Date: 2022-03-04T17:39:26+00:00

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: */

     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).

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

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

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

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

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

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

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

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

    111:   /*
    112:       Create distributed array object to manage parallel grid and vectors
    113:       for principal unknowns (x) and governing residuals (f)
    114:   */
    115:   DMDACreate2d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,4,4,PETSC_DECIDE,PETSC_DECIDE,4,1,0,0,&da);
    116:   DMSetFromOptions(da);
    117:   DMSetUp(da);
    118:   SNESSetDM(snes,(DM)da);
    119:   SNESSetNGS(snes, NonlinearGS, (void*)&user);

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

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

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

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

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

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

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

    159:   SNESSolve(snes,NULL,x);

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

    164:   /*
    165:      Visualize solution
    166:   */
    167:   if (user.draw_contours) {
    168:     VecView(x,PETSC_VIEWER_DRAW_WORLD);
    169:   }

    171:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    172:      Free work space.  All PETSc objects should be destroyed when they
    173:      are no longer needed.
    174:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    175:   VecDestroy(&x);
    176:   DMDestroy(&da);
    177:   SNESDestroy(&snes);
    178:   PetscFinalize();
    179:   return ierr;
    180: }

    182: /* ------------------------------------------------------------------- */

    184: /*
    185:    FormInitialGuess - Forms initial approximation.

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

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

    202:   grashof = user->grashof;

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

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

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

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

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

    243: PetscErrorCode FormFunctionLocal(DMDALocalInfo *info,Field **x,Field **f,void *ptr)
    244: {
    245:   AppCtx         *user = (AppCtx*)ptr;
    247:   PetscInt       xints,xinte,yints,yinte,i,j;
    248:   PetscReal      hx,hy,dhx,dhy,hxdhy,hydhx;
    249:   PetscReal      grashof,prandtl,lid;
    250:   PetscScalar    u,uxx,uyy,vx,vy,avx,avy,vxp,vxm,vyp,vym;

    253:   grashof = user->grashof;
    254:   prandtl = user->prandtl;
    255:   lid     = user->lidvelocity;

    257:   /*
    258:      Define mesh intervals ratios for uniform grid.

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

    263:   */
    264:   dhx   = (PetscReal)(info->mx-1);  dhy = (PetscReal)(info->my-1);
    265:   hx    = 1.0/dhx;                   hy = 1.0/dhy;
    266:   hxdhy = hx*dhy;                 hydhx = hy*dhx;

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

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

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

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

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

    322:   /* Compute over the interior points */
    323:   for (j=yints; j<yinte; j++) {
    324:     for (i=xints; i<xinte; i++) {

    326:       /*
    327:        convective coefficients for upwinding
    328:       */
    329:       vx  = x[j][i].u; avx = PetscAbsScalar(vx);
    330:       vxp = .5*(vx+avx); vxm = .5*(vx-avx);
    331:       vy  = x[j][i].v; avy = PetscAbsScalar(vy);
    332:       vyp = .5*(vy+avy); vym = .5*(vy-avy);

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

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

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

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

    363:   /*
    364:      Flop count (multiply-adds are counted as 2 operations)
    365:   */
    366:   PetscLogFlops(84.0*info->ym*info->xm);
    367:   return(0);
    368: }

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

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

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

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

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

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

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

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

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

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

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

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

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

    572:             /* invert the system:
    573:              [ dfu / du     0        0        0    ][yu] = [fu]
    574:              [     0    dfv / dv     0        0    ][yv]   [fv]
    575:              [ dfo / du dfo / dv dfo / do     0    ][yo]   [fo]
    576:              [ dft / du dft / dv     0    dft / dt ][yt]   [ft]
    577:              by simple back-substitution
    578:            */
    579:             yu = fu / dfudu;
    580:             yv = fv / dfvdv;
    581:             yo = (fomega - (dfodu*yu + dfodv*yv)) / dfodo;
    582:             yt = (ftemp - (dftdu*yu + dftdv*yv)) / dftdt;

