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: /*
10: Concepts: TS^time-dependent linear problems
11: Concepts: TS^heat equation
12: Concepts: TS^diffusion equation
13: Processors: 1
14: */
16: /* ------------------------------------------------------------------------
18: This program solves the one-dimensional heat equation (also called the
19: diffusion equation),
20: u_t = u_xx,
21: on the domain 0 <= x <= 1, with the boundary conditions
22: u(t,0) = 1, u(t,1) = 1,
23: and the initial condition
24: u(0,x) = cos(6*pi*x) + 3*cos(2*pi*x).
25: This is a linear, second-order, parabolic equation.
27: We discretize the right-hand side using finite differences with
28: uniform grid spacing h:
29: u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
30: We then demonstrate time evolution using the various TS methods by
31: running the program via
32: ex3 -ts_type <timestepping solver>
34: We compare the approximate solution with the exact solution, given by
35: u_exact(x,t) = exp(-36*pi*pi*t) * cos(6*pi*x) +
36: 3*exp(-4*pi*pi*t) * cos(2*pi*x)
38: Notes:
39: This code demonstrates the TS solver interface to two variants of
40: linear problems, u_t = f(u,t), namely
41: - time-dependent f: f(u,t) is a function of t
42: - time-independent f: f(u,t) is simply just f(u)
44: The parallel version of this code is ts/tutorials/ex4.c
46: ------------------------------------------------------------------------- */
48: /*
49: Include "petscts.h" so that we can use TS solvers. Note that this file
50: automatically includes:
51: petscsys.h - base PETSc routines petscvec.h - vectors
52: petscmat.h - matrices
53: petscis.h - index sets petscksp.h - Krylov subspace methods
54: petscviewer.h - viewers petscpc.h - preconditioners
55: petscksp.h - linear solvers petscsnes.h - nonlinear solvers
56: */
57: #include <petscts.h>
58: #include <petscdraw.h>
60: /*
61: User-defined application context - contains data needed by the
62: application-provided call-back routines.
63: */
64: typedef struct {
65: Vec solution; /* global exact solution vector */
66: PetscInt m; /* total number of grid points */
67: PetscReal h; /* mesh width h = 1/(m-1) */
68: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
69: PetscViewer viewer1,viewer2; /* viewers for the solution and error */
70: PetscReal norm_2,norm_max; /* error norms */
71: } AppCtx;
73: /*
74: User-defined routines
75: */
76: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
77: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat,Mat,void*);
78: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
79: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);
81: int main(int argc,char **argv)
82: {
83: AppCtx appctx; /* user-defined application context */
84: TS ts; /* timestepping context */
85: Mat A; /* matrix data structure */
86: Vec u; /* approximate solution vector */
87: PetscReal time_total_max = 100.0; /* default max total time */
88: PetscInt time_steps_max = 100; /* default max timesteps */
89: PetscDraw draw; /* drawing context */
90: PetscInt steps,m;
91: PetscMPIInt size;
92: PetscBool flg;
93: PetscReal dt,ftime;
95: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
96: Initialize program and set problem parameters
97: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
99: PetscInitialize(&argc,&argv,(char*)0,help);
100: MPI_Comm_size(PETSC_COMM_WORLD,&size);
103: m = 60;
104: PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
105: PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);
106: appctx.m = m;
107: appctx.h = 1.0/(m-1.0);
108: appctx.norm_2 = 0.0;
109: appctx.norm_max = 0.0;
111: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");
113: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
114: Create vector data structures
115: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
117: /*
118: Create vector data structures for approximate and exact solutions
119: */
120: VecCreateSeq(PETSC_COMM_SELF,m,&u);
121: VecDuplicate(u,&appctx.solution);
123: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
124: Set up displays to show graphs of the solution and error
125: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
127: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
128: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
129: PetscDrawSetDoubleBuffer(draw);
130: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
131: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
132: PetscDrawSetDoubleBuffer(draw);
134: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
135: Create timestepping solver context
136: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
138: TSCreate(PETSC_COMM_SELF,&ts);
139: TSSetProblemType(ts,TS_LINEAR);
141: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
142: Set optional user-defined monitoring routine
143: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
145: TSMonitorSet(ts,Monitor,&appctx,NULL);
147: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
149: Create matrix data structure; set matrix evaluation routine.
