Actual source code: ex3.c
petsc-3.8.4 2018-03-24
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) = 0, u(t,1) = 0,
23: and the initial condition
24: u(0,x) = sin(6*pi*x) + 3*sin(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) * sin(6*pi*x) +
36: 3*exp(-4*pi*pi*t) * sin(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 f(u)
44: The parallel version of this code is ts/examples/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: */
58: #include <petscts.h>
59: #include <petscdraw.h>
61: /*
62: User-defined application context - contains data needed by the
63: application-provided call-back routines.
64: */
65: typedef struct {
66: Vec solution; /* global exact solution vector */
67: PetscInt m; /* total number of grid points */
68: PetscReal h; /* mesh width h = 1/(m-1) */
69: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
70: PetscViewer viewer1,viewer2; /* viewers for the solution and error */
71: PetscReal norm_2,norm_max; /* error norms */
72: } AppCtx;
74: /*
75: User-defined routines
76: */
77: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
78: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat,Mat,void*);
79: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
80: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);
82: int main(int argc,char **argv)
83: {
84: AppCtx appctx; /* user-defined application context */
85: TS ts; /* timestepping context */
86: Mat A; /* matrix data structure */
87: Vec u; /* approximate solution vector */
88: PetscReal time_total_max = 100.0; /* default max total time */
89: PetscInt time_steps_max = 100; /* default max timesteps */
90: PetscDraw draw; /* drawing context */
92: PetscInt steps,m;
93: PetscMPIInt size;
94: PetscReal dt;
95: PetscBool flg;
97: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
98: Initialize program and set problem parameters
99: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
101: PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
102: MPI_Comm_size(PETSC_COMM_WORLD,&size);
103: if (size != 1) SETERRQ(PETSC_COMM_SELF,1,"This is a uniprocessor example only!");
105: m = 60;
106: PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
107: PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);
109: appctx.m = m;
110: appctx.h = 1.0/(m-1.0);
111: appctx.norm_2 = 0.0;
112: appctx.norm_max = 0.0;
114: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");
116: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
117: Create vector data structures
118: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
120: /*
121: Create vector data structures for approximate and exact solutions
122: */
123: VecCreateSeq(PETSC_COMM_SELF,m,&u);
124: VecDuplicate(u,&appctx.solution);
126: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
127: Set up displays to show graphs of the solution and error
128: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
130: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
131: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
132: PetscDrawSetDoubleBuffer(draw);
133: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
134: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
135: PetscDrawSetDoubleBuffer(draw);
137: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
138: Create timestepping solver context
139: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
141: TSCreate(PETSC_COMM_SELF,&ts);
142: TSSetProblemType(ts,TS_LINEAR);
144: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
145: Set optional user-defined monitoring routine
146: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
148: TSMonitorSet(ts,Monitor,&appctx,NULL);
150: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
152: Create matrix data structure; set matrix evaluation routine.
153: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
155: MatCreate(PETSC_COMM_SELF,&A);
156: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
157: MatSetFromOptions(A);
158: MatSetUp(A);
160: flg = PETSC_FALSE;
161: PetscOptionsGetBool(NULL,NULL,"-time_dependent_rhs",&flg,NULL);
162: if (flg) {
163: /*
164: For linear problems with a time-dependent f(u,t) in the equation
165: u_t = f(u,t), the user provides the discretized right-hand-side
166: as a time-dependent matrix.
167: */
168: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
169: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
170: } else {
171: /*
172: For linear problems with a time-independent f(u) in the equation
173: u_t = f(u), the user provides the discretized right-hand-side
174: as a matrix only once, and then sets the special Jacobian evaluation
175: routine TSComputeRHSJacobianConstant() which will NOT recompute the Jacobian.
176: */
177: RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
178: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
179: TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
180: }
182: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
183: Set solution vector and initial timestep
184: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
186: dt = appctx.h*appctx.h/2.0;
187: TSSetTimeStep(ts,dt);
189: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
190: Customize timestepping solver:
191: - Set the solution method to be the Backward Euler method.
