Actual source code: ex6.c
petsc-3.14.6 2021-03-30
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: Routines: TSCreate(); TSSetSolution(); TSSetRHSJacobian(), TSSetIJacobian();
14: Routines: TSSetTimeStep(); TSSetMaxTime(); TSMonitorSet();
15: Routines: TSSetFromOptions(); TSStep(); TSDestroy();
16: Routines: TSSetTimeStep(); TSGetTimeStep();
17: Processors: 1
18: */
20: /* ------------------------------------------------------------------------
22: This program solves the one-dimensional heat equation (also called the
23: diffusion equation),
24: u_t = u_xx,
25: on the domain 0 <= x <= 1, with the boundary conditions
26: u(t,0) = 0, u(t,1) = 0,
27: and the initial condition
28: u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
29: This is a linear, second-order, parabolic equation.
31: We discretize the right-hand side using finite differences with
32: uniform grid spacing h:
33: u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
34: We then demonstrate time evolution using the various TS methods by
35: running the program via
36: ex3 -ts_type <timestepping solver>
38: We compare the approximate solution with the exact solution, given by
39: u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
40: 3*exp(-4*pi*pi*t) * sin(2*pi*x)
42: Notes:
43: This code demonstrates the TS solver interface to two variants of
44: linear problems, u_t = f(u,t), namely
45: - time-dependent f: f(u,t) is a function of t
46: - time-independent f: f(u,t) is simply f(u)
48: The parallel version of this code is ts/tutorials/ex4.c
50: ------------------------------------------------------------------------- */
52: /*
53: Include "ts.h" so that we can use TS solvers. Note that this file
54: automatically includes:
55: petscsys.h - base PETSc routines vec.h - vectors
56: sys.h - system routines mat.h - matrices
57: is.h - index sets ksp.h - Krylov subspace methods
58: viewer.h - viewers pc.h - preconditioners
59: snes.h - nonlinear solvers
60: */
62: #include <petscts.h>
63: #include <petscdraw.h>
65: /*
66: User-defined application context - contains data needed by the
67: application-provided call-back routines.
68: */
69: typedef struct {
70: Vec solution; /* global exact solution vector */
71: PetscInt m; /* total number of grid points */
72: PetscReal h; /* mesh width h = 1/(m-1) */
73: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
74: PetscViewer viewer1, viewer2; /* viewers for the solution and error */
75: PetscReal norm_2, norm_max; /* error norms */
76: } AppCtx;
78: /*
79: User-defined routines
80: */
81: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
82: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat,Mat,void*);
83: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
84: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);
85: extern PetscErrorCode MyBCRoutine(TS,PetscReal,Vec,void*);
87: int main(int argc,char **argv)
88: {
89: AppCtx appctx; /* user-defined application context */
90: TS ts; /* timestepping context */
91: Mat A; /* matrix data structure */
92: Vec u; /* approximate solution vector */
93: PetscReal time_total_max = 100.0; /* default max total time */
94: PetscInt time_steps_max = 100; /* default max timesteps */
95: PetscDraw draw; /* drawing context */
97: PetscInt steps, m;
98: PetscMPIInt size;
99: PetscReal dt;
100: PetscReal ftime;
101: PetscBool flg;
102: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
103: Initialize program and set problem parameters
104: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
106: PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
107: MPI_Comm_size(PETSC_COMM_WORLD,&size);
108: if (size != 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_WRONG_MPI_SIZE,"This is a uniprocessor example only!");
110: m = 60;
111: PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
112: PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);
114: appctx.m = m;
115: appctx.h = 1.0/(m-1.0);
116: appctx.norm_2 = 0.0;
117: appctx.norm_max = 0.0;
119: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");
121: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
122: Create vector data structures
123: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
125: /*
126: Create vector data structures for approximate and exact solutions
127: */
128: VecCreateSeq(PETSC_COMM_SELF,m,&u);
129: VecDuplicate(u,&appctx.solution);
131: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
132: Set up displays to show graphs of the solution and error
133: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
135: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
136: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
137: PetscDrawSetDoubleBuffer(draw);
138: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
139: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
140: PetscDrawSetDoubleBuffer(draw);
142: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
143: Create timestepping solver context
144: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
146: TSCreate(PETSC_COMM_SELF,&ts);
147: TSSetProblemType(ts,TS_LINEAR);
149: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
150: Set optional user-defined monitoring routine
151: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
153: TSMonitorSet(ts,Monitor,&appctx,NULL);
155: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
157: Create matrix data structure; set matrix evaluation routine.
158: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
160: MatCreate(PETSC_COMM_SELF,&A);
161: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
162: MatSetFromOptions(A);
163: MatSetUp(A);
165: PetscOptionsHasName(NULL,NULL,"-time_dependent_rhs",&flg);
166: if (flg) {
167: /*
168: For linear problems with a time-dependent f(u,t) in the equation
169: u_t = f(u,t), the user provides the discretized right-hand-side
170: as a time-dependent matrix.
