Date: 2013-07-02T14:50:03+00:00

Actual source code: ex19.c
petsc-3.4.2 2013-07-02

      2: static char help[] = "Nonlinear driven cavity with multigrid in 2d.\n\
      3:   \n\
      4: The 2D driven cavity problem is solved in a velocity-vorticity formulation.\n\
      5: The flow can be driven with the lid or with bouyancy or both:\n\
      6:   -lidvelocity <lid>, where <lid> = dimensionless velocity of lid\n\
      7:   -grashof <gr>, where <gr> = dimensionless temperature gradent\n\
      8:   -prandtl <pr>, where <pr> = dimensionless thermal/momentum diffusity ratio\n\
      9:   -contours : draw contour plots of solution\n\n";
     10: /*
     11:       See src/ksp/ksp/examples/tutorials/ex45.c
     12: */

     14: /*T
     15:    Concepts: SNES^solving a system of nonlinear equations (parallel multicomponent example);
     16:    Concepts: DMDA^using distributed arrays;
     17:    Concepts: multicomponent
     18:    Processors: n
     19: T*/

We thank David E. Keyes for contributing the driven cavity discretization within this example code.
This problem is modeled by the partial differential equation system
\begin{eqnarray}
- \triangle U - \nabla_y \Omega & = & 0 \\
- \triangle V + \nabla_x\Omega & = & 0 \\
- \triangle \Omega + \nabla \cdot ([U*\Omega,V*\Omega]) - GR* \nabla_x T & = & 0 \\
- \triangle T + PR* \nabla \cdot ([U*T,V*T]) & = & 0
\end{eqnarray}
in the unit square, which is uniformly discretized in each of x and y in this simple encoding.
No-slip, rigid-wall Dirichlet conditions are used for $ [U,V]$.
Dirichlet conditions are used for Omega, based on the definition of
vorticity: $ \Omega = - \nabla_y U + \nabla_x V$, where along each
constant coordinate boundary, the tangential derivative is zero.
Dirichlet conditions are used for T on the left and right walls,
and insulation homogeneous Neumann conditions are used for T on
the top and bottom walls.
A finite difference approximation with the usual 5-point stencil
is used to discretize the boundary value problem to obtain a
nonlinear system of equations. Upwinding is used for the divergence
(convective) terms and central for the gradient (source) terms.
The Jacobian can be either
* formed via finite differencing using coloring (the default), or
* applied matrix-free via the option -snes_mf
(for larger grid problems this variant may not converge
without a preconditioner due to ill-conditioning).

     56: /*
     57:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
     58:    Include "petscsnes.h" so that we can use SNES solvers.  Note that this
     59:    file automatically includes:
     60:      petscsys.h       - base PETSc routines   petscvec.h - vectors
     61:      petscmat.h - matrices
     62:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
     63:      petscviewer.h - viewers               petscpc.h  - preconditioners
     64:      petscksp.h   - linear solvers
     65: */
     66: #if defined(PETSC_APPLE_FRAMEWORK)
     67: #import <PETSc/petscsnes.h>
     68: #import <PETSc/petscdmda.h>
     69: #else
     70: #include <petscsnes.h>
     71: #include <petscdmda.h>
     72: #endif

     74: /*
     75:    User-defined routines and data structures
     76: */
     77: typedef struct {
     78:   PetscScalar u,v,omega,temp;
     79: } Field;

     81: PetscErrorCode FormFunctionLocal(DMDALocalInfo*,Field**,Field**,void*);

     83: typedef struct {
     84:   PassiveReal lidvelocity,prandtl,grashof;  /* physical parameters */
     85:   PetscBool   draw_contours;                /* flag - 1 indicates drawing contours */
     86: } AppCtx;

     88: extern PetscErrorCode FormInitialGuess(AppCtx*,DM,Vec);
     89: extern PetscErrorCode NonlinearGS(SNES,Vec,Vec,void*);

