Actual source code: rk.c

  1: /*$Id: rk.c,v 0.1 2003/06/03 Asbjorn Hoiland Aarrestad$*/
  2: /*
  3:  * Code for Timestepping with Runge Kutta
  4:  *
  5:  * Written by
  6:  * Asbjorn Hoiland Aarrestad
  7:  * asbjorn@aarrestad.com
  8:  * http://asbjorn.aarrestad.com/
  9:  * 
 10:  */
 11:  #include src/ts/tsimpl.h
 12: #include "time.h"

 14: typedef struct {
 15:    Vec                y1,y2;  /* work wectors for the two rk permuations */
 16:    int                nok,nnok; /* counters for ok and not ok steps */
 17:    PetscReal        maxerror; /* variable to tell the maxerror allowed */
 18:    PetscReal    ferror; /* variable to tell (global maxerror)/(total time) */
 19:    PetscReal    tolerance; /* initial value set for maxerror by user */
 20:    Vec                tmp,tmp_y,*k; /* two temp vectors and the k vectors for rk */
 21:    PetscScalar        a[7][6]; /* rk scalars */
 22:    PetscScalar        b1[7],b2[7]; /* rk scalars */
 23:    PetscReal        c[7]; /* rk scalars */
 24:    int          p,s; /* variables to tell the size of the runge-kutta solver */
 25:    clock_t      start,end; /* variables to mesure cpu time */
 26: } TS_Rk;

 28: EXTERN_C_BEGIN
 31: int TSRKSetTolerance_RK(TS ts,PetscReal aabs)
 32: {
 33:   TS_Rk *rk = (TS_Rk*)ts->data;
 34: 
 36:   rk->tolerance = aabs;
 37:   return(0);
 38: }
 39: EXTERN_C_END

 43: /*@
 44:    TSRKSetTolerance - Sets the total error the RK explicit time integrators 
 45:                       will allow over the given time interval.

 47:    Collective on TS

 49:    Input parameters:
 50: +    ts  - the time-step context
 51: -    aabs - the absolute tolerance  

 53:    Level: intermediate

 55: .keywords: RK, tolerance

 57: .seealso: TSPVodeSetTolerance()

 59: @*/
 60: int TSRKSetTolerance(TS ts,PetscReal aabs)
 61: {
 62:   int ierr,(*f)(TS,PetscReal);
 63: 
 65:   PetscObjectQueryFunction((PetscObject)ts,"TSRKSetTolerance_C",(void (**)(void))&f);
 66:   if (f) {
 67:     (*f)(ts,aabs);
 68:   }
 69:   return(0);
 70: }


 75: static int TSSetUp_Rk(TS ts)
 76: {
 77:   TS_Rk        *rk = (TS_Rk*)ts->data;
 78:   int        ierr;

 81:   rk->nok      = 0;
 82:   rk->nnok     = 0;
 83:   rk->maxerror = rk->tolerance;

 85:   /* fixing maxerror: global vs local */
 86:   rk->ferror = rk->maxerror / (ts->max_time - ts->ptime);

 88:   /* 34.0/45.0 gives double precision division */
 89:   /* defining variables needed for Runge-Kutta computing */
 90:   /* when changing below, please remember to change a, b1, b2 and c above! */
 91:   /* Found in table on page 171: Dormand-Prince 5(4) */

 93:   /* are these right? */
 94:   rk->p=6;
 95:   rk->s=7;

 97:   rk->a[1][0]=1.0/5.0;
 98:   rk->a[2][0]=3.0/40.0;
 99:   rk->a[2][1]=9.0/40.0;
100:   rk->a[3][0]=44.0/45.0;
101:   rk->a[3][1]=-56.0/15.0;
102:   rk->a[3][2]=32.0/9.0;
103:   rk->a[4][0]=19372.0/6561.0;
104:   rk->a[4][1]=-25360.0/2187.0;
105:   rk->a[4][2]=64448.0/6561.0;
106:   rk->a[4][3]=-212.0/729.0;
107:   rk->a[5][0]=9017.0/3168.0;
108:   rk->a[5][1]=-355.0/33.0;
109:   rk->a[5][2]=46732.0/5247.0;
110:   rk->a[5][3]=49.0/176.0;
111:   rk->a[5][4]=-5103.0/18656.0;
112:   rk->a[6][0]=35.0/384.0;
113:   rk->a[6][1]=0.0;
114:   rk->a[6][2]=500.0/1113.0;
115:   rk->a[6][3]=125.0/192.0;
116:   rk->a[6][4]=-2187.0/6784.0;
117:   rk->a[6][5]=11.0/84.0;


