Actual source code: ex5.c
petsc-3.6.1 2015-08-06
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/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: */
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*);
83: int main(int argc,char **argv)
84: {
85: AppCtx appctx; /* user-defined application context */
86: TS ts; /* timestepping context */
87: Mat A; /* matrix data structure */
88: Vec u; /* approximate solution vector */
89: PetscReal time_total_max = 100.0; /* default max total time */
90: PetscInt time_steps_max = 100; /* default max timesteps */
91: PetscDraw draw; /* drawing context */
93: PetscInt steps,m;
94: PetscMPIInt size;
95: PetscBool flg;
96: PetscReal dt,ftime;
98: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
99: Initialize program and set problem parameters
100: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
102: PetscInitialize(&argc,&argv,(char*)0,help);
103: MPI_Comm_size(PETSC_COMM_WORLD,&size);
104: if (size != 1) SETERRQ(PETSC_COMM_SELF,1,"This is a uniprocessor example only!");
106: m = 60;
107: PetscOptionsGetInt(NULL,"-m",&m,NULL);
108: PetscOptionsHasName(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: PetscOptionsHasName(NULL,"-time_dependent_rhs",&flg);
161: if (flg) {
162: /*
163: For linear problems with a time-dependent f(u,t) in the equation
164: u_t = f(u,t), the user provides the discretized right-hand-side
165: as a time-dependent matrix.
166: */
167: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
168: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
169: } else {
170: /*
171: For linear problems with a time-independent f(u) in the equation
172: u_t = f(u), the user provides the discretized right-hand-side
173: as a matrix only once, and then sets a null matrix evaluation
174: routine.
175: */
176: RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
177: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
178: TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
179: }
181: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
182: Set solution vector and initial timestep
183: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
185: dt = appctx.h*appctx.h/2.0;
186: TSSetInitialTimeStep(ts,0.0,dt);
187: TSSetSolution(ts,u);
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 TSSetDuration().
197: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
199: TSSetDuration(ts,time_steps_max,time_total_max);
200: TSSetFromOptions(ts);
202: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
203: Solve the problem
204: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
206: /*
207: Evaluate initial conditions
208: */
209: InitialConditions(u,&appctx);
211: /*
212: Run the timestepping solver
213: */
214: TSSolve(ts,u);
215: TSGetSolveTime(ts,&ftime);
216: TSGetTimeStepNumber(ts,&steps);
218: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
219: View timestepping solver info
220: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
222: 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));
223: TSView(ts,PETSC_VIEWER_STDOUT_SELF);
225: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
226: Free work space. All PETSc objects should be destroyed when they
227: are no longer needed.
228: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
230: TSDestroy(&ts);
231: MatDestroy(&A);
232: VecDestroy(&u);
233: PetscViewerDestroy(&appctx.viewer1);
234: PetscViewerDestroy(&appctx.viewer2);
235: VecDestroy(&appctx.solution);
237: /*
238: Always call PetscFinalize() before exiting a program. This routine
239: - finalizes the PETSc libraries as well as MPI
240: - provides summary and diagnostic information if certain runtime
241: options are chosen (e.g., -log_summary).
242: */
243: PetscFinalize();
244: return 0;
245: }
246: /* --------------------------------------------------------------------- */
249: /*
250: InitialConditions - Computes the solution at the initial time.
252: Input Parameter:
253: u - uninitialized solution vector (global)
254: appctx - user-defined application context
256: Output Parameter:
257: u - vector with solution at initial time (global)
258: */
259: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
260: {
261: PetscScalar *u_localptr,h = appctx->h;
262: PetscInt i;
265: /*
266: Get a pointer to vector data.
267: - For default PETSc vectors, VecGetArray() returns a pointer to
268: the data array. Otherwise, the routine is implementation dependent.
269: - You MUST call VecRestoreArray() when you no longer need access to
270: the array.
271: - Note that the Fortran interface to VecGetArray() differs from the
272: C version. See the users manual for details.
273: */
274: VecGetArray(u,&u_localptr);
276: /*
277: We initialize the solution array by simply writing the solution
278: directly into the array locations. Alternatively, we could use
279: VecSetValues() or VecSetValuesLocal().
280: */
281: for (i=0; i<appctx->m; i++) u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);
283: /*
284: Restore vector
285: */
286: VecRestoreArray(u,&u_localptr);
288: /*
289: Print debugging information if desired
290: */
291: if (appctx->debug) {
292: printf("initial guess vector\n");
293: VecView(u,PETSC_VIEWER_STDOUT_SELF);
294: }
296: return 0;
297: }
298: /* --------------------------------------------------------------------- */
301: /*
302: ExactSolution - Computes the exact solution at a given time.
