Parameters:
400: ts - the TS context
401: t - current time
402: global_in - global input vector
403: dummy - optional user-defined context, as set by TSetRHSJacobian()
405: Output Parameters:
406: AA - Jacobian matrix
407: BB - optionally different preconditioning matrix
408: str - flag indicating matrix structure
410: Notes:
411: Recall that MatSetValues() uses 0-based row and column numbers
412: in Fortran as well as in C.
413: */
414: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
415: {
416: Mat A = AA; /* Jacobian matrix */
417: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
418: PetscInt mstart = 0;
419: PetscInt mend = appctx->m;
420: PetscInt i,idx[3];
421: PetscScalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;
423: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
424: Compute entries for the locally owned part of the matrix
425: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
426: /*
427: Set matrix rows corresponding to boundary data
428: */
430: mstart = 0;
431: v[0] = 1.0;
432: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
433: mstart++;
435: mend--;
436: v[0] = 1.0;
437: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
439: /*
440: Set matrix rows corresponding to interior data. We construct the
441: matrix one row at a time.
442: */
443: v[0] = sone; v[1] = stwo; v[2] = sone;
444: for (i=mstart; i<mend; i++) {
445: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
446: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
447: }
449: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
450: Complete the matrix assembly process and set some options
451: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
452: /*
453: Assemble matrix, using the 2-step process:
454: MatAssemblyBegin(), MatAssemblyEnd()
455: Computations can be done while messages are in transition
456: by placing code between these two statements.
457: */
458: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
459: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
461: /*
462: Set and option to indicate that we will never add a new nonzero location
463: to the matrix. If we do, it will generate an error.
464: */
465: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
467: return 0;
468: }
470: /*TEST
472: test:
473: requires: x
475: test:
476: suffix: nox
477: args: -nox
479: TEST*/