120:   rk->c[0]=0.0;
121:   rk->c[1]=1.0/5.0;
122:   rk->c[2]=3.0/10.0;
123:   rk->c[3]=4.0/5.0;
124:   rk->c[4]=8.0/9.0;
125:   rk->c[5]=1.0;
126:   rk->c[6]=1.0;
127: 
128:   rk->b1[0]=35.0/384.0;
129:   rk->b1[1]=0.0;
130:   rk->b1[2]=500.0/1113.0;
131:   rk->b1[3]=125.0/192.0;
132:   rk->b1[4]=-2187.0/6784.0;
133:   rk->b1[5]=11.0/84.0;
134:   rk->b1[6]=0.0;

136:   rk->b2[0]=5179.0/57600.0;
137:   rk->b2[1]=0.0;
138:   rk->b2[2]=7571.0/16695.0;
139:   rk->b2[3]=393.0/640.0;
140:   rk->b2[4]=-92097.0/339200.0;
141:   rk->b2[5]=187.0/2100.0;
142:   rk->b2[6]=1.0/40.0;
143: 
144: 
145:   /* Found in table on page 170: Fehlberg 4(5) */
146:   /*  
147:   rk->p=5;
148:   rk->s=6;

150:   rk->a[1][0]=1.0/4.0;
151:   rk->a[2][0]=3.0/32.0;
152:   rk->a[2][1]=9.0/32.0;
153:   rk->a[3][0]=1932.0/2197.0;
154:   rk->a[3][1]=-7200.0/2197.0;
155:   rk->a[3][2]=7296.0/2197.0;
156:   rk->a[4][0]=439.0/216.0;
157:   rk->a[4][1]=-8.0;
158:   rk->a[4][2]=3680.0/513.0;
159:   rk->a[4][3]=-845.0/4104.0;
160:   rk->a[5][0]=-8.0/27.0;
161:   rk->a[5][1]=2.0;
162:   rk->a[5][2]=-3544.0/2565.0;
163:   rk->a[5][3]=1859.0/4104.0;
164:   rk->a[5][4]=-11.0/40.0;

166:   rk->c[0]=0.0;
167:   rk->c[1]=1.0/4.0;
168:   rk->c[2]=3.0/8.0;
169:   rk->c[3]=12.0/13.0;
170:   rk->c[4]=1.0;
171:   rk->c[5]=1.0/2.0;

173:   rk->b1[0]=25.0/216.0;
174:   rk->b1[1]=0.0;
175:   rk->b1[2]=1408.0/2565.0;
176:   rk->b1[3]=2197.0/4104.0;
177:   rk->b1[4]=-1.0/5.0;
178:   rk->b1[5]=0.0;
179:   
180:   rk->b2[0]=16.0/135.0;
181:   rk->b2[1]=0.0;
182:   rk->b2[2]=6656.0/12825.0;
183:   rk->b2[3]=28561.0/56430.0;
184:   rk->b2[4]=-9.0/50.0;
185:   rk->b2[5]=2.0/55.0;
186:   */
187:   /* Found in table on page 169: Merson 4("5") */
188:   /*
189:   rk->p=4;
190:   rk->s=5;
191:   rk->a[1][0] = 1.0/3.0;
192:   rk->a[2][0] = 1.0/6.0;
193:   rk->a[2][1] = 1.0/6.0;
194:   rk->a[3][0] = 1.0/8.0;
195:   rk->a[3][1] = 0.0;
196:   rk->a[3][2] = 3.0/8.0;
197:   rk->a[4][0] = 1.0/2.0;
198:   rk->a[4][1] = 0.0;
199:   rk->a[4][2] = -3.0/2.0;
200:   rk->a[4][3] = 2.0;