192: - Set timestepping duration info
193: Then set runtime options, which can override these defaults.
194: For example,
195: -ts_max_steps <maxsteps> -ts_final_time <maxtime>
196: to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
197: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
199: TSSetMaxSteps(ts,time_steps_max);
200: TSSetMaxTime(ts,time_total_max);
201: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
202: TSSetFromOptions(ts);
204: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
205: Solve the problem
206: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
208: /*
209: Evaluate initial conditions
210: */
211: InitialConditions(u,&appctx);
213: /*
214: Run the timestepping solver
215: */
216: TSSolve(ts,u);
217: TSGetStepNumber(ts,&steps);
219: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
220: View timestepping solver info
221: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
223: 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));
224: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
226: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
227: Free work space. All PETSc objects should be destroyed when they
228: are no longer needed.
229: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
231: TSDestroy(&ts);
232: MatDestroy(&A);
233: VecDestroy(&u);
234: PetscViewerDestroy(&appctx.viewer1);
235: PetscViewerDestroy(&appctx.viewer2);
236: VecDestroy(&appctx.solution);
238: /*
239: Always call PetscFinalize() before exiting a program. This routine
240: - finalizes the PETSc libraries as well as MPI
241: - provides summary and diagnostic information if certain runtime
242: options are chosen (e.g., -log_view).
243: */
244: PetscFinalize();
245: return ierr;
246: }
247: /* --------------------------------------------------------------------- */
248: /*
249: InitialConditions - Computes the solution at the initial time.
251: Input Parameter:
252: u - uninitialized solution vector (global)
253: appctx - user-defined application context
255: Output Parameter:
256: u - vector with solution at initial time (global)
257: */
258: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
259: {
260: PetscScalar *u_localptr,h = appctx->h;
262: PetscInt i;
264: /*
265: Get a pointer to vector data.
266: - For default PETSc vectors, VecGetArray() returns a pointer to
267: the data array. Otherwise, the routine is implementation dependent.
268: - You MUST call VecRestoreArray() when you no longer need access to
269: the array.
270: - Note that the Fortran interface to VecGetArray() differs from the
271: C version. See the users manual for details.
272: */
273: VecGetArray(u,&u_localptr);
275: /*
276: We initialize the solution array by simply writing the solution
277: directly into the array locations. Alternatively, we could use
278: VecSetValues() or VecSetValuesLocal().
279: */
280: for (i=0; i<appctx->m; i++) u_localptr[i] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);
282: /*
283: Restore vector
284: */
285: VecRestoreArray(u,&u_localptr);
287: /*
288: Print debugging information if desired
289: */
290: if (appctx->debug) {
291: PetscPrintf(PETSC_COMM_WORLD,"Initial guess vector\n");
292: VecView(u,PETSC_VIEWER_STDOUT_SELF);
293: }
295: return 0;
296: }
297: /* --------------------------------------------------------------------- */
298: /*
299: ExactSolution - Computes the exact solution at a given time.
301: Input Parameters:
302: t - current time
303: solution - vector in which exact solution will be computed
304: appctx - user-defined application context
306: Output Parameter:
307: solution - vector with the newly computed exact solution
308: */
309: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
310: {
311: PetscScalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
313: PetscInt i;
315: /*
316: Get a pointer to vector data.
317: */
318: VecGetArray(solution,&s_localptr);
320: /*
321: Simply write the solution directly into the array locations.
322: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
323: */
324: ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc);
325: ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
326: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
327: for (i=0; i<appctx->m; i++) s_localptr[i] = PetscSinScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i)*ex2;
329: /*
330: Restore vector
331: */
332: VecRestoreArray(solution,&s_localptr);
333: return 0;
334: }
335: /* --------------------------------------------------------------------- */
336: /*
337: Monitor - User-provided routine to monitor the solution computed at
338: each timestep. This example plots the solution and computes the
339: error in two different norms.
341: This example also demonstrates changing the timestep via TSSetTimeStep().
