Actual source code: ex19.c

      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: */

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).

     52: /*
     53:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
     54:    Include "petscsnes.h" so that we can use SNES solvers.  Note that this
     55:    file automatically includes:
     56:      petscsys.h       - base PETSc routines   petscvec.h - vectors
     57:      petscmat.h - matrices
     58:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
     59:      petscviewer.h - viewers               petscpc.h  - preconditioners
     60:      petscksp.h   - linear solvers
     61: */
     62: #if defined(PETSC_APPLE_FRAMEWORK)
     63: #import <PETSc/petscsnes.h>
     64: #import <PETSc/petscdmda.h>
     65: #else
     66: #include <petscsnes.h>
     67: #include <petscdm.h>
     68: #include <petscdmda.h>
     69: #endif

     71: /*
     72:    User-defined routines and data structures
     73: */
     74: typedef struct {
     75:   PetscScalar u,v,omega,temp;
     76: } Field;

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

     80: typedef struct {
     81:   PetscReal   lidvelocity,prandtl,grashof;  /* physical parameters */
     82:   PetscBool   draw_contours;                /* flag - 1 indicates drawing contours */
     83: } AppCtx;

     85: extern PetscErrorCode FormInitialGuess(AppCtx*,DM,Vec);
     86: extern PetscErrorCode NonlinearGS(SNES,Vec,Vec,void*);

     88: int main(int argc,char **argv)
     89: {
     90:   AppCtx         user;                /* user-defined work context */
     91:   PetscInt       mx,my,its;
     92:   MPI_Comm       comm;
     93:   SNES           snes;
     94:   DM             da;
     95:   Vec            x;

     97:   PetscInitialize(&argc,&argv,(char*)0,help);

    100:   comm = PETSC_COMM_WORLD;
    101:   SNESCreate(comm,&snes);

    103:   /*
    104:       Create distributed array object to manage parallel grid and vectors
    105:       for principal unknowns (x) and governing residuals (f)
    106:   */
    107:   DMDACreate2d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,4,4,PETSC_DECIDE,PETSC_DECIDE,4,1,0,0,&da);
    108:   DMSetFromOptions(da);
    109:   DMSetUp(da);
    110:   SNESSetDM(snes,(DM)da);
    111:   SNESSetNGS(snes, NonlinearGS, (void*)&user);

    113:   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);
    114:   /*
    115:      Problem parameters (velocity of lid, prandtl, and grashof numbers)
    116:   */
    117:   user.lidvelocity = 1.0/(mx*my);
    118:   user.prandtl     = 1.0;
    119:   user.grashof     = 1.0;

    121:   PetscOptionsGetReal(NULL,NULL,"-lidvelocity",&user.lidvelocity,NULL);
    122:   PetscOptionsGetReal(NULL,NULL,"-prandtl",&user.prandtl,NULL);
    123:   PetscOptionsGetReal(NULL,NULL,"-grashof",&user.grashof,NULL);
    124:   PetscOptionsHasName(NULL,NULL,"-contours",&user.draw_contours);

    126:   DMDASetFieldName(da,0,"x_velocity");
    127:   DMDASetFieldName(da,1,"y_velocity");
    128:   DMDASetFieldName(da,2,"Omega");
    129:   DMDASetFieldName(da,3,"temperature");

    131:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    132:      Create user context, set problem data, create vector data structures.
    133:      Also, compute the initial guess.
    134:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    136:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    137:      Create nonlinear solver context

    139:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    140:   DMSetApplicationContext(da,&user);
    141:   DMDASNESSetFunctionLocal(da,INSERT_VALUES,(PetscErrorCode (*)(DMDALocalInfo*,void*,void*,void*))FormFunctionLocal,&user);
    142:   SNESSetFromOptions(snes);
    143:   PetscPrintf(comm,"lid velocity = %g, prandtl # = %g, grashof # = %g\n",(double)user.lidvelocity,(double)user.prandtl,(double)user.grashof);

    145:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    146:      Solve the nonlinear system
    147:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    148:   DMCreateGlobalVector(da,&x);
    149:   FormInitialGuess(&user,da,x);

    151:   SNESSolve(snes,NULL,x);

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

    156:   /*
    157:      Visualize solution
    158:   */
    159:   if (user.draw_contours) {
    160:     VecView(x,PETSC_VIEWER_DRAW_WORLD);
    161:   }

    163:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    164:      Free work space.  All PETSc objects should be destroyed when they
    165:      are no longer needed.
    166:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    167:   VecDestroy(&x);
    168:   DMDestroy(&da);
    169:   SNESDestroy(&snes);
    170:   PetscFinalize();
    171:   return 0;
    172: }

    174: /* ------------------------------------------------------------------- */

    176: /*
    177:    FormInitialGuess - Forms initial approximation.

