Actual source code: ex7.c

petsc-3.7.1 2016-05-15
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  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);

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

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

 66:   /* Get physics and time parameters */
 67:   Parameter_settings(&user);
 68:   /* Create a 2D DA with dof = 1 */
 69:   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);
 70:   /* Set x and y coordinates */
 71:   DMDASetUniformCoordinates(user.da,user.xmin,user.xmax,user.ymin,user.ymax,0,0);
 72:   DMDASetCoordinateName(user.da,0,"X - the angle");
 73:   DMDASetCoordinateName(user.da,1,"Y - the speed");

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

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

 83:   TSCreate(PETSC_COMM_WORLD,&ts);
 84:   TSSetDM(ts,user.da);
 85:   TSSetProblemType(ts,TS_NONLINEAR);
 86:   TSSetType(ts,TSARKIMEX);
 87:   TSSetIFunction(ts,NULL,IFunction,&user);
 88:   /*  TSSetIJacobian(ts,NULL,NULL,IJacobian,&user);  */
 89:   TSSetApplicationContext(ts,&user);
 90:   TSSetInitialTimeStep(ts,0.0,.005);
 91:   TSSetFromOptions(ts);
 92:   TSSetPostStep(ts,PostStep);
 93:   TSSolve(ts,x);

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

 99:   VecDestroy(&x);
100:   DMDestroy(&user.da);
101:   TSDestroy(&ts);
102:   PetscFinalize();
103:   return 0;
104: }

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

120:   TSGetApplicationContext(ts,&user);
121:   TSGetTime(ts,&t);
122:   TSGetSolution(ts,&X);

124:   DMGetCoordinateDM(user->da,&cda);
125:   DMDAGetCorners(cda,&xs,&ys,0,&xm,&ym,0);
126:   DMGetCoordinates(user->da,&gc);
127:   DMDAVecGetArrayRead(cda,gc,&coors);
128:   DMDAVecGetArrayRead(user->da,X,&p);
129:   for (i=xs; i < xs+xm; i++) {
130:     for (j=ys; j < ys+ym; j++) {
131:       if (coors[j][i].y < 5) sum += p[j][i];
132:     }
133:   }
134:   DMDAVecRestoreArrayRead(cda,gc,&coors);
135:   DMDAVecRestoreArrayRead(user->da,X,&p);
136:   MPI_Allreduce(&sum,&asum,1,MPIU_SCALAR,MPIU_SUM,PetscObjectComm((PetscObject)ts));
137:   PetscPrintf(PETSC_COMM_WORLD,"sum(p)*dw*dtheta at t = %f = %f\n",(double)t,(double)(asum));
138:   if (sum  < 1.0e-2) {
139:     TSSetConvergedReason(ts,TS_CONVERGED_USER);
140:     PetscPrintf(PETSC_COMM_WORLD,"Exiting TS as the integral of PDF is almost zero\n");
141:   }
142:   return(0);
143: }

147: PetscErrorCode ini_bou(Vec X,AppCtx* user)
148: {
150:   DM             cda;
151:   DMDACoor2d     **coors;
152:   PetscScalar    **p;
153:   Vec            gc;
154:   PetscInt       i,j;
155:   PetscInt       xs,ys,xm,ym,M,N;
156:   PetscScalar    xi,yi;
157:   PetscScalar    sigmax=user->sigmax,sigmay=user->sigmay;
158:   PetscScalar    rho   =user->rho;
159:   PetscScalar    mux   =user->mux,muy=user->muy;
160:   PetscMPIInt    rank;
161:   PetscScalar    sum;

164:   MPI_Comm_rank(PETSC_COMM_WORLD,&rank);
165:   DMDAGetInfo(user->da,NULL,&M,&N,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL);
166:   user->dx = (user->xmax - user->xmin)/(M-1); user->dy = (user->ymax - user->ymin)/(N-1);
167:   DMGetCoordinateDM(user->da,&cda);
168:   DMGetCoordinates(user->da,&gc);
169:   DMDAVecGetArray(cda,gc,&coors);
170:   DMDAVecGetArray(user->da,X,&p);
171:   DMDAGetCorners(cda,&xs,&ys,0,&xm,&ym,0);