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

    648: /*TEST

    650:    test:
    651:       nsize: 2
    652:       args: -da_refine 3 -snes_monitor_short -pc_type mg -ksp_type fgmres -pc_mg_type full
    653:       requires: !single

    655:    test:
    656:       suffix: 10
    657:       nsize: 3
    658:       args: -snes_monitor_short -ksp_monitor_short -pc_type fieldsplit -pc_fieldsplit_type symmetric_multiplicative -snes_view -da_refine 1 -ksp_type fgmres
    659:       requires: !single

    661:    test:
    662:       suffix: 11
    663:       nsize: 4
    664:       requires: pastix
    665:       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

    667:    test:
    668:       suffix: 12
    669:       nsize: 12
    670:       requires: pastix
    671:       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

    673:    test:
    674:       suffix: 13
    675:       nsize: 3
    676:       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
    677:       requires: !single

    679:    test:
    680:       suffix: 14
    681:       nsize: 4
    682:       args: -snes_monitor_short -pc_type mg -dm_mat_type baij -mg_coarse_pc_type bjacobi -da_refine 3 -ksp_type fgmres
    683:       requires: !single

    685:    test:
    686:       suffix: 14_ds
    687:       nsize: 4
    688:       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
    689:       output_file: output/ex19_2.out
    690:       requires: !single

    692:    test:
    693:       suffix: 17
    694:       args: -snes_monitor_short -ksp_pc_side right
    695:       requires: !single

    697:    test:
    698:       suffix: 18
    699:       args: -snes_monitor_ksp draw::draw_lg -ksp_pc_side right
    700:       requires: x !single

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    1003:    test:
    1004:       suffix: superlu_dist_2
    1005:       nsize: 2
    1006:       requires: superlu_dist
    1007:       args: -da_grid_x 20 -da_grid_y 20 -pc_type lu -pc_factor_mat_solver_type superlu_dist
    1008:       output_file: output/ex19_superlu.out

    1010:    test:
    1011:       suffix: superlu_equil
    1012:       requires: superlu
    1013:       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

    1015:    test:
    1016:       suffix: superlu_equil_sell
    1017:       requires: superlu
    1018:       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
    1019:       output_file: output/ex19_superlu_equil.out

    1021:    test:
    1022:       suffix: tcqmr
    1023:       args: -da_refine 1 -ksp_monitor_short -ksp_type tcqmr
    1024:       requires: !single

    1026:    test:
    1027:       suffix: tfqmr
    1028:       args: -da_refine 1 -ksp_monitor_short -ksp_type tfqmr
    1029:       requires: !single

    1031:    test:
    1032:       suffix: umfpack
    1033:       requires: suitesparse
    1034:       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

    1036:    test:
    1037:       suffix: tut_1
    1038:       nsize: 4
    1039:       requires: !single
    1040:       args: -da_refine 5 -snes_monitor -ksp_monitor -snes_view

    1042:    test:
    1043:       suffix: tut_2
    1044:       nsize: 4
    1045:       requires: !single
    1046:       args: -da_refine 5 -snes_monitor -ksp_monitor -snes_view -pc_type mg

    1048:    # HYPRE PtAP broken with complex numbers
    1049:    test:
    1050:       suffix: tut_3
    1051:       nsize: 4
    1052:       requires: hypre !single !complex !defined(PETSC_HAVE_HYPRE_DEVICE)
    1053:       args: -da_refine 5 -snes_monitor -ksp_monitor -snes_view -pc_type hypre

    1055:    test:
    1056:       suffix: tut_8
    1057:       nsize: 4
    1058:       requires: ml !single
    1059:       args: -da_refine 5 -snes_monitor -ksp_monitor -snes_view -pc_type ml

    1061:    test:
    1062:       suffix: tut_4
    1063:       nsize: 1
    1064:       requires: !single
    1065:       args: -da_refine 5 -log_view
    1066:       filter: head -n 2
    1067:       filter_output: head -n 2