    179:    Input Parameters:
    180:    user - user-defined application context
    181:    X - vector

    183:    Output Parameter:
    184:    X - vector
    185: */
    186: PetscErrorCode FormInitialGuess(AppCtx *user,DM da,Vec X)
    187: {
    188:   PetscInt       i,j,mx,xs,ys,xm,ym;
    189:   PetscReal      grashof,dx;
    190:   Field          **x;

    193:   grashof = user->grashof;

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

    198:   /*
    199:      Get local grid boundaries (for 2-dimensional DMDA):
    200:        xs, ys   - starting grid indices (no ghost points)
    201:        xm, ym   - widths of local grid (no ghost points)
    202:   */
    203:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

    205:   /*
    206:      Get a pointer to vector data.
    207:        - For default PETSc vectors, VecGetArray() returns a pointer to
    208:          the data array.  Otherwise, the routine is implementation dependent.
    209:        - You MUST call VecRestoreArray() when you no longer need access to
    210:          the array.
    211:   */
    212:   DMDAVecGetArrayWrite(da,X,&x);

    214:   /*
    215:      Compute initial guess over the locally owned part of the grid
    216:      Initial condition is motionless fluid and equilibrium temperature
    217:   */
    218:   for (j=ys; j<ys+ym; j++) {
    219:     for (i=xs; i<xs+xm; i++) {
    220:       x[j][i].u     = 0.0;
    221:       x[j][i].v     = 0.0;
    222:       x[j][i].omega = 0.0;
    223:       x[j][i].temp  = (grashof>0)*i*dx;
    224:     }
    225:   }

    227:   /*
    228:      Restore vector
    229:   */
    230:   DMDAVecRestoreArrayWrite(da,X,&x);
    231:   return 0;
    232: }

    234: PetscErrorCode FormFunctionLocal(DMDALocalInfo *info,Field **x,Field **f,void *ptr)
    235: {
    236:   AppCtx         *user = (AppCtx*)ptr;
    237:   PetscInt       xints,xinte,yints,yinte,i,j;
    238:   PetscReal      hx,hy,dhx,dhy,hxdhy,hydhx;
    239:   PetscReal      grashof,prandtl,lid;
    240:   PetscScalar    u,uxx,uyy,vx,vy,avx,avy,vxp,vxm,vyp,vym;

    243:   grashof = user->grashof;
    244:   prandtl = user->prandtl;
    245:   lid     = user->lidvelocity;

    247:   /*
    248:      Define mesh intervals ratios for uniform grid.

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

    253:   */
    254:   dhx   = (PetscReal)(info->mx-1);  dhy = (PetscReal)(info->my-1);
    255:   hx    = 1.0/dhx;                   hy = 1.0/dhy;
    256:   hxdhy = hx*dhy;                 hydhx = hy*dhx;

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

    260:   /* Test whether we are on the bottom edge of the global array */
    261:   if (yints == 0) {
    262:     j     = 0;
    263:     yints = yints + 1;
    264:     /* bottom edge */
    265:     for (i=info->xs; i<info->xs+info->xm; i++) {
    266:       f[j][i].u     = x[j][i].u;
    267:       f[j][i].v     = x[j][i].v;
    268:       f[j][i].omega = x[j][i].omega + (x[j+1][i].u - x[j][i].u)*dhy;
    269:       f[j][i].temp  = x[j][i].temp-x[j+1][i].temp;
    270:     }
    271:   }

    273:   /* Test whether we are on the top edge of the global array */
    274:   if (yinte == info->my) {
    275:     j     = info->my - 1;
    276:     yinte = yinte - 1;
    277:     /* top edge */
    278:     for (i=info->xs; i<info->xs+info->xm; i++) {
    279:       f[j][i].u     = x[j][i].u - lid;
    280:       f[j][i].v     = x[j][i].v;
    281:       f[j][i].omega = x[j][i].omega + (x[j][i].u - x[j-1][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 left edge of the global array */
    287:   if (xints == 0) {
    288:     i     = 0;
    289:     xints = xints + 1;
    290:     /* left edge */
    291:     for (j=info->ys; j<info->ys+info->ym; j++) {
    292:       f[j][i].u     = x[j][i].u;
    293:       f[j][i].v     = x[j][i].v;
    294:       f[j][i].omega = x[j][i].omega - (x[j][i+1].v - x[j][i].v)*dhx;
    295:       f[j][i].temp  = x[j][i].temp;
    296:     }
    297:   }

    299:   /* Test whether we are on the right edge of the global array */
    300:   if (xinte == info->mx) {
    301:     i     = info->mx - 1;
    302:     xinte = xinte - 1;
    303:     /* right 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].v - x[j][i-1].v)*dhx;
    308:       f[j][i].temp  = x[j][i].temp - (PetscReal)(grashof>0);
    309:     }
    310:   }

    312:   /* Compute over the interior points */
    313:   for (j=yints; j<yinte; j++) {
    314:     for (i=xints; i<xinte; i++) {

    316:       /*
    317:        convective coefficients for upwinding
    318:       */
    319:       vx  = x[j][i].u; avx = PetscAbsScalar(vx);
    320:       vxp = .5*(vx+avx); vxm = .5*(vx-avx);
    321:       vy  = x[j][i].v; avy = PetscAbsScalar(vy);
    322:       vyp = .5*(vy+avy); vym = .5*(vy-avy);

    324:       /* U velocity */
    325:       u         = x[j][i].u;
    326:       uxx       = (2.0*u - x[j][i-1].u - x[j][i+1].u)*hydhx;
    327:       uyy       = (2.0*u - x[j-1][i].u - x[j+1][i].u)*hxdhy;
    328:       f[j][i].u = uxx + uyy - .5*(x[j+1][i].omega-x[j-1][i].omega)*hx;