     93: int main(int argc,char **argv)
     94: {
     95:   AppCtx         user;                /* user-defined work context */
     96:   PetscInt       mx,my,its;
     98:   MPI_Comm       comm;
     99:   SNES           snes;
    100:   DM             da;
    101:   Vec            x;

    103:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return(1);

    106:   comm = PETSC_COMM_WORLD;
    107:   SNESCreate(comm,&snes);

    109:   /*
    110:       Create distributed array object to manage parallel grid and vectors
    111:       for principal unknowns (x) and governing residuals (f)
    112:   */
    113:   DMDACreate2d(PETSC_COMM_WORLD,DMDA_BOUNDARY_NONE,DMDA_BOUNDARY_NONE,DMDA_STENCIL_STAR,-4,-4,PETSC_DECIDE,PETSC_DECIDE,4,1,0,0,&da);
    114:   SNESSetDM(snes,(DM)da);
    115:   SNESSetGS(snes, NonlinearGS, (void*)&user);

    117:   DMDAGetInfo(da,0,&mx,&my,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,
    118:                      PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);
    119:   /*
    120:      Problem parameters (velocity of lid, prandtl, and grashof numbers)
    121:   */
    122:   user.lidvelocity = 1.0/(mx*my);
    123:   user.prandtl     = 1.0;
    124:   user.grashof     = 1.0;

    126:   PetscOptionsGetReal(NULL,"-lidvelocity",&user.lidvelocity,NULL);
    127:   PetscOptionsGetReal(NULL,"-prandtl",&user.prandtl,NULL);
    128:   PetscOptionsGetReal(NULL,"-grashof",&user.grashof,NULL);
    129:   PetscOptionsHasName(NULL,"-contours",&user.draw_contours);

    131:   DMDASetFieldName(da,0,"x_velocity");
    132:   DMDASetFieldName(da,1,"y_velocity");
    133:   DMDASetFieldName(da,2,"Omega");
    134:   DMDASetFieldName(da,3,"temperature");

    136:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    137:      Create user context, set problem data, create vector data structures.
    138:      Also, compute the initial guess.
    139:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    141:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    142:      Create nonlinear solver context

    144:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    145:   DMSetApplicationContext(da,&user);
    146:   DMDASNESSetFunctionLocal(da,INSERT_VALUES,(PetscErrorCode (*)(DMDALocalInfo*,void*,void*,void*))FormFunctionLocal,&user);
    147:   SNESSetFromOptions(snes);
    148:   PetscPrintf(comm,"lid velocity = %G, prandtl # = %G, grashof # = %G\n",user.lidvelocity,user.prandtl,user.grashof);

    151:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    152:      Solve the nonlinear system
    153:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    154:   DMCreateGlobalVector(da,&x);
    155:   FormInitialGuess(&user,da,x);

    157:   SNESSolve(snes,NULL,x);

    159:   SNESGetIterationNumber(snes,&its);
    160:   PetscPrintf(comm,"Number of SNES iterations = %D\n", its);

    162:   /*
    163:      Visualize solution
    164:   */
    165:   if (user.draw_contours) {
    166:     VecView(x,PETSC_VIEWER_DRAW_WORLD);
    167:   }

    169:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    170:      Free work space.  All PETSc objects should be destroyed when they
    171:      are no longer needed.
    172:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    173:   VecDestroy(&x);
    174:   DMDestroy(&da);
    175:   SNESDestroy(&snes);
    176:   PetscFinalize();
    177:   return 0;
    178: }

    180: /* ------------------------------------------------------------------- */

    185: /*
    186:    FormInitialGuess - Forms initial approximation.

