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: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
94: PetscInitialize(&argc,&argv,(char*)0,help);
95: MPI_Comm_size(PETSC_COMM_WORLD,&size);
98: m = 60;
99: PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
100: PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);
102: appctx.m = m;
103: appctx.h = 1.0/(m-1.0);
104: appctx.norm_2 = 0.0;
105: appctx.norm_max = 0.0;
107: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");
109: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
110: Create vector data structures
111: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
113: /*
114: Create vector data structures for approximate and exact solutions
115: */
116: VecCreateSeq(PETSC_COMM_SELF,m,&u);
117: VecDuplicate(u,&appctx.solution);
119: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
120: Set up displays to show graphs of the solution and error
121: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
123: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
124: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
125: PetscDrawSetDoubleBuffer(draw);
126: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
127: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
128: PetscDrawSetDoubleBuffer(draw);
130: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
131: Create timestepping solver context
132: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
134: TSCreate(PETSC_COMM_SELF,&ts);
135: TSSetProblemType(ts,TS_LINEAR);
137: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
138: Set optional user-defined monitoring routine
139: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
141: TSMonitorSet(ts,Monitor,&appctx,NULL);
143: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
145: Create matrix data structure; set matrix evaluation routine.
146: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
148: MatCreate(PETSC_COMM_SELF,&A);
149: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
150: MatSetFromOptions(A);
151: MatSetUp(A);
153: PetscOptionsHasName(NULL,NULL,"-time_dependent_rhs",&flg);
154: if (flg) {
155: /*
156: For linear problems with a time-dependent f(u,t) in the equation
157: u_t = f(u,t), the user provides the discretized right-hand-side
158: as a time-dependent matrix.
159: */
160: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
161: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
162: } else {
163: /*
164: For linear problems with a time-independent f(u) in the equation
165: u_t = f(u), the user provides the discretized right-hand-side
166: as a matrix only once, and then sets a null matrix evaluation
167: routine.
168: */
169: RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
170: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
171: TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
172: }
174: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
175: Set solution vector and initial timestep
176: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
178: dt = appctx.h*appctx.h/2.0;
179: TSSetTimeStep(ts,dt);
180: TSSetSolution(ts,u);
182: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
183: Customize timestepping solver:
184: - Set the solution method to be the Backward Euler method.
185: - Set timestepping duration info
186: Then set runtime options, which can override these defaults.
187: For example,
188: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
189: to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
190: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
192: TSSetMaxSteps(ts,time_steps_max);
193: TSSetMaxTime(ts,time_total_max);
194: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
195: TSSetFromOptions(ts);
197: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
198: Solve the problem
199: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
201: /*
202: Evaluate initial conditions
203: */
204: InitialConditions(u,&appctx);
206: /*
207: Run the timestepping solver
208: */
209: TSSolve(ts,u);
210: TSGetSolveTime(ts,&ftime);
211: TSGetStepNumber(ts,&steps);
213: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
214: View timestepping solver info
215: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
217: 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));
218: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
220: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
221: Free work space. All PETSc objects should be destroyed when they
222: are no longer needed.
223: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
225: TSDestroy(&ts);
226: MatDestroy(&A);
227: VecDestroy(&u);
228: PetscViewerDestroy(&appctx.viewer1);
229: PetscViewerDestroy(&appctx.viewer2);
230: VecDestroy(&appctx.solution);
232: /*
233: Always call PetscFinalize() before exiting a program. This routine
234: - finalizes the PETSc libraries as well as MPI
235: - provides summary and diagnostic information if certain runtime
236: options are chosen (e.g., -log_view).
237: */
238: PetscFinalize();
239: return 0;
240: }
241: /* --------------------------------------------------------------------- */
242: /*
243: InitialConditions - Computes the solution at the initial time.
245: Input Parameter:
246: u - uninitialized solution vector (global)
247: appctx - user-defined application context
249: Output Parameter:
250: u - vector with solution at initial time (global)
251: */
252: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
253: {
254: PetscScalar *u_localptr;
255: PetscInt i;
257: /*
258: Get a pointer to vector data.
259: - For default PETSc vectors, VecGetArray() returns a pointer to
260: the data array. Otherwise, the routine is implementation dependent.
261: - You MUST call VecRestoreArray() when you no longer need access to
262: the array.
263: - Note that the Fortran interface to VecGetArray() differs from the
264: C version. See the users manual for details.
265: */
266: VecGetArray(u,&u_localptr);
268: /*
269: We initialize the solution array by simply writing the solution
270: directly into the array locations. Alternatively, we could use
271: VecSetValues() or VecSetValuesLocal().
272: */
273: 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);
275: /*
276: Restore vector
277: */
278: VecRestoreArray(u,&u_localptr);
280: /*
281: Print debugging information if desired
282: */
283: if (appctx->debug) {
284: VecView(u,PETSC_VIEWER_STDOUT_SELF);
285: }
287: return 0;
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: /*
307: Get a pointer to vector data.
308: */
309: VecGetArray(solution,&s_localptr);
311: /*
312: Simply write the solution directly into the array locations.
