xref: /petsc/src/ts/impls/explicit/rk/rk.c (revision 42682096d3b4a0b3863ee209ed6652af8e092da7)
1 /*
2   Code for time stepping with the Runge-Kutta method
3 
4   Notes:
5   The general system is written as
6 
7   Udot = F(t,U)
8 
9 */
10 #include <petsc-private/tsimpl.h>                /*I   "petscts.h"   I*/
11 #include <petscdm.h>
12 
13 static TSRKType  TSRKDefault = TSRK3BS;
14 static PetscBool TSRKRegisterAllCalled;
15 static PetscBool TSRKPackageInitialized;
16 static PetscInt  explicit_stage_time_id;
17 
18 typedef struct _RKTableau *RKTableau;
19 struct _RKTableau {
20   char      *name;
21   PetscInt   order;               /* Classical approximation order of the method i              */
22   PetscInt   s;                   /* Number of stages                                           */
23   PetscBool  FSAL;                /* flag to indicate if tableau is FSAL                        */
24   PetscInt   pinterp;             /* Interpolation order                                        */
25   PetscReal *A,*b,*c;             /* Tableau                                                    */
26   PetscReal *bembed;              /* Embedded formula of order one less (order-1)               */
27   PetscReal *binterp;             /* Dense output formula                                       */
28   PetscReal  ccfl;                /* Placeholder for CFL coefficient relative to forward Euler  */
29 };
30 typedef struct _RKTableauLink *RKTableauLink;
31 struct _RKTableauLink {
32   struct _RKTableau tab;
33   RKTableauLink     next;
34 };
35 static RKTableauLink RKTableauList;
36 
37 typedef struct {
38   RKTableau   tableau;
39   Vec          *Y;               /* States computed during the step */
40   Vec          *YdotRHS;         /* Function evaluations for the non-stiff part */
41   Vec          *VecDeltaLam;     /* Increment of the adjoint sensitivity w.r.t IC at stage*/
42   Vec          *VecDeltaMu;      /* Increment of the adjoint sensitivity w.r.t P at stage*/
43   Vec          *VecSensiTemp;    /* Vector to be timed with Jacobian transpose*/
44   PetscScalar  *work;            /* Scalar work */
45   PetscReal    stage_time;
46   TSStepStatus status;
47 } TS_RK;
