Actual source code: ex7.c


  2: static char help[] = "Time-dependent PDE in 2d for calculating joint PDF. \n";
  3: /*
  4:    p_t = -x_t*p_x -y_t*p_y + f(t)*p_yy
  5:    xmin < x < xmax, ymin < y < ymax;

  7:    Boundary conditions Neumman using mirror values

  9:    Note that x_t and y_t in the above are given functions of x and y; they are not derivatives of x and y.
 10:    x_t = (y - ws)  y_t = (ws/2H)*(Pm - Pmax*sin(x))

 12: */

 14: #include <petscdm.h>
 15: #include <petscdmda.h>
 16: #include <petscts.h>

 18: /*
 19:    User-defined data structures and routines
 20: */
 21: typedef struct {
 22:   PetscScalar ws;   /* Synchronous speed */
 23:   PetscScalar H;    /* Inertia constant */
 24:   PetscScalar D;    /* Damping constant */
 25:   PetscScalar Pmax; /* Maximum power output of generator */
 26:   PetscScalar PM_min; /* Mean mechanical power input */
 27:   PetscScalar lambda; /* correlation time */
 28:   PetscScalar q;      /* noise strength */
 29:   PetscScalar mux;    /* Initial average angle */
 30:   PetscScalar sigmax; /* Standard deviation of initial angle */
 31:   PetscScalar muy;    /* Average speed */
 32:   PetscScalar sigmay; /* standard deviation of initial speed */
 33:   PetscScalar rho;    /* Cross-correlation coefficient */
 34:   PetscScalar xmin;   /* left boundary of angle */
 35:   PetscScalar xmax;   /* right boundary of angle */
 36:   PetscScalar ymin;   /* bottom boundary of speed */
 37:   PetscScalar ymax;   /* top boundary of speed */
 38:   PetscScalar dx;     /* x step size */
 39:   PetscScalar dy;     /* y step size */
 40:   PetscScalar disper_coe; /* Dispersion coefficient */
 41:   DM          da;
 42:   PetscInt    st_width; /* Stencil width */
 43:   DMBoundaryType bx; /* x boundary type */
 44:   DMBoundaryType by; /* y boundary type */
 45:   PetscBool        nonoiseinitial;
 46: } AppCtx;

 48: PetscErrorCode Parameter_settings(AppCtx*);
 49: PetscErrorCode ini_bou(Vec,AppCtx*);
 50: PetscErrorCode IFunction(TS,PetscReal,Vec,Vec,Vec,void*);
 51: PetscErrorCode IJacobian(TS,PetscReal,Vec,Vec,PetscReal,Mat,Mat,void*);
 52: PetscErrorCode PostStep(TS);

 54: int main(int argc, char **argv)
 55: {
 56:   Vec            x;  /* Solution vector */
 57:   TS             ts;   /* Time-stepping context */
 58:   AppCtx         user; /* Application context */
 59:   PetscViewer    viewer;

 61:   PetscInitialize(&argc,&argv,"petscopt_ex7", help);

 63:   /* Get physics and time parameters */
 64:   Parameter_settings(&user);
 65:   /* Create a 2D DA with dof = 1 */
 66:   DMDACreate2d(PETSC_COMM_WORLD,user.bx,user.by,DMDA_STENCIL_STAR,4,4,PETSC_DECIDE,PETSC_DECIDE,1,user.st_width,NULL,NULL,&user.da);
 67:   DMSetFromOptions(user.da);
 68:   DMSetUp(user.da);
 69:   /* Set x and y coordinates */
 70:   DMDASetUniformCoordinates(user.da,user.xmin,user.xmax,user.ymin,user.ymax,0,0);
 71:   DMDASetCoordinateName(user.da,0,"X - the angle");
 72:   DMDASetCoordinateName(user.da,1,"Y - the speed");

 74:   /* Get global vector x from DM  */
 75:   DMCreateGlobalVector(user.da,&x);

 77:   ini_bou(x,&user);
 78:   PetscViewerBinaryOpen(PETSC_COMM_WORLD,"ini_x",FILE_MODE_WRITE,&viewer);
 79:   VecView(x,viewer);
 80:   PetscViewerDestroy(&viewer);

 82:   TSCreate(PETSC_COMM_WORLD,&ts);
 83:   TSSetDM(ts,user.da);
 84:   TSSetProblemType(ts,TS_NONLINEAR);
 85:   TSSetType(ts,TSARKIMEX);
 86:   TSSetIFunction(ts,NULL,IFunction,&user);
 87:   /*  TSSetIJacobian(ts,NULL,NULL,IJacobian,&user);  */
 88:   TSSetApplicationContext(ts,&user);
 89:   TSSetTimeStep(ts,.005);
 90:   TSSetFromOptions(ts);
 91:   TSSetPostStep(ts,PostStep);
 92:   TSSolve(ts,x);

