Date: 2026-06-01T23:13:50+00:00

Actual source code: biharmonic3.c

      1: static char help[] = "Solves biharmonic equation in 1d.\n";

      3: /*
      4:   Solves the equation biharmonic equation in split form

      6:     w = -kappa \Delta u
      7:     u_t =  \Delta w
      8:     -1  <= u <= 1
      9:     Periodic boundary conditions

     11: Evolve the biharmonic heat equation with bounds:  (same as biharmonic)
     12: ---------------
     13: ./biharmonic3 -ts_monitor -snes_monitor -ts_monitor_draw_solution  -pc_type lu  -draw_pause .1 -snes_converged_reason -ts_type beuler  -da_refine 5 -draw_fields 1 -ts_time_step 9.53674e-9

     15:     w = -kappa \Delta u  + u^3  - u
     16:     u_t =  \Delta w
     17:     -1  <= u <= 1
     18:     Periodic boundary conditions

     20: Evolve the Cahn-Hillard equations:
     21: ---------------
     22: ./biharmonic3 -ts_monitor -snes_monitor -ts_monitor_draw_solution  -pc_type lu  -draw_pause .1 -snes_converged_reason  -ts_type beuler    -da_refine 6  -draw_fields 1  -kappa .00001 -ts_time_step 5.96046e-06 -cahn-hillard

     24: */
     25: #include <petscdm.h>
     26: #include <petscdmda.h>
     27: #include <petscts.h>
     28: #include <petscdraw.h>

     30: /*
     31:    User-defined routines
     32: */
     33: extern PetscErrorCode FormFunction(TS, PetscReal, Vec, Vec, Vec, void *), FormInitialSolution(DM, Vec, PetscReal);
     34: typedef struct {
     35:   PetscBool cahnhillard;
     36:   PetscReal kappa;
     37:   PetscInt  energy;
     38:   PetscReal tol;
     39:   PetscReal theta;
     40:   PetscReal theta_c;
     41: } UserCtx;

     43: int main(int argc, char **argv)
     44: {
     45:   TS            ts;   /* nonlinear solver */
     46:   Vec           x, r; /* solution, residual vectors */
     47:   Mat           J;    /* Jacobian matrix */
     48:   PetscInt      steps, Mx;
     49:   DM            da;
     50:   MatFDColoring matfdcoloring;
     51:   ISColoring    iscoloring;
     52:   PetscReal     dt;
     53:   PetscReal     vbounds[] = {-100000, 100000, -1.1, 1.1};
     54:   SNES          snes;
     55:   UserCtx       ctx;

     57:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     58:      Initialize program
     59:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     60:   PetscFunctionBeginUser;
     61:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
     62:   ctx.kappa = 1.0;
     63:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-kappa", &ctx.kappa, NULL));
     64:   ctx.cahnhillard = PETSC_FALSE;
     65:   PetscCall(PetscOptionsGetBool(NULL, NULL, "-cahn-hillard", &ctx.cahnhillard, NULL));
     66:   PetscCall(PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 2, vbounds));
     67:   PetscCall(PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 600, 600));
     68:   ctx.energy = 1;
     69:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-energy", &ctx.energy, NULL));
     70:   ctx.tol = 1.0e-8;
     71:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-tol", &ctx.tol, NULL));
     72:   ctx.theta   = .001;
     73:   ctx.theta_c = 1.0;
     74:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta", &ctx.theta, NULL));
     75:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta_c", &ctx.theta_c, NULL));

     77:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     78:      Create distributed array (DMDA) to manage parallel grid and vectors
     79:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     80:   PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10, 2, 2, NULL, &da));
     81:   PetscCall(DMSetFromOptions(da));
     82:   PetscCall(DMSetUp(da));
     83:   PetscCall(DMDASetFieldName(da, 0, "Biharmonic heat equation: w = -kappa*u_xx"));
     84:   PetscCall(DMDASetFieldName(da, 1, "Biharmonic heat equation: u"));
     85:   PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
     86:   dt = 1.0 / (10. * ctx.kappa * Mx * Mx * Mx * Mx);

     88:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     89:      Extract global vectors from DMDA; then duplicate for remaining
     90:      vectors that are the same types
     91:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     92:   PetscCall(DMCreateGlobalVector(da, &x));
     93:   PetscCall(VecDuplicate(x, &r));

     95:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     96:      Create timestepping solver context
     97:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     98:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
     99:   PetscCall(TSSetDM(ts, da));
    100:   PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
    101:   PetscCall(TSSetIFunction(ts, NULL, FormFunction, &ctx));
    102:   PetscCall(TSSetMaxTime(ts, .02));
    103:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));

    105:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    106:      Create matrix data structure; set Jacobian evaluation routine

    108: <     Set Jacobian matrix data structure and default Jacobian evaluation
    109:      routine. User can override with:
    110:      -snes_mf : matrix-free Newton-Krylov method with no preconditioning
    111:                 (unless user explicitly sets preconditioner)
    112:      -snes_mf_operator : form matrix used to construct the preconditioner as set by the user,
    113:                          but use matrix-free approx for Jacobian-vector
    114:                          products within Newton-Krylov method

