Actual source code: ex5.c
petsc-3.9.4 2018-09-11
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*);
81: int main(int argc,char **argv)
82: {
83: AppCtx appctx; /* user-defined application context */
84: TS ts; /* timestepping context */
85: Mat A; /* matrix data structure */
86: Vec u; /* approximate solution vector */
87: PetscReal time_total_max = 100.0; /* default max total time */
88: PetscInt time_steps_max = 100; /* default max timesteps */
89: PetscDraw draw; /* drawing context */
91: PetscInt steps,m;
92: PetscMPIInt size;
93: PetscBool flg;
94: PetscReal dt,ftime;
96: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
97: Initialize program and set problem parameters
98: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
100: PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
101: MPI_Comm_size(PETSC_COMM_WORLD,&size);
102: if (size != 1) SETERRQ(PETSC_COMM_SELF,1,"This is a uniprocessor example only!");
104: m = 60;
105: PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
106: PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);
107: appctx.m = m;
108: appctx.h = 1.0/(m-1.0);
109: appctx.norm_2 = 0.0;
110: appctx.norm_max = 0.0;
112: PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");
114: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
115: Create vector data structures
116: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
118: /*
119: Create vector data structures for approximate and exact solutions
120: */
121: VecCreateSeq(PETSC_COMM_SELF,m,&u);
122: VecDuplicate(u,&appctx.solution);
124: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
125: Set up displays to show graphs of the solution and error
126: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
128: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
129: PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
130: PetscDrawSetDoubleBuffer(draw);
131: PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
132: PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
133: PetscDrawSetDoubleBuffer(draw);
135: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
136: Create timestepping solver context
137: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
139: TSCreate(PETSC_COMM_SELF,&ts);
140: TSSetProblemType(ts,TS_LINEAR);
142: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
143: Set optional user-defined monitoring routine
144: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
146: TSMonitorSet(ts,Monitor,&appctx,NULL);
148: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
150: Create matrix data structure; set matrix evaluation routine.
151: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
153: MatCreate(PETSC_COMM_SELF,&A);
154: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
155: MatSetFromOptions(A);
156: MatSetUp(A);
158: PetscOptionsHasName(NULL,NULL,"-time_dependent_rhs",&flg);
159: if (flg) {
160: /*
161: For linear problems with a time-dependent f(u,t) in the equation
162: u_t = f(u,t), the user provides the discretized right-hand-side
163: as a time-dependent matrix.
164: */
165: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
166: TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
167: } else {
168: /*
169: For linear problems with a time-independent f(u) in the equation
170: u_t = f(u), the user provides the discretized right-hand-side
171: as a matrix only once, and then sets a null matrix evaluation
172: routine.
173: */
174: RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
175: TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
176: TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
177: }
179: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
180: Set solution vector and initial timestep
181: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
183: dt = appctx.h*appctx.h/2.0;
184: TSSetTimeStep(ts,dt);
185: TSSetSolution(ts,u);
187: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
188: Customize timestepping solver:
189: - Set the solution method to be the Backward Euler method.
190: - Set timestepping duration info
191: Then set runtime options, which can override these defaults.
192: For example,
193: -ts_max_steps <maxsteps> -ts_final_time <maxtime>
194: to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
195: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
197: TSSetMaxSteps(ts,time_steps_max);
198: TSSetMaxTime(ts,time_total_max);
199: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
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: TSGetStepNumber(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_view).
242: */
243: PetscFinalize();
244: return ierr;
245: }
246: /* --------------------------------------------------------------------- */
247: /*
248: InitialConditions - Computes the solution at the initial time.
250: Input Parameter:
251: u - uninitialized solution vector (global)
252: appctx - user-defined application context
254: Output Parameter:
255: u - vector with solution at initial time (global)
256: */
257: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
258: {
259: PetscScalar *u_localptr,h = appctx->h;
260: PetscInt i;
263: /*
264: Get a pointer to vector data.
265: - For default PETSc vectors, VecGetArray() returns a pointer to
266: the data array. Otherwise, the routine is implementation dependent.
267: - You MUST call VecRestoreArray() when you no longer need access to
268: the array.
269: - Note that the Fortran interface to VecGetArray() differs from the
270: C version. See the users manual for details.
271: */
272: VecGetArray(u,&u_localptr);
274: /*
275: We initialize the solution array by simply writing the solution
276: directly into the array locations. Alternatively, we could use
277: VecSetValues() or VecSetValuesLocal().
278: */
279: for (i=0; i<appctx->m; i++) u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);
281: /*
282: Restore vector
283: */
284: VecRestoreArray(u,&u_localptr);
286: /*
287: Print debugging information if desired
288: */
289: if (appctx->debug) {
290: printf("initial guess vector\n");
291: VecView(u,PETSC_VIEWER_STDOUT_SELF);
292: }
294: return 0;
295: }
296: /* --------------------------------------------------------------------- */
297: /*
298: ExactSolution - Computes the exact solution at a given time.