 94:   PetscViewerBinaryOpen(PETSC_COMM_WORLD,"fin_x",FILE_MODE_WRITE,&viewer);
 95:   VecView(x,viewer);
 96:   PetscViewerDestroy(&viewer);

 98:   VecDestroy(&x);
 99:   DMDestroy(&user.da);
100:   TSDestroy(&ts);
101:   PetscFinalize();
102:   return 0;
103: }

105: PetscErrorCode PostStep(TS ts)
106: {
107:   Vec            X,gc;
108:   AppCtx         *user;
109:   PetscScalar    sum = 0,asum;
110:   PetscReal      t,**p;
111:   DMDACoor2d     **coors;
112:   DM             cda;
113:   PetscInt       i,j,xs,ys,xm,ym;

115:   TSGetApplicationContext(ts,&user);
116:   TSGetTime(ts,&t);
117:   TSGetSolution(ts,&X);

119:   DMGetCoordinateDM(user->da,&cda);
120:   DMDAGetCorners(cda,&xs,&ys,0,&xm,&ym,0);
121:   DMGetCoordinates(user->da,&gc);
122:   DMDAVecGetArrayRead(cda,gc,&coors);
123:   DMDAVecGetArrayRead(user->da,X,&p);
124:   for (i=xs; i < xs+xm; i++) {
125:     for (j=ys; j < ys+ym; j++) {
126:       if (coors[j][i].y < 5) sum += p[j][i];
127:     }
128:   }
129:   DMDAVecRestoreArrayRead(cda,gc,&coors);
130:   DMDAVecRestoreArrayRead(user->da,X,&p);
131:   MPI_Allreduce(&sum,&asum,1,MPIU_SCALAR,MPIU_SUM,PetscObjectComm((PetscObject)ts));
132:   PetscPrintf(PETSC_COMM_WORLD,"sum(p)*dw*dtheta at t = %f = %f\n",(double)t,(double)(asum));
133:   if (sum  < 1.0e-2) {
134:     TSSetConvergedReason(ts,TS_CONVERGED_USER);
135:     PetscPrintf(PETSC_COMM_WORLD,"Exiting TS as the integral of PDF is almost zero\n");
136:   }
137:   return 0;
138: }

140: PetscErrorCode ini_bou(Vec X,AppCtx* user)
141: {
142:   DM             cda;
143:   DMDACoor2d     **coors;
144:   PetscScalar    **p;
145:   Vec            gc;
146:   PetscInt       i,j;
147:   PetscInt       xs,ys,xm,ym,M,N;
148:   PetscScalar    xi,yi;
149:   PetscScalar    sigmax=user->sigmax,sigmay=user->sigmay;
150:   PetscScalar    rho   =user->rho;
151:   PetscScalar    muy=user->muy,mux;
152:   PetscMPIInt    rank;
153:   PetscScalar    sum;

156:   MPI_Comm_rank(PETSC_COMM_WORLD,&rank);
157:   DMDAGetInfo(user->da,NULL,&M,&N,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL);
158:   user->dx = (user->xmax - user->xmin)/(M-1); user->dy = (user->ymax - user->ymin)/(N-1);
159:   DMGetCoordinateDM(user->da,&cda);
160:   DMGetCoordinates(user->da,&gc);
161:   DMDAVecGetArray(cda,gc,&coors);
162:   DMDAVecGetArray(user->da,X,&p);
163:   DMDAGetCorners(cda,&xs,&ys,0,&xm,&ym,0);

165:   /* mux and muy need to be grid points in the x and y-direction otherwise the solution goes unstable
166:      muy is set by choosing the y domain, no. of grid points along y-direction so that muy is a grid point
167:      in the y-direction. We only modify mux here
168:   */
169:   mux = user->mux = coors[0][M/2+10].x; /* For -pi < x < pi, this should be some angle between 0 and pi/2 */
170:   if (user->nonoiseinitial) {
171:     for (i=xs; i < xs+xm; i++) {
172:       for (j=ys; j < ys+ym; j++) {
173:         xi = coors[j][i].x; yi = coors[j][i].y;
174:         if ((xi == mux) && (yi == muy)) {
175:           p[j][i] = 1.0;
176:         }
177:       }
178:     }
179:   } else {
180:     /* Change PM_min accordingly */
181:     user->PM_min = user->Pmax*PetscSinScalar(mux);
182:     for (i=xs; i < xs+xm; i++) {
183:       for (j=ys; j < ys+ym; j++) {
184:         xi = coors[j][i].x; yi = coors[j][i].y;
185:         p[j][i] = (0.5/(PETSC_PI*sigmax*sigmay*PetscSqrtScalar(1.0-rho*rho)))*PetscExpScalar(-0.5/(1-rho*rho)*(PetscPowScalar((xi-mux)/sigmax,2) + PetscPowScalar((yi-muy)/sigmay,2) - 2*rho*(xi-mux)*(yi-muy)/(sigmax*sigmay)));
186:       }
187:     }
188:   }
189:   DMDAVecRestoreArray(cda,gc,&coors);
190:   DMDAVecRestoreArray(user->da,X,&p);
191:   VecSum(X,&sum);
192:   VecScale(X,1.0/sum);
193:   return 0;
194: }

