Actual source code: ex19.c
petsc-3.15.0 2021-03-30
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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/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).

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

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

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

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

     93: extern PetscErrorCode FormInitialGuess(AppCtx*,DM,Vec);
     94: 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 ierr;

    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:   DMSetFromOptions(da);
    118:   DMSetUp(da);
    119:   SNESSetDM(snes,(DM)da);
    120:   SNESSetNGS(snes, NonlinearGS, (void*)&user);

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

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

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

    140:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    141:      Create user context, set problem data, create vector data structures.
    142:      Also, compute the initial guess.
    143:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

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

    161:   SNESSolve(snes,NULL,x);

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

    166:   /*
    167:      Visualize solution
    168:   */
    169:   if (user.draw_contours) {
    170:     VecView(x,PETSC_VIEWER_DRAW_WORLD);
    171:   }

    173:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    174:      Free work space.  All PETSc objects should be destroyed when they
    175:      are no longer needed.
    176:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    177:   VecDestroy(&x);
    178:   DMDestroy(&da);
    179:   SNESDestroy(&snes);
    180:   PetscFinalize();
    181:   return ierr;
    182: }

    184: /* ------------------------------------------------------------------- */

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

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

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

    204:   grashof = user->grashof;

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

    209:   /*
    210:      Get local grid boundaries (for 2-dimensional DMDA):
    211:        xs, ys   - starting grid indices (no ghost points)
    212:        xm, ym   - widths of local grid (no ghost points)
    213:   */
    214:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

    216:   /*
    217:      Get a pointer to vector data.
    218:        - For default PETSc vectors, VecGetArray() returns a pointer to
    219:          the data array.  Otherwise, the routine is implementation dependent.
    220:        - You MUST call VecRestoreArray() when you no longer need access to
    221:          the array.
    222:   */
    223:   DMDAVecGetArrayWrite(da,X,&x);

    225:   /*
    226:      Compute initial guess over the locally owned part of the grid
    227:      Initial condition is motionless fluid and equilibrium temperature
    228:   */
    229:   for (j=ys; j<ys+ym; j++) {
    230:     for (i=xs; i<xs+xm; i++) {
    231:       x[j][i].u     = 0.0;
    232:       x[j][i].v     = 0.0;
    233:       x[j][i].omega = 0.0;
    234:       x[j][i].temp  = (grashof>0)*i*dx;
    235:     }
    236:   }

    238:   /*
    239:      Restore vector
    240:   */
    241:   DMDAVecRestoreArrayWrite(da,X,&x);
    242:   return(0);
    243: }

    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: }

    373: /*
    374:     Performs sweeps of point block nonlinear Gauss-Seidel on all the local grid points
    375: */
    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:   SNESNGSGetTolerances(snes,&rtol,&atol,&stol,&max_its);
    407:   SNESNGSGetSweeps(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:   DMDAVecGetArrayWrite(da,localX,&x);
    425:   if (B) {
    426:     DMDAVecGetArrayRead(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:     /* bottom edge */
    440:     for (i=info.xs; i<info.xs+info.xm; i++) {

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

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

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

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

    505:   for (k=0; k < sweeps; k++) {
    506:     for (j=info.ys; j<info.ys + info.ym; j++) {
    507:       for (i=info.xs; i<info.xs + info.xm; i++) {
    508:         ptconverged = PETSC_FALSE;
    509:         pfnorm0     = 0.0;
    510:         fu          = 0.0;
    511:         fv          = 0.0;
    512:         fomega      = 0.0;
    513:         ftemp       = 0.0;
    514:         /*  Run Newton's method on a single grid point */
    515:         for (l = 0; l < max_its && !ptconverged; l++) {
    516:           if (B) {
    517:             bjiu     = b[j][i].u;
    518:             bjiv     = b[j][i].v;
    519:             bjiomega = b[j][i].omega;
    520:             bjitemp  = b[j][i].temp;
    521:           } else {
    522:             bjiu     = 0.0;
    523:             bjiv     = 0.0;
    524:             bjiomega = 0.0;
    525:             bjitemp  = 0.0;
    526:           }

