Date: 2022-08-02T03:14:55+00:00

Actual source code: ex18.c

      2: static char help[] ="Nonlinear Radiative Transport PDE with multigrid in 2d.\n\
      3: Uses 2-dimensional distributed arrays.\n\
      4: A 2-dim simplified Radiative Transport test problem is used, with analytic Jacobian. \n\
      5: \n\
      6:   Solves the linear systems via multilevel methods \n\
      7: \n\
      8: The command line\n\
      9: options are:\n\
     10:   -tleft <tl>, where <tl> indicates the left Diriclet BC \n\
     11:   -tright <tr>, where <tr> indicates the right Diriclet BC \n\
     12:   -beta <beta>, where <beta> indicates the exponent in T \n\n";

     14: /*

     16:     This example models the partial differential equation

     18:          - Div(alpha* T^beta (GRAD T)) = 0.

     20:     where beta = 2.5 and alpha = 1.0

     22:     BC: T_left = 1.0, T_right = 0.1, dT/dn_top = dTdn_bottom = 0.

     24:     in the unit square, which is uniformly discretized in each of x and
     25:     y in this simple encoding.  The degrees of freedom are cell centered.

     27:     A finite volume approximation with the usual 5-point stencil
     28:     is used to discretize the boundary value problem to obtain a
     29:     nonlinear system of equations.

     31:     This code was contributed by David Keyes

     33: */

     35: #include <petscsnes.h>
     36: #include <petscdm.h>
     37: #include <petscdmda.h>

     39: /* User-defined application context */

     41: typedef struct {
     42:   PetscReal tleft,tright;    /* Dirichlet boundary conditions */
     43:   PetscReal beta,bm1,coef;   /* nonlinear diffusivity parameterizations */
     44: } AppCtx;

     46: #define POWFLOP 5 /* assume a pow() takes five flops */

     48: extern PetscErrorCode FormInitialGuess(SNES,Vec,void*);
     49: extern PetscErrorCode FormFunction(SNES,Vec,Vec,void*);
     50: extern PetscErrorCode FormJacobian(SNES,Vec,Mat,Mat,void*);

     52: int main(int argc,char **argv)
     53: {
     54:   SNES           snes;
     55:   AppCtx         user;
     56:   PetscInt       its,lits;
     57:   PetscReal      litspit;
     58:   DM             da;

     60:   PetscInitialize(&argc,&argv,NULL,help);

     62:   /* set problem parameters */
     63:   user.tleft  = 1.0;
     64:   user.tright = 0.1;
     65:   user.beta   = 2.5;
     66:   PetscOptionsGetReal(NULL,NULL,"-tleft",&user.tleft,NULL);
     67:   PetscOptionsGetReal(NULL,NULL,"-tright",&user.tright,NULL);
     68:   PetscOptionsGetReal(NULL,NULL,"-beta",&user.beta,NULL);
     69:   user.bm1    = user.beta - 1.0;
     70:   user.coef   = user.beta/2.0;

     72:   /*
     73:       Create the multilevel DM data structure
     74:   */
     75:   SNESCreate(PETSC_COMM_WORLD,&snes);

     77:   /*
     78:       Set the DMDA (grid structure) for the grids.
     79:   */
     80:   DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,5,5,PETSC_DECIDE,PETSC_DECIDE,1,1,0,0,&da);
     81:   DMSetFromOptions(da);
     82:   DMSetUp(da);
     83:   DMSetApplicationContext(da,&user);
     84:   SNESSetDM(snes,(DM)da);

     86:   /*
     87:      Create the nonlinear solver, and tell it the functions to use
     88:   */
     89:   SNESSetFunction(snes,NULL,FormFunction,&user);
     90:   SNESSetJacobian(snes,NULL,NULL,FormJacobian,&user);
     91:   SNESSetFromOptions(snes);
     92:   SNESSetComputeInitialGuess(snes,FormInitialGuess,NULL);

