Actual source code: ex3.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: -use_ifunc : Use IFunction/IJacobian interface\n\
7: -debug : Activate debugging printouts\n\
8: -nox : Deactivate x-window graphics\n\n";
10: /*
11: Concepts: TS^time-dependent linear problems
12: Concepts: TS^heat equation
13: Concepts: TS^diffusion equation
14: Processors: 1
15: */
17: /* ------------------------------------------------------------------------
19: This program solves the one-dimensional heat equation (also called the
20: diffusion equation),
21: u_t = u_xx,
22: on the domain 0 <= x <= 1, with the boundary conditions
23: u(t,0) = 0, u(t,1) = 0,
24: and the initial condition
25: u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
26: This is a linear, second-order, parabolic equation.
28: We discretize the right-hand side using finite differences with
29: uniform grid spacing h:
30: u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
31: We then demonstrate time evolution using the various TS methods by
32: running the program via
33: ex3 -ts_type <timestepping solver>
35: We compare the approximate solution with the exact solution, given by
36: u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
37: 3*exp(-4*pi*pi*t) * sin(2*pi*x)
39: Notes:
40: This code demonstrates the TS solver interface to two variants of
41: linear problems, u_t = f(u,t), namely
42: - time-dependent f: f(u,t) is a function of t
43: - time-independent f: f(u,t) is simply f(u)
45: The parallel version of this code is ts/tutorials/ex4.c
47: ------------------------------------------------------------------------- */
49: /*
50: Include "petscts.h" so that we can use TS solvers. Note that this file
51: automatically includes:
52: petscsys.h - base PETSc routines petscvec.h - vectors
53: petscmat.h - matrices
54: petscis.h - index sets petscksp.h - Krylov subspace methods
55: petscviewer.h - viewers petscpc.h - preconditioners
56: petscksp.h - linear solvers petscsnes.h - nonlinear solvers
57: */
59: #include <petscts.h>
60: #include <petscdraw.h>
62: /*
63: User-defined application context - contains data needed by the
64: application-provided call-back routines.
65: */
66: typedef struct {
67: Vec solution; /* global exact solution vector */
68: PetscInt m; /* total number of grid points */
69: PetscReal h; /* mesh width h = 1/(m-1) */
70: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
71: PetscViewer viewer1,viewer2; /* viewers for the solution and error */
72: PetscReal norm_2,norm_max; /* error norms */
73: Mat A; /* RHS mat, used with IFunction interface */
74: PetscReal oshift; /* old shift applied, prevent to recompute the IJacobian */
75: } AppCtx;
77: /*
78: User-defined routines
79: */
80: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
81: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat,Mat,void*);
82: extern PetscErrorCode IFunctionHeat(TS,PetscReal,Vec,Vec,Vec,void*);
83: extern PetscErrorCode IJacobianHeat(TS,PetscReal,Vec,Vec,PetscReal,Mat,Mat,void*);
84: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
85: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);
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: PetscBool flg,flg_string;
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);
113: flg_string = PETSC_FALSE;
114: PetscOptionsGetBool(NULL,NULL,"-test_string_viewer",&flg_string,NULL);
116: appctx.m = m;
117: appctx.h = 1.0/(m-1.0);
118: appctx.norm_2 = 0.0;
119: appctx.norm_max = 0.0;
121: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");
123: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
124: Create vector data structures
125: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
127: /*
128: Create vector data structures for approximate and exact solutions
129: */
130: VecCreateSeq(PETSC_COMM_SELF,m,&u);
131: VecDuplicate(u,&appctx.solution);
133: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
134: Set up displays to show graphs of the solution and error
135: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
137: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
138: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
139: PetscDrawSetDoubleBuffer(draw);
140: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
141: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
142: PetscDrawSetDoubleBuffer(draw);
144: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
145: Create timestepping solver context
146: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
148: TSCreate(PETSC_COMM_SELF,&ts);
149: TSSetProblemType(ts,TS_LINEAR);
151: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
152: Set optional user-defined monitoring routine
153: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
155: if (!flg_string) {
156: TSMonitorSet(ts,Monitor,&appctx,NULL);
157: }
159: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
161: Create matrix data structure; set matrix evaluation routine.
