Actual source code: ex5.c
petsc-3.4.2 2013-07-02

      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*,MatStructure*,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:     MatStructure A_structure;
    177:     RHSMatrixHeat(ts,0.0,u,&A,&A,&A_structure,&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:   TSSetInitialTimeStep(ts,0.0,dt);
    188:   TSSetSolution(ts,u);

    190:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    191:      Customize timestepping solver:
    192:        - Set the solution method to be the Backward Euler method.
    193:        - Set timestepping duration info
    194:      Then set runtime options, which can override these defaults.
    195:      For example,
    196:           -ts_max_steps <maxsteps> -ts_final_time <maxtime>
    197:      to override the defaults set by TSSetDuration().
    198:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    200:   TSSetDuration(ts,time_steps_max,time_total_max);
    201:   TSSetFromOptions(ts);

    203:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    204:      Solve the problem
    205:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    207:   /*
    208:      Evaluate initial conditions
    209:   */
    210:   InitialConditions(u,&appctx);

    212:   /*
    213:      Run the timestepping solver
    214:   */
    215:   TSSolve(ts,u);
    216:   TSGetSolveTime(ts,&ftime);
    217:   TSGetTimeStepNumber(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",
    224:                      appctx.norm_2/steps,appctx.norm_max/steps);
    225:   TSView(ts,PETSC_VIEWER_STDOUT_SELF);

    227:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    228:      Free work space.  All PETSc objects should be destroyed when they
    229:      are no longer needed.
    230:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    232:   TSDestroy(&ts);
    233:   MatDestroy(&A);
    234:   VecDestroy(&u);
    235:   PetscViewerDestroy(&appctx.viewer1);
    236:   PetscViewerDestroy(&appctx.viewer2);
    237:   VecDestroy(&appctx.solution);

    239:   /*
    240:      Always call PetscFinalize() before exiting a program.  This routine
    241:        - finalizes the PETSc libraries as well as MPI
    242:        - provides summary and diagnostic information if certain runtime
    243:          options are chosen (e.g., -log_summary).
    244:   */
    245:   PetscFinalize();
    246:   return 0;
    247: }
    248: /* --------------------------------------------------------------------- */
    251: /*
    252:    InitialConditions - Computes the solution at the initial time.

    254:    Input Parameter:
    255:    u - uninitialized solution vector (global)
    256:    appctx - user-defined application context

    258:    Output Parameter:
    259:    u - vector with solution at initial time (global)
    260: */
    261: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
    262: {
    263:   PetscScalar    *u_localptr,h = appctx->h;
    264:   PetscInt       i;

    267:   /*
    268:     Get a pointer to vector data.
    269:     - For default PETSc vectors, VecGetArray() returns a pointer to
    270:       the data array.  Otherwise, the routine is implementation dependent.
    271:     - You MUST call VecRestoreArray() when you no longer need access to
    272:       the array.
    273:     - Note that the Fortran interface to VecGetArray() differs from the
    274:       C version.  See the users manual for details.
    275:   */
    276:   VecGetArray(u,&u_localptr);

    278:   /*
    279:      We initialize the solution array by simply writing the solution
    280:      directly into the array locations.  Alternatively, we could use
    281:      VecSetValues() or VecSetValuesLocal().
    282:   */
    283:   for (i=0; i<appctx->m; i++) u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);

    285:   /*
    286:      Restore vector
    287:   */
    288:   VecRestoreArray(u,&u_localptr);

    290:   /*
    291:      Print debugging information if desired
    292:   */
    293:   if (appctx->debug) {
    294:     printf("initial guess vector\n");
    295:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
    296:   }

    298:   return 0;
    299: }
    300: /* --------------------------------------------------------------------- */
    303: /*
    304:    ExactSolution - Computes the exact solution at a given time.

    306:    Input Parameters:
    307:    t - current time
    308:    solution - vector in which exact solution will be computed
    309:    appctx - user-defined application context

    311:    Output Parameter:
    312:    solution - vector with the newly computed exact solution
    313: */
    314: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
    315: {
    316:   PetscScalar    *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
    317:   PetscInt       i;

    320:   /*
    321:      Get a pointer to vector data.
    322:   */
    323:   VecGetArray(solution,&s_localptr);

    325:   /*
    326:      Simply write the solution directly into the array locations.
    327:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
    328:   */
    329:   ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc); ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
    330:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
    331:   for (i=0; i<appctx->m; i++) s_localptr[i] = PetscCosScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscCosScalar(sc2*(PetscReal)i)*ex2;

    333:   /*
    334:      Restore vector
    335:   */
    336:   VecRestoreArray(solution,&s_localptr);
    337:   return 0;
    338: }
    339: /* --------------------------------------------------------------------- */
    342: /*
    343:    Monitor - User-provided routine to monitor the solution computed at
    344:    each timestep.  This example plots the solution and computes the
    345:    error in two different norms.