     94:   SNESSolve(snes,NULL,NULL);
     95:   SNESGetIterationNumber(snes,&its);
     96:   SNESGetLinearSolveIterations(snes,&lits);
     97:   litspit = ((PetscReal)lits)/((PetscReal)its);
     98:   PetscPrintf(PETSC_COMM_WORLD,"Number of SNES iterations = %D\n",its);
     99:   PetscPrintf(PETSC_COMM_WORLD,"Number of Linear iterations = %D\n",lits);
    100:   PetscPrintf(PETSC_COMM_WORLD,"Average Linear its / SNES = %e\n",(double)litspit);

    102:   DMDestroy(&da);
    103:   SNESDestroy(&snes);
    104:   PetscFinalize();
    105:   return 0;
    106: }
    107: /* --------------------  Form initial approximation ----------------- */
    108: PetscErrorCode FormInitialGuess(SNES snes,Vec X,void *ctx)
    109: {
    110:   AppCtx         *user;
    111:   PetscInt       i,j,xs,ys,xm,ym;
    112:   PetscReal      tleft;
    113:   PetscScalar    **x;
    114:   DM             da;

    117:   SNESGetDM(snes,&da);
    118:   DMGetApplicationContext(da,&user);
    119:   tleft = user->tleft;
    120:   /* Get ghost points */
    121:   DMDAGetCorners(da,&xs,&ys,0,&xm,&ym,0);
    122:   DMDAVecGetArray(da,X,&x);

    124:   /* Compute initial guess */
    125:   for (j=ys; j<ys+ym; j++) {
    126:     for (i=xs; i<xs+xm; i++) {
    127:       x[j][i] = tleft;
    128:     }
    129:   }
    130:   DMDAVecRestoreArray(da,X,&x);
    131:   return 0;
    132: }
    133: /* --------------------  Evaluate Function F(x) --------------------- */
    134: PetscErrorCode FormFunction(SNES snes,Vec X,Vec F,void *ptr)
    135: {
    136:   AppCtx         *user = (AppCtx*)ptr;
    137:   PetscInt       i,j,mx,my,xs,ys,xm,ym;
    138:   PetscScalar    zero = 0.0,one = 1.0;
    139:   PetscScalar    hx,hy,hxdhy,hydhx;
    140:   PetscScalar    t0,tn,ts,te,tw,an,as,ae,aw,dn,ds,de,dw,fn = 0.0,fs = 0.0,fe =0.0,fw = 0.0;
    141:   PetscScalar    tleft,tright,beta;
    142:   PetscScalar    **x,**f;
    143:   Vec            localX;
    144:   DM             da;

    147:   SNESGetDM(snes,&da);
    148:   DMGetLocalVector(da,&localX);
    149:   DMDAGetInfo(da,NULL,&mx,&my,0,0,0,0,0,0,0,0,0,0);
    150:   hx    = one/(PetscReal)(mx-1);  hy    = one/(PetscReal)(my-1);
    151:   hxdhy = hx/hy;               hydhx = hy/hx;
    152:   tleft = user->tleft;         tright = user->tright;
    153:   beta  = user->beta;

    155:   /* Get ghost points */
    156:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
    157:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);
    158:   DMDAGetCorners(da,&xs,&ys,0,&xm,&ym,0);
    159:   DMDAVecGetArray(da,localX,&x);
    160:   DMDAVecGetArray(da,F,&f);

    162:   /* Evaluate function */
    163:   for (j=ys; j<ys+ym; j++) {
    164:     for (i=xs; i<xs+xm; i++) {
    165:       t0 = x[j][i];

    167:       if (i > 0 && i < mx-1 && j > 0 && j < my-1) {

    169:         /* general interior volume */

    171:         tw = x[j][i-1];
    172:         aw = 0.5*(t0 + tw);
    173:         dw = PetscPowScalar(aw,beta);
    174:         fw = dw*(t0 - tw);