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

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

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

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

    575:             /* invert the system:
    576:              [ dfu / du     0        0        0    ][yu] = [fu]
    577:              [     0    dfv / dv     0        0    ][yv]   [fv]
    578:              [ dfo / du dfo / dv dfo / do     0    ][yo]   [fo]
    579:              [ dft / du dft / dv     0    dft / dt ][yt]   [ft]
    580:              by simple back-substitution
    581:            */
    582:             yu = fu / dfudu;
    583:             yv = fv / dfvdv;
    584:             yo = (fomega - (dfodu*yu + dfodv*yv)) / dfodo;
    585:             yt = (ftemp - (dftdu*yu + dftdv*yv)) / dftdt;

    587:             x[j][i].u     = x[j][i].u - yu;
    588:             x[j][i].v     = x[j][i].v - yv;
    589:             x[j][i].temp  = x[j][i].temp - yt;
    590:             x[j][i].omega = x[j][i].omega - yo;
    591:           }
    592:           if (i == 0) {
    593:             fomega        = x[j][i].omega - (x[j][i+1].v - x[j][i].v)*dhx - bjiomega;
    594:             ftemp         = x[j][i].temp - bjitemp;
    595:             yo            = fomega;
    596:             yt            = ftemp;
    597:             x[j][i].omega = x[j][i].omega - fomega;
    598:             x[j][i].temp  = x[j][i].temp - ftemp;
    599:           }
    600:           if (i == info.mx - 1) {
    601:             fomega        = x[j][i].omega - (x[j][i].v - x[j][i-1].v)*dhx - bjiomega;
    602:             ftemp         = x[j][i].temp - (PetscReal)(grashof>0) - bjitemp;
    603:             yo            = fomega;
    604:             yt            = ftemp;
    605:             x[j][i].omega = x[j][i].omega - fomega;
    606:             x[j][i].temp  = x[j][i].temp - ftemp;
    607:           }
    608:           if (j == 0) {
    609:             fomega        = x[j][i].omega + (x[j+1][i].u - x[j][i].u)*dhy - bjiomega;
    610:             ftemp         = x[j][i].temp-x[j+1][i].temp - bjitemp;
    611:             yo            = fomega;
    612:             yt            = ftemp;
    613:             x[j][i].omega = x[j][i].omega - fomega;
    614:             x[j][i].temp  = x[j][i].temp - ftemp;
    615:           }
    616:           if (j == info.my - 1) {
    617:             fomega        = x[j][i].omega + (x[j][i].u - x[j-1][i].u)*dhy - bjiomega;
    618:             ftemp         = x[j][i].temp-x[j-1][i].temp - bjitemp;
    619:             yo            = fomega;
    620:             yt            = ftemp;
    621:             x[j][i].omega = x[j][i].omega - fomega;
    622:             x[j][i].temp  = x[j][i].temp - ftemp;
    623:           }
    624:           tot_its++;
    625:           pfnorm = PetscRealPart(fu*fu + fv*fv + fomega*fomega + ftemp*ftemp);
    626:           pfnorm = PetscSqrtReal(pfnorm);
    627:           pynorm = PetscRealPart(yu*yu + yv*yv + yo*yo + yt*yt);
    628:           pynorm = PetscSqrtReal(pynorm);
    629:           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);
    630:           pxnorm = PetscSqrtReal(pxnorm);
    631:           if (l == 0) pfnorm0 = pfnorm;
    632:           if (rtol*pfnorm0 >pfnorm || atol > pfnorm || pxnorm*stol > pynorm) ptconverged = PETSC_TRUE;
    633:         }
    634:       }
    635:     }
    636:   }
    637:   DMDAVecRestoreArrayWrite(da,localX,&x);
    638:   if (B) {
    639:     DMDAVecRestoreArrayRead(da,localB,&b);
    640:   }
    641:   DMLocalToGlobalBegin(da,localX,INSERT_VALUES,X);
    642:   DMLocalToGlobalEnd(da,localX,INSERT_VALUES,X);
    643:   PetscLogFlops(tot_its*(84.0 + 41.0 + 26.0));
    644:   DMRestoreLocalVector(da,&localX);
    645:   if (B) {
    646:     DMRestoreLocalVector(da,&localB);
    647:   }
    648:   return(0);
    649: }

    652: /*TEST

    654:    test:
    655:       nsize: 2
    656:       args: -da_refine 3 -snes_monitor_short -pc_type mg -ksp_type fgmres -pc_mg_type full
    657:       requires: !single

    659:    test:
    660:       suffix: 10
    661:       nsize: 3
    662:       args: -snes_monitor_short -ksp_monitor_short -pc_type fieldsplit -pc_fieldsplit_type symmetric_multiplicative -snes_view -da_refine 1 -ksp_type fgmres
    663:       requires: !single