    188:    Input Parameters:
    189:    user - user-defined application context
    190:    X - vector

    192:    Output Parameter:
    193:    X - vector
    194: */
    195: PetscErrorCode FormInitialGuess(AppCtx *user,DM da,Vec X)
    196: {
    197:   PetscInt       i,j,mx,xs,ys,xm,ym;
    199:   PetscReal      grashof,dx;
    200:   Field          **x;

    202:   grashof = user->grashof;

    204:   DMDAGetInfo(da,0,&mx,0,0,0,0,0,0,0,0,0,0,0);
    205:   dx   = 1.0/(mx-1);

    207:   /*
    208:      Get local grid boundaries (for 2-dimensional DMDA):
    209:        xs, ys   - starting grid indices (no ghost points)
    210:        xm, ym   - widths of local grid (no ghost points)
    211:   */
    212:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

    214:   /*
    215:      Get a pointer to vector data.
    216:        - For default PETSc vectors, VecGetArray() returns a pointer to
    217:          the data array.  Otherwise, the routine is implementation dependent.
    218:        - You MUST call VecRestoreArray() when you no longer need access to
    219:          the array.
    220:   */
    221:   DMDAVecGetArray(da,X,&x);

    223:   /*
    224:      Compute initial guess over the locally owned part of the grid
    225:      Initial condition is motionless fluid and equilibrium temperature
    226:   */
    227:   for (j=ys; j<ys+ym; j++) {
    228:     for (i=xs; i<xs+xm; i++) {
    229:       x[j][i].u     = 0.0;
    230:       x[j][i].v     = 0.0;
    231:       x[j][i].omega = 0.0;
    232:       x[j][i].temp  = (grashof>0)*i*dx;
    233:     }
    234:   }

    236:   /*
    237:      Restore vector
    238:   */
    239:   DMDAVecRestoreArray(da,X,&x);
    240:   return 0;
    241: }

    245: PetscErrorCode FormFunctionLocal(DMDALocalInfo *info,Field **x,Field **f,void *ptr)
    246: {
    247:   AppCtx         *user = (AppCtx*)ptr;
    249:   PetscInt       xints,xinte,yints,yinte,i,j;
    250:   PetscReal      hx,hy,dhx,dhy,hxdhy,hydhx;
    251:   PetscReal      grashof,prandtl,lid;
    252:   PetscScalar    u,uxx,uyy,vx,vy,avx,avy,vxp,vxm,vyp,vym;

    255:   grashof = user->grashof;
    256:   prandtl = user->prandtl;
    257:   lid     = user->lidvelocity;

    259:   /*
    260:      Define mesh intervals ratios for uniform grid.

    262:      Note: FD formulae below are normalized by multiplying through by
    263:      local volume element (i.e. hx*hy) to obtain coefficients O(1) in two dimensions.

    266:   */
    267:   dhx   = (PetscReal)(info->mx-1);  dhy = (PetscReal)(info->my-1);
    268:   hx    = 1.0/dhx;                   hy = 1.0/dhy;
    269:   hxdhy = hx*dhy;                 hydhx = hy*dhx;

    271:   xints = info->xs; xinte = info->xs+info->xm; yints = info->ys; yinte = info->ys+info->ym;

    273:   /* Test whether we are on the bottom edge of the global array */
    274:   if (yints == 0) {
    275:     j     = 0;
    276:     yints = yints + 1;
    277:     /* bottom edge */
    278:     for (i=info->xs; i<info->xs+info->xm; i++) {
    279:       f[j][i].u     = x[j][i].u;
    280:       f[j][i].v     = x[j][i].v;
    281:       f[j][i].omega = x[j][i].omega + (x[j+1][i].u - x[j][i].u)*dhy;
    282:       f[j][i].temp  = x[j][i].temp-x[j+1][i].temp;
    283:     }
    284:   }

    286:   /* Test whether we are on the top edge of the global array */
    287:   if (yinte == info->my) {
    288:     j     = info->my - 1;
    289:     yinte = yinte - 1;
    290:     /* top edge */
    291:     for (i=info->xs; i<info->xs+info->xm; i++) {
    292:       f[j][i].u     = x[j][i].u - lid;
    293:       f[j][i].v     = x[j][i].v;
    294:       f[j][i].omega = x[j][i].omega + (x[j][i].u - x[j-1][i].u)*dhy;
    295:       f[j][i].temp  = x[j][i].temp-x[j-1][i].temp;
    296:     }
    297:   }