48 
49 /*MC
50      TSRK1 - First order forward Euler scheme.
51 
52      This method has one stage.
53 
54      Level: advanced
55 
56 .seealso: TSRK
57 M*/
58 /*MC
59      TSRK2A - Second order RK scheme.
60 
61      This method has two stages.
62 
63      Level: advanced
64 
65 .seealso: TSRK
66 M*/
67 /*MC
68      TSRK3 - Third order RK scheme.
69 
70      This method has three stages.
71 
72      Level: advanced
73 
74 .seealso: TSRK
75 M*/
76 /*MC
77      TSRK3BS - Third order RK scheme of Bogacki-Shampine with 2nd order embedded method.
78 
79      This method has four stages.
80 
81      Level: advanced
82 
83 .seealso: TSRK
84 M*/
85 /*MC
86      TSRK4 - Fourth order RK scheme.
87 
88      This is the classical Runge-Kutta method with four stages.
89 
90      Level: advanced
91 
92 .seealso: TSRK
93 M*/
94 /*MC
95      TSRK5F - Fifth order Fehlberg RK scheme with a 4th order embedded method.
96 
97      This method has six stages.
98 
99      Level: advanced
100 
101 .seealso: TSRK
102 M*/
103 /*MC
104      TSRK5DP - Fifth order Dormand-Prince RK scheme with the 4th order embedded method.
105 
106      This method has seven stages.
107 
108      Level: advanced
109 
110 .seealso: TSRK
111 M*/
112 
113 #undef __FUNCT__
114 #define __FUNCT__ "TSRKRegisterAll"
115 /*@C
116   TSRKRegisterAll - Registers all of the Runge-Kutta explicit methods in TSRK
117 
118   Not Collective, but should be called by all processes which will need the schemes to be registered
119 
120   Level: advanced
121 
122 .keywords: TS, TSRK, register, all
123 
124 .seealso:  TSRKRegisterDestroy()
125 @*/
126 PetscErrorCode TSRKRegisterAll(void)
127 {
128   PetscErrorCode ierr;
129 
130   PetscFunctionBegin;
131   if (TSRKRegisterAllCalled) PetscFunctionReturn(0);
132   TSRKRegisterAllCalled = PETSC_TRUE;
133 
134   {
135     const PetscReal
136       A[1][1] = {{0.0}},
137       b[1]    = {1.0};
138     ierr = TSRKRegister(TSRK1FE,1,1,&A[0][0],b,NULL,NULL,1,b);CHKERRQ(ierr);
139   }
140   {
141     const PetscReal
142       A[2][2]     = {{0.0,0.0},
143                     {1.0,0.0}},
144       b[2]        = {0.5,0.5},
145       bembed[2]   = {1.0,0};
146     ierr = TSRKRegister(TSRK2A,2,2,&A[0][0],b,NULL,bembed,2,b);CHKERRQ(ierr);
147   }
148   {
149     const PetscReal
150       A[3][3] = {{0,0,0},
151                  {2.0/3.0,0,0},
152                  {-1.0/3.0,1.0,0}},
153       b[3]    = {0.25,0.5,0.25};
154     ierr = TSRKRegister(TSRK3,3,3,&A[0][0],b,NULL,NULL,3,b);CHKERRQ(ierr);
155   }
156   {
157     const PetscReal
158       A[4][4] = {{0,0,0,0},
159                  {1.0/2.0,0,0,0},
160                  {0,3.0/4.0,0,0},
161                  {2.0/9.0,1.0/3.0,4.0/9.0,0}},
162       b[4]    = {2.0/9.0,1.0/3.0,4.0/9.0,0},
163       bembed[4] = {7.0/24.0,1.0/4.0,1.0/3.0,1.0/8.0};
164     ierr = TSRKRegister(TSRK3BS,3,4,&A[0][0],b,NULL,bembed,3,b);CHKERRQ(ierr);
165   }
166   {
167     const PetscReal
168       A[4][4] = {{0,0,0,0},
169                  {0.5,0,0,0},
170                  {0,0.5,0,0},
171                  {0,0,1.0,0}},
172       b[4]    = {1.0/6.0,1.0/3.0,1.0/3.0,1.0/6.0};
173     ierr = TSRKRegister(TSRK4,4,4,&A[0][0],b,NULL,NULL,4,b);CHKERRQ(ierr);
174   }
175   {
176     const PetscReal
177       A[6][6]   = {{0,0,0,0,0,0},
178                    {0.25,0,0,0,0,0},
179                    {3.0/32.0,9.0/32.0,0,0,0,0},
180                    {1932.0/2197.0,-7200.0/2197.0,7296.0/2197.0,0,0,0},
181                    {439.0/216.0,-8.0,3680.0/513.0,-845.0/4104.0,0,0},
182                    {-8.0/27.0,2.0,-3544.0/2565.0,1859.0/4104.0,-11.0/40.0,0}},
183       b[6]      = {16.0/135.0,0,6656.0/12825.0,28561.0/56430.0,-9.0/50.0,2.0/55.0},
184       bembed[6] = {25.0/216.0,0,1408.0/2565.0,2197.0/4104.0,-1.0/5.0,0};
185     ierr = TSRKRegister(TSRK5F,5,6,&A[0][0],b,NULL,bembed,5,b);CHKERRQ(ierr);
186   }
187   {
188     const PetscReal
189       A[7][7]   = {{0,0,0,0,0,0,0},
190                    {1.0/5.0,0,0,0,0,0,0},
191                    {3.0/40.0,9.0/40.0,0,0,0,0,0},
192                    {44.0/45.0,-56.0/15.0,32.0/9.0,0,0,0,0},
193                    {19372.0/6561.0,-25360.0/2187.0,64448.0/6561.0,-212.0/729.0,0,0,0},
194                    {9017.0/3168.0,-355.0/33.0,46732.0/5247.0,49.0/176.0,-5103.0/18656.0,0,0},
195                    {35.0/384.0,0,500.0/1113.0,125.0/192.0,-2187.0/6784.0,11.0/84.0,0}},
196       b[7]      = {35.0/384.0,0,500.0/1113.0,125.0/192.0,-2187.0/6784.0,11.0/84.0,0},
197       bembed[7] = {5179.0/57600.0,0,7571.0/16695.0,393.0/640.0,-92097.0/339200.0,187.0/2100.0,1.0/40.0};
198     ierr = TSRKRegister(TSRK5DP,5,7,&A[0][0],b,NULL,bembed,5,b);CHKERRQ(ierr);
199   }
200   PetscFunctionReturn(0);
201 }
202 
203 #undef __FUNCT__
204 #define __FUNCT__ "TSRKRegisterDestroy"