    665:    test:
    666:       suffix: 11
    667:       nsize: 4
    668:       requires: pastix
    669:       args: -snes_monitor_short -pc_type redundant -dm_mat_type mpiaij -redundant_pc_factor_mat_solver_type pastix -pc_redundant_number 2 -da_refine 4 -ksp_type fgmres

    671:    test:
    672:       suffix: 12
    673:       nsize: 12
    674:       requires: pastix
    675:       args: -snes_monitor_short -pc_type redundant -dm_mat_type mpiaij -redundant_pc_factor_mat_solver_type pastix -pc_redundant_number 5 -da_refine 4 -ksp_type fgmres

    677:    test:
    678:       suffix: 13
    679:       nsize: 3
    680:       args: -snes_monitor_short -ksp_monitor_short -pc_type fieldsplit -pc_fieldsplit_type multiplicative -snes_view -da_refine 1 -ksp_type fgmres -snes_mf_operator
    681:       requires: !single

    683:    test:
    684:       suffix: 14
    685:       nsize: 4
    686:       args: -snes_monitor_short -pc_type mg -dm_mat_type baij -mg_coarse_pc_type bjacobi -da_refine 3 -ksp_type fgmres
    687:       requires: !single

    689:    test:
    690:       suffix: 14_ds
    691:       nsize: 4
    692:       args: -snes_converged_reason -pc_type mg -dm_mat_type baij -mg_coarse_pc_type bjacobi -da_refine 3 -ksp_type fgmres -mat_fd_type ds
    693:       output_file: output/ex19_2.out
    694:       requires: !single

    696:    test:
    697:       suffix: 17
    698:       args: -snes_monitor_short -ksp_pc_side right
    699:       requires: !single

    701:    test:
    702:       suffix: 18
    703:       args: -snes_monitor_ksp draw::draw_lg -ksp_pc_side right
    704:       requires: x !single

    706:    test:
    707:       suffix: 2
    708:       nsize: 4
    709:       args: -da_refine 3 -snes_converged_reason -pc_type mg -mat_fd_type ds
    710:       requires: !single

    712:    test:
    713:       suffix: 2_bcols1
    714:       nsize: 4
    715:       args: -da_refine 3 -snes_converged_reason -pc_type mg -mat_fd_type ds -mat_fd_coloring_bcols
    716:       output_file: output/ex19_2.out
    717:       requires: !single

    719:    test:
    720:       suffix: 3
    721:       nsize: 4
    722:       requires: mumps
    723:       args: -da_refine 3 -snes_monitor_short -pc_type redundant -dm_mat_type mpiaij -redundant_ksp_type preonly -redundant_pc_factor_mat_solver_type mumps -pc_redundant_number 2

    725:    test:
    726:       suffix: 4
    727:       nsize: 12
    728:       requires: mumps
    729:       args: -da_refine 3 -snes_monitor_short -pc_type redundant -dm_mat_type mpiaij -redundant_ksp_type preonly -redundant_pc_factor_mat_solver_type mumps -pc_redundant_number 5
    730:       output_file: output/ex19_3.out

    732:    test:
    733:       suffix: 6
    734:       args: -snes_monitor_short -ksp_monitor_short -pc_type fieldsplit -snes_view -ksp_type fgmres -da_refine 1
    735:       requires: !single

    737:    test:
    738:       suffix: 7
    739:       nsize: 3
    740:       args: -snes_monitor_short -ksp_monitor_short -pc_type fieldsplit -snes_view -da_refine 1 -ksp_type fgmres

    742:       requires: !single
    743:    test:
    744:       suffix: 8
    745:       args: -snes_monitor_short -ksp_monitor_short -pc_type fieldsplit -pc_fieldsplit_block_size 2 -pc_fieldsplit_0_fields 0,1 -pc_fieldsplit_1_fields 0,1 -pc_fieldsplit_type multiplicative -snes_view -fieldsplit_pc_type lu -da_refine 1 -ksp_type fgmres
    746:       requires: !single

    748:    test:
    749:       suffix: 9
    750:       nsize: 3
    751:       args: -snes_monitor_short -ksp_monitor_short -pc_type fieldsplit -pc_fieldsplit_type multiplicative -snes_view -da_refine 1 -ksp_type fgmres
    752:       requires: !single