    1069:    test:
    1070:       suffix: tut_5
    1071:       nsize: 1
    1072:       requires: !single
    1073:       args: -da_refine 5 -log_view -pc_type mg
    1074:       filter: head -n 2
    1075:       filter_output: head -n 2

    1077:    test:
    1078:       suffix: tut_6
    1079:       nsize: 4
    1080:       requires: !single
    1081:       args: -da_refine 5 -log_view
    1082:       filter: head -n 2
    1083:       filter_output: head -n 2

    1085:    test:
    1086:       suffix: tut_7
    1087:       nsize: 4
    1088:       requires: !single
    1089:       args: -da_refine 5 -log_view -pc_type mg
    1090:       filter: head -n 2
    1091:       filter_output: head -n 2

    1093:    test:
    1094:       suffix: cuda_1
    1095:       nsize: 1
    1096:       requires: cuda
    1097:       args: -snes_monitor -dm_mat_type seqaijcusparse -dm_vec_type seqcuda -pc_type gamg -ksp_monitor -mg_levels_ksp_max_it 3

    1099:    test:
    1100:       suffix: cuda_2
    1101:       nsize: 3
    1102:       requires: cuda !single
    1103:       args: -snes_monitor -dm_mat_type mpiaijcusparse -dm_vec_type mpicuda -pc_type gamg -ksp_monitor  -mg_levels_ksp_max_it 3

    1105:    test:
    1106:       suffix: seqbaijmkl
    1107:       nsize: 1
    1108:       requires: defined(PETSC_HAVE_MKL_SPARSE_OPTIMIZE)
    1109:       args: -dm_mat_type baij -snes_monitor -ksp_monitor -snes_view

    1111:    test:
    1112:       suffix: mpibaijmkl
    1113:       nsize: 2
    1114:       requires:  defined(PETSC_HAVE_MKL_SPARSE_OPTIMIZE)
    1115:       args: -dm_mat_type baij -snes_monitor -ksp_monitor -snes_view

    1117:    test:
    1118:      suffix: cpardiso
    1119:      nsize: 4
    1120:      requires: mkl_cpardiso
    1121:      args: -pc_type lu -pc_factor_mat_solver_type mkl_cpardiso -ksp_monitor

    1123:    test:
    1124:      suffix: logviewmemory
    1125:      requires: defined(PETSC_USE_LOG) !defined(PETSCTEST_VALGRIND)
    1126:      args: -log_view -log_view_memory -da_refine 4
    1127:      filter: grep MatFDColorSetUp | wc -w | xargs  -I % sh -c "expr % \> 21"

    1129:    test:
    1130:      suffix: fs
    1131:      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

    1133:    test:
    1134:      suffix: asm_matconvert
    1135:      args: -mat_type aij -pc_type asm -pc_asm_sub_mat_type dense -snes_view

    1137:    test:
    1138:       suffix: euclid
    1139:       nsize: 2
    1140:       requires: hypre !single !complex !defined(PETSC_HAVE_HYPRE_MIXEDINT) !defined(PETSC_HAVE_HYPRE_DEVICE)
    1141:       args: -da_refine 2 -ksp_monitor -snes_monitor -snes_view -pc_type hypre -pc_hypre_type euclid

    1143:    test:
    1144:       suffix: euclid_bj
    1145:       nsize: 2
    1146:       requires: hypre !single !complex !defined(PETSC_HAVE_HYPRE_MIXEDINT) !defined(PETSC_HAVE_HYPRE_DEVICE)
    1147:       args: -da_refine 2 -ksp_monitor -snes_monitor -snes_view -pc_type hypre -pc_hypre_type euclid -pc_hypre_euclid_bj

    1149:    test:
    1150:       suffix: euclid_droptolerance
    1151:       nsize: 1
    1152:       requires: hypre !single !complex !defined(PETSC_HAVE_HYPRE_MIXEDINT) !defined(PETSC_HAVE_HYPRE_DEVICE)
    1153:       args: -da_refine 2 -ksp_monitor -snes_monitor -snes_view -pc_type hypre -pc_hypre_type euclid -pc_hypre_euclid_droptolerance .1

    1155: TEST*/