162: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
164: MatCreate(PETSC_COMM_SELF,&A);
165: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
166: MatSetFromOptions(A);
167: MatSetUp(A);
169: flg = PETSC_FALSE;
170: PetscOptionsGetBool(NULL,NULL,"-use_ifunc",&flg,NULL);
171: if (!flg) {
172: appctx.A = NULL;
173: PetscOptionsGetBool(NULL,NULL,"-time_dependent_rhs",&flg,NULL);
174: if (flg) {
175: /*
176: For linear problems with a time-dependent f(u,t) in the equation
177: u_t = f(u,t), the user provides the discretized right-hand-side
178: as a time-dependent matrix.
179: */
180: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
181: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
182: } else {
183: /*
184: For linear problems with a time-independent f(u) in the equation
185: u_t = f(u), the user provides the discretized right-hand-side
186: as a matrix only once, and then sets the special Jacobian evaluation
187: routine TSComputeRHSJacobianConstant() which will NOT recompute the Jacobian.
188: */
189: RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
190: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
191: TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
192: }
193: } else {
194: Mat J;
196: RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
197: MatDuplicate(A,MAT_DO_NOT_COPY_VALUES,&J);
198: TSSetIFunction(ts,NULL,IFunctionHeat,&appctx);
199: TSSetIJacobian(ts,J,J,IJacobianHeat,&appctx);
200: MatDestroy(&J);
202: PetscObjectReference((PetscObject)A);
203: appctx.A = A;
204: appctx.oshift = PETSC_MIN_REAL;
205: }
206: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
207: Set solution vector and initial timestep
208: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
210: dt = appctx.h*appctx.h/2.0;
211: TSSetTimeStep(ts,dt);
213: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
214: Customize timestepping solver:
215: - Set the solution method to be the Backward Euler method.
216: - Set timestepping duration info
217: Then set runtime options, which can override these defaults.
218: For example,
219: -ts_max_steps <maxsteps> -ts_max_time <maxtime>
220: to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
221: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
223: TSSetMaxSteps(ts,time_steps_max);
224: TSSetMaxTime(ts,time_total_max);
225: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
226: TSSetFromOptions(ts);
228: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
229: Solve the problem
230: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
232: /*
233: Evaluate initial conditions
234: */
235: InitialConditions(u,&appctx);
237: /*
238: Run the timestepping solver
239: */
240: TSSolve(ts,u);
241: TSGetStepNumber(ts,&steps);
243: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
244: View timestepping solver info
245: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
247: 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));
248: if (!flg_string) {
249: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
250: } else {
251: PetscViewer stringviewer;
252: char string[512];
253: const char *outstring;
255: PetscViewerStringOpen(PETSC_COMM_WORLD,string,sizeof(string),&stringviewer);
256: TSView(ts,stringviewer);
257: PetscViewerStringGetStringRead(stringviewer,&outstring,NULL);
258: if ((char*)outstring != (char*)string) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_PLIB,"String returned from viewer does not equal original string");
259: PetscPrintf(PETSC_COMM_WORLD,"Output from string viewer:%s\n",outstring);
260: PetscViewerDestroy(&stringviewer);
261: }
263: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
264: Free work space. All PETSc objects should be destroyed when they
265: are no longer needed.
266: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
268: TSDestroy(&ts);
269: MatDestroy(&A);
270: VecDestroy(&u);
271: PetscViewerDestroy(&appctx.viewer1);
272: PetscViewerDestroy(&appctx.viewer2);
273: VecDestroy(&appctx.solution);
274: MatDestroy(&appctx.A);
276: /*
277: Always call PetscFinalize() before exiting a program. This routine
278: - finalizes the PETSc libraries as well as MPI
279: - provides summary and diagnostic information if certain runtime
280: options are chosen (e.g., -log_view).
281: */
282: PetscFinalize();
283: return ierr;
284: }
285: /* --------------------------------------------------------------------- */
286: /*
287: InitialConditions - Computes the solution at the initial time.
289: Input Parameter:
290: u - uninitialized solution vector (global)
291: appctx - user-defined application context
293: Output Parameter:
294: u - vector with solution at initial time (global)
295: */
296: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
297: {
298: PetscScalar *u_localptr,h = appctx->h;
300: PetscInt i;
302: /*
303: Get a pointer to vector data.
304: - For default PETSc vectors, VecGetArray() returns a pointer to
305: the data array. Otherwise, the routine is implementation dependent.
306: - You MUST call VecRestoreArray() when you no longer need access to
307: the array.
308: - Note that the Fortran interface to VecGetArray() differs from the
309: C version. See the users manual for details.
