Actual source code: ex19.c

petsc-master 2015-07-31
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  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 &ltlid&gt, where &ltlid&gt = dimensionless velocity of lid\n\
  7:   -grashof &ltgr&gt, where &ltgr&gt = dimensionless temperature gradent\n\
  8:   -prandtl &ltpr&gt, where &ltpr&gt = dimensionless thermal/momentum diffusity ratio\n\
  9:  -contours : draw contour plots of solution\n\n";
 10: /* in HTML, '&lt' = '<' and '&gt' = '>' */

 12: /*
 13:       See src/ksp/ksp/examples/tutorials/ex45.c
 14: */

 16: /*T
 17:    Concepts: SNES^solving a system of nonlinear equations (parallel multicomponent example);
 18:    Concepts: DMDA^using distributed arrays;
 19:    Concepts: multicomponent
 20:    Processors: n
 21: 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).

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

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

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

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

 91: extern PetscErrorCode FormInitialGuess(AppCtx*,DM,Vec);
 92: extern PetscErrorCode NonlinearGS(SNES,Vec,Vec,void*);

 96: int main(int argc,char **argv)
 97: {
 98:   AppCtx         user;                /* user-defined work context */
 99:   PetscInt       mx,my,its;
101:   MPI_Comm       comm;
102:   SNES           snes;
103:   DM             da;
104:   Vec            x;

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

109:   comm = PETSC_COMM_WORLD;
110:   SNESCreate(comm,&snes);

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

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

129:   PetscOptionsGetReal(NULL,"-lidvelocity",&user.lidvelocity,NULL);
130:   PetscOptionsGetReal(NULL,"-prandtl",&user.prandtl,NULL);
131:   PetscOptionsGetReal(NULL,"-grashof",&user.grashof,NULL);
132:   PetscOptionsHasName(NULL,"-contours",&user.draw_contours);

134:   DMDASetFieldName(da,0,"x_velocity");
135:   DMDASetFieldName(da,1,"y_velocity");
136:   DMDASetFieldName(da,2,"Omega");
137:   DMDASetFieldName(da,3,"temperature");

139:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
140:      Create user context, set problem data, create vector data structures.
141:      Also, compute the initial guess.
142:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

144:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
145:      Create nonlinear solver context

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


154:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
155:      Solve the nonlinear system
156:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
157:   DMCreateGlobalVector(da,&x);
158:   FormInitialGuess(&user,da,x);

160:   SNESSolve(snes,NULL,x);

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

165:   /*
166:      Visualize solution
167:   */
168:   if (user.draw_contours) {
169:     VecView(x,PETSC_VIEWER_DRAW_WORLD);
170:   }

172:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
173:      Free work space.  All PETSc objects should be destroyed when they
174:      are no longer needed.
175:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
176:   VecDestroy(&x);
177:   DMDestroy(&da);
178:   SNESDestroy(&snes);
179:   PetscFinalize();
180:   return 0;
181: }

183: /* ------------------------------------------------------------------- */


188: /*
189:    FormInitialGuess - Forms initial approximation.

191:    Input Parameters:
192:    user - user-defined application context
193:    X - vector

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

206:   grashof = user->grashof;

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

211:   /*
212:      Get local grid boundaries (for 2-dimensional DMDA):
213:        xs, ys   - starting grid indices (no ghost points)
214:        xm, ym   - widths of local grid (no ghost points)
215:   */
216:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

218:   /*
219:      Get a pointer to vector data.
220:        - For default PETSc vectors, VecGetArray() returns a pointer to
221:          the data array.  Otherwise, the routine is implementation dependent.
222:        - You MUST call VecRestoreArray() when you no longer need access to
223:          the array.
224:   */
225:   DMDAVecGetArray(da,X,&x);

227:   /*
228:      Compute initial guess over the locally owned part of the grid
229:      Initial condition is motionless fluid and equilibrium temperature
230:   */
231:   for (j=ys; j<ys+ym; j++) {
232:     for (i=xs; i<xs+xm; i++) {
233:       x[j][i].u     = 0.0;
234:       x[j][i].v     = 0.0;
235:       x[j][i].omega = 0.0;
236:       x[j][i].temp  = (grashof>0)*i*dx;
237:     }
238:   }

240:   /*
241:      Restore vector
242:   */
243:   DMDAVecRestoreArray(da,X,&x);
244:   return(0);
245: }

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

259:   grashof = user->grashof;
260:   prandtl = user->prandtl;
261:   lid     = user->lidvelocity;

263:   /*
264:      Define mesh intervals ratios for uniform grid.