205 /*@C
206    TSRKRegisterDestroy - Frees the list of schemes that were registered by TSRKRegister().
207 
208    Not Collective
209 
210    Level: advanced
211 
212 .keywords: TSRK, register, destroy
213 .seealso: TSRKRegister(), TSRKRegisterAll()
214 @*/
215 PetscErrorCode TSRKRegisterDestroy(void)
216 {
217   PetscErrorCode ierr;
218   RKTableauLink link;
219 
220   PetscFunctionBegin;
221   while ((link = RKTableauList)) {
222     RKTableau t = &link->tab;
223     RKTableauList = link->next;
224     ierr = PetscFree3(t->A,t->b,t->c);  CHKERRQ(ierr);
225     ierr = PetscFree (t->bembed);       CHKERRQ(ierr);
226     ierr = PetscFree (t->binterp);      CHKERRQ(ierr);
227     ierr = PetscFree (t->name);         CHKERRQ(ierr);
228     ierr = PetscFree (link);            CHKERRQ(ierr);
229   }
230   TSRKRegisterAllCalled = PETSC_FALSE;
231   PetscFunctionReturn(0);
232 }
233 
234 #undef __FUNCT__
235 #define __FUNCT__ "TSRKInitializePackage"
236 /*@C
237   TSRKInitializePackage - This function initializes everything in the TSRK package. It is called
238   from PetscDLLibraryRegister() when using dynamic libraries, and on the first call to TSCreate_RK()
239   when using static libraries.
240 
241   Level: developer
242 
243 .keywords: TS, TSRK, initialize, package
244 .seealso: PetscInitialize()
245 @*/
246 PetscErrorCode TSRKInitializePackage(void)
247 {
248   PetscErrorCode ierr;
249 
250   PetscFunctionBegin;
251   if (TSRKPackageInitialized) PetscFunctionReturn(0);
252   TSRKPackageInitialized = PETSC_TRUE;
253   ierr = TSRKRegisterAll();CHKERRQ(ierr);
254   ierr = PetscObjectComposedDataRegister(&explicit_stage_time_id);CHKERRQ(ierr);
255   ierr = PetscRegisterFinalize(TSRKFinalizePackage);CHKERRQ(ierr);
256   PetscFunctionReturn(0);
257 }
258 
259 #undef __FUNCT__
260 #define __FUNCT__ "TSRKFinalizePackage"
261 /*@C
262   TSRKFinalizePackage - This function destroys everything in the TSRK package. It is
263   called from PetscFinalize().
264 
265   Level: developer
266 
267 .keywords: Petsc, destroy, package
268 .seealso: PetscFinalize()
269 @*/
270 PetscErrorCode TSRKFinalizePackage(void)
271 {
272   PetscErrorCode ierr;
273 
274   PetscFunctionBegin;
275   TSRKPackageInitialized = PETSC_FALSE;
276   ierr = TSRKRegisterDestroy();CHKERRQ(ierr);
277   PetscFunctionReturn(0);
278 }
279 
280 #undef __FUNCT__
281 #define __FUNCT__ "TSRKRegister"
282 /*@C
283    TSRKRegister - register an RK scheme by providing the entries in the Butcher tableau and optionally embedded approximations and interpolation
284 
285    Not Collective, but the same schemes should be registered on all processes on which they will be used
286 
287    Input Parameters:
288 +  name - identifier for method
289 .  order - approximation order of method
290 .  s - number of stages, this is the dimension of the matrices below
291 .  A - stage coefficients (dimension s*s, row-major)
292 .  b - step completion table (dimension s; NULL to use last row of A)
293 .  c - abscissa (dimension s; NULL to use row sums of A)
294 .  bembed - completion table for embedded method (dimension s; NULL if not available)
295 .  pinterp - Order of the interpolation scheme, equal to the number of columns of binterp
296 -  binterp - Coefficients of the interpolation formula (dimension s*pinterp; NULL to reuse binterpt)