150: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
152: MatCreate(PETSC_COMM_SELF,&A);
153: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
154: MatSetFromOptions(A);
155: MatSetUp(A);
157: PetscOptionsHasName(NULL,NULL,"-time_dependent_rhs",&flg);
158: if (flg) {
159: /*
160: For linear problems with a time-dependent f(u,t) in the equation
161: u_t = f(u,t), the user provides the discretized right-hand-side
162: as a time-dependent matrix.
163: */
164: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
165: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
166: } else {
167: /*
168: For linear problems with a time-independent f(u) in the equation
169: u_t = f(u), the user provides the discretized right-hand-side
170: as a matrix only once, and then sets a null matrix evaluation
171: routine.
172: */
173: RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
174: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
175: TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
176: }
178: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
179: Set solution vector and initial timestep
180: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
182: dt = appctx.h*appctx.h/2.0;
183: TSSetTimeStep(ts,dt);
184: TSSetSolution(ts,u);
186: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
187: Customize timestepping solver:
188: - Set the solution method to be the Backward Euler method.
189: - Set timestepping duration info
190: Then set runtime options, which can override these defaults.
191: For example,
192: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
193: to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
194: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
196: TSSetMaxSteps(ts,time_steps_max);
197: TSSetMaxTime(ts,time_total_max);
198: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
199: TSSetFromOptions(ts);
201: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
202: Solve the problem
203: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
205: /*
206: Evaluate initial conditions
207: */
208: InitialConditions(u,&appctx);
210: /*
211: Run the timestepping solver
212: */
213: TSSolve(ts,u);
214: TSGetSolveTime(ts,&ftime);
215: TSGetStepNumber(ts,&steps);
217: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
218: View timestepping solver info
219: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
221: 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));
222: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
224: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
225: Free work space. All PETSc objects should be destroyed when they
226: are no longer needed.
227: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
229: TSDestroy(&ts);
230: MatDestroy(&A);
231: VecDestroy(&u);
232: PetscViewerDestroy(&appctx.viewer1);
233: PetscViewerDestroy(&appctx.viewer2);
234: VecDestroy(&appctx.solution);
236: /*
237: Always call PetscFinalize() before exiting a program. This routine
238: - finalizes the PETSc libraries as well as MPI
239: - provides summary and diagnostic information if certain runtime
240: options are chosen (e.g., -log_view).
241: */
242: PetscFinalize();
243: return 0;
244: }
245: /* --------------------------------------------------------------------- */
246: /*
247: InitialConditions - Computes the solution at the initial time.
249: Input Parameter:
250: u - uninitialized solution vector (global)
251: appctx - user-defined application context
253: Output Parameter:
254: u - vector with solution at initial time (global)
255: */
256: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
257: {
258: PetscScalar *u_localptr,h = appctx->h;
259: PetscInt i;
261: /*
262: Get a pointer to vector data.
263: - For default PETSc vectors, VecGetArray() returns a pointer to
264: the data array. Otherwise, the routine is implementation dependent.
265: - You MUST call VecRestoreArray() when you no longer need access to
266: the array.
267: - Note that the Fortran interface to VecGetArray() differs from the
268: C version. See the users manual for details.
269: */
270: VecGetArray(u,&u_localptr);
272: /*
273: We initialize the solution array by simply writing the solution
274: directly into the array locations. Alternatively, we could use
275: VecSetValues() or VecSetValuesLocal().
276: */
277: for (i=0; i<appctx->m; i++) u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);
279: /*
280: Restore vector
281: */
282: VecRestoreArray(u,&u_localptr);
284: /*
285: Print debugging information if desired
286: */
287: if (appctx->debug) {
288: printf("initial guess vector\n");
289: VecView(u,PETSC_VIEWER_STDOUT_SELF);
290: }
292: return 0;
293: }
294: /* --------------------------------------------------------------------- */
295: /*
296: ExactSolution - Computes the exact solution at a given time.
