Actual source code: ex4.c

      2: /* Program usage:  mpirun -np <procs> ex4 [-help] [all PETSc options] */

      4: static char help[] ="Solves a simple time-dependent linear PDE (the heat equation).\n\
      5: Input parameters include:\n\
      6:   -m <points>, where <points> = number of grid points\n\
      7:   -time_dependent_rhs : Treat the problem as having a time-dependent right-hand side\n\
      8:   -debug              : Activate debugging printouts\n\
      9:   -nox                : Deactivate x-window graphics\n\n";

     11: /*
     12:    Concepts: TS^time-dependent linear problems
     13:    Concepts: TS^heat equation
     14:    Concepts: TS^diffusion equation
     15:    Processors: n
     16: */

     18: /* ------------------------------------------------------------------------

     20:    This program solves the one-dimensional heat equation (also called the
     21:    diffusion equation),
     22:        u_t = u_xx,
     23:    on the domain 0 <= x <= 1, with the boundary conditions
     24:        u(t,0) = 0, u(t,1) = 0,
     25:    and the initial condition
     26:        u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
     27:    This is a linear, second-order, parabolic equation.

     29:    We discretize the right-hand side using finite differences with
     30:    uniform grid spacing h:
     31:        u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
     32:    We then demonstrate time evolution using the various TS methods by
     33:    running the program via
     34:        mpirun -np <procs> ex3 -ts_type <timestepping solver>

     36:    We compare the approximate solution with the exact solution, given by
     37:        u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
     38:                       3*exp(-4*pi*pi*t) * sin(2*pi*x)

     40:    Notes:
     41:    This code demonstrates the TS solver interface to two variants of
     42:    linear problems, u_t = f(u,t), namely
     43:      - time-dependent f:   f(u,t) is a function of t
     44:      - time-independent f: f(u,t) is simply f(u)

     46:     The uniprocessor version of this code is ts/examples/tutorials/ex3.c

     48:   ------------------------------------------------------------------------- */

     50: /*
     51:    Include "petscda.h" so that we can use distributed arrays (DAs) to manage
     52:    the parallel grid.  Include "petscts.h" so that we can use TS solvers.
     53:    Note that this file automatically includes:
     54:      petsc.h       - base PETSc routines   petscvec.h  - vectors
     55:      petscsys.h    - system routines       petscmat.h  - matrices
     56:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
     57:      petscviewer.h - viewers               petscpc.h   - preconditioners
     58:      petscksp.h   - linear solvers        petscsnes.h - nonlinear solvers
     59: */

     61:  #include petscda.h
     62:  #include petscts.h

     64: /*
     65:    User-defined application context - contains data needed by the
     66:    application-provided call-back routines.
     67: */
     68: typedef struct {
     69:   MPI_Comm    comm;              /* communicator */
     70:   DA          da;                /* distributed array data structure */
     71:   Vec         localwork;         /* local ghosted work vector */
     72:   Vec         u_local;           /* local ghosted approximate solution vector */
     73:   Vec         solution;          /* global exact solution vector */
     74:   PetscInt    m;                 /* total number of grid points */
     75:   PetscReal   h;                 /* mesh width h = 1/(m-1) */
     76:   PetscTruth  debug;             /* flag (1 indicates activation of debugging printouts) */
     77:   PetscViewer viewer1,viewer2;  /* viewers for the solution and error */
     78:   PetscReal   norm_2,norm_max;  /* error norms */
     79: } AppCtx;

     81: /*
     82:    User-defined routines
     83: */

     91: int main(int argc,char **argv)
     92: {
     93:   AppCtx         appctx;                 /* user-defined application context */
     94:   TS             ts;                     /* timestepping context */
     95:   Mat            A;                      /* matrix data structure */
     96:   Vec            u;                      /* approximate solution vector */
     97:   PetscReal      time_total_max = 100.0; /* default max total time */
     98:   PetscInt       time_steps_max = 100;   /* default max timesteps */
     99:   PetscDraw      draw;                   /* drawing context */
    101:   PetscInt       steps,m;
    102:   PetscMPIInt    size;
    103:   PetscReal      dt,ftime;
    104:   PetscTruth     flg;

    106:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    107:      Initialize program and set problem parameters
    108:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    109:
    110:   PetscInitialize(&argc,&argv,(char*)0,help);
    111:   appctx.comm = PETSC_COMM_WORLD;

    113:   m    = 60;
    114:   PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
    115:   PetscOptionsHasName(PETSC_NULL,"-debug",&appctx.debug);
    116:   appctx.m        = m;
    117:   appctx.h        = 1.0/(m-1.0);
    118:   appctx.norm_2   = 0.0;
    119:   appctx.norm_max = 0.0;
    120:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
    121:   PetscPrintf(PETSC_COMM_WORLD,"Solving a linear TS problem, number of processors = %d\n",size);