    116:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    117:   PetscCall(TSGetSNES(ts, &snes));
    118:   PetscCall(SNESSetType(snes, SNESVINEWTONRSLS));
    119:   PetscCall(DMCreateColoring(da, IS_COLORING_GLOBAL, &iscoloring));
    120:   PetscCall(DMSetMatType(da, MATAIJ));
    121:   PetscCall(DMCreateMatrix(da, &J));
    122:   PetscCall(MatFDColoringCreate(J, iscoloring, &matfdcoloring));
    123:   PetscCall(MatFDColoringSetFunction(matfdcoloring, (MatFDColoringFn *)SNESTSFormFunction, ts));
    124:   PetscCall(MatFDColoringSetFromOptions(matfdcoloring));
    125:   PetscCall(MatFDColoringSetUp(J, iscoloring, matfdcoloring));
    126:   PetscCall(ISColoringDestroy(&iscoloring));
    127:   PetscCall(SNESSetJacobian(snes, J, J, SNESComputeJacobianDefaultColor, matfdcoloring));

    129:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    130:      Customize nonlinear solver
    131:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    132:   PetscCall(TSSetType(ts, TSBEULER));

    134:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    135:      Set initial conditions
    136:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    137:   PetscCall(FormInitialSolution(da, x, ctx.kappa));
    138:   PetscCall(TSSetTimeStep(ts, dt));
    139:   PetscCall(TSSetSolution(ts, x));

    141:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    142:      Set runtime options
    143:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    144:   PetscCall(TSSetFromOptions(ts));

    146:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    147:      Solve nonlinear system
    148:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    149:   PetscCall(TSSolve(ts, x));
    150:   PetscCall(TSGetStepNumber(ts, &steps));

    152:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    153:      Free work space.  All PETSc objects should be destroyed when they
    154:      are no longer needed.
    155:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    156:   PetscCall(MatDestroy(&J));
    157:   PetscCall(MatFDColoringDestroy(&matfdcoloring));
    158:   PetscCall(VecDestroy(&x));
    159:   PetscCall(VecDestroy(&r));
    160:   PetscCall(TSDestroy(&ts));
    161:   PetscCall(DMDestroy(&da));

    163:   PetscCall(PetscFinalize());
    164:   return 0;
    165: }

    167: typedef struct {
    168:   PetscScalar w, u;
    169: } Field;
    170: /* ------------------------------------------------------------------- */
    171: /*
    172:    FormFunction - Evaluates nonlinear function, F(x).

    174:    Input Parameters:
    175: .  ts - the TS context
    176: .  X - input vector
    177: .  ptr - optional user-defined context, as set by SNESSetFunction()

    179:    Output Parameter:
    180: .  F - function vector
    181:  */
    182: PetscErrorCode FormFunction(TS ts, PetscReal ftime, Vec X, Vec Xdot, Vec F, void *ptr)
    183: {
    184:   DM          da;
    185:   PetscInt    i, Mx, xs, xm;
    186:   PetscReal   hx, sx;
    187:   PetscScalar r, l;
    188:   Field      *x, *xdot, *f;
    189:   Vec         localX, localXdot;
    190:   UserCtx    *ctx = (UserCtx *)ptr;

    192:   PetscFunctionBegin;
    193:   PetscCall(TSGetDM(ts, &da));
    194:   PetscCall(DMGetLocalVector(da, &localX));
    195:   PetscCall(DMGetLocalVector(da, &localXdot));
    196:   PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE));

    198:   hx = 1.0 / (PetscReal)Mx;
    199:   sx = 1.0 / (hx * hx);

    201:   /*
    202:      Scatter ghost points to local vector,using the 2-step process
    203:         DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
    204:      By placing code between these two statements, computations can be
    205:      done while messages are in transition.
    206:   */
    207:   PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX));
    208:   PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX));
    209:   PetscCall(DMGlobalToLocalBegin(da, Xdot, INSERT_VALUES, localXdot));
    210:   PetscCall(DMGlobalToLocalEnd(da, Xdot, INSERT_VALUES, localXdot));

    212:   /*
    213:      Get pointers to vector data
    214:   */
    215:   PetscCall(DMDAVecGetArrayRead(da, localX, &x));
    216:   PetscCall(DMDAVecGetArrayRead(da, localXdot, &xdot));
    217:   PetscCall(DMDAVecGetArray(da, F, &f));

    219:   /*
    220:      Get local grid boundaries
    221:   */
    222:   PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL));