    330:       /* V velocity */
    331:       u         = x[j][i].v;
    332:       uxx       = (2.0*u - x[j][i-1].v - x[j][i+1].v)*hydhx;
    333:       uyy       = (2.0*u - x[j-1][i].v - x[j+1][i].v)*hxdhy;
    334:       f[j][i].v = uxx + uyy + .5*(x[j][i+1].omega-x[j][i-1].omega)*hy;

    336:       /* Omega */
    337:       u             = x[j][i].omega;
    338:       uxx           = (2.0*u - x[j][i-1].omega - x[j][i+1].omega)*hydhx;
    339:       uyy           = (2.0*u - x[j-1][i].omega - x[j+1][i].omega)*hxdhy;
    340:       f[j][i].omega = uxx + uyy + (vxp*(u - x[j][i-1].omega) + vxm*(x[j][i+1].omega - u))*hy +
    341:                       (vyp*(u - x[j-1][i].omega) + vym*(x[j+1][i].omega - u))*hx -
    342:                       .5*grashof*(x[j][i+1].temp - x[j][i-1].temp)*hy;

    344:       /* Temperature */
    345:       u            = x[j][i].temp;
    346:       uxx          = (2.0*u - x[j][i-1].temp - x[j][i+1].temp)*hydhx;
    347:       uyy          = (2.0*u - x[j-1][i].temp - x[j+1][i].temp)*hxdhy;
    348:       f[j][i].temp =  uxx + uyy  + prandtl*((vxp*(u - x[j][i-1].temp) + vxm*(x[j][i+1].temp - u))*hy +
    349:                                             (vyp*(u - x[j-1][i].temp) + vym*(x[j+1][i].temp - u))*hx);
    350:     }
    351:   }

    353:   /*
    354:      Flop count (multiply-adds are counted as 2 operations)
    355:   */
    356:   PetscLogFlops(84.0*info->ym*info->xm);
    357:   return 0;
    358: }

    360: /*
    361:     Performs sweeps of point block nonlinear Gauss-Seidel on all the local grid points
    362: */
    363: PetscErrorCode NonlinearGS(SNES snes, Vec X, Vec B, void *ctx)
    364: {
    365:   DMDALocalInfo  info;
    366:   Field          **x,**b;
    367:   Vec            localX, localB;
    368:   DM             da;
    369:   PetscInt       xints,xinte,yints,yinte,i,j,k,l;
    370:   PetscInt       max_its,tot_its;
    371:   PetscInt       sweeps;
    372:   PetscReal      rtol,atol,stol;
    373:   PetscReal      hx,hy,dhx,dhy,hxdhy,hydhx;
    374:   PetscReal      grashof,prandtl,lid;
    375:   PetscScalar    u,uxx,uyy,vx,vy,avx,avy,vxp,vxm,vyp,vym;
    376:   PetscScalar    fu, fv, fomega, ftemp;
    377:   PetscScalar    dfudu;
    378:   PetscScalar    dfvdv;
    379:   PetscScalar    dfodu, dfodv, dfodo;
    380:   PetscScalar    dftdu, dftdv, dftdt;
    381:   PetscScalar    yu=0, yv=0, yo=0, yt=0;
    382:   PetscScalar    bjiu, bjiv, bjiomega, bjitemp;
    383:   PetscBool      ptconverged;
    384:   PetscReal      pfnorm,pfnorm0,pynorm,pxnorm;
    385:   AppCtx         *user = (AppCtx*)ctx;

    388:   grashof = user->grashof;
    389:   prandtl = user->prandtl;
    390:   lid     = user->lidvelocity;
    391:   tot_its = 0;
    392:   SNESNGSGetTolerances(snes,&rtol,&atol,&stol,&max_its);
    393:   SNESNGSGetSweeps(snes,&sweeps);
    394:   SNESGetDM(snes,(DM*)&da);
    395:   DMGetLocalVector(da,&localX);
    396:   if (B) {
    397:     DMGetLocalVector(da,&localB);
    398:   }
    399:   /*
    400:      Scatter ghost points to local vector, using the 2-step process
    401:         DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
    402:   */
    403:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
    404:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);
    405:   if (B) {
    406:     DMGlobalToLocalBegin(da,B,INSERT_VALUES,localB);
    407:     DMGlobalToLocalEnd(da,B,INSERT_VALUES,localB);
    408:   }
    409:   DMDAGetLocalInfo(da,&info);
    410:   DMDAVecGetArrayWrite(da,localX,&x);
    411:   if (B) {
    412:     DMDAVecGetArrayRead(da,localB,&b);
    413:   }
    414:   /* looks like a combination of the formfunction / formjacobian routines */
    415:   dhx   = (PetscReal)(info.mx-1);dhy   = (PetscReal)(info.my-1);
    416:   hx    = 1.0/dhx;               hy    = 1.0/dhy;
    417:   hxdhy = hx*dhy;                hydhx = hy*dhx;

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

    421:   /* Set the boundary conditions on the momentum equations */
    422:   /* Test whether we are on the bottom edge of the global array */
    423:   if (yints == 0) {
    424:     j     = 0;
    425:     /* bottom edge */
    426:     for (i=info.xs; i<info.xs+info.xm; i++) {

