Date: 2026-03-01T05:25:19+00:00

Actual source code: ex15.c

      1: static char help[] = "Time-dependent PDE in 2d. Modified from ex13.c for illustrating how to solve DAEs. \n";
      2: /*
      3:    u_t = uxx + uyy
      4:    0 < x < 1, 0 < y < 1;
      5:    At t=0: u(x,y) = exp(c*r*r*r), if r=PetscSqrtReal((x-.5)*(x-.5) + (y-.5)*(y-.5)) < .125
      6:            u(x,y) = 0.0           if r >= .125

      8:    Boundary conditions:
      9:    Drichlet BC:
     10:    At x=0, x=1, y=0, y=1: u = 0.0

     12:    Neumann BC:
     13:    At x=0, x=1: du(x,y,t)/dx = 0
     14:    At y=0, y=1: du(x,y,t)/dy = 0

     16:    mpiexec -n 2 ./ex15 -da_grid_x 40 -da_grid_y 40 -ts_max_steps 2 -snes_monitor -ksp_monitor
     17:          ./ex15 -da_grid_x 40 -da_grid_y 40  -draw_pause .1 -boundary 1 -ts_monitor_draw_solution
     18:          ./ex15 -da_grid_x 40 -da_grid_y 40  -draw_pause .1 -boundary 1 -Jtype 2 -nstencilpts 9

     20: */

     22: #include <petscdm.h>
     23: #include <petscdmda.h>
     24: #include <petscts.h>

     26: /*
     27:    User-defined data structures and routines
     28: */

     30: /* AppCtx: used by FormIFunction() and FormIJacobian() */
     31: typedef struct {
     32:   DM        da;
     33:   PetscInt  nstencilpts; /* number of stencil points: 5 or 9 */
     34:   PetscReal c;
     35:   PetscInt  boundary; /* Type of boundary condition */
     36:   PetscBool viewJacobian;
     37: } AppCtx;

     39: extern PetscErrorCode FormIFunction(TS, PetscReal, Vec, Vec, Vec, void *);
     40: extern PetscErrorCode FormIJacobian(TS, PetscReal, Vec, Vec, PetscReal, Mat, Mat, void *);
     41: extern PetscErrorCode FormInitialSolution(Vec, void *);

     43: int main(int argc, char **argv)
     44: {
     45:   TS        ts;            /* nonlinear solver */
     46:   Vec       u, r;          /* solution, residual vectors */
     47:   Mat       J, Jmf = NULL; /* Jacobian matrices */
     48:   DM        da;
     49:   PetscReal dt;
     50:   AppCtx    user; /* user-defined work context */
     51:   SNES      snes;
     52:   PetscInt  Jtype; /* Jacobian type
     53:                             0: user provide Jacobian;
     54:                             1: slow finite difference;
     55:                             2: fd with coloring; */

     57:   PetscFunctionBeginUser;
     58:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
     59:   /* Initialize user application context */
     60:   user.da           = NULL;
     61:   user.nstencilpts  = 5;
     62:   user.c            = -30.0;
     63:   user.boundary     = 0; /* 0: Drichlet BC; 1: Neumann BC */
     64:   user.viewJacobian = PETSC_FALSE;

     66:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-nstencilpts", &user.nstencilpts, NULL));
     67:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-boundary", &user.boundary, NULL));
     68:   PetscCall(PetscOptionsHasName(NULL, NULL, "-viewJacobian", &user.viewJacobian));

     70:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     71:      Create distributed array (DMDA) to manage parallel grid and vectors
     72:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     73:   if (user.nstencilpts == 5) {
     74:     PetscCall(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_STAR, 11, 11, PETSC_DECIDE, PETSC_DECIDE, 1, 1, NULL, NULL, &da));
     75:   } else if (user.nstencilpts == 9) {
     76:     PetscCall(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_BOX, 11, 11, PETSC_DECIDE, PETSC_DECIDE, 1, 1, NULL, NULL, &da));
     77:   } else SETERRQ(PETSC_COMM_WORLD, PETSC_ERR_SUP, "nstencilpts %" PetscInt_FMT " is not supported", user.nstencilpts);
     78:   PetscCall(DMSetFromOptions(da));
     79:   PetscCall(DMSetUp(da));
     80:   user.da = da;