    224:   /*
    225:      Compute function over the locally owned part of the grid
    226:   */
    227:   for (i = xs; i < xs + xm; i++) {
    228:     f[i].w = x[i].w + ctx->kappa * (x[i - 1].u + x[i + 1].u - 2.0 * x[i].u) * sx;
    229:     if (ctx->cahnhillard) {
    230:       switch (ctx->energy) {
    231:       case 1: /* double well */
    232:         f[i].w += -x[i].u * x[i].u * x[i].u + x[i].u;
    233:         break;
    234:       case 2: /* double obstacle */
    235:         f[i].w += x[i].u;
    236:         break;
    237:       case 3: /* logarithmic */
    238:         if (x[i].u < -1.0 + 2.0 * ctx->tol) f[i].w += .5 * ctx->theta * (-PetscLogScalar(ctx->tol) + PetscLogScalar((1.0 - x[i].u) / 2.0)) + ctx->theta_c * x[i].u;
    239:         else if (x[i].u > 1.0 - 2.0 * ctx->tol) f[i].w += .5 * ctx->theta * (-PetscLogScalar((1.0 + x[i].u) / 2.0) + PetscLogScalar(ctx->tol)) + ctx->theta_c * x[i].u;
    240:         else f[i].w += .5 * ctx->theta * (-PetscLogScalar((1.0 + x[i].u) / 2.0) + PetscLogScalar((1.0 - x[i].u) / 2.0)) + ctx->theta_c * x[i].u;
    241:         break;
    242:       case 4:
    243:         break;
    244:       }
    245:     }
    246:     f[i].u = xdot[i].u - (x[i - 1].w + x[i + 1].w - 2.0 * x[i].w) * sx;
    247:     if (ctx->energy == 4) {
    248:       f[i].u = xdot[i].u;
    249:       /* approximation of \grad (M(u) \grad w), where M(u) = (1-u^2) */
    250:       r = (1.0 - x[i + 1].u * x[i + 1].u) * (x[i + 2].w - x[i].w) * .5 / hx;
    251:       l = (1.0 - x[i - 1].u * x[i - 1].u) * (x[i].w - x[i - 2].w) * .5 / hx;
    252:       f[i].u -= (r - l) * .5 / hx;
    253:       f[i].u += 2.0 * ctx->theta_c * x[i].u * (x[i + 1].u - x[i - 1].u) * (x[i + 1].u - x[i - 1].u) * .25 * sx - (ctx->theta - ctx->theta_c * (1 - x[i].u * x[i].u)) * (x[i + 1].u + x[i - 1].u - 2.0 * x[i].u) * sx;
    254:     }
    255:   }

    257:   /*
    258:      Restore vectors
    259:   */
    260:   PetscCall(DMDAVecRestoreArrayRead(da, localXdot, &xdot));
    261:   PetscCall(DMDAVecRestoreArrayRead(da, localX, &x));
    262:   PetscCall(DMDAVecRestoreArray(da, F, &f));
    263:   PetscCall(DMRestoreLocalVector(da, &localX));
    264:   PetscCall(DMRestoreLocalVector(da, &localXdot));
    265:   PetscFunctionReturn(PETSC_SUCCESS);
    266: }

    268: /* ------------------------------------------------------------------- */
    269: PetscErrorCode FormInitialSolution(DM da, Vec X, PetscReal kappa)
    270: {
    271:   PetscInt  i, xs, xm, Mx, xgs, xgm;
    272:   Field    *x;
    273:   PetscReal hx, xx, r, sx;
    274:   Vec       Xg;

    276:   PetscFunctionBegin;
    277:   PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE));

    279:   hx = 1.0 / (PetscReal)Mx;
    280:   sx = 1.0 / (hx * hx);

    282:   /*
    283:      Get pointers to vector data
    284:   */
    285:   PetscCall(DMCreateLocalVector(da, &Xg));
    286:   PetscCall(DMDAVecGetArray(da, Xg, &x));

    288:   /*
    289:      Get local grid boundaries
    290:   */
    291:   PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL));
    292:   PetscCall(DMDAGetGhostCorners(da, &xgs, NULL, NULL, &xgm, NULL, NULL));

    294:   /*
    295:      Compute u function over the locally owned part of the grid including ghost points
    296:   */
    297:   for (i = xgs; i < xgs + xgm; i++) {
    298:     xx = i * hx;
    299:     r  = PetscSqrtReal((xx - .5) * (xx - .5));
    300:     if (r < .125) x[i].u = 1.0;
    301:     else x[i].u = -.50;
    302:     /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */
    303:     x[i].w = 0;
    304:   }
    305:   for (i = xs; i < xs + xm; i++) x[i].w = -kappa * (x[i - 1].u + x[i + 1].u - 2.0 * x[i].u) * sx;

    307:   /*
    308:      Restore vectors
    309:   */
    310:   PetscCall(DMDAVecRestoreArray(da, Xg, &x));

    312:   /* Grab only the global part of the vector */
    313:   PetscCall(VecSet(X, 0));
    314:   PetscCall(DMLocalToGlobalBegin(da, Xg, ADD_VALUES, X));
    315:   PetscCall(DMLocalToGlobalEnd(da, Xg, ADD_VALUES, X));
    316:   PetscCall(VecDestroy(&Xg));
    317:   PetscFunctionReturn(PETSC_SUCCESS);
    318: }

    320: /*TEST

    322:    build:
    323:      requires: !complex !single

    325:    test:
    326:      args: -ts_monitor -snes_monitor -pc_type lu -snes_converged_reason -ts_type beuler -da_refine 5 -ts_time_step 9.53674e-9 -ts_max_steps 50
    327:      requires: x

    329: TEST*/