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


270:   */
271:   dhx   = (PetscReal)(info->mx-1);  dhy = (PetscReal)(info->my-1);
272:   hx    = 1.0/dhx;                   hy = 1.0/dhy;
273:   hxdhy = hx*dhy;                 hydhx = hy*dhx;

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

277:   /* Test whether we are on the bottom edge of the global array */
278:   if (yints == 0) {
279:     j     = 0;
280:     yints = yints + 1;
281:     /* bottom edge */
282:     for (i=info->xs; i<info->xs+info->xm; i++) {
283:       f[j][i].u     = x[j][i].u;
284:       f[j][i].v     = x[j][i].v;
285:       f[j][i].omega = x[j][i].omega + (x[j+1][i].u - x[j][i].u)*dhy;
286:       f[j][i].temp  = x[j][i].temp-x[j+1][i].temp;
287:     }
288:   }

290:   /* Test whether we are on the top edge of the global array */
291:   if (yinte == info->my) {
292:     j     = info->my - 1;
293:     yinte = yinte - 1;
294:     /* top edge */
295:     for (i=info->xs; i<info->xs+info->xm; i++) {
296:       f[j][i].u     = x[j][i].u - lid;
297:       f[j][i].v     = x[j][i].v;
298:       f[j][i].omega = x[j][i].omega + (x[j][i].u - x[j-1][i].u)*dhy;
299:       f[j][i].temp  = x[j][i].temp-x[j-1][i].temp;
300:     }
301:   }

303:   /* Test whether we are on the left edge of the global array */
304:   if (xints == 0) {
305:     i     = 0;
306:     xints = xints + 1;
307:     /* left edge */
308:     for (j=info->ys; j<info->ys+info->ym; j++) {
309:       f[j][i].u     = x[j][i].u;
310:       f[j][i].v     = x[j][i].v;
311:       f[j][i].omega = x[j][i].omega - (x[j][i+1].v - x[j][i].v)*dhx;
312:       f[j][i].temp  = x[j][i].temp;
313:     }
314:   }

316:   /* Test whether we are on the right edge of the global array */
317:   if (xinte == info->mx) {
318:     i     = info->mx - 1;
319:     xinte = xinte - 1;
320:     /* right edge */
321:     for (j=info->ys; j<info->ys+info->ym; j++) {
322:       f[j][i].u     = x[j][i].u;
323:       f[j][i].v     = x[j][i].v;
324:       f[j][i].omega = x[j][i].omega - (x[j][i].v - x[j][i-1].v)*dhx;
325:       f[j][i].temp  = x[j][i].temp - (PetscReal)(grashof>0);
326:     }
327:   }

329:   /* Compute over the interior points */
330:   for (j=yints; j<yinte; j++) {
331:     for (i=xints; i<xinte; i++) {

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

341:       /* U velocity */
342:       u         = x[j][i].u;
343:       uxx       = (2.0*u - x[j][i-1].u - x[j][i+1].u)*hydhx;
344:       uyy       = (2.0*u - x[j-1][i].u - x[j+1][i].u)*hxdhy;
345:       f[j][i].u = uxx + uyy - .5*(x[j+1][i].omega-x[j-1][i].omega)*hx;

347:       /* V velocity */
348:       u         = x[j][i].v;
349:       uxx       = (2.0*u - x[j][i-1].v - x[j][i+1].v)*hydhx;
350:       uyy       = (2.0*u - x[j-1][i].v - x[j+1][i].v)*hxdhy;
351:       f[j][i].v = uxx + uyy + .5*(x[j][i+1].omega-x[j][i-1].omega)*hy;

353:       /* Omega */
354:       u             = x[j][i].omega;
355:       uxx           = (2.0*u - x[j][i-1].omega - x[j][i+1].omega)*hydhx;
356:       uyy           = (2.0*u - x[j-1][i].omega - x[j+1][i].omega)*hxdhy;
357:       f[j][i].omega = uxx + uyy + (vxp*(u - x[j][i-1].omega) + vxm*(x[j][i+1].omega - u))*hy +
358:                       (vyp*(u - x[j-1][i].omega) + vym*(x[j+1][i].omega - u))*hx -
359:                       .5*grashof*(x[j][i+1].temp - x[j][i-1].temp)*hy;