    299:   /* Test whether we are on the left edge of the global array */
    300:   if (xints == 0) {
    301:     i     = 0;
    302:     xints = xints + 1;
    303:     /* left edge */
    304:     for (j=info->ys; j<info->ys+info->ym; j++) {
    305:       f[j][i].u     = x[j][i].u;
    306:       f[j][i].v     = x[j][i].v;
    307:       f[j][i].omega = x[j][i].omega - (x[j][i+1].v - x[j][i].v)*dhx;
    308:       f[j][i].temp  = x[j][i].temp;
    309:     }
    310:   }

    312:   /* Test whether we are on the right edge of the global array */
    313:   if (xinte == info->mx) {
    314:     i     = info->mx - 1;
    315:     xinte = xinte - 1;
    316:     /* right edge */
    317:     for (j=info->ys; j<info->ys+info->ym; j++) {
    318:       f[j][i].u     = x[j][i].u;
    319:       f[j][i].v     = x[j][i].v;
    320:       f[j][i].omega = x[j][i].omega - (x[j][i].v - x[j][i-1].v)*dhx;
    321:       f[j][i].temp  = x[j][i].temp - (PetscReal)(grashof>0);
    322:     }
    323:   }

    325:   /* Compute over the interior points */
    326:   for (j=yints; j<yinte; j++) {
    327:     for (i=xints; i<xinte; i++) {

    329:       /*
    330:        convective coefficients for upwinding
    331:       */
    332:       vx  = x[j][i].u; avx = PetscAbsScalar(vx);
    333:       vxp = .5*(vx+avx); vxm = .5*(vx-avx);
    334:       vy  = x[j][i].v; avy = PetscAbsScalar(vy);
    335:       vyp = .5*(vy+avy); vym = .5*(vy-avy);

    337:       /* U velocity */
    338:       u         = x[j][i].u;
    339:       uxx       = (2.0*u - x[j][i-1].u - x[j][i+1].u)*hydhx;
    340:       uyy       = (2.0*u - x[j-1][i].u - x[j+1][i].u)*hxdhy;
    341:       f[j][i].u = uxx + uyy - .5*(x[j+1][i].omega-x[j-1][i].omega)*hx;

    343:       /* V velocity */
    344:       u         = x[j][i].v;
    345:       uxx       = (2.0*u - x[j][i-1].v - x[j][i+1].v)*hydhx;
    346:       uyy       = (2.0*u - x[j-1][i].v - x[j+1][i].v)*hxdhy;
    347:       f[j][i].v = uxx + uyy + .5*(x[j][i+1].omega-x[j][i-1].omega)*hy;

    349:       /* Omega */
    350:       u             = x[j][i].omega;
    351:       uxx           = (2.0*u - x[j][i-1].omega - x[j][i+1].omega)*hydhx;
    352:       uyy           = (2.0*u - x[j-1][i].omega - x[j+1][i].omega)*hxdhy;
    353:       f[j][i].omega = uxx + uyy + (vxp*(u - x[j][i-1].omega) + vxm*(x[j][i+1].omega - u))*hy +
    354:                       (vyp*(u - x[j-1][i].omega) + vym*(x[j+1][i].omega - u))*hx -
    355:                       .5*grashof*(x[j][i+1].temp - x[j][i-1].temp)*hy;

    357:       /* Temperature */
    358:       u            = x[j][i].temp;
    359:       uxx          = (2.0*u - x[j][i-1].temp - x[j][i+1].temp)*hydhx;
    360:       uyy          = (2.0*u - x[j-1][i].temp - x[j+1][i].temp)*hxdhy;
    361:       f[j][i].temp =  uxx + uyy  + prandtl*((vxp*(u - x[j][i-1].temp) + vxm*(x[j][i+1].temp - u))*hy +
    362:                                             (vyp*(u - x[j-1][i].temp) + vym*(x[j+1][i].temp - u))*hx);
    363:     }
    364:   }