    428:       if (B) {
    429:         bjiu = b[j][i].u;
    430:         bjiv = b[j][i].v;
    431:       } else {
    432:         bjiu = 0.0;
    433:         bjiv = 0.0;
    434:       }
    435:       x[j][i].u = 0.0 + bjiu;
    436:       x[j][i].v = 0.0 + bjiv;
    437:     }
    438:   }

    440:   /* Test whether we are on the top edge of the global array */
    441:   if (yinte == info.my) {
    442:     j     = info.my - 1;
    443:     /* top edge */
    444:     for (i=info.xs; i<info.xs+info.xm; i++) {
    445:       if (B) {
    446:         bjiu = b[j][i].u;
    447:         bjiv = b[j][i].v;
    448:       } else {
    449:         bjiu = 0.0;
    450:         bjiv = 0.0;
    451:       }
    452:       x[j][i].u = lid + bjiu;
    453:       x[j][i].v = bjiv;
    454:     }
    455:   }

    457:   /* Test whether we are on the left edge of the global array */
    458:   if (xints == 0) {
    459:     i     = 0;
    460:     /* left edge */
    461:     for (j=info.ys; j<info.ys+info.ym; j++) {
    462:       if (B) {
    463:         bjiu = b[j][i].u;
    464:         bjiv = b[j][i].v;
    465:       } else {
    466:         bjiu = 0.0;
    467:         bjiv = 0.0;
    468:       }
    469:       x[j][i].u = 0.0 + bjiu;
    470:       x[j][i].v = 0.0 + bjiv;
    471:     }
    472:   }

    474:   /* Test whether we are on the right edge of the global array */
    475:   if (xinte == info.mx) {
    476:     i     = info.mx - 1;
    477:     /* right 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:   for (k=0; k < sweeps; k++) {
    492:     for (j=info.ys; j<info.ys + info.ym; j++) {
    493:       for (i=info.xs; i<info.xs + info.xm; i++) {
    494:         ptconverged = PETSC_FALSE;
    495:         pfnorm0     = 0.0;
    496:         fu          = 0.0;
    497:         fv          = 0.0;
    498:         fomega      = 0.0;
    499:         ftemp       = 0.0;
    500:         /*  Run Newton's method on a single grid point */
    501:         for (l = 0; l < max_its && !ptconverged; l++) {
    502:           if (B) {
    503:             bjiu     = b[j][i].u;
    504:             bjiv     = b[j][i].v;
    505:             bjiomega = b[j][i].omega;
    506:             bjitemp  = b[j][i].temp;
    507:           } else {
    508:             bjiu     = 0.0;
    509:             bjiv     = 0.0;
    510:             bjiomega = 0.0;
    511:             bjitemp  = 0.0;
    512:           }

    514:           if (i != 0 && i != info.mx - 1 && j != 0 && j != info.my-1) {
    515:             /* U velocity */
    516:             u     = x[j][i].u;
    517:             uxx   = (2.0*u - x[j][i-1].u - x[j][i+1].u)*hydhx;
    518:             uyy   = (2.0*u - x[j-1][i].u - x[j+1][i].u)*hxdhy;
    519:             fu    = uxx + uyy - .5*(x[j+1][i].omega-x[j-1][i].omega)*hx - bjiu;
    520:             dfudu = 2.0*(hydhx + hxdhy);
    521:             /* V velocity */
    522:             u     = x[j][i].v;
    523:             uxx   = (2.0*u - x[j][i-1].v - x[j][i+1].v)*hydhx;
    524:             uyy   = (2.0*u - x[j-1][i].v - x[j+1][i].v)*hxdhy;
    525:             fv    = uxx + uyy + .5*(x[j][i+1].omega-x[j][i-1].omega)*hy - bjiv;
    526:             dfvdv = 2.0*(hydhx + hxdhy);
    527:             /*
    528:              convective coefficients for upwinding
    529:              */
    530:             vx  = x[j][i].u; avx = PetscAbsScalar(vx);
    531:             vxp = .5*(vx+avx); vxm = .5*(vx-avx);
    532:             vy  = x[j][i].v; avy = PetscAbsScalar(vy);
    533:             vyp = .5*(vy+avy); vym = .5*(vy-avy);
    534:             /* Omega */
    535:             u      = x[j][i].omega;
    536:             uxx    = (2.0*u - x[j][i-1].omega - x[j][i+1].omega)*hydhx;
    537:             uyy    = (2.0*u - x[j-1][i].omega - x[j+1][i].omega)*hxdhy;
    538:             fomega = uxx + uyy +  (vxp*(u - x[j][i-1].omega) + vxm*(x[j][i+1].omega - u))*hy +
    539:                      (vyp*(u - x[j-1][i].omega) + vym*(x[j+1][i].omega - u))*hx -
    540:                      .5*grashof*(x[j][i+1].temp - x[j][i-1].temp)*hy - bjiomega;
    541:             /* convective coefficient derivatives */
    542:             dfodo = 2.0*(hydhx + hxdhy) + ((vxp - vxm)*hy + (vyp - vym)*hx);
    543:             if (PetscRealPart(vx) > 0.0) dfodu = (u - x[j][i-1].omega)*hy;
    544:             else dfodu = (x[j][i+1].omega - u)*hy;