    754:    test:
    755:       suffix: aspin
    756:       nsize: 4
    757:       args: -da_refine 3 -da_overlap 2 -snes_monitor_short -snes_type aspin -grashof 4e4 -lidvelocity 100 -ksp_monitor_short
    758:       requires: !single

    760:    test:
    761:       suffix: bcgsl
    762:       nsize: 2
    763:       args: -ksp_type bcgsl -ksp_monitor_short -da_refine 2 -ksp_bcgsl_ell 3 -snes_view
    764:       requires: !single

    766:    test:
    767:       suffix: bcols1
    768:       nsize: 2
    769:       args: -da_refine 3 -snes_monitor_short -pc_type mg -ksp_type fgmres -pc_mg_type full -mat_fd_coloring_bcols 1
    770:       output_file: output/ex19_1.out
    771:       requires: !single

    773:    test:
    774:       suffix: bjacobi
    775:       nsize: 4
    776:       args: -da_refine 4 -ksp_type fgmres -pc_type bjacobi -pc_bjacobi_blocks 2 -sub_ksp_type gmres -sub_ksp_max_it 2 -sub_pc_type bjacobi -sub_sub_ksp_type preonly -sub_sub_pc_type ilu -snes_monitor_short
    777:       requires: !single

    779:    test:
    780:       suffix: cgne
    781:       args: -da_refine 2 -pc_type lu -ksp_type cgne -ksp_monitor_short -ksp_converged_reason -ksp_view -ksp_norm_type unpreconditioned
    782:       filter: grep -v HERMITIAN
    783:       requires: !single

    785:    test:
    786:       suffix: cgs
    787:       args: -da_refine 1 -ksp_monitor_short -ksp_type cgs
    788:       requires: !single

    790:    test:
    791:       suffix: composite_fieldsplit
    792:       args: -ksp_type fgmres -pc_type composite -pc_composite_type MULTIPLICATIVE -pc_composite_pcs fieldsplit,none -sub_0_pc_fieldsplit_block_size 4 -sub_0_pc_fieldsplit_type additive -sub_0_pc_fieldsplit_0_fields 0,1,2 -sub_0_pc_fieldsplit_1_fields 3 -snes_monitor_short -ksp_monitor_short
    793:       requires: !single

    795:    test:
    796:       suffix: composite_fieldsplit_bjacobi
    797:       args: -ksp_type fgmres -pc_type composite -pc_composite_type MULTIPLICATIVE -pc_composite_pcs fieldsplit,bjacobi -sub_0_pc_fieldsplit_block_size 4 -sub_0_pc_fieldsplit_type additive -sub_0_pc_fieldsplit_0_fields 0,1,2 -sub_0_pc_fieldsplit_1_fields 3 -sub_1_pc_bjacobi_blocks 16 -sub_1_sub_pc_type lu -snes_monitor_short -ksp_monitor_short
    798:       requires: !single

    800:    test:
    801:       suffix: composite_fieldsplit_bjacobi_2
    802:       nsize: 4
    803:       args: -ksp_type fgmres -pc_type composite -pc_composite_type MULTIPLICATIVE -pc_composite_pcs fieldsplit,bjacobi -sub_0_pc_fieldsplit_block_size 4 -sub_0_pc_fieldsplit_type additive -sub_0_pc_fieldsplit_0_fields 0,1,2 -sub_0_pc_fieldsplit_1_fields 3 -sub_1_pc_bjacobi_blocks 16 -sub_1_sub_pc_type lu -snes_monitor_short -ksp_monitor_short
    804:       requires: !single

    806:    test:
    807:       suffix: composite_gs_newton
    808:       nsize: 2
    809:       args: -da_refine 3 -grashof 4e4 -lidvelocity 100 -snes_monitor_short -snes_type composite -snes_composite_type additiveoptimal -snes_composite_sneses ngs,newtonls -sub_0_snes_max_it 20 -sub_1_pc_type mg
    810:       requires: !single

    812:    test:
    813:       suffix: cuda
    814:       requires: cuda !single
    815:       args: -dm_vec_type cuda -dm_mat_type aijcusparse -pc_type none -ksp_type fgmres -snes_monitor_short -snes_rtol 1.e-5

    817:    test:
    818:       suffix: draw
    819:       args: -pc_type fieldsplit -snes_view draw -fieldsplit_x_velocity_pc_type mg -fieldsplit_x_velocity_pc_mg_galerkin pmat -fieldsplit_x_velocity_pc_mg_levels 2 -da_refine 1 -fieldsplit_x_velocity_mg_coarse_pc_type svd
    820:       requires: x !single