196: /* First advection term */
197: PetscErrorCode adv1(PetscScalar **p,PetscScalar y,PetscInt i,PetscInt j,PetscInt M,PetscScalar *p1,AppCtx *user)
198: {
199:   PetscScalar f,fpos,fneg;
200:   f   =  (y - user->ws);
201:   fpos = PetscMax(f,0);
202:   fneg = PetscMin(f,0);
203:   if (user->st_width == 1) {
204:     *p1 = fpos*(p[j][i] - p[j][i-1])/user->dx + fneg*(p[j][i+1] - p[j][i])/user->dx;
205:   } else if (user->st_width == 2) {
206:     *p1 = fpos*(3*p[j][i] - 4*p[j][i-1] + p[j][i-2])/(2*user->dx) + fneg*(-p[j][i+2] + 4*p[j][i+1] - 3*p[j][i])/(2*user->dx);
207:   } else if (user->st_width == 3) {
208:     *p1 = fpos*(2*p[j][i+1] + 3*p[j][i] - 6*p[j][i-1] + p[j][i-2])/(6*user->dx) + fneg*(-p[j][i+2] + 6*p[j][i+1] - 3*p[j][i] - 2*p[j][i-1])/(6*user->dx);
209:   } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No support for wider stencils");
210:   return 0;
211: }

213: /* Second advection term */
214: PetscErrorCode adv2(PetscScalar **p,PetscScalar x,PetscInt i,PetscInt j,PetscInt N,PetscScalar *p2,AppCtx *user)
215: {
216:   PetscScalar f,fpos,fneg;
217:   f   = (user->ws/(2*user->H))*(user->PM_min - user->Pmax*PetscSinScalar(x));
218:   fpos = PetscMax(f,0);
219:   fneg = PetscMin(f,0);
220:   if (user->st_width == 1) {
221:     *p2 = fpos*(p[j][i] - p[j-1][i])/user->dy + fneg*(p[j+1][i] - p[j][i])/user->dy;
222:   } else if (user->st_width ==2) {
223:     *p2 = fpos*(3*p[j][i] - 4*p[j-1][i] + p[j-2][i])/(2*user->dy) + fneg*(-p[j+2][i] + 4*p[j+1][i] - 3*p[j][i])/(2*user->dy);
224:   } else if (user->st_width == 3) {
225:     *p2 = fpos*(2*p[j+1][i] + 3*p[j][i] - 6*p[j-1][i] + p[j-2][i])/(6*user->dy) + fneg*(-p[j+2][i] + 6*p[j+1][i] - 3*p[j][i] - 2*p[j-1][i])/(6*user->dy);
226:   } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No support for wider stencils");
227:   return 0;
228: }

230: /* Diffusion term */
231: PetscErrorCode diffuse(PetscScalar **p,PetscInt i,PetscInt j,PetscReal t,PetscScalar *p_diff,AppCtx * user)
232: {
234:   if (user->st_width == 1) {
235:     *p_diff = user->disper_coe*((p[j-1][i] - 2*p[j][i] + p[j+1][i])/(user->dy*user->dy));
236:   } else if (user->st_width == 2) {
237:     *p_diff = user->disper_coe*((-p[j-2][i] + 16*p[j-1][i] - 30*p[j][i] + 16*p[j+1][i] - p[j+2][i])/(12.0*user->dy*user->dy));
238:   } else if (user->st_width == 3) {
239:     *p_diff = user->disper_coe*((2*p[j-3][i] - 27*p[j-2][i] + 270*p[j-1][i] - 490*p[j][i] + 270*p[j+1][i] - 27*p[j+2][i] + 2*p[j+3][i])/(180.0*user->dy*user->dy));
240:   } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"No support for wider stencils");
241:   return 0;
242: }