173:   /* mux and muy need to be grid points in the x and y-direction otherwise the solution goes unstable
174:      muy is set by choosing the y domain, no. of grid points along y-direction so that muy is a grid point
175:      in the y-direction. We only modify mux here
176:   */
177:   mux = user->mux = coors[0][M/2+10].x; /* For -pi < x < pi, this should be some angle between 0 and pi/2 */
178:   if (user->nonoiseinitial) {
179:     for (i=xs; i < xs+xm; i++) {
180:       for (j=ys; j < ys+ym; j++) {
181:         xi = coors[j][i].x; yi = coors[j][i].y;
182:         if ((xi == mux) && (yi == muy)) {
183:           p[j][i] = 1.0;
184:         }
185:       }
186:     }
187:   } else {
188:     /* Change PM_min accordingly */
189:     user->PM_min = user->Pmax*PetscSinScalar(mux);
190:     for (i=xs; i < xs+xm; i++) {
191:       for (j=ys; j < ys+ym; j++) {
192:         xi = coors[j][i].x; yi = coors[j][i].y;
193:         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)));
194:       }
195:     }
196:   }
197:   DMDAVecRestoreArray(cda,gc,&coors);
198:   DMDAVecRestoreArray(user->da,X,&p);
199:   VecSum(X,&sum);
200:   VecScale(X,1.0/sum);
201:   return(0);
202: }

204: /* First advection term */
207: PetscErrorCode adv1(PetscScalar **p,PetscScalar y,PetscInt i,PetscInt j,PetscInt M,PetscScalar *p1,AppCtx *user)
208: {
209:   PetscScalar f,fpos,fneg;
211:   f   =  (y - user->ws);
212:   fpos = PetscMax(f,0);
213:   fneg = PetscMin(f,0);
214:   if (user->st_width == 1) {
215:     *p1 = fpos*(p[j][i] - p[j][i-1])/user->dx + fneg*(p[j][i+1] - p[j][i])/user->dx;
216:   } else if (user->st_width == 2) {
217:     *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);
218:   } else if (user->st_width == 3) {
219:     *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);
220:   }
221:   /* *p1 = f*(p[j][i+1] - p[j][i-1])/user->dx;*/
222:   return(0);
223: }

225: /* Second advection term */
228: PetscErrorCode adv2(PetscScalar **p,PetscScalar x,PetscInt i,PetscInt j,PetscInt N,PetscScalar *p2,AppCtx *user)
229: {
230:   PetscScalar f,fpos,fneg;
232:   f   = (user->ws/(2*user->H))*(user->PM_min - user->Pmax*PetscSinScalar(x));
233:   fpos = PetscMax(f,0);
234:   fneg = PetscMin(f,0);
235:   if (user->st_width == 1) {
236:     *p2 = fpos*(p[j][i] - p[j-1][i])/user->dy + fneg*(p[j+1][i] - p[j][i])/user->dy;
237:   } else if (user->st_width ==2) {
238:     *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);
239:   } else if (user->st_width == 3) {
240:     *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);
241:   }

243:   /* *p2 = f*(p[j+1][i] - p[j-1][i])/user->dy;*/
244:   return(0);
245: }

247: /* Diffusion term */
250: PetscErrorCode diffuse(PetscScalar **p,PetscInt i,PetscInt j,PetscReal t,PetscScalar *p_diff,AppCtx * user)
251: {
253:   if (user->st_width == 1) {
254:     *p_diff = user->disper_coe*((p[j-1][i] - 2*p[j][i] + p[j+1][i])/(user->dy*user->dy));
255:   } else if (user->st_width == 2) {
256:     *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));
257:   } else if (user->st_width == 3) {
258:     *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));
259:   }
260:   return(0);
261: }

265: PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx)
266: {
268:   AppCtx         *user=(AppCtx*)ctx;
269:   DM             cda;
270:   DMDACoor2d     **coors;
271:   PetscScalar    **p,**f,**pdot;
272:   PetscInt       i,j;
273:   PetscInt       xs,ys,xm,ym,M,N;
274:   Vec            localX,gc,localXdot;
275:   PetscScalar    p_adv1,p_adv2,p_diff;

