Actual source code: ex26.c

```  1: static char help[] = "Transient nonlinear driven cavity in 2d.\n\
2:   \n\
3: The 2D driven cavity problem is solved in a velocity-vorticity formulation.\n\
4: The flow can be driven with the lid or with buoyancy or both:\n\
5:   -lidvelocity <lid>, where <lid> = dimensionless velocity of lid\n\
6:   -grashof <gr>, where <gr> = dimensionless temperature gradent\n\
7:   -prandtl <pr>, where <pr> = dimensionless thermal/momentum diffusity ratio\n\
8:   -contours : draw contour plots of solution\n\n";
9: /*
10:       See src/snes/tutorials/ex19.c for the steady-state version.

12:       There used to be a SNES example (src/snes/tutorials/ex27.c) that
13:       implemented this algorithm without using TS and was used for the numerical
14:       results in the paper

16:         Todd S. Coffey and C. T. Kelley and David E. Keyes, Pseudotransient
17:         Continuation and Differential-Algebraic Equations, 2003.

19:       That example was removed because it used obsolete interfaces, but the
20:       algorithms from the paper can be reproduced using this example.

22:       Note: The paper describes the algorithm as being linearly implicit but the
23:       numerical results were created using nonlinearly implicit Euler.  The
24:       algorithm as described (linearly implicit) is more efficient and is the
25:       default when using TSPSEUDO.  If you want to reproduce the numerical
26:       results from the paper, you'll have to change the SNES to converge the
27:       nonlinear solve (e.g., -snes_type newtonls).  The DAE versus ODE variants
28:       are controlled using the -parabolic option.

30:       Comment preserved from snes/tutorials/ex27.c, since removed:

32:         [H]owever Figure 3.1 was generated with a slightly different algorithm
33:         (see targets runex27 and runex27_p) than described in the paper.  In
34:         particular, the described algorithm is linearly implicit, advancing to
35:         the next step after one Newton step, so that the steady state residual
36:         is always used, but the figure was generated by converging each step to
37:         a relative tolerance of 1.e-3.  On the example problem, setting
38:         -snes_type ksponly has only minor impact on number of steps, but
39:         significantly reduces the required number of linear solves.

42: */

44: /* ------------------------------------------------------------------------

46:     We thank David E. Keyes for contributing the driven cavity discretization
47:     within this example code.

49:     This problem is modeled by the partial differential equation system

51:         - Lap(U) - Grad_y(Omega) = 0
52:         - Lap(V) + Grad_x(Omega) = 0
53:         Omega_t - Lap(Omega) + Div([U*Omega,V*Omega]) - GR*Grad_x(T) = 0
54:         T_t - Lap(T) + PR*Div([U*T,V*T]) = 0

56:     in the unit square, which is uniformly discretized in each of x and
57:     y in this simple encoding.

59:     No-slip, rigid-wall Dirichlet conditions are used for [U,V].
60:     Dirichlet conditions are used for Omega, based on the definition of
62:     constant coordinate boundary, the tangential derivative is zero.
63:     Dirichlet conditions are used for T on the left and right walls,
64:     and insulation homogeneous Neumann conditions are used for T on
65:     the top and bottom walls.

67:     A finite difference approximation with the usual 5-point stencil
68:     is used to discretize the boundary value problem to obtain a
69:     nonlinear system of equations.  Upwinding is used for the divergence
70:     (convective) terms and central for the gradient (source) terms.

72:     The Jacobian can be either
73:       * formed via finite differencing using coloring (the default), or
74:       * applied matrix-free via the option -snes_mf
75:         (for larger grid problems this variant may not converge
76:         without a preconditioner due to ill-conditioning).