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

    549:             /* Temperature */
    550:             u     = x[j][i].temp;
    551:             uxx   = (2.0*u - x[j][i-1].temp - x[j][i+1].temp)*hydhx;
    552:             uyy   = (2.0*u - x[j-1][i].temp - x[j+1][i].temp)*hxdhy;
    553:             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;
    554:             dftdt = 2.0*(hydhx + hxdhy) + prandtl*((vxp - vxm)*hy + (vyp - vym)*hx);
    555:             if (PetscRealPart(vx) > 0.0) dftdu = prandtl*(u - x[j][i-1].temp)*hy;
    556:             else dftdu = prandtl*(x[j][i+1].temp - u)*hy;

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

    561:             /* invert the system:
    562:              [ dfu / du     0        0        0    ][yu] = [fu]
    563:              [     0    dfv / dv     0        0    ][yv]   [fv]
    564:              [ dfo / du dfo / dv dfo / do     0    ][yo]   [fo]
    565:              [ dft / du dft / dv     0    dft / dt ][yt]   [ft]
    566:              by simple back-substitution
    567:            */
    568:             yu = fu / dfudu;
    569:             yv = fv / dfvdv;
    570:             yo = (fomega - (dfodu*yu + dfodv*yv)) / dfodo;
    571:             yt = (ftemp - (dftdu*yu + dftdv*yv)) / dftdt;

    573:             x[j][i].u     = x[j][i].u - yu;
    574:             x[j][i].v     = x[j][i].v - yv;
    575:             x[j][i].temp  = x[j][i].temp - yt;
    576:             x[j][i].omega = x[j][i].omega - yo;
    577:           }
    578:           if (i == 0) {
    579:             fomega        = x[j][i].omega - (x[j][i+1].v - x[j][i].v)*dhx - bjiomega;
    580:             ftemp         = x[j][i].temp - bjitemp;
    581:             yo            = fomega;
    582:             yt            = ftemp;
    583:             x[j][i].omega = x[j][i].omega - fomega;
    584:             x[j][i].temp  = x[j][i].temp - ftemp;
    585:           }
    586:           if (i == info.mx - 1) {
    587:             fomega        = x[j][i].omega - (x[j][i].v - x[j][i-1].v)*dhx - bjiomega;
    588:             ftemp         = x[j][i].temp - (PetscReal)(grashof>0) - bjitemp;
    589:             yo            = fomega;
    590:             yt            = ftemp;
    591:             x[j][i].omega = x[j][i].omega - fomega;
    592:             x[j][i].temp  = x[j][i].temp - ftemp;
    593:           }
    594:           if (j == 0) {
    595:             fomega        = x[j][i].omega + (x[j+1][i].u - x[j][i].u)*dhy - bjiomega;
    596:             ftemp         = x[j][i].temp-x[j+1][i].temp - bjitemp;
    597:             yo            = fomega;
    598:             yt            = ftemp;
    599:             x[j][i].omega = x[j][i].omega - fomega;
    600:             x[j][i].temp  = x[j][i].temp - ftemp;
    601:           }
    602:           if (j == info.my - 1) {
    603:             fomega        = x[j][i].omega + (x[j][i].u - x[j-1][i].u)*dhy - bjiomega;
    604:             ftemp         = x[j][i].temp-x[j-1][i].temp - bjitemp;
    605:             yo            = fomega;
    606:             yt            = ftemp;
    607:             x[j][i].omega = x[j][i].omega - fomega;
    608:             x[j][i].temp  = x[j][i].temp - ftemp;
    609:           }
    610:           tot_its++;
    611:           pfnorm = PetscRealPart(fu*fu + fv*fv + fomega*fomega + ftemp*ftemp);
    612:           pfnorm = PetscSqrtReal(pfnorm);
    613:           pynorm = PetscRealPart(yu*yu + yv*yv + yo*yo + yt*yt);
    614:           pynorm = PetscSqrtReal(pynorm);
    615:           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);
    616:           pxnorm = PetscSqrtReal(pxnorm);
    617:           if (l == 0) pfnorm0 = pfnorm;
    618:           if (rtol*pfnorm0 >pfnorm || atol > pfnorm || pxnorm*stol > pynorm) ptconverged = PETSC_TRUE;
    619:         }
    620:       }
    621:     }
    622:   }
    623:   DMDAVecRestoreArrayWrite(da,localX,&x);
    624:   if (B) {
    625:     DMDAVecRestoreArrayRead(da,localB,&b);
    626:   }
    627:   DMLocalToGlobalBegin(da,localX,INSERT_VALUES,X);
    628:   DMLocalToGlobalEnd(da,localX,INSERT_VALUES,X);
    629:   PetscLogFlops(tot_its*(84.0 + 41.0 + 26.0));
    630:   DMRestoreLocalVector(da,&localX);
    631:   if (B) {
    632:     DMRestoreLocalVector(da,&localB);
    633:   }
    634:   return 0;
    635: }