297 
298    Notes:
299    Several RK methods are provided, this function is only needed to create new methods.
300 
301    Level: advanced
302 
303 .keywords: TS, register
304 
305 .seealso: TSRK
306 @*/
307 PetscErrorCode TSRKRegister(TSRKType name,PetscInt order,PetscInt s,
308                                  const PetscReal A[],const PetscReal b[],const PetscReal c[],
309                                  const PetscReal bembed[],
310                                  PetscInt pinterp,const PetscReal binterp[])
311 {
312   PetscErrorCode  ierr;
313   RKTableauLink   link;
314   RKTableau       t;
315   PetscInt        i,j;
316 
317   PetscFunctionBegin;
318   ierr     = PetscMalloc(sizeof(*link),&link);CHKERRQ(ierr);
319   ierr     = PetscMemzero(link,sizeof(*link));CHKERRQ(ierr);
320   t        = &link->tab;
321   ierr     = PetscStrallocpy(name,&t->name);CHKERRQ(ierr);
322   t->order = order;
323   t->s     = s;
324   ierr     = PetscMalloc3(s*s,&t->A,s,&t->b,s,&t->c);CHKERRQ(ierr);
325   ierr     = PetscMemcpy(t->A,A,s*s*sizeof(A[0]));CHKERRQ(ierr);
326   if (b)  { ierr = PetscMemcpy(t->b,b,s*sizeof(b[0]));CHKERRQ(ierr); }
327   else for (i=0; i<s; i++) t->b[i] = A[(s-1)*s+i];
328   if (c)  { ierr = PetscMemcpy(t->c,c,s*sizeof(c[0]));CHKERRQ(ierr); }
329   else for (i=0; i<s; i++) for (j=0,t->c[i]=0; j<s; j++) t->c[i] += A[i*s+j];
330   t->FSAL = PETSC_TRUE;
331   for (i=0; i<s; i++) if (t->A[(s-1)*s+i] != t->b[i]) t->FSAL = PETSC_FALSE;
332   if (bembed) {
333     ierr = PetscMalloc1(s,&t->bembed);CHKERRQ(ierr);
334     ierr = PetscMemcpy(t->bembed,bembed,s*sizeof(bembed[0]));CHKERRQ(ierr);
335   }
336 
337   t->pinterp     = pinterp;
338   ierr           = PetscMalloc1(s*pinterp,&t->binterp);CHKERRQ(ierr);
339   ierr           = PetscMemcpy(t->binterp,binterp,s*pinterp*sizeof(binterp[0]));CHKERRQ(ierr);
340   link->next     = RKTableauList;
341   RKTableauList = link;
342   PetscFunctionReturn(0);
343 }
344 
345 #undef __FUNCT__
346 #define __FUNCT__ "TSEvaluateStep_RK"
347 /*
348  The step completion formula is
349 
350  x1 = x0 + h b^T YdotRHS
351 
352  This function can be called before or after ts->vec_sol has been updated.
353  Suppose we have a completion formula (b) and an embedded formula (be) of different order.
354  We can write
355 
356  x1e = x0 + h be^T YdotRHS
357      = x1 - h b^T YdotRHS + h be^T YdotRHS
358      = x1 + h (be - b)^T YdotRHS
359 
360  so we can evaluate the method with different order even after the step has been optimistically completed.
361 */
362 static PetscErrorCode TSEvaluateStep_RK(TS ts,PetscInt order,Vec X,PetscBool *done)
363 {
364   TS_RK         *rk   = (TS_RK*)ts->data;
365   RKTableau      tab  = rk->tableau;
366   PetscScalar   *w    = rk->work;
367   PetscReal      h;
368   PetscInt       s    = tab->s,j;
369   PetscErrorCode ierr;
370 
371   PetscFunctionBegin;
372   switch (rk->status) {
373   case TS_STEP_INCOMPLETE:
374   case TS_STEP_PENDING:
375     h = ts->time_step; break;
376   case TS_STEP_COMPLETE:
377     h = ts->time_step_prev; break;
378   default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
379   }
380   if (order == tab->order) {
381     if (rk->status == TS_STEP_INCOMPLETE) {
382       ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
383       for (j=0; j<s; j++) w[j] = h*tab->b[j];
384       ierr = VecMAXPY(X,s,w,rk->YdotRHS);CHKERRQ(ierr);
385     } else {ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);}
386     PetscFunctionReturn(0);
387   } else if (order == tab->order-1) {
388     if (!tab->bembed) goto unavailable;