313: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
314: */
315: ex1 = PetscExpReal(-36.*PETSC_PI*PETSC_PI*t); ex2 = PetscExpReal(-4.*PETSC_PI*PETSC_PI*t);
316: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
317: for (i=0; i<appctx->m; i++) s_localptr[i] = PetscSinReal(PetscRealPart(sc1)*(PetscReal)i)*ex1 + 3.*PetscSinReal(PetscRealPart(sc2)*(PetscReal)i)*ex2;
319: /*
320: Restore vector
321: */
322: VecRestoreArray(solution,&s_localptr);
323: return 0;
324: }
325: /* --------------------------------------------------------------------- */
326: /*
327: Monitor - User-provided routine to monitor the solution computed at
328: each timestep. This example plots the solution and computes the
329: error in two different norms.
331: This example also demonstrates changing the timestep via TSSetTimeStep().
333: Input Parameters:
334: ts - the timestep context
335: step - the count of the current step (with 0 meaning the
336: initial condition)
337: crtime - the current time
338: u - the solution at this timestep
339: ctx - the user-provided context for this monitoring routine.
340: In this case we use the application context which contains
341: information about the problem size, workspace and the exact
342: solution.
343: */
344: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal crtime,Vec u,void *ctx)
345: {
346: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
347: PetscReal norm_2, norm_max, dt, dttol;
348: PetscBool flg;
350: /*
351: View a graph of the current iterate
352: */
353: VecView(u,appctx->viewer2);
355: /*
356: Compute the exact solution
357: */
358: ExactSolution(crtime,appctx->solution,appctx);
360: /*
361: Print debugging information if desired
362: */
363: if (appctx->debug) {
364: PetscPrintf(PETSC_COMM_SELF,"Computed solution vector\n");
365: VecView(u,PETSC_VIEWER_STDOUT_SELF);
366: PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");
367: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
368: }
370: /*
371: Compute the 2-norm and max-norm of the error
372: */
373: VecAXPY(appctx->solution,-1.0,u);
374: VecNorm(appctx->solution,NORM_2,&norm_2);
375: norm_2 = PetscSqrtReal(appctx->h)*norm_2;
376: VecNorm(appctx->solution,NORM_MAX,&norm_max);
378: TSGetTimeStep(ts,&dt);
379: if (norm_2 > 1.e-2) {
380: PetscPrintf(PETSC_COMM_SELF,"Timestep %D: 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: }
382: appctx->norm_2 += norm_2;
383: appctx->norm_max += norm_max;
385: dttol = .0001;
386: PetscOptionsGetReal(NULL,NULL,"-dttol",&dttol,&flg);
387: if (dt < dttol) {
388: dt *= .999;
389: TSSetTimeStep(ts,dt);
390: }
392: /*
393: View a graph of the error
394: */
395: VecView(appctx->solution,appctx->viewer1);
397: /*
398: Print debugging information if desired
399: */
400: if (appctx->debug) {
401: PetscPrintf(PETSC_COMM_SELF,"Error vector\n");
402: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
403: }
405: return 0;
406: }
407: /* --------------------------------------------------------------------- */
408: /*
409: RHSMatrixHeat - User-provided routine to compute the right-hand-side
410: matrix for the heat equation.
412: Input Parameters:
413: ts - the TS context
414: t - current time
415: global_in - global input vector
416: dummy - optional user-defined context, as set by TSetRHSJacobian()
418: Output Parameters:
419: AA - Jacobian matrix
420: BB - optionally different preconditioning matrix
421: str - flag indicating matrix structure
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,void *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: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
437: Compute entries for the locally owned part of the matrix
438: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
439: /*
440: Set matrix rows corresponding to boundary data
441: */
443: mstart = 0;
444: v[0] = 1.0;
445: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
446: mstart++;
448: mend--;
449: v[0] = 1.0;
450: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
452: /*
453: Set matrix rows corresponding to interior data. We construct the
454: matrix one row at a time.
455: */
456: v[0] = sone; v[1] = stwo; v[2] = sone;
457: for (i=mstart; i<mend; i++) {
458: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
459: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
460: }
462: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
463: Complete the matrix assembly process and set some options
464: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
465: /*
466: Assemble matrix, using the 2-step process:
467: MatAssemblyBegin(), MatAssemblyEnd()
468: Computations can be done while messages are in transition
469: by placing code between these two statements.
470: */
471: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
472: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
474: /*
475: Set and option to indicate that we will never add a new nonzero location
476: to the matrix. If we do, it will generate an error.
477: */
478: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
480: return 0;
481: }
482: /* --------------------------------------------------------------------- */
483: /*
484: Input Parameters:
485: ts - the TS context
486: t - current time
487: f - function
488: ctx - optional user-defined context, as set by TSetBCFunction()
489: */
490: PetscErrorCode MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
491: {
492: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
493: PetscInt m = appctx->m;
494: PetscScalar *fa;
496: VecGetArray(f,&fa);
497: fa[0] = 0.0;
498: fa[m-1] = 1.0;
499: VecRestoreArray(f,&fa);
500: PetscPrintf(PETSC_COMM_SELF,"t=%g\n",(double)t);
502: return 0;
503: }
505: /*TEST
507: test:
508: args: -nox -ts_max_steps 4
510: TEST*/