171: */
172: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
173: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
174: } else {
175: /*
176: For linear problems with a time-independent f(u) in the equation
177: u_t = f(u), the user provides the discretized right-hand-side
178: as a matrix only once, and then sets a null matrix evaluation
179: routine.
180: */
181: RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
182: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
183: TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
184: }
186: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
187: Set solution vector and initial timestep
188: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
190: dt = appctx.h*appctx.h/2.0;
191: TSSetTimeStep(ts,dt);
192: TSSetSolution(ts,u);
194: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
195: Customize timestepping solver:
196: - Set the solution method to be the Backward Euler method.
197: - Set timestepping duration info
198: Then set runtime options, which can override these defaults.
199: For example,
200: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
201: to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
202: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
204: TSSetMaxSteps(ts,time_steps_max);
205: TSSetMaxTime(ts,time_total_max);
206: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
207: TSSetFromOptions(ts);
209: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
210: Solve the problem
211: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
213: /*
214: Evaluate initial conditions
215: */
216: InitialConditions(u,&appctx);
218: /*
219: Run the timestepping solver
220: */
221: TSSolve(ts,u);
222: TSGetSolveTime(ts,&ftime);
223: TSGetStepNumber(ts,&steps);
225: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
226: View timestepping solver info
227: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
229: 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));
230: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
232: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
233: Free work space. All PETSc objects should be destroyed when they
234: are no longer needed.
235: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
237: TSDestroy(&ts);
238: MatDestroy(&A);
239: VecDestroy(&u);
240: PetscViewerDestroy(&appctx.viewer1);
241: PetscViewerDestroy(&appctx.viewer2);
242: VecDestroy(&appctx.solution);
244: /*
245: Always call PetscFinalize() before exiting a program. This routine
246: - finalizes the PETSc libraries as well as MPI
247: - provides summary and diagnostic information if certain runtime
248: options are chosen (e.g., -log_view).
249: */
250: PetscFinalize();
251: return ierr;
252: }
253: /* --------------------------------------------------------------------- */
254: /*
255: InitialConditions - Computes the solution at the initial time.
257: Input Parameter:
258: u - uninitialized solution vector (global)
259: appctx - user-defined application context
261: Output Parameter:
262: u - vector with solution at initial time (global)
263: */
264: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
265: {
266: PetscScalar *u_localptr;
267: PetscInt i;
270: /*
271: Get a pointer to vector data.
272: - For default PETSc vectors, VecGetArray() returns a pointer to
273: the data array. Otherwise, the routine is implementation dependent.
274: - You MUST call VecRestoreArray() when you no longer need access to
275: the array.
276: - Note that the Fortran interface to VecGetArray() differs from the
277: C version. See the users manual for details.
278: */
279: VecGetArray(u,&u_localptr);
281: /*
282: We initialize the solution array by simply writing the solution
283: directly into the array locations. Alternatively, we could use
284: VecSetValues() or VecSetValuesLocal().
285: */
286: 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);
288: /*
289: Restore vector
290: */
291: VecRestoreArray(u,&u_localptr);
293: /*
294: Print debugging information if desired
295: */
296: if (appctx->debug) {
297: VecView(u,PETSC_VIEWER_STDOUT_SELF);
298: }
300: return 0;
301: }
302: /* --------------------------------------------------------------------- */
303: /*
304: ExactSolution - Computes the exact solution at a given time.
306: Input Parameters:
307: t - current time
308: solution - vector in which exact solution will be computed
309: appctx - user-defined application context
311: Output Parameter:
312: solution - vector with the newly computed exact solution
313: */
314: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
315: {
316: PetscScalar *s_localptr, h = appctx->h, ex1, ex2, sc1, sc2;
317: PetscInt i;
320: /*
321: Get a pointer to vector data.
322: */
323: VecGetArray(solution,&s_localptr);
325: /*
326: Simply write the solution directly into the array locations.
327: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
328: */
329: ex1 = PetscExpReal(-36.*PETSC_PI*PETSC_PI*t); ex2 = PetscExpReal(-4.*PETSC_PI*PETSC_PI*t);
330: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
331: for (i=0; i<appctx->m; i++) s_localptr[i] = PetscSinReal(PetscRealPart(sc1)*(PetscReal)i)*ex1 + 3.*PetscSinReal(PetscRealPart(sc2)*(PetscReal)i)*ex2;
333: /*
334: Restore vector
335: */
336: VecRestoreArray(solution,&s_localptr);
337: return 0;
338: }
339: /* --------------------------------------------------------------------- */
340: /*
341: Monitor - User-provided routine to monitor the solution computed at
342: each timestep. This example plots the solution and computes the
343: error in two different norms.