78:   ------------------------------------------------------------------------- */

80: /*
81:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
82:    Include "petscts.h" so that we can use TS solvers.  Note that this
83:    file automatically includes:
84:      petscsys.h       - base PETSc routines   petscvec.h - vectors
85:      petscmat.h - matrices
86:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
87:      petscviewer.h - viewers               petscpc.h  - preconditioners
88:      petscksp.h   - linear solvers         petscsnes.h - nonlinear solvers
89: */
90: #include <petscts.h>
91: #include <petscdm.h>
92: #include <petscdmda.h>

94: /*
95:    User-defined routines and data structures
96: */
97: typedef struct {
98:   PetscScalar u, v, omega, temp;
99: } Field;

101: PetscErrorCode FormIFunctionLocal(DMDALocalInfo *, PetscReal, Field **, Field **, Field **, void *);

103: typedef struct {
104:   PetscReal lidvelocity, prandtl, grashof; /* physical parameters */
105:   PetscBool parabolic;                     /* allow a transient term corresponding roughly to artificial compressibility */
106:   PetscReal cfl_initial;                   /* CFL for first time step */
107: } AppCtx;

109: PetscErrorCode FormInitialSolution(TS, Vec, AppCtx *);

111: int main(int argc, char **argv)
112: {
113:   AppCtx            user; /* user-defined work context */
114:   PetscInt          mx, my, steps;
115:   TS                ts;
116:   DM                da;
117:   Vec               X;
118:   PetscReal         ftime;
119:   TSConvergedReason reason;

121:   PetscFunctionBeginUser;
122:   PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));
123:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
124:   PetscCall(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_STAR, 4, 4, PETSC_DECIDE, PETSC_DECIDE, 4, 1, 0, 0, &da));
125:   PetscCall(DMSetFromOptions(da));
126:   PetscCall(DMSetUp(da));
127:   PetscCall(TSSetDM(ts, (DM)da));

129:   PetscCall(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));
130:   /*
131:      Problem parameters (velocity of lid, prandtl, and grashof numbers)
132:   */
133:   user.lidvelocity = 1.0 / (mx * my);
134:   user.prandtl     = 1.0;
135:   user.grashof     = 1.0;
136:   user.parabolic   = PETSC_FALSE;
137:   user.cfl_initial = 50.;

139:   PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Driven cavity/natural convection options", "");
140:   PetscCall(PetscOptionsReal("-lidvelocity", "Lid velocity, related to Reynolds number", "", user.lidvelocity, &user.lidvelocity, NULL));
141:   PetscCall(PetscOptionsReal("-prandtl", "Ratio of viscous to thermal diffusivity", "", user.prandtl, &user.prandtl, NULL));
142:   PetscCall(PetscOptionsReal("-grashof", "Ratio of buoyant to viscous forces", "", user.grashof, &user.grashof, NULL));
143:   PetscCall(PetscOptionsBool("-parabolic", "Relax incompressibility to make the system parabolic instead of differential-algebraic", "", user.parabolic, &user.parabolic, NULL));
144:   PetscCall(PetscOptionsReal("-cfl_initial", "Advective CFL for the first time step", "", user.cfl_initial, &user.cfl_initial, NULL));
145:   PetscOptionsEnd();

147:   PetscCall(DMDASetFieldName(da, 0, "x-velocity"));
148:   PetscCall(DMDASetFieldName(da, 1, "y-velocity"));
149:   PetscCall(DMDASetFieldName(da, 2, "Omega"));
150:   PetscCall(DMDASetFieldName(da, 3, "temperature"));

152:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
153:      Create user context, set problem data, create vector data structures.
154:      Also, compute the initial guess.
155:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

157:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
158:      Create time integration context
159:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
160:   PetscCall(DMSetApplicationContext(da, &user));
161:   PetscCall(DMDATSSetIFunctionLocal(da, INSERT_VALUES, (DMDATSIFunctionLocalFn *)FormIFunctionLocal, &user));
162:   PetscCall(TSSetMaxSteps(ts, 10000));
163:   PetscCall(TSSetMaxTime(ts, 1e12));
164:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
165:   PetscCall(TSSetTimeStep(ts, user.cfl_initial / (user.lidvelocity * mx)));
166:   PetscCall(TSSetFromOptions(ts));

168:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "%" PetscInt_FMT "x%" PetscInt_FMT " grid, lid velocity = %g, prandtl # = %g, grashof # = %g\n", mx, my, (double)user.lidvelocity, (double)user.prandtl, (double)user.grashof));

170:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
171:      Solve the nonlinear system
172:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

174:   PetscCall(DMCreateGlobalVector(da, &X));
175:   PetscCall(FormInitialSolution(ts, X, &user));

177:   PetscCall(TSSolve(ts, X));
178:   PetscCall(TSGetSolveTime(ts, &ftime));
179:   PetscCall(TSGetStepNumber(ts, &steps));
180:   PetscCall(TSGetConvergedReason(ts, &reason));

182:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "%s at time %g after %" PetscInt_FMT " steps\n", TSConvergedReasons[reason], (double)ftime, steps));

184:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
185:      Free work space.  All PETSc objects should be destroyed when they
186:      are no longer needed.
187:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
188:   PetscCall(VecDestroy(&X));
189:   PetscCall(DMDestroy(&da));
190:   PetscCall(TSDestroy(&ts));

192:   PetscCall(PetscFinalize());
193:   return 0;
194: }

196: /* ------------------------------------------------------------------- */

198: /*
199:    FormInitialSolution - Forms initial approximation.