    123:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    124:      Create vector data structures
    125:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    126:   /*
    127:      Create distributed array (DA) to manage parallel grid and vectors
    128:      and to set up the ghost point communication pattern.  There are M
    129:      total grid values spread equally among all the processors.
    130:   */

    132:   DACreate1d(PETSC_COMM_WORLD,DA_NONPERIODIC,m,1,1,PETSC_NULL,&appctx.da);

    134:   /*
    135:      Extract global and local vectors from DA; we use these to store the
    136:      approximate solution.  Then duplicate these for remaining vectors that
    137:      have the same types.
    138:   */
    139:   DACreateGlobalVector(appctx.da,&u);
    140:   DACreateLocalVector(appctx.da,&appctx.u_local);

    142:   /*
    143:      Create local work vector for use in evaluating right-hand-side function;
    144:      create global work vector for storing exact solution.
    145:   */
    146:   VecDuplicate(appctx.u_local,&appctx.localwork);
    147:   VecDuplicate(u,&appctx.solution);

    149:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    150:      Set up displays to show graphs of the solution and error
    151:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    153:   PetscViewerDrawOpen(PETSC_COMM_WORLD,0,"",80,380,400,160,&appctx.viewer1);
    154:   PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
    155:   PetscDrawSetDoubleBuffer(draw);
    156:   PetscViewerDrawOpen(PETSC_COMM_WORLD,0,"",80,0,400,160,&appctx.viewer2);
    157:   PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
    158:   PetscDrawSetDoubleBuffer(draw);

    160:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    161:      Create timestepping solver context
    162:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    164:   TSCreate(PETSC_COMM_WORLD,&ts);
    165:   TSSetProblemType(ts,TS_LINEAR);

    167:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    168:      Set optional user-defined monitoring routine
    169:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    171:   TSSetMonitor(ts,Monitor,&appctx,PETSC_NULL);

    173:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    175:      Create matrix data structure; set matrix evaluation routine.
    176:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    178:   MatCreate(PETSC_COMM_WORLD,&A);
    179:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
    180:   MatSetFromOptions(A);

    182:   PetscOptionsHasName(PETSC_NULL,"-time_dependent_rhs",&flg);
    183:   if (flg) {
    184:     /*
    185:        For linear problems with a time-dependent f(u,t) in the equation
    186:        u_t = f(u,t), the user provides the discretized right-hand-side
    187:        as a time-dependent matrix.
    188:     */
    189:     TSSetRHSMatrix(ts,A,A,RHSMatrixHeat,&appctx);
    190:   } else {
    191:     /*
    192:        For linear problems with a time-independent f(u) in the equation
    193:        u_t = f(u), the user provides the discretized right-hand-side
    194:        as a matrix only once, and then sets a null matrix evaluation
    195:        routine.
    196:     */
    197:     MatStructure A_structure;
    198:     RHSMatrixHeat(ts,0.0,&A,&A,&A_structure,&appctx);
    199:     TSSetRHSMatrix(ts,A,A,PETSC_NULL,&appctx);
    200:   }

    202:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    203:      Set solution vector and initial timestep
    204:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    206:   dt = appctx.h*appctx.h/2.0;
    207:   TSSetInitialTimeStep(ts,0.0,dt);
    208:   TSSetSolution(ts,u);

    210:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    211:      Customize timestepping solver:
    212:        - Set the solution method to be the Backward Euler method.
    213:        - Set timestepping duration info
    214:      Then set runtime options, which can override these defaults.
    215:      For example,
    216:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
    217:      to override the defaults set by TSSetDuration().
    218:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    220:   TSSetDuration(ts,time_steps_max,time_total_max);
    221:   TSSetFromOptions(ts);

    223:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    224:      Solve the problem
    225:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    227:   /*
    228:      Evaluate initial conditions
    229:   */
    230:   InitialConditions(u,&appctx);

    232:   /*
    233:      Run the timestepping solver
    234:   */
    235:   TSStep(ts,&steps,&ftime);

    237:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    238:      View timestepping solver info
    239:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    241:   PetscPrintf(PETSC_COMM_WORLD,"avg. error (2 norm) = %g, avg. error (max norm) = %g\n",
    242:               appctx.norm_2/steps,appctx.norm_max/steps);
    243:   TSView(ts,PETSC_VIEWER_STDOUT_WORLD);

    245:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    246:      Free work space.  All PETSc objects should be destroyed when they
    247:      are no longer needed.
    248:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    250:   TSDestroy(ts);
    251:   MatDestroy(A);
    252:   VecDestroy(u);
    253:   PetscViewerDestroy(appctx.viewer1);
    254:   PetscViewerDestroy(appctx.viewer2);
    255:   VecDestroy(appctx.localwork);
    256:   VecDestroy(appctx.solution);
    257:   VecDestroy(appctx.u_local);
    258:   DADestroy(appctx.da);

    260:   /*
    261:      Always call PetscFinalize() before exiting a program.  This routine
    262:        - finalizes the PETSc libraries as well as MPI
    263:        - provides summary and diagnostic information if certain runtime
    264:          options are chosen (e.g., -log_summary).
    265:   */
    266:   PetscFinalize();
    267:   return 0;
    268: }
    269: /* --------------------------------------------------------------------- */
    272: /*
    273:    InitialConditions - Computes the solution at the initial time.