    347:    Input Parameters:
    348:    ts     - the timestep context
    349:    step   - the count of the current step (with 0 meaning the
    350:              initial condition)
    351:    time   - the current time
    352:    u      - the solution at this timestep
    353:    ctx    - the user-provided context for this monitoring routine.
    354:             In this case we use the application context which contains
    355:             information about the problem size, workspace and the exact
    356:             solution.
    357: */
    358: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
    359: {
    360:   AppCtx         *appctx = (AppCtx*) ctx;   /* user-defined application context */
    362:   PetscReal      norm_2,norm_max;

    364:   /*
    365:      View a graph of the current iterate
    366:   */
    367:   VecView(u,appctx->viewer2);

    369:   /*
    370:      Compute the exact solution
    371:   */
    372:   ExactSolution(time,appctx->solution,appctx);

    374:   /*
    375:      Print debugging information if desired
    376:   */
    377:   if (appctx->debug) {
    378:     printf("Computed solution vector\n");
    379:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
    380:     printf("Exact solution vector\n");
    381:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    382:   }

    384:   /*
    385:      Compute the 2-norm and max-norm of the error
    386:   */
    387:   VecAXPY(appctx->solution,-1.0,u);
    388:   VecNorm(appctx->solution,NORM_2,&norm_2);
    389:   norm_2 = PetscSqrtReal(appctx->h)*norm_2;
    390:   VecNorm(appctx->solution,NORM_MAX,&norm_max);

    392:   PetscPrintf(PETSC_COMM_WORLD,"Timestep %D: time = %G, 2-norm error = %G, max norm error = %G\n",
    393:                      step,time,norm_2,norm_max);
    394:   appctx->norm_2   += norm_2;
    395:   appctx->norm_max += norm_max;

    397:   /*
    398:      View a graph of the error
    399:   */
    400:   VecView(appctx->solution,appctx->viewer1);

    402:   /*
    403:      Print debugging information if desired
    404:   */
    405:   if (appctx->debug) {
    406:     printf("Error vector\n");
    407:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    408:   }

    410:   return 0;
    411: }
    412: /* --------------------------------------------------------------------- */
    415: /*
    416:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
    417:    matrix for the heat equation.

    419:    Input Parameters:
    420:    ts - the TS context
    421:    t - current time
    422:    global_in - global input vector
    423:    dummy - optional user-defined context, as set by TSetRHSJacobian()

    425:    Output Parameters:
    426:    AA - Jacobian matrix
    427:    BB - optionally different preconditioning matrix
    428:    str - flag indicating matrix structure

    430:   Notes:
    431:   Recall that MatSetValues() uses 0-based row and column numbers
    432:   in Fortran as well as in C.
    433: */
    434: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
    435: {
    436:   Mat            A       = *AA;                /* Jacobian matrix */
    437:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined application context */
    438:   PetscInt       mstart  = 0;
    439:   PetscInt       mend    = appctx->m;
    441:   PetscInt       i,idx[3];
    442:   PetscScalar    v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;

    444:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    445:      Compute entries for the locally owned part of the matrix
    446:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    447:   /*
    448:      Set matrix rows corresponding to boundary data
    449:   */

    451:   mstart = 0;
    452:   v[0]   = 1.0;
    453:   MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
    454:   mstart++;

    456:   mend--;
    457:   v[0] = 1.0;
    458:   MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);

    460:   /*
    461:      Set matrix rows corresponding to interior data.  We construct the
    462:      matrix one row at a time.
    463:   */
    464:   v[0] = sone; v[1] = stwo; v[2] = sone;
    465:   for (i=mstart; i<mend; i++) {
    466:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
    467:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
    468:   }

    470:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    471:      Complete the matrix assembly process and set some options
    472:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    473:   /*
    474:      Assemble matrix, using the 2-step process:
    475:        MatAssemblyBegin(), MatAssemblyEnd()
    476:      Computations can be done while messages are in transition
    477:      by placing code between these two statements.
    478:   */
    479:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    480:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

    482:   /*
    483:      Set flag to indicate that the Jacobian matrix retains an identical
    484:      nonzero structure throughout all timestepping iterations (although the
    485:      values of the entries change). Thus, we can save some work in setting
    486:      up the preconditioner (e.g., no need to redo symbolic factorization for
    487:      ILU/ICC preconditioners).
    488:       - If the nonzero structure of the matrix is different during
    489:         successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
    490:         must be used instead.  If you are unsure whether the matrix
    491:         structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
    492:       - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc
    493:         believes your assertion and does not check the structure
    494:         of the matrix.  If you erroneously claim that the structure
    495:         is the same when it actually is not, the new preconditioner
    496:         will not function correctly.  Thus, use this optimization
    497:         feature with caution!
    498:   */
    499:   *str = SAME_NONZERO_PATTERN;

    501:   /*
    502:      Set and option to indicate that we will never add a new nonzero location
    503:      to the matrix. If we do, it will generate an error.
    504:   */
    505:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

    507:   return 0;
    508: }