310: */
311: VecGetArrayWrite(u,&u_localptr);
313: /*
314: We initialize the solution array by simply writing the solution
315: directly into the array locations. Alternatively, we could use
316: VecSetValues() or VecSetValuesLocal().
317: */
318: for (i=0; i<appctx->m; i++) u_localptr[i] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);
320: /*
321: Restore vector
322: */
323: VecRestoreArrayWrite(u,&u_localptr);
325: /*
326: Print debugging information if desired
327: */
328: if (appctx->debug) {
329: PetscPrintf(PETSC_COMM_WORLD,"Initial guess vector\n");
330: VecView(u,PETSC_VIEWER_STDOUT_SELF);
331: }
333: return 0;
334: }
335: /* --------------------------------------------------------------------- */
336: /*
337: ExactSolution - Computes the exact solution at a given time.
339: Input Parameters:
340: t - current time
341: solution - vector in which exact solution will be computed
342: appctx - user-defined application context
344: Output Parameter:
345: solution - vector with the newly computed exact solution
346: */
347: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
348: {
349: PetscScalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
351: PetscInt i;
353: /*
354: Get a pointer to vector data.
355: */
356: VecGetArrayWrite(solution,&s_localptr);
358: /*
359: Simply write the solution directly into the array locations.
360: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
361: */
362: ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc);
363: ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
364: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
365: for (i=0; i<appctx->m; i++) s_localptr[i] = PetscSinScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i)*ex2;
367: /*
368: Restore vector
369: */
370: VecRestoreArrayWrite(solution,&s_localptr);
371: return 0;
372: }
373: /* --------------------------------------------------------------------- */
374: /*
375: Monitor - User-provided routine to monitor the solution computed at
376: each timestep. This example plots the solution and computes the
377: error in two different norms.
379: This example also demonstrates changing the timestep via TSSetTimeStep().
381: Input Parameters:
382: ts - the timestep context
383: step - the count of the current step (with 0 meaning the
384: initial condition)
385: time - the current time
386: u - the solution at this timestep
387: ctx - the user-provided context for this monitoring routine.
388: In this case we use the application context which contains
389: information about the problem size, workspace and the exact
390: solution.
391: */
392: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
393: {
394: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
396: PetscReal norm_2,norm_max,dt,dttol;
398: /*
399: View a graph of the current iterate
400: */
401: VecView(u,appctx->viewer2);
403: /*
404: Compute the exact solution
405: */
406: ExactSolution(time,appctx->solution,appctx);
408: /*
409: Print debugging information if desired
410: */
411: if (appctx->debug) {
412: PetscPrintf(PETSC_COMM_SELF,"Computed solution vector\n");
413: VecView(u,PETSC_VIEWER_STDOUT_SELF);
414: PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");
415: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
416: }
418: /*
419: Compute the 2-norm and max-norm of the error
420: */
421: VecAXPY(appctx->solution,-1.0,u);
422: VecNorm(appctx->solution,NORM_2,&norm_2);
423: norm_2 = PetscSqrtReal(appctx->h)*norm_2;
424: VecNorm(appctx->solution,NORM_MAX,&norm_max);
426: TSGetTimeStep(ts,&dt);
427: PetscPrintf(PETSC_COMM_WORLD,"Timestep %3D: step size = %g, time = %g, 2-norm error = %g, max norm error = %g\n",step,(double)dt,(double)time,(double)norm_2,(double)norm_max);
429: appctx->norm_2 += norm_2;
430: appctx->norm_max += norm_max;
432: dttol = .0001;
433: PetscOptionsGetReal(NULL,NULL,"-dttol",&dttol,NULL);
434: if (dt < dttol) {
435: dt *= .999;
436: TSSetTimeStep(ts,dt);
437: }
439: /*
440: View a graph of the error
441: */
442: VecView(appctx->solution,appctx->viewer1);
444: /*
445: Print debugging information if desired
446: */
447: if (appctx->debug) {
448: PetscPrintf(PETSC_COMM_SELF,"Error vector\n");
449: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
450: }
452: return 0;
453: }
454: /* --------------------------------------------------------------------- */
455: /*
456: RHSMatrixHeat - User-provided routine to compute the right-hand-side
457: matrix for the heat equation.