201:    Input Parameters:
202:    user - user-defined application context
203:    X - vector

205:    Output Parameter:
206:    X - vector
207:  */
208: PetscErrorCode FormInitialSolution(TS ts, Vec X, AppCtx *user)
209: {
210:   DM        da;
211:   PetscInt  i, j, mx, xs, ys, xm, ym;
212:   PetscReal grashof, dx;
213:   Field   **x;

215:   PetscFunctionBeginUser;
216:   grashof = user->grashof;
217:   PetscCall(TSGetDM(ts, &da));
218:   PetscCall(DMDAGetInfo(da, 0, &mx, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
219:   dx = 1.0 / (mx - 1);

221:   /*
222:      Get local grid boundaries (for 2-dimensional DMDA):
223:        xs, ys   - starting grid indices (no ghost points)
224:        xm, ym   - widths of local grid (no ghost points)
225:   */
226:   PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL));

228:   /*
229:      Get a pointer to vector data.
230:        - For default PETSc vectors, VecGetArray() returns a pointer to
231:          the data array.  Otherwise, the routine is implementation dependent.
232:        - You MUST call VecRestoreArray() when you no longer need access to
233:          the array.
234:   */
235:   PetscCall(DMDAVecGetArray(da, X, &x));

237:   /*
238:      Compute initial guess over the locally owned part of the grid
239:      Initial condition is motionless fluid and equilibrium temperature
240:   */
241:   for (j = ys; j < ys + ym; j++) {
242:     for (i = xs; i < xs + xm; i++) {
243:       x[j][i].u     = 0.0;
244:       x[j][i].v     = 0.0;
245:       x[j][i].omega = 0.0;
246:       x[j][i].temp  = (grashof > 0) * i * dx;
247:     }
248:   }

250:   /*
251:      Restore vector
252:   */
253:   PetscCall(DMDAVecRestoreArray(da, X, &x));
254:   PetscFunctionReturn(PETSC_SUCCESS);
255: }

257: PetscErrorCode FormIFunctionLocal(DMDALocalInfo *info, PetscReal ptime, Field **x, Field **xdot, Field **f, void *ptr)
258: {
259:   AppCtx     *user = (AppCtx *)ptr;
260:   PetscInt    xints, xinte, yints, yinte, i, j;
261:   PetscReal   hx, hy, dhx, dhy, hxdhy, hydhx;
262:   PetscReal   grashof, prandtl, lid;
263:   PetscScalar u, udot, uxx, uyy, vx, vy, avx, avy, vxp, vxm, vyp, vym;

265:   PetscFunctionBeginUser;
266:   grashof = user->grashof;
267:   prandtl = user->prandtl;
268:   lid     = user->lidvelocity;

270:   /*
271:      Define mesh intervals ratios for uniform grid.

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

276:   */
277:   dhx   = (PetscReal)(info->mx - 1);
278:   dhy   = (PetscReal)(info->my - 1);
279:   hx    = 1.0 / dhx;
280:   hy    = 1.0 / dhy;
281:   hxdhy = hx * dhy;
282:   hydhx = hy * dhx;

284:   xints = info->xs;
285:   xinte = info->xs + info->xm;
286:   yints = info->ys;
287:   yinte = info->ys + info->ym;

289:   /* Test whether we are on the bottom edge of the global array */
290:   if (yints == 0) {
291:     j     = 0;
292:     yints = yints + 1;
293:     /* bottom edge */
294:     for (i = info->xs; i < info->xs + info->xm; i++) {
295:       f[j][i].u     = x[j][i].u;
296:       f[j][i].v     = x[j][i].v;
297:       f[j][i].omega = x[j][i].omega + (x[j + 1][i].u - x[j][i].u) * dhy;
298:       f[j][i].temp  = x[j][i].temp - x[j + 1][i].temp;
299:     }
300:   }

302:   /* Test whether we are on the top edge of the global array */
303:   if (yinte == info->my) {
304:     j     = info->my - 1;
305:     yinte = yinte - 1;
306:     /* top edge */
307:     for (i = info->xs; i < info->xs + info->xm; i++) {
308:       f[j][i].u     = x[j][i].u - lid;
309:       f[j][i].v     = x[j][i].v;
310:       f[j][i].omega = x[j][i].omega + (x[j][i].u - x[j - 1][i].u) * dhy;
311:       f[j][i].temp  = x[j][i].temp - x[j - 1][i].temp;
312:     }
313:   }

315:   /* Test whether we are on the left edge of the global array */
316:   if (xints == 0) {
317:     i     = 0;
318:     xints = xints + 1;
319:     /* left edge */
320:     for (j = info->ys; j < info->ys + info->ym; j++) {
321:       f[j][i].u     = x[j][i].u;
322:       f[j][i].v     = x[j][i].v;
323:       f[j][i].omega = x[j][i].omega - (x[j][i + 1].v - x[j][i].v) * dhx;