    366:   /*
    367:      Flop count (multiply-adds are counted as 2 operations)
    368:   */
    369:   PetscLogFlops(84.0*info->ym*info->xm);
    370:   return(0);
    371: }

    376: PetscErrorCode NonlinearGS(SNES snes, Vec X, Vec B, void *ctx)
    377: {
    378:   DMDALocalInfo  info;
    379:   Field          **x,**b;
    381:   Vec            localX, localB;
    382:   DM             da;
    383:   PetscInt       xints,xinte,yints,yinte,i,j,k,l;
    384:   PetscInt       max_its,tot_its;
    385:   PetscInt       sweeps;
    386:   PetscReal      rtol,atol,stol;
    387:   PetscReal      hx,hy,dhx,dhy,hxdhy,hydhx;
    388:   PetscReal      grashof,prandtl,lid;
    389:   PetscScalar    u,uxx,uyy,vx,vy,avx,avy,vxp,vxm,vyp,vym;
    390:   PetscScalar    fu, fv, fomega, ftemp;
    391:   PetscScalar    dfudu;
    392:   PetscScalar    dfvdv;
    393:   PetscScalar    dfodu, dfodv, dfodo;
    394:   PetscScalar    dftdu, dftdv, dftdt;
    395:   PetscScalar    yu=0, yv=0, yo=0, yt=0;
    396:   PetscScalar    bjiu, bjiv, bjiomega, bjitemp;
    397:   PetscBool      ptconverged;
    398:   PetscReal      pfnorm,pfnorm0,pynorm,pxnorm;
    399:   AppCtx         *user = (AppCtx*)ctx;

    402:   grashof = user->grashof;
    403:   prandtl = user->prandtl;
    404:   lid     = user->lidvelocity;
    405:   tot_its = 0;
    406:   SNESGSGetTolerances(snes,&rtol,&atol,&stol,&max_its);
    407:   SNESGSGetSweeps(snes,&sweeps);
    408:   SNESGetDM(snes,(DM*)&da);
    409:   DMGetLocalVector(da,&localX);
    410:   if (B) {
    411:     DMGetLocalVector(da,&localB);
    412:   }
    413:   /*
    414:      Scatter ghost points to local vector, using the 2-step process
    415:         DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
    416:   */
    417:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
    418:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);
    419:   if (B) {
    420:     DMGlobalToLocalBegin(da,B,INSERT_VALUES,localB);
    421:     DMGlobalToLocalEnd(da,B,INSERT_VALUES,localB);
    422:   }
    423:   DMDAGetLocalInfo(da,&info);
    424:   DMDAVecGetArray(da,localX,&x);
    425:   if (B) {
    426:     DMDAVecGetArray(da,localB,&b);
    427:   }
    428:   /* looks like a combination of the formfunction / formjacobian routines */
    429:   dhx   = (PetscReal)(info.mx-1);dhy   = (PetscReal)(info.my-1);
    430:   hx    = 1.0/dhx;               hy    = 1.0/dhy;
    431:   hxdhy = hx*dhy;                hydhx = hy*dhx;

    433:   xints = info.xs; xinte = info.xs+info.xm; yints = info.ys; yinte = info.ys+info.ym;

    435:   /* Set the boundary conditions on the momentum equations */
    436:   /* Test whether we are on the bottom edge of the global array */
    437:   if (yints == 0) {
    438:     j     = 0;
    439:     yints = yints + 1;
    440:     /* bottom edge */
    441:     for (i=info.xs; i<info.xs+info.xm; i++) {

    443:       if (B) {
    444:         bjiu = b[j][i].u;
    445:         bjiv = b[j][i].v;
    446:       } else {
    447:         bjiu = 0.0;
    448:         bjiv = 0.0;
    449:       }
    450:       x[j][i].u = 0.0 + bjiu;
    451:       x[j][i].v = 0.0 + bjiv;
    452:     }
    453:   }

