Actual source code: ex5.c

      2: /* Program usage:  ex3 [-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: 1
     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) = 1, u(t,1) = 1,
     25:    and the initial condition
     26:        u(0,x) = cos(6*pi*x) + 3*cos(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:        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) * cos(6*pi*x) +
     38:                       3*exp(-4*pi*pi*t) * cos(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 just f(u)

     46:     The parallel version of this code is ts/examples/tutorials/ex4.c

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

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

     60:  #include petscts.h

     62: /*
     63:    User-defined application context - contains data needed by the
     64:    application-provided call-back routines.
     65: */
     66: typedef struct {
     67:   Vec         solution;          /* global exact solution vector */
     68:   PetscInt         m;                 /* total number of grid points */
     69:   PetscReal   h;                 /* mesh width h = 1/(m-1) */
     70:   PetscTruth  debug;             /* flag (1 indicates activation of debugging printouts) */
     71:   PetscViewer viewer1,viewer2;  /* viewers for the solution and error */
     72:   PetscReal   norm_2,norm_max;  /* error norms */
     73: } AppCtx;

     75: /*
     76:    User-defined routines
     77: */

     85: int main(int argc,char **argv)
     86: {
     87:   AppCtx         appctx;                 /* user-defined application context */
     88:   TS             ts;                     /* timestepping context */
     89:   Mat            A;                      /* matrix data structure */
     90:   Vec            u;                      /* approximate solution vector */
     91:   PetscReal      time_total_max = 100.0; /* default max total time */
     92:   PetscInt       time_steps_max = 100;   /* default max timesteps */
     93:   PetscDraw      draw;                   /* drawing context */
     95:   PetscInt       steps,m;
     96:   PetscMPIInt    size;
     97:   PetscTruth     flg;
     98:   PetscReal      dt,ftime;

    100:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    101:      Initialize program and set problem parameters
    102:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    103:
    104:   PetscInitialize(&argc,&argv,(char*)0,help);
    105:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
    106:   if (size != 1) SETERRQ(1,"This is a uniprocessor example only!");

    108:   m    = 60;
    109:   PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
    110:   PetscOptionsHasName(PETSC_NULL,"-debug",&appctx.debug);
    111:   appctx.m        = m;
    112:   appctx.h        = 1.0/(m-1.0);
    113:   appctx.norm_2   = 0.0;
    114:   appctx.norm_max = 0.0;
    115:   PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");

    117:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    118:      Create vector data structures
    119:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    121:   /*
    122:      Create vector data structures for approximate and exact solutions
    123:   */
    124:   VecCreateSeq(PETSC_COMM_SELF,m,&u);
    125:   VecDuplicate(u,&appctx.solution);

    127:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    128:      Set up displays to show graphs of the solution and error
    129:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    131:   PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
    132:   PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
    133:   PetscDrawSetDoubleBuffer(draw);
    134:   PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
    135:   PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
    136:   PetscDrawSetDoubleBuffer(draw);

    138:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    139:      Create timestepping solver context
    140:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    142:   TSCreate(PETSC_COMM_SELF,&ts);
    143:   TSSetProblemType(ts,TS_LINEAR);

    145:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    146:      Set optional user-defined monitoring routine
    147:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    151:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    153:      Create matrix data structure; set matrix evaluation routine.
    154:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    156:   MatCreate(PETSC_COMM_SELF,&A);
    157:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
    158:   MatSetFromOptions(A);

    160:   PetscOptionsHasName(PETSC_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:     TSSetRHSMatrix(ts,A,A,RHSMatrixHeat,&appctx);
    168:   } else {
    169:     /*
    170:        For linear problems with a time-independent f(u) in the equation
    171:        u_t = f(u), the user provides the discretized right-hand-side
    172:        as a matrix only once, and then sets a null matrix evaluation
    173:        routine.
    174:     */
    175:     MatStructure A_structure;
    176:     RHSMatrixHeat(ts,0.0,&A,&A,&A_structure,&appctx);
    177:     TSSetRHSMatrix(ts,A,A,PETSC_NULL,&appctx);
    178:   }

    180:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    181:      Set solution vector and initial timestep
    182:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    184:   dt = appctx.h*appctx.h/2.0;
    185:   TSSetInitialTimeStep(ts,0.0,dt);
    186:   TSSetSolution(ts,u);

    188:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    189:      Customize timestepping solver:
    190:        - Set the solution method to be the Backward Euler method.
    191:        - Set timestepping duration info
    192:      Then set runtime options, which can override these defaults.
    193:      For example,
    194:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
    195:      to override the defaults set by TSSetDuration().
    196:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    198:   TSSetDuration(ts,time_steps_max,time_total_max);
    199:   TSSetFromOptions(ts);

    201:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    202:      Solve the problem
    203:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    205:   /*
    206:      Evaluate initial conditions
    207:   */
    208:   InitialConditions(u,&appctx);

    210:   /*
    211:      Run the timestepping solver
    212:   */
    213:   TSStep(ts,&steps,&ftime);

    215:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    216:      View timestepping solver info
    217:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    219:   PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %g, avg. error (max norm) = %g\n",
    220:               appctx.norm_2/steps,appctx.norm_max/steps);
    221:   TSView(ts,PETSC_VIEWER_STDOUT_SELF);

    223:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    224:      Free work space.  All PETSc objects should be destroyed when they
    225:      are no longer needed.
    226:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    228:   TSDestroy(ts);
    229:   MatDestroy(A);
    230:   VecDestroy(u);
    231:   PetscViewerDestroy(appctx.viewer1);
    232:   PetscViewerDestroy(appctx.viewer2);
    233:   VecDestroy(appctx.solution);

    235:   /*
    236:      Always call PetscFinalize() before exiting a program.  This routine
    237:        - finalizes the PETSc libraries as well as MPI
    238:        - provides summary and diagnostic information if certain runtime
    239:          options are chosen (e.g., -log_summary).
    240:   */
    241:   PetscFinalize();
    242:   return 0;
    243: }
    244: /* --------------------------------------------------------------------- */
    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++) {
    280:     u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);
    281:   }

    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++) {
    330:     s_localptr[i] = PetscCosScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscCosScalar(sc2*(PetscReal)i)*ex2;
    331:   }

    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;
    363:   PetscScalar    mone = -1.0;

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

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

    375:   /*
    376:      Print debugging information if desired
    377:   */
    378:   if (appctx->debug) {
    379:      printf("Computed solution vector\n");
    380:      VecView(u,PETSC_VIEWER_STDOUT_SELF);
    381:      printf("Exact solution vector\n");
    382:      VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    383:   }

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

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

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

    403:   /*
    404:      Print debugging information if desired
    405:   */
    406:   if (appctx->debug) {
    407:      printf("Error vector\n");
    408:      VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    409:   }

    411:   return 0;
    412: }
    413: /* --------------------------------------------------------------------- */
    416: /*
    417:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
    418:    matrix for the heat equation.

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

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

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

    445:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    446:      Compute entries for the locally owned part of the matrix
    447:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    448:   /*
    449:      Set matrix rows corresponding to boundary data
    450:   */

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

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

    461:   /*
    462:      Set matrix rows corresponding to interior data.  We construct the
    463:      matrix one row at a time.
    464:   */
    465:   v[0] = sone; v[1] = stwo; v[2] = sone;
    466:   for (i=mstart; i<mend; i++) {
    467:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
    468:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
    469:   }

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

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

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

    508:   return 0;
    509: }