Date: 2013-07-02T14:51:53+00:00

Actual source code: ex3.c
petsc-3.4.2 2013-07-02

      2: static char help[] ="Model Equations for Advection-Diffusion\n";

      4: /*
      5:     Page 9, Section 1.2 Model Equations for Advection-Diffusion

      7:           u_t = a u_x + d u_xx

      9:    The initial conditions used here different then in the book.

     11: */

     13: /*
     14:      Helpful runtime linear solver options:
     15:            -pc_type mg -da_refine 2 -snes_monitor -ksp_monitor -ts_view   (geometric multigrid with three levels)

     17: */

     19: /*
     20:    Include "petscts.h" so that we can use TS solvers.  Note that this file
     21:    automatically includes:
     22:      petscsys.h       - base PETSc routines   petscvec.h  - vectors
     23:      petscmat.h  - matrices
     24:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
     25:      petscviewer.h - viewers               petscpc.h   - preconditioners
     26:      petscksp.h   - linear solvers        petscsnes.h - nonlinear solvers
     27: */

     29: #include <petscts.h>
     30: #include <petscdmda.h>

     32: /*
     33:    User-defined application context - contains data needed by the
     34:    application-provided call-back routines.
     35: */
     36: typedef struct {
     37:   PetscScalar a,d;   /* advection and diffusion strength */
     38:   PetscBool   upwind;
     39: } AppCtx;

     41: /*
     42:    User-defined routines
     43: */
     44: extern PetscErrorCode InitialConditions(TS,Vec,AppCtx*);
     45: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat*,Mat*,MatStructure*,void*);
     46: extern PetscErrorCode Solution(TS,PetscReal,Vec,AppCtx*);

     50: int main(int argc,char **argv)
     51: {
     52:   AppCtx         appctx;                 /* user-defined application context */
     53:   TS             ts;                     /* timestepping context */
     54:   Vec            U;                      /* approximate solution vector */
     56:   PetscReal      dt;
     57:   DM             da;
     58:   PetscInt       M;

     60:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     61:      Initialize program and set problem parameters
     62:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     64:   PetscInitialize(&argc,&argv,(char*)0,help);
     65:   appctx.a      = 1.0;
     66:   appctx.d      = 0.0;
     67:   PetscOptionsGetScalar(NULL,"-a",&appctx.a,NULL);
     68:   PetscOptionsGetScalar(NULL,"-d",&appctx.d,NULL);
     69:   appctx.upwind = PETSC_TRUE;
     70:   PetscOptionsGetBool(NULL,"-upwind",&appctx.upwind,NULL);

     72:   DMDACreate1d(PETSC_COMM_WORLD,DMDA_BOUNDARY_PERIODIC, -60, 1, 1,NULL,&da);
     73:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     74:      Create vector data structures
     75:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     77:   /*
     78:      Create vector data structures for approximate and exact solutions
     79:   */
     80:   DMCreateGlobalVector(da,&U);

     82:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     83:      Create timestepping solver context
     84:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     86:   TSCreate(PETSC_COMM_WORLD,&ts);
     87:   TSSetDM(ts,da);

     89:   /*
     90:       For linear problems with a time-dependent f(U,t) in the equation
     91:      u_t = f(u,t), the user provides the discretized right-hand-side
     92:       as a time-dependent matrix.
     93:   */
     94:   TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
     95:   TSSetRHSJacobian(ts,NULL,NULL,RHSMatrixHeat,&appctx);
     96:   TSSetSolutionFunction(ts,(PetscErrorCode (*)(TS,PetscReal,Vec,void*))Solution,&appctx);

     98:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     99:      Customize timestepping solver:
    100:        - Set timestepping duration info
    101:      Then set runtime options, which can override these defaults.
    102:      For example,
    103:           -ts_max_steps <maxsteps> -ts_final_time <maxtime>
    104:      to override the defaults set by TSSetDuration().
    105:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    107:   DMDAGetInfo(da,PETSC_IGNORE,&M,0,0,0,0,0,0,0,0,0,0,0);
    108:   dt   = .48/(M*M);
    109:   TSSetInitialTimeStep(ts,0.0,dt);
    110:   TSSetDuration(ts,1000,100.0);
    111:   TSSetType(ts,TSARKIMEX);
    112:   TSSetFromOptions(ts);

    114:   /*
    115:      Evaluate initial conditions
    116:   */
    117:   InitialConditions(ts,U,&appctx);

    119:   /*
    120:      Run the timestepping solver
    121:   */
    122:   TSSolve(ts,U);

    125:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    126:      Free work space.  All PETSc objects should be destroyed when they
    127:      are no longer needed.
    128:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    130:   TSDestroy(&ts);
    131:   VecDestroy(&U);
    132:   DMDestroy(&da);

    134:   /*
    135:      Always call PetscFinalize() before exiting a program.  This routine
    136:        - finalizes the PETSc libraries as well as MPI
    137:        - provides summary and diagnostic information if certain runtime
    138:          options are chosen (e.g., -log_summary).
    139:   */
    140:   PetscFinalize();
    141:   return 0;
    142: }
    143: /* --------------------------------------------------------------------- */
    146: /*
    147:    InitialConditions - Computes the solution at the initial time.