    637: /*TEST

    639:    test:
    640:       nsize: 2
    641:       args: -da_refine 3 -snes_monitor_short -pc_type mg -ksp_type fgmres -pc_mg_type full
    642:       requires: !single

    644:    test:
    645:       suffix: 10
    646:       nsize: 3
    647:       args: -snes_monitor_short -ksp_monitor_short -pc_type fieldsplit -pc_fieldsplit_type symmetric_multiplicative -snes_view -da_refine 1 -ksp_type fgmres
    648:       requires: !single

    650:    test:
    651:       suffix: 11
    652:       nsize: 4
    653:       requires: pastix
    654:       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

    656:    test:
    657:       suffix: 12
    658:       nsize: 12
    659:       requires: pastix
    660:       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

    662:    test:
    663:       suffix: 13
    664:       nsize: 3
    665:       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
    666:       requires: !single

    668:    test:
    669:       suffix: 14
    670:       nsize: 4
    671:       args: -snes_monitor_short -pc_type mg -dm_mat_type baij -mg_coarse_pc_type bjacobi -da_refine 3 -ksp_type fgmres
    672:       requires: !single

    674:    test:
    675:       suffix: 14_ds
    676:       nsize: 4
    677:       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
    678:       output_file: output/ex19_2.out
    679:       requires: !single

    681:    test:
    682:       suffix: 17
    683:       args: -snes_monitor_short -ksp_pc_side right
    684:       requires: !single

    686:    test:
    687:       suffix: 18
    688:       args: -snes_monitor_ksp draw::draw_lg -ksp_pc_side right
    689:       requires: x !single

    691:    test:
    692:       suffix: 19
    693:       nsize: 2
    694:       args: -da_refine 3 -snes_monitor_short -pc_type mg -ksp_type fgmres -pc_mg_type full -snes_type newtontrdc
    695:       requires: !single

    697:    test:
    698:       suffix: 20
    699:       nsize: 2
    700:       args: -da_refine 3 -snes_monitor_short -pc_type mg -ksp_type fgmres -pc_mg_type full -snes_type newtontrdc -snes_trdc_use_cauchy false
    701:       requires: !single

    703:    test:
    704:       suffix: 2
    705:       nsize: 4
    706:       args: -da_refine 3 -snes_converged_reason -pc_type mg -mat_fd_type ds
    707:       requires: !single

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

    716:    test:
    717:       suffix: 3
    718:       nsize: 4
    719:       requires: mumps
    720:       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

    722:    test:
    723:       suffix: 4
    724:       nsize: 12
    725:       requires: mumps
    726:       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
    727:       output_file: output/ex19_3.out

    729:    test:
    730:       suffix: 6
    731:       args: -snes_monitor_short -ksp_monitor_short -pc_type fieldsplit -snes_view -ksp_type fgmres -da_refine 1
    732:       requires: !single

    734:    test:
    735:       suffix: 7
    736:       nsize: 3
    737:       args: -snes_monitor_short -ksp_monitor_short -pc_type fieldsplit -snes_view -da_refine 1 -ksp_type fgmres

    739:       requires: !single
    740:    test:
    741:       suffix: 8
    742:       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
    743:       requires: !single

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

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

    757:    test:
    758:       suffix: bcgsl
    759:       nsize: 2
    760:       args: -ksp_type bcgsl -ksp_monitor_short -da_refine 2 -ksp_bcgsl_ell 3 -snes_view
    761:       requires: !single

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

    770:    test:
    771:       suffix: bjacobi
    772:       nsize: 4
    773:       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
    774:       requires: !single

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

    782:    test:
    783:       suffix: cgs
    784:       args: -da_refine 1 -ksp_monitor_short -ksp_type cgs
    785:       requires: !single

    787:    test:
    788:       suffix: composite_fieldsplit
    789:       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
    790:       requires: !single

    792:    test:
    793:       suffix: composite_fieldsplit_bjacobi
    794:       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
    795:       requires: !single

    797:    test:
    798:       suffix: composite_fieldsplit_bjacobi_2
    799:       nsize: 4
    800:       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
    801:       requires: !single

    803:    test:
    804:       suffix: composite_gs_newton
    805:       nsize: 2
    806:       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
    807:       requires: !single

    809:    test:
    810:       suffix: cuda
    811:       requires: cuda !single
    812:       args: -dm_vec_type cuda -dm_mat_type aijcusparse -pc_type none -ksp_type fgmres -snes_monitor_short -snes_rtol 1.e-5

    814:    test:
    815:       suffix: draw
    816:       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
    817:       requires: x !single

    819:    test:
    820:       suffix: drawports
    821:       args: -snes_monitor_solution draw::draw_ports -da_refine 1
    822:       output_file: output/ex19_draw.out
    823:       requires: x !single

    825:    test:
    826:       suffix: fas
    827:       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
    828:       requires: !single

    830:    test:
    831:       suffix: fas_full
    832:       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
    833:       requires: !single

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

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

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

    853:    test:
    854:       suffix: fdcoloring_wp
    855:       args: -da_refine 3 -snes_monitor_short -pc_type mg
    856:       requires: !single

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

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

    870:    test:
    871:       suffix: fieldsplit_2
    872:       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
    873:       requires: !single

    875:    test:
    876:       suffix: fieldsplit_3
    877:       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
    878:       requires: !single