459: Input Parameters:
460: ts - the TS context
461: t - current time
462: global_in - global input vector
463: dummy - optional user-defined context, as set by TSetRHSJacobian()
465: Output Parameters:
466: AA - Jacobian matrix
467: BB - optionally different preconditioning matrix
468: str - flag indicating matrix structure
470: Notes:
471: Recall that MatSetValues() uses 0-based row and column numbers
472: in Fortran as well as in C.
473: */
474: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
475: {
476: Mat A = AA; /* Jacobian matrix */
477: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
478: PetscInt mstart = 0;
479: PetscInt mend = appctx->m;
481: PetscInt i,idx[3];
482: PetscScalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;
484: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
485: Compute entries for the locally owned part of the matrix
486: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
487: /*
488: Set matrix rows corresponding to boundary data
489: */
491: mstart = 0;
492: v[0] = 1.0;
493: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
494: mstart++;
496: mend--;
497: v[0] = 1.0;
498: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
500: /*
501: Set matrix rows corresponding to interior data. We construct the
502: matrix one row at a time.
503: */
504: v[0] = sone; v[1] = stwo; v[2] = sone;
505: for (i=mstart; i<mend; i++) {
506: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
507: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
508: }
510: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
511: Complete the matrix assembly process and set some options
512: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
513: /*
514: Assemble matrix, using the 2-step process:
515: MatAssemblyBegin(), MatAssemblyEnd()
516: Computations can be done while messages are in transition
517: by placing code between these two statements.
518: */
519: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
520: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
522: /*
523: Set and option to indicate that we will never add a new nonzero location
524: to the matrix. If we do, it will generate an error.
525: */
526: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
528: return 0;
529: }
531: PetscErrorCode IFunctionHeat(TS ts,PetscReal t,Vec X,Vec Xdot,Vec r,void *ctx)
532: {
533: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
536: MatMult(appctx->A,X,r);
537: VecAYPX(r,-1.0,Xdot);
538: return 0;
539: }
541: PetscErrorCode IJacobianHeat(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal s,Mat A,Mat B,void *ctx)
542: {
543: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
546: if (appctx->oshift == s) return 0;
547: MatCopy(appctx->A,A,SAME_NONZERO_PATTERN);
548: MatScale(A,-1);
549: MatShift(A,s);
550: MatCopy(A,B,SAME_NONZERO_PATTERN);
551: appctx->oshift = s;
552: return 0;
553: }
555: /*TEST
557: test:
558: args: -nox -ts_type ssp -ts_dt 0.0005
560: test:
561: suffix: 2
562: args: -nox -ts_type ssp -ts_dt 0.0005 -time_dependent_rhs 1
564: test:
565: suffix: 3
566: args: -nox -ts_type rosw -ts_max_steps 3 -ksp_converged_reason
567: filter: sed "s/ATOL/RTOL/g"
568: requires: !single
570: test:
571: suffix: 4
572: args: -nox -ts_type beuler -ts_max_steps 3 -ksp_converged_reason
573: filter: sed "s/ATOL/RTOL/g"
575: test:
576: suffix: 5
577: args: -nox -ts_type beuler -ts_max_steps 3 -ksp_converged_reason -time_dependent_rhs
578: filter: sed "s/ATOL/RTOL/g"
580: test:
581: requires: !single
582: suffix: pod_guess
583: args: -nox -ts_type beuler -use_ifunc -ts_dt 0.0005 -ksp_guess_type pod -pc_type none -ksp_converged_reason
585: test:
586: requires: !single
587: suffix: pod_guess_Ainner
588: args: -nox -ts_type beuler -use_ifunc -ts_dt 0.0005 -ksp_guess_type pod -ksp_guess_pod_Ainner -pc_type none -ksp_converged_reason
590: test:
591: requires: !single
592: suffix: fischer_guess
593: args: -nox -ts_type beuler -use_ifunc -ts_dt 0.0005 -ksp_guess_type fischer -pc_type none -ksp_converged_reason
595: test:
596: requires: !single
597: suffix: fischer_guess_2
598: args: -nox -ts_type beuler -use_ifunc -ts_dt 0.0005 -ksp_guess_type fischer -ksp_guess_fischer_model 2,10 -pc_type none -ksp_converged_reason
600: test:
601: requires: !single
602: suffix: stringview
603: args: -nox -ts_type rosw -test_string_viewer
605: test:
606: requires: !single
607: suffix: stringview_euler
608: args: -nox -ts_type euler -test_string_viewer
610: TEST*/