    149:    Input Parameter:
    150:    u - uninitialized solution vector (global)
    151:    appctx - user-defined application context

    153:    Output Parameter:
    154:    u - vector with solution at initial time (global)
    155: */
    156: PetscErrorCode InitialConditions(TS ts,Vec U,AppCtx *appctx)
    157: {
    158:   PetscScalar    *u,h;
    160:   PetscInt       i,mstart,mend,xm,M;
    161:   DM             da;

    163:   TSGetDM(ts,&da);
    164:   DMDAGetCorners(da,&mstart,0,0,&xm,0,0);
    165:   DMDAGetInfo(da,PETSC_IGNORE,&M,0,0,0,0,0,0,0,0,0,0,0);
    166:   h    = 1.0/M;
    167:   mend = mstart + xm;
    168:   /*
    169:     Get a pointer to vector data.
    170:     - For default PETSc vectors, VecGetArray() returns a pointer to
    171:       the data array.  Otherwise, the routine is implementation dependent.
    172:     - You MUST call VecRestoreArray() when you no longer need access to
    173:       the array.
    174:     - Note that the Fortran interface to VecGetArray() differs from the
    175:       C version.  See the users manual for details.
    176:   */
    177:   DMDAVecGetArray(da,U,&u);

    179:   /*
    180:      We initialize the solution array by simply writing the solution
    181:      directly into the array locations.  Alternatively, we could use
    182:      VecSetValues() or VecSetValuesLocal().
    183:   */
    184:   for (i=mstart; i<mend; i++) u[i] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);

    186:   /*
    187:      Restore vector
    188:   */
    189:   DMDAVecRestoreArray(da,U,&u);
    190:   return 0;
    191: }
    192: /* --------------------------------------------------------------------- */
    195: /*
    196:    Solution - Computes the exact solution at a given time.

    198:    Input Parameters:
    199:    t - current time
    200:    solution - vector in which exact solution will be computed
    201:    appctx - user-defined application context

    203:    Output Parameter:
    204:    solution - vector with the newly computed exact solution
    205: */
    206: PetscErrorCode Solution(TS ts,PetscReal t,Vec U,AppCtx *appctx)
    207: {
    208:   PetscScalar    *u,ex1,ex2,sc1,sc2,h;
    210:   PetscInt       i,mstart,mend,xm,M;
    211:   DM             da;

    213:   TSGetDM(ts,&da);
    214:   DMDAGetCorners(da,&mstart,0,0,&xm,0,0);
    215:   DMDAGetInfo(da,PETSC_IGNORE,&M,0,0,0,0,0,0,0,0,0,0,0);
    216:   h    = 1.0/M;
    217:   mend = mstart + xm;
    218:   /*
    219:      Get a pointer to vector data.
    220:   */
    221:   DMDAVecGetArray(da,U,&u);

    223:   /*
    224:      Simply write the solution directly into the array locations.
    225:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
    226:   */
    227:   ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*appctx->d*t);
    228:   ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*appctx->d*t);
    229:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
    230:   for (i=mstart; i<mend; i++) u[i] = PetscSinScalar(sc1*(PetscReal)i + appctx->a*PETSC_PI*6.*t)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i + appctx->a*PETSC_PI*2.*t)*ex2;

    232:   /*
    233:      Restore vector
    234:   */
    235:   DMDAVecRestoreArray(da,U,&u);
    236:   return 0;
    237: }

    239: /* --------------------------------------------------------------------- */
    242: /*
    243:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
    244:    matrix for the heat equation.

    246:    Input Parameters:
    247:    ts - the TS context
    248:    t - current time
    249:    global_in - global input vector
    250:    dummy - optional user-defined context, as set by TSetRHSJacobian()

    252:    Output Parameters:
    253:    AA - Jacobian matrix
    254:    BB - optionally different preconditioning matrix
    255:    str - flag indicating matrix structure

    257:    Notes:
    258:    Recall that MatSetValues() uses 0-based row and column numbers
    259:    in Fortran as well as in C.
    260: */
    261: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec U,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
    262: {
    263:   Mat            A       = *AA;                /* Jacobian matrix */
    264:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined application context */
    265:   PetscInt       mstart, mend;
    267:   PetscInt       i,idx[3],M,xm;
    268:   PetscScalar    v[3],h;
    269:   DM             da;