    880:    test:
    881:       suffix: fieldsplit_4
    882:       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
    883:       requires: !single

    885:    # HYPRE PtAP broken with complex numbers
    886:    test:
    887:       suffix: fieldsplit_hypre
    888:       nsize: 2
    889:       requires: hypre mumps !complex !defined(PETSC_HAVE_HYPRE_DEVICE)
    890:       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

    892:    test:
    893:       suffix: fieldsplit_mumps
    894:       nsize: 2
    895:       requires: mumps
    896:       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
    897:       output_file: output/ex19_fieldsplit_5.out

    899:    test:
    900:       suffix: greedy_coloring
    901:       nsize: 2
    902:       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
    903:       requires: !single

    905:    # HYPRE PtAP broken with complex numbers
    906:    test:
    907:       suffix: hypre
    908:       nsize: 2
    909:       requires: hypre !complex !defined(PETSC_HAVE_HYPRE_DEVICE)
    910:       args: -da_refine 3 -snes_monitor_short -pc_type hypre -ksp_norm_type unpreconditioned

    912:    # ibcgs is broken when using device vectors
    913:    test:
    914:       suffix: ibcgs
    915:       nsize: 2
    916:       args: -ksp_type ibcgs -ksp_monitor_short -da_refine 2 -snes_view
    917:       requires: !complex !single

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

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

    931:    test:
    932:       suffix: klu_2
    933:       requires: suitesparse
    934:       args: -da_grid_x 20 -da_grid_y 20 -pc_type lu -pc_factor_mat_solver_type klu -pc_factor_mat_ordering_type nd
    935:       output_file: output/ex19_superlu.out

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

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

    949:    test:
    950:       suffix: ngmres_fas
    951:       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
    952:       requires: !single

    954:    test:
    955:       suffix: ngmres_fas_gssecant
    956:       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
    957:       requires: !single

    959:    test:
    960:       suffix: ngmres_fas_ms
    961:       nsize: 2
    962:       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
    963:       requires: !single

    965:    test:
    966:       suffix: ngmres_nasm
    967:       nsize: 4
    968:       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
    969:       requires: !single

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

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

    981:    test:
    982:       suffix: parms
    983:       nsize: 2
    984:       requires: parms
    985:       args: -pc_type parms -ksp_monitor_short -snes_view

    987:    test:
    988:       suffix: superlu
    989:       requires: superlu
    990:       args: -da_grid_x 20 -da_grid_y 20 -pc_type lu -pc_factor_mat_solver_type superlu

    992:    test:
    993:       suffix: superlu_sell
    994:       requires: superlu
    995:       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
    996:       output_file: output/ex19_superlu.out

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

    1004:    test:
    1005:       suffix: superlu_dist_2
    1006:       nsize: 2
    1007:       requires: superlu_dist
    1008:       args: -da_grid_x 20 -da_grid_y 20 -pc_type lu -pc_factor_mat_solver_type superlu_dist
    1009:       output_file: output/ex19_superlu.out

    1011:    test:
    1012:       suffix: superlu_dist_3d
    1013:       nsize: 4
    1014:       requires: superlu_dist !defined(PETSCTEST_VALGRIND)
    1015:       filter: grep -v iam | grep -v openMP
    1016:       args: -da_grid_x 20 -da_grid_y 20 -pc_type lu -pc_factor_mat_solver_type superlu_dist -mat_superlu_dist_3d -mat_superlu_dist_d 2 -snes_view -snes_monitor -ksp_monitor

    1018:    test:
    1019:       suffix: superlu_equil
    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

    1023:    test:
    1024:       suffix: superlu_equil_sell
    1025:       requires: superlu
    1026:       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
    1027:       output_file: output/ex19_superlu_equil.out

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

    1034:    test:
    1035:       suffix: tfqmr
    1036:       args: -da_refine 1 -ksp_monitor_short -ksp_type tfqmr
    1037:       requires: !single

    1039:    test:
    1040:       suffix: umfpack
    1041:       requires: suitesparse
    1042:       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

    1044:    test:
    1045:       suffix: tut_1
    1046:       nsize: 4
    1047:       requires: !single
    1048:       args: -da_refine 5 -snes_monitor -ksp_monitor -snes_view

    1050:    test:
    1051:       suffix: tut_2
    1052:       nsize: 4
    1053:       requires: !single
    1054:       args: -da_refine 5 -snes_monitor -ksp_monitor -snes_view -pc_type mg

    1056:    # HYPRE PtAP broken with complex numbers
    1057:    test:
    1058:       suffix: tut_3
    1059:       nsize: 4
    1060:       requires: hypre !single !complex !defined(PETSC_HAVE_HYPRE_DEVICE)
    1061:       args: -da_refine 5 -snes_monitor -ksp_monitor -snes_view -pc_type hypre

    1063:    test:
    1064:       suffix: tut_8
    1065:       nsize: 4
    1066:       requires: ml !single
    1067:       args: -da_refine 5 -snes_monitor -ksp_monitor -snes_view -pc_type ml

    1069:    test:
    1070:       suffix: tut_4
    1071:       nsize: 1
    1072:       requires: !single
    1073:       args: -da_refine 5 -log_view
    1074:       filter: head -n 2
    1075:       filter_output: head -n 2