324:       f[j][i].temp  = x[j][i].temp;
325:     }
326:   }

328:   /* Test whether we are on the right edge of the global array */
329:   if (xinte == info->mx) {
330:     i     = info->mx - 1;
331:     xinte = xinte - 1;
332:     /* right edge */
333:     for (j = info->ys; j < info->ys + info->ym; j++) {
334:       f[j][i].u     = x[j][i].u;
335:       f[j][i].v     = x[j][i].v;
336:       f[j][i].omega = x[j][i].omega - (x[j][i].v - x[j][i - 1].v) * dhx;
337:       f[j][i].temp  = x[j][i].temp - (PetscReal)(grashof > 0);
338:     }
339:   }

341:   /* Compute over the interior points */
342:   for (j = yints; j < yinte; j++) {
343:     for (i = xints; i < xinte; i++) {
344:       /*
345:         convective coefficients for upwinding
346:       */
347:       vx  = x[j][i].u;
348:       avx = PetscAbsScalar(vx);
349:       vxp = .5 * (vx + avx);
350:       vxm = .5 * (vx - avx);
351:       vy  = x[j][i].v;
352:       avy = PetscAbsScalar(vy);
353:       vyp = .5 * (vy + avy);
354:       vym = .5 * (vy - avy);

356:       /* U velocity */
357:       u         = x[j][i].u;
358:       udot      = user->parabolic ? xdot[j][i].u : 0.;
359:       uxx       = (2.0 * u - x[j][i - 1].u - x[j][i + 1].u) * hydhx;
360:       uyy       = (2.0 * u - x[j - 1][i].u - x[j + 1][i].u) * hxdhy;
361:       f[j][i].u = udot + uxx + uyy - .5 * (x[j + 1][i].omega - x[j - 1][i].omega) * hx;

363:       /* V velocity */
364:       u         = x[j][i].v;
365:       udot      = user->parabolic ? xdot[j][i].v : 0.;
366:       uxx       = (2.0 * u - x[j][i - 1].v - x[j][i + 1].v) * hydhx;
367:       uyy       = (2.0 * u - x[j - 1][i].v - x[j + 1][i].v) * hxdhy;
368:       f[j][i].v = udot + uxx + uyy + .5 * (x[j][i + 1].omega - x[j][i - 1].omega) * hy;

370:       /* Omega */
371:       u   = x[j][i].omega;
372:       uxx = (2.0 * u - x[j][i - 1].omega - x[j][i + 1].omega) * hydhx;
373:       uyy = (2.0 * u - x[j - 1][i].omega - x[j + 1][i].omega) * hxdhy;
374:       f[j][i].omega = (xdot[j][i].omega + uxx + uyy + (vxp * (u - x[j][i - 1].omega) + vxm * (x[j][i + 1].omega - u)) * hy + (vyp * (u - x[j - 1][i].omega) + vym * (x[j + 1][i].omega - u)) * hx - .5 * grashof * (x[j][i + 1].temp - x[j][i - 1].temp) * hy);

376:       /* Temperature */
377:       u            = x[j][i].temp;
378:       uxx          = (2.0 * u - x[j][i - 1].temp - x[j][i + 1].temp) * hydhx;
379:       uyy          = (2.0 * u - x[j - 1][i].temp - x[j + 1][i].temp) * hxdhy;
380:       f[j][i].temp = (xdot[j][i].temp + 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));
381:     }
382:   }

384:   /*
385:      Flop count (multiply-adds are counted as 2 operations)
386:   */
387:   PetscCall(PetscLogFlops(84.0 * info->ym * info->xm));
388:   PetscFunctionReturn(PETSC_SUCCESS);
389: }

391: /*TEST

393:     test:
394:       args: -da_grid_x 20 -da_grid_y 20 -lidvelocity 100 -grashof 1e3 -ts_max_steps 100 -ts_rtol 1e-3 -ts_atol 1e-3 -ts_type rosw -ts_rosw_type ra3pw -ts_monitor -ts_monitor_solution_vtk 'foo-%03d.vts'
395:       requires: !complex !single

397:     test:
398:       suffix: 2
399:       nsize: 4
400:       args: -da_grid_x 20 -da_grid_y 20 -lidvelocity 100 -grashof 1e3 -ts_max_steps 100 -ts_rtol 1e-3 -ts_atol 1e-3 -ts_type rosw -ts_rosw_type ra3pw -ts_monitor -ts_monitor_solution_vtk 'foo-%03d.vts'
401:       requires: !complex !single

403:     test:
404:       suffix: 3
405:       nsize: 4
406:       args: -da_refine 2 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3 -pc_type none -ts_type beuler -ts_monitor -snes_monitor_short -snes_type aspin -da_overlap 4 -npc_sub_ksp_type preonly -npc_sub_pc_type lu
407:       requires: !complex !single

409:     test:
410:       suffix: 4
411:       nsize: 2
412:       args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3
413:       requires: !complex !single

415:     test:
416:       suffix: asm
417:       nsize: 4
418:       args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3
419:       requires: !complex !single

421: TEST*/
```