    822:    test:
    823:       suffix: drawports
    824:       args: -snes_monitor_solution draw::draw_ports -da_refine 1
    825:       output_file: output/ex19_draw.out
    826:       requires: x !single

    828:    test:
    829:       suffix: fas
    830:       args: -da_refine 4 -snes_monitor_short -snes_type fas -fas_levels_snes_type ngs -fas_levels_snes_ngs_sweeps 3 -fas_levels_snes_ngs_atol 0.0 -fas_levels_snes_ngs_stol 0.0 -grashof 4e4 -snes_fas_smoothup 6 -snes_fas_smoothdown 6 -lidvelocity 100
    831:       requires: !single

    833:    test:
    834:       suffix: fas_full
    835:       args: -da_refine 4 -snes_monitor_short -snes_type fas -snes_fas_type full -snes_fas_full_downsweep -fas_levels_snes_type ngs -fas_levels_snes_ngs_sweeps 3 -fas_levels_snes_ngs_atol 0.0 -fas_levels_snes_ngs_stol 0.0 -grashof 4e4 -snes_fas_smoothup 6 -snes_fas_smoothdown 6 -lidvelocity 100
    836:       requires: !single

    838:    test:
    839:       suffix: fdcoloring_ds
    840:       args: -da_refine 3 -snes_converged_reason -pc_type mg -mat_fd_type ds
    841:       output_file: output/ex19_2.out
    842:       requires: !single

    844:    test:
    845:       suffix: fdcoloring_ds_baij
    846:       args: -da_refine 3 -snes_converged_reason -pc_type mg -mat_fd_type ds -dm_mat_type baij
    847:       output_file: output/ex19_2.out
    848:       requires: !single

    850:    test:
    851:       suffix: fdcoloring_ds_bcols1
    852:       args: -da_refine 3 -snes_converged_reason -pc_type mg -mat_fd_type ds -mat_fd_coloring_bcols 1
    853:       output_file: output/ex19_2.out
    854:       requires: !single

    856:    test:
    857:       suffix: fdcoloring_wp
    858:       args: -da_refine 3 -snes_monitor_short -pc_type mg
    859:       requires: !single

    861:    test:
    862:       suffix: fdcoloring_wp_baij
    863:       args: -da_refine 3 -snes_monitor_short -pc_type mg -dm_mat_type baij
    864:       output_file: output/ex19_fdcoloring_wp.out
    865:       requires: !single

    867:    test:
    868:       suffix: fdcoloring_wp_bcols1
    869:       args: -da_refine 3 -snes_monitor_short -pc_type mg -mat_fd_coloring_bcols 1
    870:       output_file: output/ex19_fdcoloring_wp.out
    871:       requires: !single

    873:    test:
    874:       suffix: fieldsplit_2
    875:       args: -ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_block_size 4 -pc_fieldsplit_type additive -pc_fieldsplit_0_fields 0,1,2 -pc_fieldsplit_1_fields 3 -snes_monitor_short -ksp_monitor_short
    876:       requires: !single

    878:    test:
    879:       suffix: fieldsplit_3
    880:       args: -ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_block_size 4 -pc_fieldsplit_type additive -pc_fieldsplit_0_fields 0,1,2 -pc_fieldsplit_1_fields 3 -fieldsplit_0_pc_type lu -fieldsplit_1_pc_type lu -snes_monitor_short -ksp_monitor_short
    881:       requires: !single

    883:    test:
    884:       suffix: fieldsplit_4
    885:       args: -ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_block_size 4 -pc_fieldsplit_type SCHUR -pc_fieldsplit_0_fields 0,1,2 -pc_fieldsplit_1_fields 3 -fieldsplit_0_pc_type lu -fieldsplit_1_pc_type lu -snes_monitor_short -ksp_monitor_short
    886:       requires: !single

    888:    # HYPRE PtAP broken with complex numbers
    889:    test:
    890:       suffix: fieldsplit_hypre
    891:       nsize: 2
    892:       requires: hypre mumps !complex
    893:       args: -pc_type fieldsplit -pc_fieldsplit_block_size 4 -pc_fieldsplit_type SCHUR -pc_fieldsplit_0_fields 0,1,2 -pc_fieldsplit_1_fields 3 -fieldsplit_0_pc_type lu -fieldsplit_0_pc_factor_mat_solver_type mumps -fieldsplit_1_pc_type hypre -fieldsplit_1_pc_hypre_type boomeramg -snes_monitor_short -ksp_monitor_short