    176:         te = x[j][i+1];
    177:         ae = 0.5*(t0 + te);
    178:         de = PetscPowScalar(ae,beta);
    179:         fe = de*(te - t0);

    181:         ts = x[j-1][i];
    182:         as = 0.5*(t0 + ts);
    183:         ds = PetscPowScalar(as,beta);
    184:         fs = ds*(t0 - ts);

    186:         tn = x[j+1][i];
    187:         an = 0.5*(t0 + tn);
    188:         dn = PetscPowScalar(an,beta);
    189:         fn = dn*(tn - t0);

    191:       } else if (i == 0) {

    193:         /* left-hand boundary */
    194:         tw = tleft;
    195:         aw = 0.5*(t0 + tw);
    196:         dw = PetscPowScalar(aw,beta);
    197:         fw = dw*(t0 - tw);

    199:         te = x[j][i+1];
    200:         ae = 0.5*(t0 + te);
    201:         de = PetscPowScalar(ae,beta);
    202:         fe = de*(te - t0);

    204:         if (j > 0) {
    205:           ts = x[j-1][i];
    206:           as = 0.5*(t0 + ts);
    207:           ds = PetscPowScalar(as,beta);
    208:           fs = ds*(t0 - ts);
    209:         } else fs = zero;

    211:         if (j < my-1) {
    212:           tn = x[j+1][i];
    213:           an = 0.5*(t0 + tn);
    214:           dn = PetscPowScalar(an,beta);
    215:           fn = dn*(tn - t0);
    216:         } else fn = zero;

    218:       } else if (i == mx-1) {

    220:         /* right-hand boundary */
    221:         tw = x[j][i-1];
    222:         aw = 0.5*(t0 + tw);
    223:         dw = PetscPowScalar(aw,beta);
    224:         fw = dw*(t0 - tw);

    226:         te = tright;
    227:         ae = 0.5*(t0 + te);
    228:         de = PetscPowScalar(ae,beta);
    229:         fe = de*(te - t0);

    231:         if (j > 0) {
    232:           ts = x[j-1][i];
    233:           as = 0.5*(t0 + ts);
    234:           ds = PetscPowScalar(as,beta);
    235:           fs = ds*(t0 - ts);
    236:         } else fs = zero;

    238:         if (j < my-1) {
    239:           tn = x[j+1][i];
    240:           an = 0.5*(t0 + tn);
    241:           dn = PetscPowScalar(an,beta);
    242:           fn = dn*(tn - t0);
    243:         } else fn = zero;

    245:       } else if (j == 0) {

    247:         /* bottom boundary,and i <> 0 or mx-1 */
    248:         tw = x[j][i-1];
    249:         aw = 0.5*(t0 + tw);
    250:         dw = PetscPowScalar(aw,beta);
    251:         fw = dw*(t0 - tw);

    253:         te = x[j][i+1];
    254:         ae = 0.5*(t0 + te);
    255:         de = PetscPowScalar(ae,beta);
    256:         fe = de*(te - t0);

    258:         fs = zero;

    260:         tn = x[j+1][i];
    261:         an = 0.5*(t0 + tn);
    262:         dn = PetscPowScalar(an,beta);
    263:         fn = dn*(tn - t0);

    265:       } else if (j == my-1) {

    267:         /* top boundary,and i <> 0 or mx-1 */
    268:         tw = x[j][i-1];
    269:         aw = 0.5*(t0 + tw);
    270:         dw = PetscPowScalar(aw,beta);
    271:         fw = dw*(t0 - tw);

    273:         te = x[j][i+1];
    274:         ae = 0.5*(t0 + te);
    275:         de = PetscPowScalar(ae,beta);
    276:         fe = de*(te - t0);

    278:         ts = x[j-1][i];
    279:         as = 0.5*(t0 + ts);
    280:         ds = PetscPowScalar(as,beta);
    281:         fs = ds*(t0 - ts);

    283:         fn = zero;

    285:       }

    287:       f[j][i] = -hydhx*(fe-fw) - hxdhy*(fn-fs);