343: Input Parameters:
344: ts - the timestep context
345: step - the count of the current step (with 0 meaning the
346: initial condition)
347: time - the current time
348: u - the solution at this timestep
349: ctx - the user-provided context for this monitoring routine.
350: In this case we use the application context which contains
351: information about the problem size, workspace and the exact
352: solution.
353: */
354: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
355: {
356: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
358: PetscReal norm_2,norm_max,dt,dttol;
360: /*
361: View a graph of the current iterate
362: */
363: VecView(u,appctx->viewer2);
365: /*
366: Compute the exact solution
367: */
368: ExactSolution(time,appctx->solution,appctx);
370: /*
371: Print debugging information if desired
372: */
373: if (appctx->debug) {
374: PetscPrintf(PETSC_COMM_SELF,"Computed solution vector\n");
375: VecView(u,PETSC_VIEWER_STDOUT_SELF);
376: PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");
377: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
378: }
380: /*
381: Compute the 2-norm and max-norm of the error
382: */
383: VecAXPY(appctx->solution,-1.0,u);
384: VecNorm(appctx->solution,NORM_2,&norm_2);
385: norm_2 = PetscSqrtReal(appctx->h)*norm_2;
386: VecNorm(appctx->solution,NORM_MAX,&norm_max);
387: if (norm_2 < 1e-14) norm_2 = 0;
388: if (norm_max < 1e-14) norm_max = 0;
390: TSGetTimeStep(ts,&dt);
391: 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);
393: appctx->norm_2 += norm_2;
394: appctx->norm_max += norm_max;
396: dttol = .0001;
397: PetscOptionsGetReal(NULL,NULL,"-dttol",&dttol,NULL);
398: if (dt < dttol) {
399: dt *= .999;
400: TSSetTimeStep(ts,dt);
401: }
403: /*
404: View a graph of the error
405: */
406: VecView(appctx->solution,appctx->viewer1);
408: /*
409: Print debugging information if desired
410: */
411: if (appctx->debug) {
412: PetscPrintf(PETSC_COMM_SELF,"Error vector\n");
413: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
414: }
416: return 0;
417: }
418: /* --------------------------------------------------------------------- */
419: /*
420: RHSMatrixHeat - User-provided routine to compute the right-hand-side
421: matrix for the heat equation.
423: Input Parameters:
424: ts - the TS context
425: t - current time
426: global_in - global input vector
427: dummy - optional user-defined context, as set by TSetRHSJacobian()
429: Output Parameters:
430: AA - Jacobian matrix
431: BB - optionally different preconditioning matrix
432: str - flag indicating matrix structure
434: Notes:
435: Recall that MatSetValues() uses 0-based row and column numbers
436: in Fortran as well as in C.
437: */
438: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
439: {
440: Mat A = AA; /* Jacobian matrix */
441: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
442: PetscInt mstart = 0;
443: PetscInt mend = appctx->m;
445: PetscInt i,idx[3];
446: PetscScalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;
448: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
449: Compute entries for the locally owned part of the matrix
450: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
451: /*
452: Set matrix rows corresponding to boundary data
453: */
455: mstart = 0;
456: v[0] = 1.0;
457: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
458: mstart++;
460: mend--;
461: v[0] = 1.0;
462: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
464: /*
465: Set matrix rows corresponding to interior data. We construct the
466: matrix one row at a time.
467: */
468: v[0] = sone; v[1] = stwo; v[2] = sone;
469: for (i=mstart; i<mend; i++) {
470: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
471: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
472: }
474: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
475: Complete the matrix assembly process and set some options
476: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
477: /*
478: Assemble matrix, using the 2-step process:
479: MatAssemblyBegin(), MatAssemblyEnd()
480: Computations can be done while messages are in transition
481: by placing code between these two statements.
482: */
483: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
484: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
486: /*
487: Set and option to indicate that we will never add a new nonzero location
488: to the matrix. If we do, it will generate an error.
489: */
490: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
492: return 0;
493: }