278:   DMDAGetInfo(user->da,NULL,&M,&N,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL);
279:   DMGetCoordinateDM(user->da,&cda);
280:   DMDAGetCorners(cda,&xs,&ys,0,&xm,&ym,0);

282:   DMGetLocalVector(user->da,&localX);
283:   DMGetLocalVector(user->da,&localXdot);

285:   DMGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
286:   DMGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);
287:   DMGlobalToLocalBegin(user->da,Xdot,INSERT_VALUES,localXdot);
288:   DMGlobalToLocalEnd(user->da,Xdot,INSERT_VALUES,localXdot);

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

292:   DMDAVecGetArrayRead(cda,gc,&coors);
293:   DMDAVecGetArrayRead(user->da,localX,&p);
294:   DMDAVecGetArrayRead(user->da,localXdot,&pdot);
295:   DMDAVecGetArray(user->da,F,&f);

297:   user->disper_coe = PetscPowScalar((user->lambda*user->ws)/(2*user->H),2)*user->q*(1.0-PetscExpScalar(-t/user->lambda));
298:   for (i=xs; i < xs+xm; i++) {
299:     for (j=ys; j < ys+ym; j++) {
300:       adv1(p,coors[j][i].y,i,j,M,&p_adv1,user);
301:       adv2(p,coors[j][i].x,i,j,N,&p_adv2,user);
302:       diffuse(p,i,j,t,&p_diff,user);
303:       f[j][i] = -p_adv1 - p_adv2  + p_diff - pdot[j][i];
304:     }
305:   }
306:   DMDAVecRestoreArrayRead(user->da,localX,&p);
307:   DMDAVecRestoreArrayRead(user->da,localX,&pdot);
308:   DMRestoreLocalVector(user->da,&localX);
309:   DMRestoreLocalVector(user->da,&localXdot);
310:   DMDAVecRestoreArray(user->da,F,&f);
311:   DMDAVecRestoreArrayRead(cda,gc,&coors);

313:   return(0);
314: }

318: PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat J,Mat Jpre,void *ctx)
319: {
321:   AppCtx         *user=(AppCtx*)ctx;
322:   DM             cda;
323:   DMDACoor2d     **coors;
324:   PetscInt       i,j;
325:   PetscInt       xs,ys,xm,ym,M,N;
326:   Vec            gc;
327:   PetscScalar    val[5],xi,yi;
328:   MatStencil     row,col[5];
329:   PetscScalar    c1,c3,c5,c1pos,c1neg,c3pos,c3neg;

332:   DMDAGetInfo(user->da,NULL,&M,&N,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL);
333:   DMGetCoordinateDM(user->da,&cda);
334:   DMDAGetCorners(cda,&xs,&ys,0,&xm,&ym,0);

336:   DMGetCoordinatesLocal(user->da,&gc);
337:   DMDAVecGetArrayRead(cda,gc,&coors);
338:   for (i=xs; i < xs+xm; i++) {
339:     for (j=ys; j < ys+ym; j++) {
340:       PetscInt nc = 0;
341:       xi = coors[j][i].x; yi = coors[j][i].y;
342:       row.i = i; row.j = j;
343:       c1        = (yi-user->ws)/user->dx;
344:       c1pos    = PetscMax(c1,0);
345:       c1neg    = PetscMin(c1,0);
346:       c3        = (user->ws/(2.0*user->H))*(user->PM_min - user->Pmax*PetscSinScalar(xi))/user->dy;
347:       c3pos    = PetscMax(c3,0);
348:       c3neg    = PetscMin(c3,0);
349:       c5        = (PetscPowScalar((user->lambda*user->ws)/(2*user->H),2)*user->q*(1.0-PetscExpScalar(-t/user->lambda)))/(user->dy*user->dy);
350:       col[nc].i = i-1; col[nc].j = j;   val[nc++] = c1pos;
351:       col[nc].i = i+1; col[nc].j = j;   val[nc++] = -c1neg;
352:       col[nc].i = i;   col[nc].j = j-1; val[nc++] = c3pos + c5;
353:       col[nc].i = i;   col[nc].j = j+1; val[nc++] = -c3neg + c5;
354:       col[nc].i = i;   col[nc].j = j;   val[nc++] = -c1pos + c1neg -c3pos + c3neg -2*c5 -a;
355:       MatSetValuesStencil(Jpre,1,&row,nc,col,val,INSERT_VALUES);
356:     }
357:   }
358:   DMDAVecRestoreArrayRead(cda,gc,&coors);