    289:     }
    290:   }
    291:   DMDAVecRestoreArray(da,localX,&x);
    292:   DMDAVecRestoreArray(da,F,&f);
    293:   DMRestoreLocalVector(da,&localX);
    294:   PetscLogFlops((22.0 + 4.0*POWFLOP)*ym*xm);
    295:   return 0;
    296: }
    297: /* --------------------  Evaluate Jacobian F(x) --------------------- */
    298: PetscErrorCode FormJacobian(SNES snes,Vec X,Mat jac,Mat B,void *ptr)
    299: {
    300:   AppCtx         *user = (AppCtx*)ptr;
    301:   PetscInt       i,j,mx,my,xs,ys,xm,ym;
    302:   PetscScalar    one = 1.0,hx,hy,hxdhy,hydhx,t0,tn,ts,te,tw;
    303:   PetscScalar    dn,ds,de,dw,an,as,ae,aw,bn,bs,be,bw,gn,gs,ge,gw;
    304:   PetscScalar    tleft,tright,beta,bm1,coef;
    305:   PetscScalar    v[5],**x;
    306:   Vec            localX;
    307:   MatStencil     col[5],row;
    308:   DM             da;

    311:   SNESGetDM(snes,&da);
    312:   DMGetLocalVector(da,&localX);
    313:   DMDAGetInfo(da,NULL,&mx,&my,0,0,0,0,0,0,0,0,0,0);
    314:   hx    = one/(PetscReal)(mx-1);  hy     = one/(PetscReal)(my-1);
    315:   hxdhy = hx/hy;               hydhx  = hy/hx;
    316:   tleft = user->tleft;         tright = user->tright;
    317:   beta  = user->beta;          bm1    = user->bm1;          coef = user->coef;

    319:   /* Get ghost points */
    320:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
    321:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);
    322:   DMDAGetCorners(da,&xs,&ys,0,&xm,&ym,0);
    323:   DMDAVecGetArray(da,localX,&x);

    325:   /* Evaluate Jacobian of function */
    326:   for (j=ys; j<ys+ym; j++) {
    327:     for (i=xs; i<xs+xm; i++) {
    328:       t0 = x[j][i];

    330:       if (i > 0 && i < mx-1 && j > 0 && j < my-1) {

    332:         /* general interior volume */

    334:         tw = x[j][i-1];
    335:         aw = 0.5*(t0 + tw);
    336:         bw = PetscPowScalar(aw,bm1);
    337:         /* dw = bw * aw */
    338:         dw = PetscPowScalar(aw,beta);
    339:         gw = coef*bw*(t0 - tw);

    341:         te = x[j][i+1];
    342:         ae = 0.5*(t0 + te);
    343:         be = PetscPowScalar(ae,bm1);
    344:         /* de = be * ae; */
    345:         de = PetscPowScalar(ae,beta);
    346:         ge = coef*be*(te - t0);

    348:         ts = x[j-1][i];
    349:         as = 0.5*(t0 + ts);
    350:         bs = PetscPowScalar(as,bm1);
    351:         /* ds = bs * as; */
    352:         ds = PetscPowScalar(as,beta);
    353:         gs = coef*bs*(t0 - ts);

    355:         tn = x[j+1][i];
    356:         an = 0.5*(t0 + tn);
    357:         bn = PetscPowScalar(an,bm1);
    358:         /* dn = bn * an; */
    359:         dn = PetscPowScalar(an,beta);
    360:         gn = coef*bn*(tn - t0);

    362:         v[0] = -hxdhy*(ds - gs);                                       col[0].j = j-1;       col[0].i = i;
    363:         v[1] = -hydhx*(dw - gw);                                       col[1].j = j;         col[1].i = i-1;
    364:         v[2] = hxdhy*(ds + dn + gs - gn) + hydhx*(dw + de + gw - ge);  col[2].j = row.j = j; col[2].i = row.i = i;
    365:         v[3] = -hydhx*(de + ge);                                       col[3].j = j;         col[3].i = i+1;
    366:         v[4] = -hxdhy*(dn + gn);                                       col[4].j = j+1;       col[4].i = i;
    367:         MatSetValuesStencil(B,1,&row,5,col,v,INSERT_VALUES);