    455:   /* Test whether we are on the top edge of the global array */
    456:   if (yinte == info.my) {
    457:     j     = info.my - 1;
    458:     yinte = yinte - 1;
    459:     /* top edge */
    460:     for (i=info.xs; i<info.xs+info.xm; i++) {
    461:       if (B) {
    462:         bjiu = b[j][i].u;
    463:         bjiv = b[j][i].v;
    464:       } else {
    465:         bjiu = 0.0;
    466:         bjiv = 0.0;
    467:       }
    468:       x[j][i].u = lid + bjiu;
    469:       x[j][i].v = bjiv;
    470:     }
    471:   }

    473:   /* Test whether we are on the left edge of the global array */
    474:   if (xints == 0) {
    475:     i     = 0;
    476:     xints = xints + 1;
    477:     /* left edge */
    478:     for (j=info.ys; j<info.ys+info.ym; j++) {
    479:       if (B) {
    480:         bjiu = b[j][i].u;
    481:         bjiv = b[j][i].v;
    482:       } else {
    483:         bjiu = 0.0;
    484:         bjiv = 0.0;
    485:       }
    486:       x[j][i].u = 0.0 + bjiu;
    487:       x[j][i].v = 0.0 + bjiv;
    488:     }
    489:   }

    491:   /* Test whether we are on the right edge of the global array */
    492:   if (xinte == info.mx) {
    493:     i     = info.mx - 1;
    494:     xinte = xinte - 1;
    495:     /* right edge */
    496:     for (j=info.ys; j<info.ys+info.ym; j++) {
    497:       if (B) {
    498:         bjiu = b[j][i].u;
    499:         bjiv = b[j][i].v;
    500:       } else {
    501:         bjiu = 0.0;
    502:         bjiv = 0.0;
    503:       }
    504:       x[j][i].u = 0.0 + bjiu;
    505:       x[j][i].v = 0.0 + bjiv;
    506:     }
    507:   }

    509:   for (k=0; k < sweeps; k++) {
    510:     for (j=info.ys; j<info.ys + info.ym; j++) {
    511:       for (i=info.xs; i<info.xs + info.xm; i++) {
    512:         ptconverged = PETSC_FALSE;
    513:         pfnorm0     = 0.0;
    514:         pfnorm      = 0.0;
    515:         fu          = 0.0;
    516:         fv          = 0.0;
    517:         fomega      = 0.0;
    518:         ftemp       = 0.0;
    519:         for (l = 0; l < max_its && !ptconverged; l++) {
    520:           if (B) {
    521:             bjiu     = b[j][i].u;
    522:             bjiv     = b[j][i].v;
    523:             bjiomega = b[j][i].omega;
    524:             bjitemp  = b[j][i].temp;
    525:           } else {
    526:             bjiu     = 0.0;
    527:             bjiv     = 0.0;
    528:             bjiomega = 0.0;
    529:             bjitemp  = 0.0;
    530:           }