202:   rk->c[0] = 0.0;
203:   rk->c[1] = 1.0/3.0;
204:   rk->c[2] = 1.0/3.0;
205:   rk->c[3] = 0.5;
206:   rk->c[4] = 1.0;

208:   rk->b1[0] = 1.0/2.0;
209:   rk->b1[1] = 0.0;
210:   rk->b1[2] = -3.0/2.0;
211:   rk->b1[3] = 2.0;
212:   rk->b1[4] = 0.0;

214:   rk->b2[0] = 1.0/6.0;
215:   rk->b2[1] = 0.0;
216:   rk->b2[2] = 0.0;
217:   rk->b2[3] = 2.0/3.0;
218:   rk->b2[4] = 1.0/6.0;
219:   */

221:   /* making b2 -> e=b1-b2 */
222:   /*
223:     for(i=0;i<rk->s;i++){
224:      rk->b2[i] = (rk->b1[i]) - (rk->b2[i]);
225:   }
226:   */
227:   rk->b2[0]=71.0/57600.0;
228:   rk->b2[1]=0.0;
229:   rk->b2[2]=-71.0/16695.0;
230:   rk->b2[3]=71.0/1920.0;
231:   rk->b2[4]=-17253.0/339200.0;
232:   rk->b2[5]=22.0/525.0;
233:   rk->b2[6]=-1.0/40.0;

235:   /* initializing vectors */
236:   VecDuplicate(ts->vec_sol,&rk->y1);
237:   VecDuplicate(ts->vec_sol,&rk->y2);
238:   VecDuplicate(rk->y1,&rk->tmp);
239:   VecDuplicate(rk->y1,&rk->tmp_y);
240:   VecDuplicateVecs(rk->y1,rk->s,&rk->k);

242:   return(0);
243: }

245: /*------------------------------------------------------------*/
248: int TSRkqs(TS ts,PetscReal t,PetscReal h)
249: {
250:   TS_Rk                *rk = (TS_Rk*)ts->data;
251:   int                ierr,j,l;
252:   PetscReal        tmp_t=t;
253:   PetscScalar        null=0.0,hh=h;

255:   /*  printf("h: %f, hh: %f",h,hh); */
256: 
258: 
259:   /* k[0]=0  */
260:   VecSet(&null,rk->k[0]);
261: 
262:   /* k[0] = derivs(t,y1) */
263:   TSComputeRHSFunction(ts,t,rk->y1,rk->k[0]);
264:   /* looping over runge-kutta variables */
265:   /* building the k - array of vectors */
266:   for(j = 1 ; j < rk->s ; j++){

268:      /* rk->tmp = 0 */
269:      VecSet(&null,rk->tmp);

271:      for(l=0;l<j;l++){
272:         /* tmp += a(j,l)*k[l] */
273:         /* PetscPrintf(PETSC_COMM_WORLD,"a(%i,%i)=%f \n",j,l,rk->a[j][l]); */
274:         VecAXPY(&rk->a[j][l],rk->k[l],rk->tmp);
275:      }

277:      /* VecView(rk->tmp,PETSC_VIEWER_STDOUT_WORLD); */
278: 
279:      /* k[j] = derivs(t+c(j)*h,y1+h*tmp,k(j)) */
280:      /* I need the following helpers:
281:         PetscScalar  tmp_t=t+c(j)*h
282:         Vec          tmp_y=h*tmp+y1
283:      */

285:      tmp_t = t + rk->c[j] * h;

287:      /* tmp_y = h * tmp + y1 */
288:      VecWAXPY(&hh,rk->tmp,rk->y1,rk->tmp_y);

290:      /* rk->k[j]=0 */
291:      VecSet(&null,rk->k[j]);
292:      TSComputeRHSFunction(ts,tmp_t,rk->tmp_y,rk->k[j]);
293:   }