    895:    test:
    896:       suffix: fieldsplit_mumps
    897:       nsize: 2
    898:       requires: mumps
    899:       args: -pc_type fieldsplit -pc_fieldsplit_block_size 4 -pc_fieldsplit_type SCHUR -pc_fieldsplit_0_fields 0,1,2 -pc_fieldsplit_1_fields 3 -fieldsplit_0_pc_type lu -fieldsplit_1_pc_type lu -snes_monitor_short -ksp_monitor_short -fieldsplit_0_pc_factor_mat_solver_type mumps -fieldsplit_1_pc_factor_mat_solver_type mumps
    900:       output_file: output/ex19_fieldsplit_5.out

    902:    test:
    903:       suffix: greedy_coloring
    904:       nsize: 2
    905:       args: -da_refine 3 -snes_monitor_short -snes_fd_color -snes_fd_color_use_mat -mat_coloring_type greedy -mat_coloring_weight_type lf -mat_coloring_view> ex19_greedy_coloring.tmp 2>&1
    906:       requires: !single

    908:    # HYPRE PtAP broken with complex numbers
    909:    test:
    910:       suffix: hypre
    911:       nsize: 2
    912:       requires: hypre !complex
    913:       args: -da_refine 3 -snes_monitor_short -pc_type hypre

    915:    test:
    916:       suffix: ibcgs
    917:       nsize: 2
    918:       args: -ksp_type ibcgs -ksp_monitor_short -da_refine 2 -snes_view
    919:       requires: !complex !single

    921:    test:
    922:       suffix: kaczmarz
    923:       nsize: 2
    924:       args: -pc_type kaczmarz -ksp_monitor_short -snes_monitor_short -snes_view
    925:       requires: !single

    927:    test:
    928:       suffix: klu
    929:       requires: suitesparse
    930:       args: -da_grid_x 20 -da_grid_y 20 -pc_type lu -pc_factor_mat_solver_type klu
    931:       output_file: output/ex19_superlu.out

    933:    test:
    934:       suffix: klu_2
    935:       requires: suitesparse
    936:       args: -da_grid_x 20 -da_grid_y 20 -pc_type lu -pc_factor_mat_solver_type klu -mat_klu_ordering PETSC
    937:       output_file: output/ex19_superlu.out

    939:    test:
    940:       suffix: klu_3
    941:       requires: suitesparse
    942:       args: -da_grid_x 20 -da_grid_y 20 -pc_type lu -pc_factor_mat_solver_type klu -mat_klu_use_btf 0
    943:       output_file: output/ex19_superlu.out

    945:    test:
    946:       suffix: ml
    947:       nsize: 2
    948:       requires: ml
    949:       args: -da_refine 3 -snes_monitor_short -pc_type ml

    951:    test:
    952:       suffix: ngmres_fas
    953:       args: -da_refine 4 -snes_monitor_short -snes_type ngmres -npc_fas_levels_snes_type ngs -npc_fas_levels_snes_ngs_sweeps 3 -npc_fas_levels_snes_ngs_atol 0.0 -npc_fas_levels_snes_ngs_stol 0.0 -npc_snes_type fas -npc_fas_levels_snes_type ngs -npc_snes_max_it 1 -npc_snes_fas_smoothup 6 -npc_snes_fas_smoothdown 6 -lidvelocity 100 -grashof 4e4
    954:       requires: !single

    956:    test:
    957:       suffix: ngmres_fas_gssecant
    958:       args: -da_refine 3 -snes_monitor_short -snes_type ngmres -npc_snes_type fas -npc_fas_levels_snes_type ngs -npc_fas_levels_snes_max_it 6 -npc_fas_levels_snes_ngs_secant -npc_fas_levels_snes_ngs_max_it 1 -npc_fas_coarse_snes_max_it 1 -lidvelocity 100 -grashof 4e4
    959:       requires: !single

    961:    test:
    962:       suffix: ngmres_fas_ms
    963:       nsize: 2
    964:       args: -snes_grid_sequence 2 -lidvelocity 200 -grashof 1e4 -snes_monitor_short -snes_view -snes_converged_reason -snes_type ngmres -npc_snes_type fas -npc_fas_coarse_snes_type newtonls -npc_fas_coarse_ksp_type preonly -npc_snes_max_it 1
    965:       requires: !single