     82:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     83:      Extract global vectors from DMDA;
     84:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     85:   PetscCall(DMCreateGlobalVector(da, &u));
     86:   PetscCall(VecDuplicate(u, &r));

     88:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     89:      Create timestepping solver context
     90:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     91:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
     92:   PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
     93:   PetscCall(TSSetType(ts, TSBEULER));
     94:   PetscCall(TSSetDM(ts, da));
     95:   PetscCall(TSSetIFunction(ts, r, FormIFunction, &user));
     96:   PetscCall(TSSetMaxTime(ts, 1.0));
     97:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));

     99:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    100:      Set initial conditions
    101:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    102:   PetscCall(FormInitialSolution(u, &user));
    103:   PetscCall(TSSetSolution(ts, u));
    104:   dt = .01;
    105:   PetscCall(TSSetTimeStep(ts, dt));

    107:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    108:    Set Jacobian evaluation routine
    109:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    110:   PetscCall(DMSetMatType(da, MATAIJ));
    111:   PetscCall(DMCreateMatrix(da, &J));
    112:   Jtype = 0;
    113:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-Jtype", &Jtype, NULL));
    114:   if (Jtype == 0) { /* use user provided Jacobian evaluation routine */
    115:     PetscCheck(user.nstencilpts == 5, PETSC_COMM_WORLD, PETSC_ERR_SUP, "user Jacobian routine FormIJacobian() does not support nstencilpts=%" PetscInt_FMT, user.nstencilpts);
    116:     PetscCall(TSSetIJacobian(ts, J, J, FormIJacobian, &user));
    117:   } else { /* use finite difference Jacobian J as preconditioner and '-snes_mf_operator' for Mat*vec */
    118:     PetscCall(TSGetSNES(ts, &snes));
    119:     PetscCall(MatCreateSNESMF(snes, &Jmf));
    120:     if (Jtype == 1) { /* slow finite difference J; */
    121:       PetscCall(SNESSetJacobian(snes, Jmf, J, SNESComputeJacobianDefault, NULL));
    122:     } else if (Jtype == 2) { /* Use coloring to compute  finite difference J efficiently */
    123:       PetscCall(SNESSetJacobian(snes, Jmf, J, SNESComputeJacobianDefaultColor, 0));
    124:     } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Jtype is not supported");
    125:   }

    127:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    128:    Sets various TS parameters from user options
    129:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    130:   PetscCall(TSSetFromOptions(ts));

    132:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    133:      Solve nonlinear system
    134:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    135:   PetscCall(TSSolve(ts, u));

    137:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    138:      Free work space.
    139:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    140:   PetscCall(MatDestroy(&J));
    141:   PetscCall(MatDestroy(&Jmf));
    142:   PetscCall(VecDestroy(&u));
    143:   PetscCall(VecDestroy(&r));
    144:   PetscCall(TSDestroy(&ts));
    145:   PetscCall(DMDestroy(&da));

    147:   PetscCall(PetscFinalize());
    148:   return 0;
    149: }

    151: /* --------------------------------------------------------------------- */
    152: /*
    153:   FormIFunction = Udot - RHSFunction
    154: */
    155: PetscErrorCode FormIFunction(TS ts, PetscReal t, Vec U, Vec Udot, Vec F, void *ctx)
    156: {
    157:   AppCtx     *user = (AppCtx *)ctx;
    158:   DM          da   = user->da;
    159:   PetscInt    i, j, Mx, My, xs, ys, xm, ym;
    160:   PetscReal   hx, hy, sx, sy;
    161:   PetscScalar u, uxx, uyy, **uarray, **f, **udot;
    162:   Vec         localU;