295:   /* tmp=0 and tmp_y=0 */
296:   VecSet(&null,rk->tmp);
297:   VecSet(&null,rk->tmp_y);
298: 
299:   for(j = 0 ; j < rk->s ; j++){
300:      /* tmp=b1[j]*k[j]+tmp  */
301:      VecAXPY(&rk->b1[j],rk->k[j],rk->tmp);
302:      /* tmp_y=b2[j]*k[j]+tmp_y */
303:      VecAXPY(&rk->b2[j],rk->k[j],rk->tmp_y);
304:   }

306:   /* y2 = hh * tmp_y */
307:   VecSet(&null,rk->y2);
308:   VecAXPY(&hh,rk->tmp_y,rk->y2);
309:   /* y1 = hh*tmp + y1 */
310:   VecAXPY(&hh,rk->tmp,rk->y1);
311:   /* Finding difference between y1 and y2 */

313:   return(0);
314: }

318: static int TSStep_Rk(TS ts,int *steps,PetscReal *ptime)
319: {
320:   TS_Rk                *rk = (TS_Rk*)ts->data;
321:   int                ierr;
322:   PetscReal        dt = 0.001; /* fixed first step guess */
323:   PetscReal        norm=0.0,dt_fac=0.0,fac = 0.0/*,ttmp=0.0*/;

326:   rk->start=clock();
327:   ierr=VecCopy(ts->vec_sol,rk->y1);
328:   *steps = -ts->steps;
329:   /* trying to save the vector */
330:   TSMonitor(ts,ts->steps,ts->ptime,ts->vec_sol);
331:   /* while loop to get from start to stop */
332:   while (ts->ptime < ts->max_time){
333:    /* calling rkqs */
334:      /*
335:        -- input
336:        ts        - pointer to ts
337:        ts->ptime - current time
338:        dt        - try this timestep
339:        y1        - solution for this step

341:        --output
342:        y1        - suggested solution
343:        y2        - check solution (runge - kutta second permutation)
344:      */
345:      TSRkqs(ts,ts->ptime,dt);
346:    /* checking for maxerror */
347:      /* comparing difference to maxerror */
348:      VecNorm(rk->y2,NORM_2,&norm);
349:      /* modifying maxerror to satisfy this timestep */
350:      rk->maxerror = rk->ferror * dt;
351:      /* PetscPrintf(PETSC_COMM_WORLD,"norm err: %f maxerror: %f dt: %f",norm,rk->maxerror,dt); */

353:    /* handling ok and not ok */
354:      if(norm < rk->maxerror){
355:         /* if ok: */
356:         ierr=VecCopy(rk->y1,ts->vec_sol); /* saves the suggested solution to current solution */
357:         ts->ptime += dt; /* storing the new current time */
358:         rk->nok++;
359:         fac=5.0;
360:         /* trying to save the vector */
361:        /* calling monitor */
362:        TSMonitor(ts,ts->steps,ts->ptime,ts->vec_sol);
363:      }else{
364:         /* if not OK */
365:         rk->nnok++;
366:         fac=1.0;
367:         ierr=VecCopy(ts->vec_sol,rk->y1);  /* restores old solution */
368:      }

370:      /*Computing next stepsize. See page 167 in Solving ODE 1
371:       *
372:       * h_new = h * min( facmax , max( facmin , fac * (tol/err)^(1/(p+1)) ) )
373:       * facmax set above
374:       * facmin
375:       */
376:      dt_fac = exp(log((rk->maxerror) / norm) / ((rk->p) + 1) ) * 0.9 ;

378:      if(dt_fac > fac){
379:         /*PetscPrintf(PETSC_COMM_WORLD,"changing fac %f\n",fac);*/
380:         dt_fac = fac;
381:      }

383:      /* computing new dt */
384:      dt = dt * dt_fac;

386:      if(ts->ptime+dt > ts->max_time){
387:         dt = ts->max_time - ts->ptime;
388:      }