    369:       } else if (i == 0) {

    371:         /* left-hand boundary */
    372:         tw = tleft;
    373:         aw = 0.5*(t0 + tw);
    374:         bw = PetscPowScalar(aw,bm1);
    375:         /* dw = bw * aw */
    376:         dw = PetscPowScalar(aw,beta);
    377:         gw = coef*bw*(t0 - tw);

    379:         te = x[j][i + 1];
    380:         ae = 0.5*(t0 + te);
    381:         be = PetscPowScalar(ae,bm1);
    382:         /* de = be * ae; */
    383:         de = PetscPowScalar(ae,beta);
    384:         ge = coef*be*(te - t0);

    386:         /* left-hand bottom boundary */
    387:         if (j == 0) {

    389:           tn = x[j+1][i];
    390:           an = 0.5*(t0 + tn);
    391:           bn = PetscPowScalar(an,bm1);
    392:           /* dn = bn * an; */
    393:           dn = PetscPowScalar(an,beta);
    394:           gn = coef*bn*(tn - t0);

    396:           v[0] = hxdhy*(dn - gn) + hydhx*(dw + de + gw - ge); col[0].j = row.j = j; col[0].i = row.i = i;
    397:           v[1] = -hydhx*(de + ge);                            col[1].j = j;         col[1].i = i+1;
    398:           v[2] = -hxdhy*(dn + gn);                            col[2].j = j+1;       col[2].i = i;
    399:           MatSetValuesStencil(B,1,&row,3,col,v,INSERT_VALUES);

    401:           /* left-hand interior boundary */
    402:         } else if (j < my-1) {

    404:           ts = x[j-1][i];
    405:           as = 0.5*(t0 + ts);
    406:           bs = PetscPowScalar(as,bm1);
    407:           /* ds = bs * as; */
    408:           ds = PetscPowScalar(as,beta);
    409:           gs = coef*bs*(t0 - ts);

    411:           tn = x[j+1][i];
    412:           an = 0.5*(t0 + tn);
    413:           bn = PetscPowScalar(an,bm1);
    414:           /* dn = bn * an; */
    415:           dn = PetscPowScalar(an,beta);
    416:           gn = coef*bn*(tn - t0);

    418:           v[0] = -hxdhy*(ds - gs);                                       col[0].j = j-1;       col[0].i = i;
    419:           v[1] = hxdhy*(ds + dn + gs - gn) + hydhx*(dw + de + gw - ge);  col[1].j = row.j = j; col[1].i = row.i = i;
    420:           v[2] = -hydhx*(de + ge);                                       col[2].j = j;         col[2].i = i+1;
    421:           v[3] = -hxdhy*(dn + gn);                                       col[3].j = j+1;       col[3].i = i;
    422:           MatSetValuesStencil(B,1,&row,4,col,v,INSERT_VALUES);
    423:           /* left-hand top boundary */
    424:         } else {

    426:           ts = x[j-1][i];
    427:           as = 0.5*(t0 + ts);
    428:           bs = PetscPowScalar(as,bm1);
    429:           /* ds = bs * as; */
    430:           ds = PetscPowScalar(as,beta);
    431:           gs = coef*bs*(t0 - ts);

    433:           v[0] = -hxdhy*(ds - gs);                             col[0].j = j-1;       col[0].i = i;
    434:           v[1] = hxdhy*(ds + gs) + hydhx*(dw + de + gw - ge);  col[1].j = row.j = j; col[1].i = row.i = i;
    435:           v[2] = -hydhx*(de + ge);                             col[2].j = j;         col[2].i = i+1;
    436:           MatSetValuesStencil(B,1,&row,3,col,v,INSERT_VALUES);
    437:         }