298: Input Parameters:
299: t - current time
300: solution - vector in which exact solution will be computed
301: appctx - user-defined application context
303: Output Parameter:
304: solution - vector with the newly computed exact solution
305: */
306: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
307: {
308: PetscScalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
309: PetscInt i;
311: /*
312: Get a pointer to vector data.
313: */
314: VecGetArray(solution,&s_localptr);
316: /*
317: Simply write the solution directly into the array locations.
318: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
319: */
320: ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc); ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
321: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
322: for (i=0; i<appctx->m; i++) s_localptr[i] = PetscCosScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscCosScalar(sc2*(PetscReal)i)*ex2;
324: /*
325: Restore vector
326: */
327: VecRestoreArray(solution,&s_localptr);
328: return 0;
329: }
330: /* --------------------------------------------------------------------- */
331: /*
332: Monitor - User-provided routine to monitor the solution computed at
333: each timestep. This example plots the solution and computes the
334: error in two different norms.
336: Input Parameters:
337: ts - the timestep context
338: step - the count of the current step (with 0 meaning the
339: initial condition)
340: time - 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 time,Vec u,void *ctx)
348: {
349: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
350: PetscReal norm_2,norm_max;
352: /*
353: View a graph of the current iterate
354: */
355: VecView(u,appctx->viewer2);
357: /*
358: Compute the exact solution
359: */
360: ExactSolution(time,appctx->solution,appctx);
362: /*
363: Print debugging information if desired
364: */
365: if (appctx->debug) {
366: printf("Computed solution vector\n");
367: VecView(u,PETSC_VIEWER_STDOUT_SELF);
368: printf("Exact solution vector\n");
369: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
370: }
372: /*
373: Compute the 2-norm and max-norm of the error
374: */
375: VecAXPY(appctx->solution,-1.0,u);
376: VecNorm(appctx->solution,NORM_2,&norm_2);
377: norm_2 = PetscSqrtReal(appctx->h)*norm_2;
378: VecNorm(appctx->solution,NORM_MAX,&norm_max);
379: if (norm_2 < 1e-14) norm_2 = 0;
380: if (norm_max < 1e-14) norm_max = 0;
382: 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);
383: appctx->norm_2 += norm_2;
384: appctx->norm_max += norm_max;
386: /*
387: View a graph of the error
388: */
389: VecView(appctx->solution,appctx->viewer1);
391: /*
392: Print debugging information if desired
393: */
394: if (appctx->debug) {
395: printf("Error vector\n");
396: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
397: }
399: return 0;
400: }
401: /* --------------------------------------------------------------------- */
402: /*
403: RHSMatrixHeat - User-provided routine to compute the right-hand-side
404: matrix for the heat equation.
406: Input Parameters:
407: ts - the TS context
408: t - current time
409: global_in - global input vector
410: dummy - optional user-defined context, as set by TSetRHSJacobian()
412: Output Parameters:
413: AA - Jacobian matrix
414: BB - optionally different preconditioning matrix
415: str - flag indicating matrix structure
417: Notes:
418: Recall that MatSetValues() uses 0-based row and column numbers
419: in Fortran as well as in C.
420: */
421: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
422: {
423: Mat A = AA; /* Jacobian matrix */
424: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
425: PetscInt mstart = 0;
426: PetscInt mend = appctx->m;
427: PetscInt i,idx[3];
428: PetscScalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;
430: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
431: Compute entries for the locally owned part of the matrix
432: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
433: /*
434: Set matrix rows corresponding to boundary data
435: */
437: mstart = 0;
438: v[0] = 1.0;
439: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
440: mstart++;
442: mend--;
443: v[0] = 1.0;
444: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
446: /*
447: Set matrix rows corresponding to interior data. We construct the
448: matrix one row at a time.
449: */
450: v[0] = sone; v[1] = stwo; v[2] = sone;
451: for (i=mstart; i<mend; i++) {
452: idx[0] = i-1; idx[1] = i; 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*/