361:       /* Temperature */
362:       u            = x[j][i].temp;
363:       uxx          = (2.0*u - x[j][i-1].temp - x[j][i+1].temp)*hydhx;
364:       uyy          = (2.0*u - x[j-1][i].temp - x[j+1][i].temp)*hxdhy;
365:       f[j][i].temp =  uxx + uyy  + prandtl*((vxp*(u - x[j][i-1].temp) + vxm*(x[j][i+1].temp - u))*hy +
366:                                             (vyp*(u - x[j-1][i].temp) + vym*(x[j+1][i].temp - u))*hx);
367:     }
368:   }

370:   /*
371:      Flop count (multiply-adds are counted as 2 operations)
372:   */
373:   PetscLogFlops(84.0*info->ym*info->xm);
374:   return(0);
375: }


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

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

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

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

447:       if (B) {
448:         bjiu = b[j][i].u;
449:         bjiv = b[j][i].v;
450:       } else {
451:         bjiu = 0.0;
452:         bjiv = 0.0;
453:       }
454:       x[j][i].u = 0.0 + bjiu;
455:       x[j][i].v = 0.0 + bjiv;
456:     }
457:   }

459:   /* Test whether we are on the top edge of the global array */
460:   if (yinte == info.my) {
461:     j     = info.my - 1;
462:     yinte = yinte - 1;
463:     /* top edge */
464:     for (i=info.xs; i<info.xs+info.xm; i++) {
465:       if (B) {
466:         bjiu = b[j][i].u;
467:         bjiv = b[j][i].v;
468:       } else {
469:         bjiu = 0.0;
470:         bjiv = 0.0;
471:       }
472:       x[j][i].u = lid + bjiu;
473:       x[j][i].v = bjiv;
474:     }
475:   }

477:   /* Test whether we are on the left edge of the global array */
478:   if (xints == 0) {
479:     i     = 0;
480:     xints = xints + 1;
481:     /* left edge */
482:     for (j=info.ys; j<info.ys+info.ym; j++) {
483:       if (B) {
484:         bjiu = b[j][i].u;
485:         bjiv = b[j][i].v;
486:       } else {
487:         bjiu = 0.0;
488:         bjiv = 0.0;
489:       }
490:       x[j][i].u = 0.0 + bjiu;
491:       x[j][i].v = 0.0 + bjiv;
492:     }
493:   }

495:   /* Test whether we are on the right edge of the global array */
496:   if (xinte == info.mx) {
497:     i     = info.mx - 1;
498:     xinte = xinte - 1;
499:     /* right edge */
500:     for (j=info.ys; j<info.ys+info.ym; j++) {
501:       if (B) {
502:         bjiu = b[j][i].u;
503:         bjiv = b[j][i].v;
504:       } else {
505:         bjiu = 0.0;
506:         bjiv = 0.0;
507:       }
508:       x[j][i].u = 0.0 + bjiu;
509:       x[j][i].v = 0.0 + bjiv;
510:     }
511:   }

513:   for (k=0; k < sweeps; k++) {
514:     for (j=info.ys; j<info.ys + info.ym; j++) {
515:       for (i=info.xs; i<info.xs + info.xm; i++) {
516:         ptconverged = PETSC_FALSE;
517:         pfnorm0     = 0.0;
518:         pfnorm      = 0.0;
519:         fu          = 0.0;
520:         fv          = 0.0;
521:         fomega      = 0.0;
522:         ftemp       = 0.0;
523:         for (l = 0; l < max_its && !ptconverged; l++) {
524:           if (B) {
525:             bjiu     = b[j][i].u;
526:             bjiv     = b[j][i].v;
527:             bjiomega = b[j][i].omega;
528:             bjitemp  = b[j][i].temp;
529:           } else {
530:             bjiu     = 0.0;
531:             bjiv     = 0.0;
532:             bjiomega = 0.0;
533:             bjitemp  = 0.0;
534:           }

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

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

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

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

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

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