    271:   TSGetDM(ts,&da);
    272:   DMDAGetInfo(da,0,&M,0,0,0,0,0,0,0,0,0,0,0);
    273:   DMDAGetCorners(da,&mstart,0,0,&xm,0,0);
    274:   h    = 1.0/M;
    275:   mend = mstart + xm;
    276:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    277:      Compute entries for the locally owned part of the matrix
    278:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    279:   /*
    280:      Set matrix rows corresponding to boundary data
    281:   */

    283:   /* diffusion */
    284:   v[0] = appctx->d/(h*h);
    285:   v[1] = -2.0*appctx->d/(h*h);
    286:   v[2] = appctx->d/(h*h);
    287:   if (!mstart) {
    288:     idx[0] = M-1; idx[1] = 0; idx[2] = 1;
    289:     MatSetValues(A,1,&mstart,3,idx,v,INSERT_VALUES);
    290:     mstart++;
    291:   }

    293:   if (mend == M) {
    294:     mend--;
    295:     idx[0] = M-2; idx[1] = M-1; idx[2] = 0;
    296:     MatSetValues(A,1,&mend,3,idx,v,INSERT_VALUES);
    297:   }

    299:   /*
    300:      Set matrix rows corresponding to interior data.  We construct the
    301:      matrix one row at a time.
    302:   */
    303:   for (i=mstart; i<mend; i++) {
    304:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
    305:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
    306:   }
    307:   MatAssemblyBegin(A,MAT_FLUSH_ASSEMBLY);
    308:   MatAssemblyEnd(A,MAT_FLUSH_ASSEMBLY);

    310:   DMDAGetCorners(da,&mstart,0,0,&xm,0,0);
    311:   mend = mstart + xm;
    312:   if (!appctx->upwind) {
    313:     /* advection -- centered differencing */
    314:     v[0] = -.5*appctx->a/(h);
    315:     v[1] = .5*appctx->a/(h);
    316:     if (!mstart) {
    317:       idx[0] = M-1; idx[1] = 1;
    318:       MatSetValues(A,1,&mstart,2,idx,v,ADD_VALUES);
    319:       mstart++;
    320:     }

    322:     if (mend == M) {
    323:       mend--;
    324:       idx[0] = M-2; idx[1] = 0;
    325:       MatSetValues(A,1,&mend,2,idx,v,ADD_VALUES);
    326:     }

    328:     for (i=mstart; i<mend; i++) {
    329:       idx[0] = i-1; idx[1] = i+1;
    330:       MatSetValues(A,1,&i,2,idx,v,ADD_VALUES);
    331:     }
    332:   } else {
    333:     /* advection -- upwinding */
    334:     v[0] = -appctx->a/(h);
    335:     v[1] = appctx->a/(h);
    336:     if (!mstart) {
    337:       idx[0] = 0; idx[1] = 1;
    338:       MatSetValues(A,1,&mstart,2,idx,v,ADD_VALUES);
    339:       mstart++;
    340:     }

    342:     if (mend == M) {
    343:       mend--;
    344:       idx[0] = M-1; idx[1] = 0;
    345:       MatSetValues(A,1,&mend,2,idx,v,ADD_VALUES);
    346:     }

    348:     for (i=mstart; i<mend; i++) {
    349:       idx[0] = i; idx[1] = i+1;
    350:       MatSetValues(A,1,&i,2,idx,v,ADD_VALUES);
    351:     }
    352:   }

    355:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    356:      Complete the matrix assembly process and set some options
    357:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    358:   /*
    359:      Assemble matrix, using the 2-step process:
    360:        MatAssemblyBegin(), MatAssemblyEnd()
    361:      Computations can be done while messages are in transition
    362:      by placing code between these two statements.
    363:   */
    364:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    365:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

    367:   /*
    368:      Set flag to indicate that the Jacobian matrix retains an identical
    369:      nonzero structure throughout all timestepping iterations (although the
    370:      values of the entries change). Thus, we can save some work in setting
    371:      up the preconditioner (e.g., no need to redo symbolic factorization for
    372:      ILU/ICC preconditioners).
    373:       - If the nonzero structure of the matrix is different during
    374:         successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
    375:         must be used instead.  If you are unsure whether the matrix
    376:         structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
    377:       - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc
    378:         believes your assertion and does not check the structure
    379:         of the matrix.  If you erroneously claim that the structure
    380:         is the same when it actually is not, the new preconditioner
    381:         will not function correctly.  Thus, use this optimization
    382:         feature with caution!
    383:   */
    384:   *str = SAME_NONZERO_PATTERN;

    386:   /*
    387:      Set and option to indicate that we will never add a new nonzero location
    388:      to the matrix. If we do, it will generate an error.
    389:   */
    390:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
    391:   return 0;
    392: }