    967:    test:
    968:       suffix: ngmres_nasm
    969:       nsize: 4
    970:       args: -da_refine 4 -da_overlap 2 -snes_monitor_short -snes_type ngmres -snes_max_it 10 -npc_snes_type nasm -npc_snes_nasm_type basic -grashof 4e4 -lidvelocity 100
    971:       requires: !single

    973:    test:
    974:       suffix: ngs
    975:       args: -snes_type ngs -snes_view -snes_monitor -snes_rtol 1e-4
    976:       requires: !single

    978:    test:
    979:       suffix: ngs_fd
    980:       args: -snes_type ngs -snes_ngs_secant -snes_view -snes_monitor -snes_rtol 1e-4
    981:       requires: !single

    983:    test:
    984:       suffix: parms
    985:       nsize: 2
    986:       requires: parms
    987:       args: -pc_type parms -ksp_monitor_short -snes_view

    989:    test:
    990:       suffix: superlu
    991:       requires: superlu
    992:       args: -da_grid_x 20 -da_grid_y 20 -pc_type lu -pc_factor_mat_solver_type superlu

    994:    test:
    995:       suffix: superlu_sell
    996:       requires: superlu
    997:       args: -da_grid_x 20 -da_grid_y 20 -pc_type lu -pc_factor_mat_solver_type superlu -dm_mat_type sell -pc_factor_mat_ordering_type natural
    998:       output_file: output/ex19_superlu.out

    1000:    test:
    1001:       suffix: superlu_dist
    1002:       requires: superlu_dist
    1003:       args: -da_grid_x 20 -da_grid_y 20 -pc_type lu -pc_factor_mat_solver_type superlu_dist
    1004:       output_file: output/ex19_superlu.out

    1006:    test:
    1007:       suffix: superlu_dist_2
    1008:       nsize: 2
    1009:       requires: superlu_dist
    1010:       args: -da_grid_x 20 -da_grid_y 20 -pc_type lu -pc_factor_mat_solver_type superlu_dist
    1011:       output_file: output/ex19_superlu.out

    1013:    test:
    1014:       suffix: superlu_equil
    1015:       requires: superlu
    1016:       args: -da_grid_x 20 -da_grid_y 20 -{snes,ksp}_monitor_short -pc_type lu -pc_factor_mat_solver_type superlu -mat_superlu_equil

    1018:    test:
    1019:       suffix: superlu_equil_sell
    1020:       requires: superlu
    1021:       args: -da_grid_x 20 -da_grid_y 20 -{snes,ksp}_monitor_short -pc_type lu -pc_factor_mat_solver_type superlu -mat_superlu_equil -dm_mat_type sell -pc_factor_mat_ordering_type natural
    1022:       output_file: output/ex19_superlu_equil.out

    1024:    test:
    1025:       suffix: tcqmr
    1026:       args: -da_refine 1 -ksp_monitor_short -ksp_type tcqmr
    1027:       requires: !single

    1029:    test:
    1030:       suffix: tfqmr
    1031:       args: -da_refine 1 -ksp_monitor_short -ksp_type tfqmr
    1032:       requires: !single

    1034:    test:
    1035:       suffix: umfpack
    1036:       requires: suitesparse
    1037:       args: -da_refine 2 -pc_type lu -pc_factor_mat_solver_type umfpack -snes_view -snes_monitor_short -ksp_monitor_short -pc_factor_mat_ordering_type external

    1039:    test:
    1040:       suffix: tut_1
    1041:       nsize: 4
    1042:       requires: !single
    1043:       args: -da_refine 5 -snes_monitor -ksp_monitor -snes_view

    1045:    test:
    1046:       suffix: tut_2
    1047:       nsize: 4
    1048:       requires: !single
    1049:       args: -da_refine 5 -snes_monitor -ksp_monitor -snes_view -pc_type mg

    1051:    # HYPRE PtAP broken with complex numbers
    1052:    test:
    1053:       suffix: tut_3
    1054:       nsize: 4
    1055:       requires: hypre !single !complex
    1056:       args: -da_refine 5 -snes_monitor -ksp_monitor -snes_view -pc_type hypre

    1058:    test:
    1059:       suffix: tut_8
    1060:       nsize: 4
    1061:       requires: ml !single
    1062:       args: -da_refine 5 -snes_monitor -ksp_monitor -snes_view -pc_type ml