    275:    Input Parameter:
    276:    u - uninitialized solution vector (global)
    277:    appctx - user-defined application context

    279:    Output Parameter:
    280:    u - vector with solution at initial time (global)
    281: */
    282: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
    283: {
    284:   PetscScalar    *u_localptr,h = appctx->h;
    285:   PetscInt       i,mybase,myend;

    288:   /*
    289:      Determine starting point of each processor's range of
    290:      grid values.
    291:   */
    292:   VecGetOwnershipRange(u,&mybase,&myend);

    294:   /*
    295:     Get a pointer to vector data.
    296:     - For default PETSc vectors, VecGetArray() returns a pointer to
    297:       the data array.  Otherwise, the routine is implementation dependent.
    298:     - You MUST call VecRestoreArray() when you no longer need access to
    299:       the array.
    300:     - Note that the Fortran interface to VecGetArray() differs from the
    301:       C version.  See the users manual for details.
    302:   */
    303:   VecGetArray(u,&u_localptr);

    305:   /*
    306:      We initialize the solution array by simply writing the solution
    307:      directly into the array locations.  Alternatively, we could use
    308:      VecSetValues() or VecSetValuesLocal().
    309:   */
    310:   for (i=mybase; i<myend; i++) {
    311:     u_localptr[i-mybase] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);
    312:   }

    314:   /*
    315:      Restore vector
    316:   */
    317:   VecRestoreArray(u,&u_localptr);

    319:   /*
    320:      Print debugging information if desired
    321:   */
    322:   if (appctx->debug) {
    323:      PetscPrintf(appctx->comm,"initial guess vector\n");
    324:      VecView(u,PETSC_VIEWER_STDOUT_WORLD);
    325:   }

    327:   return 0;
    328: }
    329: /* --------------------------------------------------------------------- */
    332: /*
    333:    ExactSolution - Computes the exact solution at a given time.

    335:    Input Parameters:
    336:    t - current time
    337:    solution - vector in which exact solution will be computed
    338:    appctx - user-defined application context

    340:    Output Parameter:
    341:    solution - vector with the newly computed exact solution
    342: */
    343: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
    344: {
    345:   PetscScalar    *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2;
    346:   PetscInt       i,mybase,myend;

    349:   /*
    350:      Determine starting and ending points of each processor's
    351:      range of grid values
    352:   */
    353:   VecGetOwnershipRange(solution,&mybase,&myend);

    355:   /*
    356:      Get a pointer to vector data.
    357:   */
    358:   VecGetArray(solution,&s_localptr);

    360:   /*
    361:      Simply write the solution directly into the array locations.
    362:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
    363:   */
    364:   ex1 = exp(-36.*PETSC_PI*PETSC_PI*t); ex2 = exp(-4.*PETSC_PI*PETSC_PI*t);
    365:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
    366:   for (i=mybase; i<myend; i++) {
    367:     s_localptr[i-mybase] = PetscSinScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i)*ex2;
    368:   }

    370:   /*
    371:      Restore vector
    372:   */
    373:   VecRestoreArray(solution,&s_localptr);
    374:   return 0;
    375: }
    376: /* --------------------------------------------------------------------- */
    379: /*
    380:    Monitor - User-provided routine to monitor the solution computed at
    381:    each timestep.  This example plots the solution and computes the
    382:    error in two different norms.

    384:    Input Parameters:
    385:    ts     - the timestep context
    386:    step   - the count of the current step (with 0 meaning the
    387:              initial condition)
    388:    time   - the current time
    389:    u      - the solution at this timestep
    390:    ctx    - the user-provided context for this monitoring routine.
    391:             In this case we use the application context which contains
    392:             information about the problem size, workspace and the exact
    393:             solution.
    394: */
    395: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
    396: {
    397:   AppCtx         *appctx = (AppCtx*) ctx;   /* user-defined application context */
    399:   PetscReal      norm_2,norm_max;
    400:   PetscScalar    mone = -1.0;

    402:   /*
    403:      View a graph of the current iterate
    404:   */
    405:   VecView(u,appctx->viewer2);

    407:   /*
    408:      Compute the exact solution
    409:   */
    410:   ExactSolution(time,appctx->solution,appctx);