    532:           if (i != 0 && i != info.mx - 1 && j != 0 && j != info.my-1) {
    533:             /* U velocity */
    534:             u     = x[j][i].u;
    535:             uxx   = (2.0*u - x[j][i-1].u - x[j][i+1].u)*hydhx;
    536:             uyy   = (2.0*u - x[j-1][i].u - x[j+1][i].u)*hxdhy;
    537:             fu    = uxx + uyy - .5*(x[j+1][i].omega-x[j-1][i].omega)*hx - bjiu;
    538:             dfudu = 2.0*(hydhx + hxdhy);
    539:             /* V velocity */
    540:             u     = x[j][i].v;
    541:             uxx   = (2.0*u - x[j][i-1].v - x[j][i+1].v)*hydhx;
    542:             uyy   = (2.0*u - x[j-1][i].v - x[j+1][i].v)*hxdhy;
    543:             fv    = uxx + uyy + .5*(x[j][i+1].omega-x[j][i-1].omega)*hy - bjiv;
    544:             dfvdv = 2.0*(hydhx + hxdhy);
    545:             /*
    546:              convective coefficients for upwinding
    547:              */
    548:             vx  = x[j][i].u; avx = PetscAbsScalar(vx);
    549:             vxp = .5*(vx+avx); vxm = .5*(vx-avx);
    550:             vy  = x[j][i].v; avy = PetscAbsScalar(vy);
    551:             vyp = .5*(vy+avy); vym = .5*(vy-avy);
    552:             /* Omega */
    553:             u      = x[j][i].omega;
    554:             uxx    = (2.0*u - x[j][i-1].omega - x[j][i+1].omega)*hydhx;
    555:             uyy    = (2.0*u - x[j-1][i].omega - x[j+1][i].omega)*hxdhy;
    556:             fomega = uxx + uyy +  (vxp*(u - x[j][i-1].omega) + vxm*(x[j][i+1].omega - u))*hy +
    557:                      (vyp*(u - x[j-1][i].omega) + vym*(x[j+1][i].omega - u))*hx -
    558:                      .5*grashof*(x[j][i+1].temp - x[j][i-1].temp)*hy - bjiomega;
    559:             /* convective coefficient derivatives */
    560:             dfodo = 2.0*(hydhx + hxdhy) + ((vxp - vxm)*hy + (vyp - vym)*hx);
    561:             if (PetscRealPart(vx) > 0.0) dfodu = (u - x[j][i-1].omega)*hy;
    562:             else dfodu = (x[j][i+1].omega - u)*hy;

    564:             if (PetscRealPart(vy) > 0.0) dfodv = (u - x[j-1][i].omega)*hx;
    565:             else dfodv = (x[j+1][i].omega - u)*hx;

    567:             /* Temperature */
    568:             u     = x[j][i].temp;
    569:             uxx   = (2.0*u - x[j][i-1].temp - x[j][i+1].temp)*hydhx;
    570:             uyy   = (2.0*u - x[j-1][i].temp - x[j+1][i].temp)*hxdhy;
    571:             ftemp =  uxx + uyy  + prandtl*((vxp*(u - x[j][i-1].temp) + vxm*(x[j][i+1].temp - u))*hy +
    572:                                            (vyp*(u - x[j-1][i].temp) + vym*(x[j+1][i].temp - u))*hx) - bjitemp;
    573:             dftdt = 2.0*(hydhx + hxdhy) + prandtl*((vxp - vxm)*hy + (vyp - vym)*hx);
    574:             if (PetscRealPart(vx) > 0.0) dftdu = prandtl*(u - x[j][i-1].temp)*hy;
    575:             else dftdu = prandtl*(x[j][i+1].temp - u)*hy;

    577:             if (PetscRealPart(vy) > 0.0) dftdv = prandtl*(u - x[j-1][i].temp)*hx;
    578:             else dftdv = prandtl*(x[j+1][i].temp - u)*hx;

    580:             /* invert the system:
    581:              [ dfu / du     0        0        0    ][yu] = [fu]
    582:              [     0    dfv / dv     0        0    ][yv]   [fv]
    583:              [ dfo / du dfo / dv dfo / do     0    ][yo]   [fo]
    584:              [ dft / du dft / dv     0    dft / dt ][yt]   [ft]
    585:              by simple back-substitution
    586:            */
    587:             yu = fu / dfudu;
    588:             yv = fv / dfvdv;
    589:             yo = fomega / dfodo;
    590:             yt = ftemp / dftdt;
    591:             yo = (fomega - (dfodu*yu + dfodv*yv)) / dfodo;
    592:             yt = (ftemp - (dftdu*yu + dftdv*yv)) / dftdt;