    1064:    test:
    1065:       suffix: tut_4
    1066:       nsize: 1
    1067:       requires: !single
    1068:       args: -da_refine 5 -log_view
    1069:       filter: head -n 2
    1070:       filter_output: head -n 2

    1072:    test:
    1073:       suffix: tut_5
    1074:       nsize: 1
    1075:       requires: !single
    1076:       args: -da_refine 5 -log_view -pc_type mg
    1077:       filter: head -n 2
    1078:       filter_output: head -n 2

    1080:    test:
    1081:       suffix: tut_6
    1082:       nsize: 4
    1083:       requires: !single
    1084:       args: -da_refine 5 -log_view
    1085:       filter: head -n 2
    1086:       filter_output: head -n 2

    1088:    test:
    1089:       suffix: tut_7
    1090:       nsize: 4
    1091:       requires: !single
    1092:       args: -da_refine 5 -log_view -pc_type mg
    1093:       filter: head -n 2
    1094:       filter_output: head -n 2

    1096:    test:
    1097:       suffix: cuda_1
    1098:       nsize: 1
    1099:       requires: cuda
    1100:       args: -snes_monitor -dm_mat_type seqaijcusparse -dm_vec_type seqcuda -pc_type gamg -ksp_monitor -mg_levels_ksp_max_it 3

    1103:    test:
    1104:       suffix: cuda_2
    1105:       nsize: 3
    1106:       requires: cuda !single
    1107:       args: -snes_monitor -dm_mat_type mpiaijcusparse -dm_vec_type mpicuda -pc_type gamg -ksp_monitor  -mg_levels_ksp_max_it 3

    1109:    test:
    1110:       suffix: seqbaijmkl
    1111:       nsize: 1
    1112:       requires: define(PETSC_HAVE_MKL_SPARSE_OPTIMIZE)
    1113:       args: -dm_mat_type baij -snes_monitor -ksp_monitor -snes_view

    1115:    test:
    1116:       suffix: mpibaijmkl
    1117:       nsize: 2
    1118:       requires:  define(PETSC_HAVE_MKL_SPARSE_OPTIMIZE)
    1119:       args: -dm_mat_type baij -snes_monitor -ksp_monitor -snes_view

    1121:    test:
    1122:      suffix: cpardiso
    1123:      nsize: 4
    1124:      requires: mkl_cpardiso
    1125:      args: -pc_type lu -pc_factor_mat_solver_type mkl_cpardiso -ksp_monitor

    1127:    test:
    1128:      suffix: logviewmemory
    1129:      requires: define(PETSC_USE_LOG) !define(PETSC_HAVE_VALGRIND)
    1130:      args: -log_view -log_view_memory -da_refine 4
    1131:      filter: grep MatFDColorSetUp | wc -w | xargs  -I % sh -c "expr % \> 21"

    1133:    test:
    1134:      suffix: fs
    1135:      args: -pc_type fieldsplit -da_refine 3  -all_ksp_monitor -fieldsplit_y_velocity_pc_type lu  -fieldsplit_temperature_pc_type lu -fieldsplit_x_velocity_pc_type lu  -snes_view

    1137:    test:
    1138:      suffix: asm_matconvert
    1139:      args: -mat_type aij -pc_type asm -pc_asm_sub_mat_type dense -snes_view

    1141:    test:
    1142:       suffix: euclid
    1143:       nsize: 2
    1144:       requires: hypre !single !complex !define(PETSC_HAVE_HYPRE_MIXEDINT)
    1145:       args: -da_refine 2 -ksp_monitor -snes_monitor -snes_view -pc_type hypre -pc_hypre_type euclid

    1147:    test:
    1148:       suffix: euclid_bj
    1149:       nsize: 2
    1150:       requires: hypre !single !complex !define(PETSC_HAVE_HYPRE_MIXEDINT)
    1151:       args: -da_refine 2 -ksp_monitor -snes_monitor -snes_view -pc_type hypre -pc_hypre_type euclid -pc_hypre_euclid_bj

    1153:    test:
    1154:       suffix: euclid_droptolerance
    1155:       nsize: 1
    1156:       requires: hypre !single !complex !define(PETSC_HAVE_HYPRE_MIXEDINT)
    1157:       args: -da_refine 2 -ksp_monitor -snes_monitor -snes_view -pc_type hypre -pc_hypre_type euclid -pc_hypre_euclid_droptolerance .1

    1159: TEST*/