    439:       } else if (i == mx-1) {

    441:         /* right-hand boundary */
    442:         tw = x[j][i-1];
    443:         aw = 0.5*(t0 + tw);
    444:         bw = PetscPowScalar(aw,bm1);
    445:         /* dw = bw * aw */
    446:         dw = PetscPowScalar(aw,beta);
    447:         gw = coef*bw*(t0 - tw);

    449:         te = tright;
    450:         ae = 0.5*(t0 + te);
    451:         be = PetscPowScalar(ae,bm1);
    452:         /* de = be * ae; */
    453:         de = PetscPowScalar(ae,beta);
    454:         ge = coef*be*(te - t0);

    456:         /* right-hand bottom boundary */
    457:         if (j == 0) {

    459:           tn = x[j+1][i];
    460:           an = 0.5*(t0 + tn);
    461:           bn = PetscPowScalar(an,bm1);
    462:           /* dn = bn * an; */
    463:           dn = PetscPowScalar(an,beta);
    464:           gn = coef*bn*(tn - t0);

    466:           v[0] = -hydhx*(dw - gw);                            col[0].j = j;         col[0].i = i-1;
    467:           v[1] = hxdhy*(dn - gn) + hydhx*(dw + de + gw - ge); col[1].j = row.j = j; col[1].i = row.i = i;
    468:           v[2] = -hxdhy*(dn + gn);                            col[2].j = j+1;       col[2].i = i;
    469:           MatSetValuesStencil(B,1,&row,3,col,v,INSERT_VALUES);

    471:           /* right-hand interior boundary */
    472:         } else if (j < my-1) {

    474:           ts = x[j-1][i];
    475:           as = 0.5*(t0 + ts);
    476:           bs = PetscPowScalar(as,bm1);
    477:           /* ds = bs * as; */
    478:           ds = PetscPowScalar(as,beta);
    479:           gs = coef*bs*(t0 - ts);

    481:           tn = x[j+1][i];
    482:           an = 0.5*(t0 + tn);
    483:           bn = PetscPowScalar(an,bm1);
    484:           /* dn = bn * an; */
    485:           dn = PetscPowScalar(an,beta);
    486:           gn = coef*bn*(tn - t0);

    488:           v[0] = -hxdhy*(ds - gs);                                       col[0].j = j-1;       col[0].i = i;
    489:           v[1] = -hydhx*(dw - gw);                                       col[1].j = j;         col[1].i = i-1;
    490:           v[2] = hxdhy*(ds + dn + gs - gn) + hydhx*(dw + de + gw - ge);  col[2].j = row.j = j; col[2].i = row.i = i;
    491:           v[3] = -hxdhy*(dn + gn);                                       col[3].j = j+1;       col[3].i = i;
    492:           MatSetValuesStencil(B,1,&row,4,col,v,INSERT_VALUES);
    493:         /* right-hand top boundary */
    494:         } else {

    496:           ts = x[j-1][i];
    497:           as = 0.5*(t0 + ts);
    498:           bs = PetscPowScalar(as,bm1);
    499:           /* ds = bs * as; */
    500:           ds = PetscPowScalar(as,beta);
    501:           gs = coef*bs*(t0 - ts);

    503:           v[0] = -hxdhy*(ds - gs);                             col[0].j = j-1;       col[0].i = i;
    504:           v[1] = -hydhx*(dw - gw);                             col[1].j = j;         col[1].i = i-1;
    505:           v[2] = hxdhy*(ds + gs) + hydhx*(dw + de + gw - ge);  col[2].j = row.j = j; col[2].i = row.i = i;
    506:           MatSetValuesStencil(B,1,&row,3,col,v,INSERT_VALUES);
    507:         }

    509:         /* bottom boundary,and i <> 0 or mx-1 */
    510:       } else if (j == 0) {