345: This example also demonstrates changing the timestep via TSSetTimeStep().
347: Input Parameters:
348: ts - the timestep context
349: step - the count of the current step (with 0 meaning the
350: initial condition)
351: crtime - the current time
352: u - the solution at this timestep
353: ctx - the user-provided context for this monitoring routine.
354: In this case we use the application context which contains
355: information about the problem size, workspace and the exact
356: solution.
357: */
358: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal crtime,Vec u,void *ctx)
359: {
360: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
362: PetscReal norm_2, norm_max, dt, dttol;
363: PetscBool flg;
365: /*
366: View a graph of the current iterate
367: */
368: VecView(u,appctx->viewer2);
370: /*
371: Compute the exact solution
372: */
373: ExactSolution(crtime,appctx->solution,appctx);
375: /*
376: Print debugging information if desired
377: */
378: if (appctx->debug) {
379: PetscPrintf(PETSC_COMM_SELF,"Computed solution vector\n");
380: VecView(u,PETSC_VIEWER_STDOUT_SELF);
381: PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");
382: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
383: }
385: /*
386: Compute the 2-norm and max-norm of the error
387: */
388: VecAXPY(appctx->solution,-1.0,u);
389: VecNorm(appctx->solution,NORM_2,&norm_2);
390: norm_2 = PetscSqrtReal(appctx->h)*norm_2;
391: VecNorm(appctx->solution,NORM_MAX,&norm_max);
393: TSGetTimeStep(ts,&dt);
394: if (norm_2 > 1.e-2) {
395: 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);
396: }
397: appctx->norm_2 += norm_2;
398: appctx->norm_max += norm_max;
400: dttol = .0001;
401: PetscOptionsGetReal(NULL,NULL,"-dttol",&dttol,&flg);
402: if (dt < dttol) {
403: dt *= .999;
404: TSSetTimeStep(ts,dt);
405: }
407: /*
408: View a graph of the error
409: */
410: VecView(appctx->solution,appctx->viewer1);
412: /*
413: Print debugging information if desired
414: */
415: if (appctx->debug) {
416: PetscPrintf(PETSC_COMM_SELF,"Error vector\n");
417: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
418: }
420: return 0;
421: }
422: /* --------------------------------------------------------------------- */
423: /*
424: RHSMatrixHeat - User-provided routine to compute the right-hand-side
425: matrix for the heat equation.
427: Input Parameters:
428: ts - the TS context
429: t - current time
430: global_in - global input vector
431: dummy - optional user-defined context, as set by TSetRHSJacobian()
433: Output Parameters:
434: AA - Jacobian matrix
435: BB - optionally different preconditioning matrix
436: str - flag indicating matrix structure
438: Notes:
439: Recall that MatSetValues() uses 0-based row and column numbers
440: in Fortran as well as in C.
441: */
442: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
443: {
444: Mat A = AA; /* Jacobian matrix */
445: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
446: PetscInt mstart = 0;
447: PetscInt mend = appctx->m;
449: PetscInt i, idx[3];
450: PetscScalar v[3], stwo = -2./(appctx->h*appctx->h), sone = -.5*stwo;
452: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
453: Compute entries for the locally owned part of the matrix
454: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
455: /*
456: Set matrix rows corresponding to boundary data
457: */
459: mstart = 0;
460: v[0] = 1.0;
461: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
462: mstart++;
464: mend--;
465: v[0] = 1.0;
466: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
468: /*
469: Set matrix rows corresponding to interior data. We construct the
470: matrix one row at a time.
471: */
472: v[0] = sone; v[1] = stwo; v[2] = sone;
473: for (i=mstart; i<mend; i++) {
474: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
475: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
476: }
478: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
479: Complete the matrix assembly process and set some options
480: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
481: /*
482: Assemble matrix, using the 2-step process:
483: MatAssemblyBegin(), MatAssemblyEnd()
484: Computations can be done while messages are in transition
485: by placing code between these two statements.
486: */
487: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
488: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
490: /*
491: Set and option to indicate that we will never add a new nonzero location
492: to the matrix. If we do, it will generate an error.
493: */
494: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
496: return 0;
497: }
498: /* --------------------------------------------------------------------- */
499: /*
500: Input Parameters:
501: ts - the TS context
502: t - current time
503: f - function
504: ctx - optional user-defined context, as set by TSetBCFunction()
505: */
506: PetscErrorCode MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
507: {
508: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
510: PetscInt m = appctx->m;
511: PetscScalar *fa;
513: VecGetArray(f,&fa);
514: fa[0] = 0.0;
515: fa[m-1] = 1.0;
516: VecRestoreArray(f,&fa);
517: PetscPrintf(PETSC_COMM_SELF,"t=%g\n",(double)t);
519: return 0;
520: }
522: /*TEST
524: test:
525: args: -nox -ts_max_steps 4
527: TEST*/