    594:             x[j][i].u     = x[j][i].u - yu;
    595:             x[j][i].v     = x[j][i].v - yv;
    596:             x[j][i].temp  = x[j][i].temp - yt;
    597:             x[j][i].omega = x[j][i].omega - yo;
    598:           }
    599:           if (i == 0) {
    600:             fomega        = x[j][i].omega - (x[j][i+1].v - x[j][i].v)*dhx - bjiomega;
    601:             ftemp         = x[j][i].temp - bjitemp;
    602:             yo            = fomega;
    603:             yt            = ftemp;
    604:             x[j][i].omega = x[j][i].omega - fomega;
    605:             x[j][i].temp  = x[j][i].temp - ftemp;
    606:           }
    607:           if (i == info.mx - 1) {
    608:             fomega        = x[j][i].omega - (x[j][i].v - x[j][i-1].v)*dhx - bjiomega;
    609:             ftemp         = x[j][i].temp - (PetscReal)(grashof>0) - bjitemp;
    610:             yo            = fomega;
    611:             yt            = ftemp;
    612:             x[j][i].omega = x[j][i].omega - fomega;
    613:             x[j][i].temp  = x[j][i].temp - ftemp;
    614:           }
    615:           if (j == 0) {
    616:             fomega        = x[j][i].omega + (x[j+1][i].u - x[j][i].u)*dhy - bjiomega;
    617:             ftemp         = x[j][i].temp-x[j+1][i].temp - bjitemp;
    618:             yo            = fomega;
    619:             yt            = ftemp;
    620:             x[j][i].omega = x[j][i].omega - fomega;
    621:             x[j][i].temp  = x[j][i].temp - ftemp;
    622:           }
    623:           if (j == info.my - 1) {
    624:             fomega        = x[j][i].omega + (x[j][i].u - x[j-1][i].u)*dhy - bjiomega;
    625:             ftemp         = x[j][i].temp-x[j-1][i].temp - bjitemp;
    626:             yo            = fomega;
    627:             yt            = ftemp;
    628:             x[j][i].omega = x[j][i].omega - fomega;
    629:             x[j][i].temp  = x[j][i].temp - ftemp;
    630:           }
    631:           tot_its++;
    632:           pfnorm = PetscRealPart(fu*fu + fv*fv + fomega*fomega + ftemp*ftemp);
    633:           pfnorm = PetscSqrtReal(pfnorm);
    634:           pynorm = PetscRealPart(yu*yu + yv*yv + yo*yo + yt*yt);
    635:           pfnorm = PetscSqrtReal(pynorm);
    636:           pxnorm = PetscRealPart(x[j][i].u*x[j][i].u + x[j][i].v*x[j][i].v + x[j][i].omega*x[j][i].omega + x[j][i].temp*x[j][i].temp);
    637:           pxnorm = PetscSqrtReal(pxnorm);
    638:           if (l == 0) pfnorm0 = pfnorm;
    639:           if (rtol*pfnorm0 > pfnorm ||
    640:               atol > pfnorm ||
    641:               pxnorm*stol > pynorm) ptconverged = PETSC_TRUE;
    642:         }
    643:       }
    644:     }
    645:   }
    646:   DMDAVecRestoreArray(da,localX,&x);
    647:   if (B) {
    648:     DMDAVecRestoreArray(da,localB,&b);
    649:   }
    650:   DMLocalToGlobalBegin(da,localX,INSERT_VALUES,X);
    651:   DMLocalToGlobalEnd(da,localX,INSERT_VALUES,X);
    652:   PetscLogFlops(tot_its*(84.0 + 41.0 + 26.0));
    653:   if (B) {
    654:     DMLocalToGlobalBegin(da,localB,INSERT_VALUES,B);
    655:     DMLocalToGlobalEnd(da,localB,INSERT_VALUES,B);
    656:   }
    657:   DMRestoreLocalVector(da,&localX);
    658:   if (B) {
    659:     DMRestoreLocalVector(da,&localB);
    660:   }
    661:   return(0);
    662: }