    512:         tw = x[j][i-1];
    513:         aw = 0.5*(t0 + tw);
    514:         bw = PetscPowScalar(aw,bm1);
    515:         /* dw = bw * aw */
    516:         dw = PetscPowScalar(aw,beta);
    517:         gw = coef*bw*(t0 - tw);

    519:         te = x[j][i+1];
    520:         ae = 0.5*(t0 + te);
    521:         be = PetscPowScalar(ae,bm1);
    522:         /* de = be * ae; */
    523:         de = PetscPowScalar(ae,beta);
    524:         ge = coef*be*(te - t0);

    526:         tn = x[j+1][i];
    527:         an = 0.5*(t0 + tn);
    528:         bn = PetscPowScalar(an,bm1);
    529:         /* dn = bn * an; */
    530:         dn = PetscPowScalar(an,beta);
    531:         gn = coef*bn*(tn - t0);

    533:         v[0] = -hydhx*(dw - gw);                            col[0].j = j;         col[0].i = i-1;
    534:         v[1] = hxdhy*(dn - gn) + hydhx*(dw + de + gw - ge); col[1].j = row.j = j; col[1].i = row.i = i;
    535:         v[2] = -hydhx*(de + ge);                            col[2].j = j;         col[2].i = i+1;
    536:         v[3] = -hxdhy*(dn + gn);                            col[3].j = j+1;       col[3].i = i;
    537:         MatSetValuesStencil(B,1,&row,4,col,v,INSERT_VALUES);

    539:         /* top boundary,and i <> 0 or mx-1 */
    540:       } else if (j == my-1) {

    542:         tw = x[j][i-1];
    543:         aw = 0.5*(t0 + tw);
    544:         bw = PetscPowScalar(aw,bm1);
    545:         /* dw = bw * aw */
    546:         dw = PetscPowScalar(aw,beta);
    547:         gw = coef*bw*(t0 - tw);

    549:         te = x[j][i+1];
    550:         ae = 0.5*(t0 + te);
    551:         be = PetscPowScalar(ae,bm1);
    552:         /* de = be * ae; */
    553:         de = PetscPowScalar(ae,beta);
    554:         ge = coef*be*(te - t0);

    556:         ts = x[j-1][i];
    557:         as = 0.5*(t0 + ts);
    558:         bs = PetscPowScalar(as,bm1);
    559:         /* ds = bs * as; */
    560:         ds = PetscPowScalar(as,beta);
    561:         gs = coef*bs*(t0 - ts);

    563:         v[0] = -hxdhy*(ds - gs);                             col[0].j = j-1;       col[0].i = i;
    564:         v[1] = -hydhx*(dw - gw);                             col[1].j = j;         col[1].i = i-1;
    565:         v[2] = hxdhy*(ds + gs) + hydhx*(dw + de + gw - ge);  col[2].j = row.j = j; col[2].i = row.i = i;
    566:         v[3] = -hydhx*(de + ge);                             col[3].j = j;         col[3].i = i+1;
    567:         MatSetValuesStencil(B,1,&row,4,col,v,INSERT_VALUES);

    569:       }
    570:     }
    571:   }
    572:   MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
    573:   DMDAVecRestoreArray(da,localX,&x);
    574:   MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
    575:   DMRestoreLocalVector(da,&localX);
    576:   if (jac != B) {
    577:     MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
    578:     MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);
    579:   }

    581:   PetscLogFlops((41.0 + 8.0*POWFLOP)*xm*ym);
    582:   return 0;
    583: }

    585: /*TEST

    587:    test:
    588:       suffix: 1
    589:       args: -pc_type mg -ksp_type fgmres -da_refine 2 -pc_mg_galerkin pmat -snes_view
    590:       requires: !single

    592:    test:
    593:       suffix: 2
    594:       args: -pc_type mg -ksp_type fgmres -da_refine 2 -pc_mg_galerkin pmat -snes_view -snes_type newtontrdc
    595:       requires: !single

    597: TEST*/