    412:   /*
    413:      Print debugging information if desired
    414:   */
    415:   if (appctx->debug) {
    416:      PetscPrintf(appctx->comm,"Computed solution vector\n");
    417:      VecView(u,PETSC_VIEWER_STDOUT_WORLD);
    418:      PetscPrintf(appctx->comm,"Exact solution vector\n");
    419:      VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
    420:   }

    422:   /*
    423:      Compute the 2-norm and max-norm of the error
    424:   */
    425:   VecAXPY(appctx->solution,mone,u);
    426:   VecNorm(appctx->solution,NORM_2,&norm_2);
    427:   norm_2 = sqrt(appctx->h)*norm_2;
    428:   VecNorm(appctx->solution,NORM_MAX,&norm_max);

    430:   /*
    431:      PetscPrintf() causes only the first processor in this
    432:      communicator to print the timestep information.
    433:   */
    434:   PetscPrintf(appctx->comm,"Timestep %D: time = %g, 2-norm error = %g, max norm error = %g\n",
    435:               step,time,norm_2,norm_max);
    436:   appctx->norm_2   += norm_2;
    437:   appctx->norm_max += norm_max;

    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(appctx->comm,"Error vector\n");
    449:      VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
    450:   }

    452:   return 0;
    453: }
    454: /* --------------------------------------------------------------------- */
    457: /*
    458:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
    459:    matrix for the heat equation.

    461:    Input Parameters:
    462:    ts - the TS context
    463:    t - current time
    464:    global_in - global input vector
    465:    dummy - optional user-defined context, as set by TSetRHSJacobian()

    467:    Output Parameters:
    468:    AA - Jacobian matrix
    469:    BB - optionally different preconditioning matrix
    470:    str - flag indicating matrix structure

    472:   Notes:
    473:   RHSMatrixHeat computes entries for the locally owned part of the system.
    474:    - Currently, all PETSc parallel matrix formats are partitioned by
    475:      contiguous chunks of rows across the processors.
    476:    - Each processor needs to insert only elements that it owns
    477:      locally (but any non-local elements will be sent to the
    478:      appropriate processor during matrix assembly).
    479:    - Always specify global row and columns of matrix entries when
    480:      using MatSetValues(); we could alternatively use MatSetValuesLocal().
    481:    - Here, we set all entries for a particular row at once.
    482:    - Note that MatSetValues() uses 0-based row and column numbers
    483:      in Fortran as well as in C.
    484: */
    485: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
    486: {
    487:   Mat            A = *AA;                      /* Jacobian matrix */
    488:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined application context */
    490:   PetscInt       i,mstart,mend,idx[3];
    491:   PetscScalar    v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;

    493:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    494:      Compute entries for the locally owned part of the matrix
    495:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    497:   MatGetOwnershipRange(A,&mstart,&mend);

    499:   /*
    500:      Set matrix rows corresponding to boundary data
    501:   */

    503:   if (mstart == 0) {  /* first processor only */
    504:     v[0] = 1.0;
    505:     MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
    506:     mstart++;
    507:   }

    509:   if (mend == appctx->m) { /* last processor only */
    510:     mend--;
    511:     v[0] = 1.0;
    512:     MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
    513:   }

    515:   /*
    516:      Set matrix rows corresponding to interior data.  We construct the
    517:      matrix one row at a time.
    518:   */
    519:   v[0] = sone; v[1] = stwo; v[2] = sone;
    520:   for (i=mstart; i<mend; i++) {
    521:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
    522:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
    523:   }

    525:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    526:      Complete the matrix assembly process and set some options
    527:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    528:   /*
    529:      Assemble matrix, using the 2-step process:
    530:        MatAssemblyBegin(), MatAssemblyEnd()
    531:      Computations can be done while messages are in transition
    532:      by placing code between these two statements.
    533:   */
    534:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    535:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

    537:   /*
    538:      Set flag to indicate that the Jacobian matrix retains an identical
    539:      nonzero structure throughout all timestepping iterations (although the
    540:      values of the entries change). Thus, we can save some work in setting
    541:      up the preconditioner (e.g., no need to redo symbolic factorization for
    542:      ILU/ICC preconditioners).
    543:       - If the nonzero structure of the matrix is different during
    544:         successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
    545:         must be used instead.  If you are unsure whether the matrix
    546:         structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
    547:       - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc
    548:         believes your assertion and does not check the structure
    549:         of the matrix.  If you erroneously claim that the structure
    550:         is the same when it actually is not, the new preconditioner
    551:         will not function correctly.  Thus, use this optimization
    552:         feature with caution!
    553:   */
    554:   *str = SAME_NONZERO_PATTERN;

    556:   /*
    557:      Set and option to indicate that we will never add a new nonzero location
    558:      to the matrix. If we do, it will generate an error.
    559:   */
    560:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR);

    562:   return 0;
    563: }