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   = (DM)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   = (DM)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*/