    164:   PetscFunctionBeginUser;
    165:   PetscCall(DMGetLocalVector(da, &localU));
    166:   PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, &My, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE));

    168:   hx = 1.0 / (PetscReal)(Mx - 1);
    169:   sx = 1.0 / (hx * hx);
    170:   hy = 1.0 / (PetscReal)(My - 1);
    171:   sy = 1.0 / (hy * hy);
    172:   PetscCheck(user->nstencilpts != 9 || hx == hy, PETSC_COMM_WORLD, PETSC_ERR_SUP, "hx must equal hy when nstencilpts = 9 for this example");

    174:   /*
    175:      Scatter ghost points to local vector,using the 2-step process
    176:         DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
    177:      By placing code between these two statements, computations can be
    178:      done while messages are in transition.
    179:   */
    180:   PetscCall(DMGlobalToLocalBegin(da, U, INSERT_VALUES, localU));
    181:   PetscCall(DMGlobalToLocalEnd(da, U, INSERT_VALUES, localU));

    183:   /* Get pointers to vector data */
    184:   PetscCall(DMDAVecGetArrayRead(da, localU, &uarray));
    185:   PetscCall(DMDAVecGetArray(da, F, &f));
    186:   PetscCall(DMDAVecGetArray(da, Udot, &udot));

    188:   /* Get local grid boundaries */
    189:   PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL));

    191:   /* Compute function over the locally owned part of the grid */
    192:   for (j = ys; j < ys + ym; j++) {
    193:     for (i = xs; i < xs + xm; i++) {
    194:       /* Boundary conditions */
    195:       if (i == 0 || j == 0 || i == Mx - 1 || j == My - 1) {
    196:         if (user->boundary == 0) { /* Drichlet BC */
    197:           f[j][i] = uarray[j][i];  /* F = U */
    198:         } else {                   /* Neumann BC */
    199:           if (i == 0 && j == 0) {  /* SW corner */
    200:             f[j][i] = uarray[j][i] - uarray[j + 1][i + 1];
    201:           } else if (i == Mx - 1 && j == 0) { /* SE corner */
    202:             f[j][i] = uarray[j][i] - uarray[j + 1][i - 1];
    203:           } else if (i == 0 && j == My - 1) { /* NW corner */
    204:             f[j][i] = uarray[j][i] - uarray[j - 1][i + 1];
    205:           } else if (i == Mx - 1 && j == My - 1) { /* NE corner */
    206:             f[j][i] = uarray[j][i] - uarray[j - 1][i - 1];
    207:           } else if (i == 0) { /* Left */
    208:             f[j][i] = uarray[j][i] - uarray[j][i + 1];
    209:           } else if (i == Mx - 1) { /* Right */
    210:             f[j][i] = uarray[j][i] - uarray[j][i - 1];
    211:           } else if (j == 0) { /* Bottom */
    212:             f[j][i] = uarray[j][i] - uarray[j + 1][i];
    213:           } else if (j == My - 1) { /* Top */
    214:             f[j][i] = uarray[j][i] - uarray[j - 1][i];
    215:           }
    216:         }
    217:       } else { /* Interior */
    218:         u = uarray[j][i];
    219:         /* 5-point stencil */
    220:         uxx = (-2.0 * u + uarray[j][i - 1] + uarray[j][i + 1]);
    221:         uyy = (-2.0 * u + uarray[j - 1][i] + uarray[j + 1][i]);
    222:         if (user->nstencilpts == 9) {
    223:           /* 9-point stencil: assume hx=hy */
    224:           uxx = 2.0 * uxx / 3.0 + (0.5 * (uarray[j - 1][i - 1] + uarray[j - 1][i + 1] + uarray[j + 1][i - 1] + uarray[j + 1][i + 1]) - 2.0 * u) / 6.0;
    225:           uyy = 2.0 * uyy / 3.0 + (0.5 * (uarray[j - 1][i - 1] + uarray[j - 1][i + 1] + uarray[j + 1][i - 1] + uarray[j + 1][i + 1]) - 2.0 * u) / 6.0;
    226:         }
    227:         f[j][i] = udot[j][i] - (uxx * sx + uyy * sy);
    228:       }
    229:     }
    230:   }