    1077:    test:
    1078:       suffix: tut_5
    1079:       nsize: 1
    1080:       requires: !single
    1081:       args: -da_refine 5 -log_view -pc_type mg
    1082:       filter: head -n 2
    1083:       filter_output: head -n 2

    1085:    test:
    1086:       suffix: tut_6
    1087:       nsize: 4
    1088:       requires: !single
    1089:       args: -da_refine 5 -log_view
    1090:       filter: head -n 2
    1091:       filter_output: head -n 2

    1093:    test:
    1094:       suffix: tut_7
    1095:       nsize: 4
    1096:       requires: !single
    1097:       args: -da_refine 5 -log_view -pc_type mg
    1098:       filter: head -n 2
    1099:       filter_output: head -n 2

    1101:    test:
    1102:       suffix: cuda_1
    1103:       nsize: 1
    1104:       requires: cuda
    1105:       args: -snes_monitor -dm_mat_type seqaijcusparse -dm_vec_type seqcuda -pc_type gamg -pc_gamg_esteig_ksp_max_it 10 -ksp_monitor -mg_levels_ksp_max_it 3

    1107:    test:
    1108:       suffix: cuda_2
    1109:       nsize: 3
    1110:       requires: cuda !single
    1111:       args: -snes_monitor -dm_mat_type mpiaijcusparse -dm_vec_type mpicuda -pc_type gamg -pc_gamg_esteig_ksp_max_it 10 -ksp_monitor  -mg_levels_ksp_max_it 3

    1113:    test:
    1114:       suffix: cuda_dm_bind_below
    1115:       nsize: 2
    1116:       requires: cuda
    1117:       args: -dm_mat_type aijcusparse -dm_vec_type cuda -da_refine 3 -pc_type mg -mg_levels_ksp_type chebyshev -mg_levels_pc_type jacobi -log_view -pc_mg_log -dm_bind_below 10000
    1118:       filter: awk "/Level/ {print \$NF}"

    1120:    test:
    1121:       suffix: viennacl_dm_bind_below
    1122:       nsize: 2
    1123:       requires: viennacl
    1124:       args: -dm_mat_type aijviennacl -dm_vec_type viennacl -da_refine 3 -pc_type mg -mg_levels_ksp_type chebyshev -mg_levels_pc_type jacobi -log_view -pc_mg_log -dm_bind_below 10000
    1125:       filter: awk "/Level/ {print \$NF}"

    1127:    test:
    1128:       suffix: seqbaijmkl
    1129:       nsize: 1
    1130:       requires: defined(PETSC_HAVE_MKL_SPARSE_OPTIMIZE)
    1131:       args: -dm_mat_type baij -snes_monitor -ksp_monitor -snes_view

    1133:    test:
    1134:       suffix: mpibaijmkl
    1135:       nsize: 2
    1136:       requires:  defined(PETSC_HAVE_MKL_SPARSE_OPTIMIZE)
    1137:       args: -dm_mat_type baij -snes_monitor -ksp_monitor -snes_view

    1139:    test:
    1140:      suffix: cpardiso
    1141:      nsize: 4
    1142:      requires: mkl_cpardiso
    1143:      args: -pc_type lu -pc_factor_mat_solver_type mkl_cpardiso -ksp_monitor

    1145:    test:
    1146:      suffix: logviewmemory
    1147:      requires: defined(PETSC_USE_LOG) !defined(PETSCTEST_VALGRIND)
    1148:      args: -log_view -log_view_memory -da_refine 4
    1149:      filter: grep MatFDColorSetUp | wc -w | xargs  -I % sh -c "expr % \> 21"

    1151:    test:
    1152:      suffix: fs
    1153:      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

    1155:    test:
    1156:      suffix: asm_matconvert
    1157:      args: -mat_type aij -pc_type asm -pc_asm_sub_mat_type dense -snes_view

    1159:    test:
    1160:       suffix: euclid
    1161:       nsize: 2
    1162:       requires: hypre !single !complex !defined(PETSC_HAVE_HYPRE_MIXEDINT) !defined(PETSC_HAVE_HYPRE_DEVICE)
    1163:       args: -da_refine 2 -ksp_monitor -snes_monitor -snes_view -pc_type hypre -pc_hypre_type euclid

    1165:    test:
    1166:       suffix: euclid_bj
    1167:       nsize: 2
    1168:       requires: hypre !single !complex !defined(PETSC_HAVE_HYPRE_MIXEDINT) !defined(PETSC_HAVE_HYPRE_DEVICE)
    1169:       args: -da_refine 2 -ksp_monitor -snes_monitor -snes_view -pc_type hypre -pc_hypre_type euclid -pc_hypre_euclid_bj

    1171:    test:
    1172:       suffix: euclid_droptolerance
    1173:       nsize: 1
    1174:       requires: hypre !single !complex !defined(PETSC_HAVE_HYPRE_MIXEDINT) !defined(PETSC_HAVE_HYPRE_DEVICE)
    1175:       args: -da_refine 2 -ksp_monitor -snes_monitor -snes_view -pc_type hypre -pc_hypre_type euclid -pc_hypre_euclid_droptolerance .1

    1177: TEST*/