300: Input Parameters:
301: t - current time
302: solution - vector in which exact solution will be computed
303: appctx - user-defined application context
305: Output Parameter:
306: solution - vector with the newly computed exact solution
307: */
308: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
309: {
310: PetscScalar *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
311: PetscInt i;
314: /*
315: Get a pointer to vector data.
316: */
317: VecGetArray(solution,&s_localptr);
319: /*
320: Simply write the solution directly into the array locations.
321: Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
322: */
323: ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc); ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
324: sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h;
325: for (i=0; i<appctx->m; i++) s_localptr[i] = PetscCosScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscCosScalar(sc2*(PetscReal)i)*ex2;
327: /*
328: Restore vector
329: */
330: VecRestoreArray(solution,&s_localptr);
331: return 0;
332: }
333: /* --------------------------------------------------------------------- */
334: /*
335: Monitor - User-provided routine to monitor the solution computed at
336: each timestep. This example plots the solution and computes the
337: error in two different norms.
339: Input Parameters:
340: ts - the timestep context
341: step - the count of the current step (with 0 meaning the
342: initial condition)
343: time - the current time
344: u - the solution at this timestep
345: ctx - the user-provided context for this monitoring routine.
346: In this case we use the application context which contains
347: information about the problem size, workspace and the exact
348: solution.
349: */
350: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
351: {
352: AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */
354: PetscReal norm_2,norm_max;
356: /*
357: View a graph of the current iterate
358: */
359: VecView(u,appctx->viewer2);
361: /*
362: Compute the exact solution
363: */
364: ExactSolution(time,appctx->solution,appctx);
366: /*
367: Print debugging information if desired
368: */
369: if (appctx->debug) {
370: printf("Computed solution vector\n");
371: VecView(u,PETSC_VIEWER_STDOUT_SELF);
372: printf("Exact solution vector\n");
373: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
374: }
376: /*
377: Compute the 2-norm and max-norm of the error
378: */
379: VecAXPY(appctx->solution,-1.0,u);
380: VecNorm(appctx->solution,NORM_2,&norm_2);
381: norm_2 = PetscSqrtReal(appctx->h)*norm_2;
382: VecNorm(appctx->solution,NORM_MAX,&norm_max);
383: if (norm_2 < 1e-14) norm_2 = 0;
384: if (norm_max < 1e-14) norm_max = 0;
386: 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);
387: appctx->norm_2 += norm_2;
388: appctx->norm_max += norm_max;
390: /*
391: View a graph of the error
392: */
393: VecView(appctx->solution,appctx->viewer1);
395: /*
396: Print debugging information if desired
397: */
398: if (appctx->debug) {
399: printf("Error vector\n");
400: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
401: }
403: return 0;
404: }
405: /* --------------------------------------------------------------------- */
406: /*
407: RHSMatrixHeat - User-provided routine to compute the right-hand-side
408: matrix for the heat equation.
410: Input Parameters:
411: ts - the TS context
412: t - current time
413: global_in - global input vector
414: dummy - optional user-defined context, as set by TSetRHSJacobian()
416: Output Parameters:
417: AA - Jacobian matrix
418: BB - optionally different preconditioning matrix
419: str - flag indicating matrix structure
421: Notes:
422: Recall that MatSetValues() uses 0-based row and column numbers
423: in Fortran as well as in C.
424: */
425: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
426: {
427: Mat A = AA; /* Jacobian matrix */
428: AppCtx *appctx = (AppCtx*)ctx; /* user-defined application context */
429: PetscInt mstart = 0;
430: PetscInt mend = appctx->m;
432: PetscInt i,idx[3];
433: PetscScalar v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;
435: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
436: Compute entries for the locally owned part of the matrix
437: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
438: /*
439: Set matrix rows corresponding to boundary data
440: */
442: mstart = 0;
443: v[0] = 1.0;
444: MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
445: mstart++;
447: mend--;
448: v[0] = 1.0;
449: MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
451: /*
452: Set matrix rows corresponding to interior data. We construct the
453: matrix one row at a time.
454: */
455: v[0] = sone; v[1] = stwo; v[2] = sone;
456: for (i=mstart; i<mend; i++) {
457: idx[0] = i-1; idx[1] = i; idx[2] = i+1;
458: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
459: }
461: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
462: Complete the matrix assembly process and set some options
463: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
464: /*
465: Assemble matrix, using the 2-step process:
466: MatAssemblyBegin(), MatAssemblyEnd()
467: Computations can be done while messages are in transition
468: by placing code between these two statements.
469: */
470: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
471: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
473: /*
474: Set and option to indicate that we will never add a new nonzero location
475: to the matrix. If we do, it will generate an error.
476: */
477: MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
479: return 0;
480: }
482: /*TEST
484: test:
485: requires: x
487: test:
488: suffix: nox
489: args: -nox
491: TEST*/