    232:   /* Restore vectors */
    233:   PetscCall(DMDAVecRestoreArrayRead(da, localU, &uarray));
    234:   PetscCall(DMDAVecRestoreArray(da, F, &f));
    235:   PetscCall(DMDAVecRestoreArray(da, Udot, &udot));
    236:   PetscCall(DMRestoreLocalVector(da, &localU));
    237:   PetscCall(PetscLogFlops(11.0 * ym * xm));
    238:   PetscFunctionReturn(PETSC_SUCCESS);
    239: }

    241: /* --------------------------------------------------------------------- */
    242: /*
    243:   FormIJacobian() - Compute IJacobian = dF/dU + a dF/dUdot
    244:   This routine is not used with option '-use_coloring'
    245: */
    246: PetscErrorCode FormIJacobian(TS ts, PetscReal t, Vec U, Vec Udot, PetscReal a, Mat J, Mat Jpre, void *ctx)
    247: {
    248:   PetscInt    i, j, Mx, My, xs, ys, xm, ym, nc;
    249:   AppCtx     *user = (AppCtx *)ctx;
    250:   DM          da   = user->da;
    251:   MatStencil  col[5], row;
    252:   PetscScalar vals[5], hx, hy, sx, sy;

    254:   PetscFunctionBeginUser;
    255:   PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, &My, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE));
    256:   PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL));

    258:   hx = 1.0 / (PetscReal)(Mx - 1);
    259:   sx = 1.0 / (hx * hx);
    260:   hy = 1.0 / (PetscReal)(My - 1);
    261:   sy = 1.0 / (hy * hy);

    263:   for (j = ys; j < ys + ym; j++) {
    264:     for (i = xs; i < xs + xm; i++) {
    265:       nc    = 0;
    266:       row.j = j;
    267:       row.i = i;
    268:       if (user->boundary == 0 && (i == 0 || i == Mx - 1 || j == 0 || j == My - 1)) {
    269:         col[nc].j  = j;
    270:         col[nc].i  = i;
    271:         vals[nc++] = 1.0;

    273:       } else if (user->boundary > 0 && i == 0) { /* Left Neumann */
    274:         col[nc].j  = j;
    275:         col[nc].i  = i;
    276:         vals[nc++] = 1.0;
    277:         col[nc].j  = j;
    278:         col[nc].i  = i + 1;
    279:         vals[nc++] = -1.0;
    280:       } else if (user->boundary > 0 && i == Mx - 1) { /* Right Neumann */
    281:         col[nc].j  = j;
    282:         col[nc].i  = i;
    283:         vals[nc++] = 1.0;
    284:         col[nc].j  = j;
    285:         col[nc].i  = i - 1;
    286:         vals[nc++] = -1.0;
    287:       } else if (user->boundary > 0 && j == 0) { /* Bottom Neumann */
    288:         col[nc].j  = j;
    289:         col[nc].i  = i;
    290:         vals[nc++] = 1.0;
    291:         col[nc].j  = j + 1;
    292:         col[nc].i  = i;
    293:         vals[nc++] = -1.0;
    294:       } else if (user->boundary > 0 && j == My - 1) { /* Top Neumann */
    295:         col[nc].j  = j;
    296:         col[nc].i  = i;
    297:         vals[nc++] = 1.0;
    298:         col[nc].j  = j - 1;
    299:         col[nc].i  = i;
    300:         vals[nc++] = -1.0;
    301:       } else { /* Interior */
    302:         col[nc].j  = j - 1;
    303:         col[nc].i  = i;
    304:         vals[nc++] = -sy;
    305:         col[nc].j  = j;
    306:         col[nc].i  = i - 1;
    307:         vals[nc++] = -sx;
    308:         col[nc].j  = j;
    309:         col[nc].i  = i;
    310:         vals[nc++] = 2.0 * (sx + sy) + a;
    311:         col[nc].j  = j;
    312:         col[nc].i  = i + 1;
    313:         vals[nc++] = -sx;
    314:         col[nc].j  = j + 1;
    315:         col[nc].i  = i;
    316:         vals[nc++] = -sy;
    317:       }
    318:       PetscCall(MatSetValuesStencil(Jpre, 1, &row, nc, col, vals, INSERT_VALUES));
    319:     }
    320:   }
    321:   PetscCall(MatAssemblyBegin(Jpre, MAT_FINAL_ASSEMBLY));
    322:   PetscCall(MatAssemblyEnd(Jpre, MAT_FINAL_ASSEMBLY));
    323:   if (J != Jpre) {
    324:     PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    325:     PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    326:   }