244: PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx)
245: {
246:   AppCtx         *user=(AppCtx*)ctx;
247:   DM             cda;
248:   DMDACoor2d     **coors;
249:   PetscScalar    **p,**f,**pdot;
250:   PetscInt       i,j;
251:   PetscInt       xs,ys,xm,ym,M,N;
252:   Vec            localX,gc,localXdot;
253:   PetscScalar    p_adv1,p_adv2,p_diff;

256:   DMDAGetInfo(user->da,NULL,&M,&N,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL);
257:   DMGetCoordinateDM(user->da,&cda);
258:   DMDAGetCorners(cda,&xs,&ys,0,&xm,&ym,0);

260:   DMGetLocalVector(user->da,&localX);
261:   DMGetLocalVector(user->da,&localXdot);

263:   DMGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
264:   DMGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);
265:   DMGlobalToLocalBegin(user->da,Xdot,INSERT_VALUES,localXdot);
266:   DMGlobalToLocalEnd(user->da,Xdot,INSERT_VALUES,localXdot);

268:   DMGetCoordinatesLocal(user->da,&gc);

270:   DMDAVecGetArrayRead(cda,gc,&coors);
271:   DMDAVecGetArrayRead(user->da,localX,&p);
272:   DMDAVecGetArrayRead(user->da,localXdot,&pdot);
273:   DMDAVecGetArray(user->da,F,&f);

275:   user->disper_coe = PetscPowScalar((user->lambda*user->ws)/(2*user->H),2)*user->q*(1.0-PetscExpScalar(-t/user->lambda));
276:   for (i=xs; i < xs+xm; i++) {
277:     for (j=ys; j < ys+ym; j++) {
278:       adv1(p,coors[j][i].y,i,j,M,&p_adv1,user);
279:       adv2(p,coors[j][i].x,i,j,N,&p_adv2,user);
280:       diffuse(p,i,j,t,&p_diff,user);
281:       f[j][i] = -p_adv1 - p_adv2  + p_diff - pdot[j][i];
282:     }
283:   }
284:   DMDAVecRestoreArrayRead(user->da,localX,&p);
285:   DMDAVecRestoreArrayRead(user->da,localX,&pdot);
286:   DMRestoreLocalVector(user->da,&localX);
287:   DMRestoreLocalVector(user->da,&localXdot);
288:   DMDAVecRestoreArray(user->da,F,&f);
289:   DMDAVecRestoreArrayRead(cda,gc,&coors);

291:   return 0;
292: }

294: PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat J,Mat Jpre,void *ctx)
295: {
296:   AppCtx         *user=(AppCtx*)ctx;
297:   DM             cda;
298:   DMDACoor2d     **coors;
299:   PetscInt       i,j;
300:   PetscInt       xs,ys,xm,ym,M,N;
301:   Vec            gc;
302:   PetscScalar    val[5],xi,yi;
303:   MatStencil     row,col[5];
304:   PetscScalar    c1,c3,c5,c1pos,c1neg,c3pos,c3neg;

307:   DMDAGetInfo(user->da,NULL,&M,&N,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL);
308:   DMGetCoordinateDM(user->da,&cda);
309:   DMDAGetCorners(cda,&xs,&ys,0,&xm,&ym,0);

311:   DMGetCoordinatesLocal(user->da,&gc);
312:   DMDAVecGetArrayRead(cda,gc,&coors);
313:   for (i=xs; i < xs+xm; i++) {
314:     for (j=ys; j < ys+ym; j++) {
315:       PetscInt nc = 0;
316:       xi = coors[j][i].x; yi = coors[j][i].y;
317:       row.i = i; row.j = j;
318:       c1        = (yi-user->ws)/user->dx;
319:       c1pos    = PetscMax(c1,0);
320:       c1neg    = PetscMin(c1,0);
321:       c3        = (user->ws/(2.0*user->H))*(user->PM_min - user->Pmax*PetscSinScalar(xi))/user->dy;
322:       c3pos    = PetscMax(c3,0);
323:       c3neg    = PetscMin(c3,0);
324:       c5        = (PetscPowScalar((user->lambda*user->ws)/(2*user->H),2)*user->q*(1.0-PetscExpScalar(-t/user->lambda)))/(user->dy*user->dy);
325:       col[nc].i = i-1; col[nc].j = j;   val[nc++] = c1pos;
326:       col[nc].i = i+1; col[nc].j = j;   val[nc++] = -c1neg;
327:       col[nc].i = i;   col[nc].j = j-1; val[nc++] = c3pos + c5;
328:       col[nc].i = i;   col[nc].j = j+1; val[nc++] = -c3neg + c5;
329:       col[nc].i = i;   col[nc].j = j;   val[nc++] = -c1pos + c1neg -c3pos + c3neg -2*c5 -a;
330:       MatSetValuesStencil(Jpre,1,&row,nc,col,val,INSERT_VALUES);
331:     }
332:   }
333:   DMDAVecRestoreArrayRead(cda,gc,&coors);