304: Input Parameters:
305: t - current time
306: solution - vector in which exact solution will be computed
307: appctx - user-defined application context
309: Output Parameter:
310: solution - vector with the newly computed exact solution
311: */
312: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
313: {
314: PetscScalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
315: PetscInt i;
318: /*
319: Get a pointer to vector data.
320: */
321: VecGetArray(solution,&s_localptr);
323: /*
324: Simply write the solution directly into the array locations.
325: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
326: */
327: ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc); ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
328: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
329: for (i=0; i<appctx->m; i++) s_localptr[i] = PetscCosScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscCosScalar(sc2*(PetscReal)i)*ex2;
331: /*
332: Restore vector
333: */
334: VecRestoreArray(solution,&s_localptr);
335: return 0;
336: }
337: /* --------------------------------------------------------------------- */
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: Input Parameters:
346: ts - the timestep context
347: step - the count of the current step (with 0 meaning the
348: initial condition)
349: time - the current time
350: u - the solution at this timestep
351: ctx - the user-provided context for this monitoring routine.
352: In this case we use the application context which contains
353: information about the problem size, workspace and the exact
354: solution.
355: */
356: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
357: {
358: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
360: PetscReal norm_2,norm_max;
362: /*
363: View a graph of the current iterate
364: */
365: VecView(u,appctx->viewer2);
367: /*
368: Compute the exact solution
369: */
370: ExactSolution(time,appctx->solution,appctx);
372: /*
373: Print debugging information if desired
374: */
375: if (appctx->debug) {
376: printf("Computed solution vector\n");
377: VecView(u,PETSC_VIEWER_STDOUT_SELF);
378: printf("Exact solution vector\n");
379: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
380: }
382: /*
383: Compute the 2-norm and max-norm of the error
384: */
385: VecAXPY(appctx->solution,-1.0,u);
386: VecNorm(appctx->solution,NORM_2,&norm_2);
387: norm_2 = PetscSqrtReal(appctx->h)*norm_2;
388: VecNorm(appctx->solution,NORM_MAX,&norm_max);
390: 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);
391: appctx->norm_2 += norm_2;
392: appctx->norm_max += norm_max;
394: /*
395: View a graph of the error
396: */
397: VecView(appctx->solution,appctx->viewer1);
399: /*
400: Print debugging information if desired
401: */
402: if (appctx->debug) {
403: printf("Error vector\n");
404: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
405: }
407: return 0;
408: }
409: /* --------------------------------------------------------------------- */
412: /*
413: RHSMatrixHeat - User-provided routine to compute the right-hand-side
414: matrix for the heat equation.
416: Input Parameters:
417: ts - the TS context
418: t - current time
419: global_in - global input vector
420: dummy - optional user-defined context, as set by TSetRHSJacobian()
422: Output Parameters:
423: AA - Jacobian matrix
424: BB - optionally different preconditioning matrix
425: str - flag indicating matrix structure
427: Notes:
428: Recall that MatSetValues() uses 0-based row and column numbers
429: in Fortran as well as in C.
430: */
431: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
432: {
433: Mat A = AA; /* Jacobian matrix */
434: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
435: PetscInt mstart = 0;
436: PetscInt mend = appctx->m;
438: PetscInt i,idx[3];
439: PetscScalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;
441: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
442: Compute entries for the locally owned part of the matrix
443: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
444: /*
445: Set matrix rows corresponding to boundary data
446: */
448: mstart = 0;
449: v[0] = 1.0;
450: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
451: mstart++;
453: mend--;
454: v[0] = 1.0;
455: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
457: /*
458: Set matrix rows corresponding to interior data. We construct the
459: matrix one row at a time.
460: */
461: v[0] = sone; v[1] = stwo; v[2] = sone;
462: for (i=mstart; i<mend; i++) {
463: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
464: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
465: }
467: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
468: Complete the matrix assembly process and set some options
469: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
470: /*
471: Assemble matrix, using the 2-step process:
472: MatAssemblyBegin(), MatAssemblyEnd()
473: Computations can be done while messages are in transition
474: by placing code between these two statements.
475: */
476: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
477: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
479: /*
480: Set and option to indicate that we will never add a new nonzero location
481: to the matrix. If we do, it will generate an error.
482: */
483: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
485: return 0;
486: }