360:    MatAssemblyBegin(Jpre,MAT_FINAL_ASSEMBLY);
361:   MatAssemblyEnd(Jpre,MAT_FINAL_ASSEMBLY);
362:   if (J != Jpre) {
363:     MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);
364:     MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);
365:   }
366:   return(0);
367: }



373: PetscErrorCode Parameter_settings(AppCtx *user)
374: {
376:   PetscBool      flg;


380:   /* Set default parameters */
381:   user->ws     = 1.0;
382:   user->H      = 5.0;
383:   user->Pmax   = 2.1;
384:   user->PM_min = 1.0;
385:   user->lambda = 0.1;
386:   user->q      = 1.0;
387:   user->mux    = PetscAsinScalar(user->PM_min/user->Pmax);
388:   user->sigmax = 0.1;
389:   user->sigmay = 0.1;
390:   user->rho    = 0.0;
391:   user->xmin   = -PETSC_PI;
392:   user->xmax   =  PETSC_PI;
393:   user->bx     = DM_BOUNDARY_PERIODIC;
394:   user->by     = DM_BOUNDARY_MIRROR;
395:   user->nonoiseinitial = PETSC_FALSE;

397:   /*
398:      ymin of -3 seems to let the unstable solution move up and leave a zero in its wake
399:      with an ymin of -1 the wake is never exactly zero
400:   */
401:   user->ymin   = -3.0;
402:   user->ymax   = 10.0;
403:   user->st_width = 1;

405:   PetscOptionsGetScalar(NULL,NULL,"-ws",&user->ws,&flg);
406:   PetscOptionsGetScalar(NULL,NULL,"-Inertia",&user->H,&flg);
407:   PetscOptionsGetScalar(NULL,NULL,"-Pmax",&user->Pmax,&flg);
408:   PetscOptionsGetScalar(NULL,NULL,"-PM_min",&user->PM_min,&flg);
409:   PetscOptionsGetScalar(NULL,NULL,"-lambda",&user->lambda,&flg);
410:   PetscOptionsGetScalar(NULL,NULL,"-q",&user->q,&flg);
411:   PetscOptionsGetScalar(NULL,NULL,"-mux",&user->mux,&flg);
412:   PetscOptionsGetScalar(NULL,NULL,"-sigmax",&user->sigmax,&flg);
413:   PetscOptionsGetScalar(NULL,NULL,"-muy",&user->muy,&flg);
414:   if (flg == 0) {
415:     user->muy = user->ws;
416:   }
417:   PetscOptionsGetScalar(NULL,NULL,"-sigmay",&user->sigmay,&flg);
418:   PetscOptionsGetScalar(NULL,NULL,"-rho",&user->rho,&flg);
419:   PetscOptionsGetScalar(NULL,NULL,"-xmin",&user->xmin,&flg);
420:   PetscOptionsGetScalar(NULL,NULL,"-xmax",&user->xmax,&flg);
421:   PetscOptionsGetScalar(NULL,NULL,"-ymin",&user->ymin,&flg);
422:   PetscOptionsGetScalar(NULL,NULL,"-ymax",&user->ymax,&flg);
423:   PetscOptionsGetInt(NULL,NULL,"-stencil_width",&user->st_width,&flg);
424:   PetscOptionsGetEnum(NULL,NULL,"-bx",DMBoundaryTypes,(PetscEnum*)&user->bx,&flg);
425:   PetscOptionsGetEnum(NULL,NULL,"-by",DMBoundaryTypes,(PetscEnum*)&user->by,&flg);
426:   PetscOptionsGetBool(NULL,NULL,"-nonoiseinitial",&user->nonoiseinitial,&flg);

428:   return(0);
429: }