335:   MatAssemblyBegin(Jpre,MAT_FINAL_ASSEMBLY);
336:   MatAssemblyEnd(Jpre,MAT_FINAL_ASSEMBLY);
337:   if (J != Jpre) {
338:     MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);
339:     MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);
340:   }
341:   return 0;
342: }

344: PetscErrorCode Parameter_settings(AppCtx *user)
345: {
346:   PetscBool      flg;


350:   /* Set default parameters */
351:   user->ws     = 1.0;
352:   user->H      = 5.0;
353:   user->Pmax   = 2.1;
354:   user->PM_min = 1.0;
355:   user->lambda = 0.1;
356:   user->q      = 1.0;
357:   user->mux    = PetscAsinScalar(user->PM_min/user->Pmax);
358:   user->sigmax = 0.1;
359:   user->sigmay = 0.1;
360:   user->rho    = 0.0;
361:   user->xmin   = -PETSC_PI;
362:   user->xmax   =  PETSC_PI;
363:   user->bx     = DM_BOUNDARY_PERIODIC;
364:   user->by     = DM_BOUNDARY_MIRROR;
365:   user->nonoiseinitial = PETSC_FALSE;

367:   /*
368:      ymin of -3 seems to let the unstable solution move up and leave a zero in its wake
369:      with an ymin of -1 the wake is never exactly zero
370:   */
371:   user->ymin   = -3.0;
372:   user->ymax   = 10.0;
373:   user->st_width = 1;

375:   PetscOptionsGetScalar(NULL,NULL,"-ws",&user->ws,&flg);
376:   PetscOptionsGetScalar(NULL,NULL,"-Inertia",&user->H,&flg);
377:   PetscOptionsGetScalar(NULL,NULL,"-Pmax",&user->Pmax,&flg);
378:   PetscOptionsGetScalar(NULL,NULL,"-PM_min",&user->PM_min,&flg);
379:   PetscOptionsGetScalar(NULL,NULL,"-lambda",&user->lambda,&flg);
380:   PetscOptionsGetScalar(NULL,NULL,"-q",&user->q,&flg);
381:   PetscOptionsGetScalar(NULL,NULL,"-mux",&user->mux,&flg);
382:   PetscOptionsGetScalar(NULL,NULL,"-sigmax",&user->sigmax,&flg);
383:   PetscOptionsGetScalar(NULL,NULL,"-muy",&user->muy,&flg);
384:   if (flg == 0) {
385:     user->muy = user->ws;
386:   }
387:   PetscOptionsGetScalar(NULL,NULL,"-sigmay",&user->sigmay,&flg);
388:   PetscOptionsGetScalar(NULL,NULL,"-rho",&user->rho,&flg);
389:   PetscOptionsGetScalar(NULL,NULL,"-xmin",&user->xmin,&flg);
390:   PetscOptionsGetScalar(NULL,NULL,"-xmax",&user->xmax,&flg);
391:   PetscOptionsGetScalar(NULL,NULL,"-ymin",&user->ymin,&flg);
392:   PetscOptionsGetScalar(NULL,NULL,"-ymax",&user->ymax,&flg);
393:   PetscOptionsGetInt(NULL,NULL,"-stencil_width",&user->st_width,&flg);
394:   PetscOptionsGetEnum(NULL,NULL,"-bx",DMBoundaryTypes,(PetscEnum*)&user->bx,&flg);
395:   PetscOptionsGetEnum(NULL,NULL,"-by",DMBoundaryTypes,(PetscEnum*)&user->by,&flg);
396:   PetscOptionsGetBool(NULL,NULL,"-nonoiseinitial",&user->nonoiseinitial,&flg);

398:   return 0;
399: }

401: /*TEST

403:    build:
404:       requires: !complex !single

406:    test:
407:       args: -ts_max_steps 2
408:       localrunfiles: petscopt_ex7

410:    test:
411:       suffix: 2
412:       args: -ts_max_steps 2 -snes_mf_operator
413:       output_file: output/ex7_1.out
414:       localrunfiles: petscopt_ex7
415:       timeoutfactor: 2

417:    test:
418:       suffix: 3
419:       args: -ts_max_steps 2 -snes_mf -pc_type none
420:       output_file: output/ex7_1.out
421:       localrunfiles: petscopt_ex7
422:       timeoutfactor: 2

424: TEST*/