    328:   if (user->viewJacobian) {
    329:     PetscCall(PetscPrintf(PetscObjectComm((PetscObject)Jpre), "Jpre:\n"));
    330:     PetscCall(MatView(Jpre, PETSC_VIEWER_STDOUT_WORLD));
    331:   }
    332:   PetscFunctionReturn(PETSC_SUCCESS);
    333: }

    335: /* ------------------------------------------------------------------- */
    336: PetscErrorCode FormInitialSolution(Vec U, void *ptr)
    337: {
    338:   AppCtx       *user = (AppCtx *)ptr;
    339:   DM            da   = user->da;
    340:   PetscReal     c    = user->c;
    341:   PetscInt      i, j, xs, ys, xm, ym, Mx, My;
    342:   PetscScalar **u;
    343:   PetscReal     hx, hy, x, y, r;

    345:   PetscFunctionBeginUser;
    346:   PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, &My, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE));

    348:   hx = 1.0 / (PetscReal)(Mx - 1);
    349:   hy = 1.0 / (PetscReal)(My - 1);

    351:   /* Get pointers to vector data */
    352:   PetscCall(DMDAVecGetArray(da, U, &u));

    354:   /* Get local grid boundaries */
    355:   PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL));

    357:   /* Compute function over the locally owned part of the grid */
    358:   for (j = ys; j < ys + ym; j++) {
    359:     y = j * hy;
    360:     for (i = xs; i < xs + xm; i++) {
    361:       x = i * hx;
    362:       r = PetscSqrtReal((x - .5) * (x - .5) + (y - .5) * (y - .5));
    363:       if (r < .125) u[j][i] = PetscExpReal(c * r * r * r);
    364:       else u[j][i] = 0.0;
    365:     }
    366:   }

    368:   /* Restore vectors */
    369:   PetscCall(DMDAVecRestoreArray(da, U, &u));
    370:   PetscFunctionReturn(PETSC_SUCCESS);
    371: }

    373: /*TEST

    375:     test:
    376:       args: -da_grid_x 20 -da_grid_y 20 -boundary 0 -ts_max_steps 10 -ts_monitor

    378:     test:
    379:       suffix: 2
    380:       args: -da_grid_x 20 -da_grid_y 20 -boundary 0 -ts_max_steps 10 -Jtype 2 -ts_monitor

    382:     test:
    383:       suffix: 3
    384:       requires: !single
    385:       args: -da_grid_x 20 -da_grid_y 20 -boundary 1 -ts_max_steps 10 -ts_monitor

    387:     test:
    388:       suffix: 4
    389:       requires: !single
    390:       nsize: 2
    391:       args: -da_grid_x 20 -da_grid_y 20 -boundary 1 -ts_max_steps 10 -ts_monitor

    393:     test:
    394:       suffix: 5
    395:       nsize: 1
    396:       args: -da_grid_x 20 -da_grid_y 20 -boundary 0 -ts_max_steps 10 -Jtype 1 -ts_monitor

    398: TEST*/