390:      if(dt < 1e-14){
391:         PetscPrintf(PETSC_COMM_WORLD,"Very small steps: %f\n",dt);
392:         dt = 1e-14;
393:      }

395:      /* trying to purify h */
396:      /* (did not give any visible result) */
397:      /* ttmp = ts->ptime + dt;
398:         dt = ttmp - ts->ptime; */
399: 
400:      /* counting steps */
401:      ts->steps++;
402:   }
403: 
404:   ierr=VecCopy(rk->y1,ts->vec_sol);
405:   *steps += ts->steps;
406:   *ptime  = ts->ptime;
407:   rk->end=clock();
408:   return(0);
409: }

411: /*------------------------------------------------------------*/
414: static int TSDestroy_Rk(TS ts)
415: {
416:   TS_Rk *rk = (TS_Rk*)ts->data;
417:   int   i,ierr;

419:   /* REMEMBER TO DESTROY ALL */
420: 
422:   if (rk->y1) {VecDestroy(rk->y1);}
423:   if (rk->y2) {VecDestroy(rk->y2);}
424:   if (rk->tmp) {VecDestroy(rk->tmp);}
425:   if (rk->tmp_y) {VecDestroy(rk->tmp_y);}
426:   for(i=0;i<rk->s;i++){
427:      if (rk->k[i]) {VecDestroy(rk->k[i]);}
428:   }
429:   PetscFree(rk);
430:   return(0);
431: }
432: /*------------------------------------------------------------*/

436: static int TSSetFromOptions_Rk(TS ts)
437: {
438:   TS_Rk *rk = (TS_Rk*)ts->data;
439:   int   ierr;

442:   PetscOptionsHead("RK ODE solver options");
443:     PetscOptionsReal("-ts_rk_tol","Tolerance for convergence","TSRKSetTolerance",rk->tolerance,&rk->tolerance,PETSC_NULL);
444:   PetscOptionsTail();
445:   return(0);
446: }

450: static int TSView_Rk(TS ts,PetscViewer viewer)
451: {
452:    TS_Rk *rk = (TS_Rk*)ts->data;
454:    /*double elapsed;*/
455: 
457:    PetscPrintf(PETSC_COMM_WORLD,"  number of ok steps: %d\n",rk->nok);
458:    PetscPrintf(PETSC_COMM_WORLD,"  number of rejected steps: %d\n",rk->nnok);
459:    /*   elapsed = ((double) (rk->end - rk->start)) / CLOCKS_PER_SEC;
460:    
461:    PetscPrintf(PETSC_COMM_WORLD,"  CPU time used (in seconds): %f\n",elapsed); */
462:    return(0);
463: }

465: /* ------------------------------------------------------------ */
466: /*MC
467:       TS_RK - ODE solver using the explicit Runge-Kutta methods

469:    Options Database:
470: .  -ts_rk_tol <tol> Tolerance for convergence

472:   Contributed by: Asbjorn Hoiland Aarrestad, asbjorn@aarrestad.com, http://asbjorn.aarrestad.com/

474: .seealso:  TSCreate(), TS, TSSetType(), TS_EULER, TSRKSetTolerance()

476: M*/

478: EXTERN_C_BEGIN
481: int TSCreate_Rk(TS ts)
482: {
483:   TS_Rk    *rk;
484:   int      ierr;

487:   ts->ops->setup           = TSSetUp_Rk;
488:   ts->ops->step            = TSStep_Rk;
489:   ts->ops->destroy         = TSDestroy_Rk;
490:   ts->ops->setfromoptions  = TSSetFromOptions_Rk;
491:   ts->ops->view            = TSView_Rk;

493:   PetscNew(TS_Rk,&rk);
494:   PetscLogObjectMemory(ts,sizeof(TS_Rk));
495:   ts->data = (void*)rk;

497:   PetscObjectComposeFunctionDynamic((PetscObject)ts,"TSRKSetTolerance_C","TSRKSetTolerance_RK",
498:                                            TSRKSetTolerance_RK);

500:   return(0);
501: }
502: EXTERN_C_END