389     if (rk->status == TS_STEP_INCOMPLETE) { /* Complete with the embedded method (be) */
390       ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
391       for (j=0; j<s; j++) w[j] = h*tab->bembed[j];
392       ierr = VecMAXPY(X,s,w,rk->YdotRHS);CHKERRQ(ierr);
393     } else {                    /* Rollback and re-complete using (be-b) */
394       ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
395       for (j=0; j<s; j++) w[j] = h*(tab->bembed[j] - tab->b[j]);
396       ierr = VecMAXPY(X,s,w,rk->YdotRHS);CHKERRQ(ierr);
397     }
398     if (done) *done = PETSC_TRUE;
399     PetscFunctionReturn(0);
400   }
401 unavailable:
402   if (done) *done = PETSC_FALSE;
403   else SETERRQ3(PetscObjectComm((PetscObject)ts),PETSC_ERR_SUP,"RK '%s' of order %D cannot evaluate step at order %D",tab->name,tab->order,order);
404   PetscFunctionReturn(0);
405 }
406 
407 #undef __FUNCT__
408 #define __FUNCT__ "TSStep_RK"
409 static PetscErrorCode TSStep_RK(TS ts)
410 {
411   TS_RK           *rk   = (TS_RK*)ts->data;
412   RKTableau        tab  = rk->tableau;
413   const PetscInt   s    = tab->s;
414   const PetscReal *A = tab->A,*b = tab->b,*c = tab->c;
415   PetscScalar     *w    = rk->work;
416   Vec             *Y    = rk->Y,*YdotRHS = rk->YdotRHS;
417   TSAdapt          adapt;
418   PetscInt         i,j,reject,next_scheme;
419   PetscReal        next_time_step;
420   PetscReal        t;
421   PetscBool        accept;
422   PetscErrorCode   ierr;
423 
424   PetscFunctionBegin;
425 
426   next_time_step = ts->time_step;
427   t              = ts->ptime;
428   accept         = PETSC_TRUE;
429   rk->status     = TS_STEP_INCOMPLETE;
430 
431 
432   for (reject=0; reject<ts->max_reject && !ts->reason; reject++,ts->reject++) {
433     PetscReal h = ts->time_step;
434     ierr = TSPreStep(ts);CHKERRQ(ierr);
435     for (i=0; i<s; i++) {
436       rk->stage_time = t + h*c[i];
437       ierr = TSPreStage(ts,rk->stage_time); CHKERRQ(ierr);
438       ierr = VecCopy(ts->vec_sol,Y[i]);CHKERRQ(ierr);
439       for (j=0; j<i; j++) w[j] = h*A[i*s+j];
440       ierr = VecMAXPY(Y[i],i,w,YdotRHS);CHKERRQ(ierr);
441       ierr = TSPostStage(ts,rk->stage_time,i,Y); CHKERRQ(ierr);
442       ierr = TSGetAdapt(ts,&adapt);CHKERRQ(ierr);
443       ierr = TSAdaptCheckStage(adapt,ts,&accept);CHKERRQ(ierr);
444       if (!accept) goto reject_step;
445       ierr = TSComputeRHSFunction(ts,t+h*c[i],Y[i],YdotRHS[i]);CHKERRQ(ierr);
446     }
447     ierr = TSEvaluateStep(ts,tab->order,ts->vec_sol,NULL);CHKERRQ(ierr);
448     rk->status = TS_STEP_PENDING;
449 
450     /* Register only the current method as a candidate because we're not supporting multiple candidates yet. */
451     ierr = TSGetAdapt(ts,&adapt);CHKERRQ(ierr);
452     ierr = TSAdaptCandidatesClear(adapt);CHKERRQ(ierr);
453     ierr = TSAdaptCandidateAdd(adapt,tab->name,tab->order,1,tab->ccfl,1.*tab->s,PETSC_TRUE);CHKERRQ(ierr);
454     ierr = TSAdaptChoose(adapt,ts,ts->time_step,&next_scheme,&next_time_step,&accept);CHKERRQ(ierr);
455     if (accept) {
456       /* ignore next_scheme for now */
457       ts->ptime    += ts->time_step;
458       ts->time_step = next_time_step;
459       ts->steps++;
460       rk->status = TS_STEP_COMPLETE;
461       ierr = PetscObjectComposedDataSetReal((PetscObject)ts->vec_sol,explicit_stage_time_id,ts->ptime);CHKERRQ(ierr);
462       break;
463     } else {                    /* Roll back the current step */
464       for (j=0; j<s; j++) w[j] = -h*b[j];
465       ierr = VecMAXPY(ts->vec_sol,s,w,rk->YdotRHS);CHKERRQ(ierr);
466       ts->time_step = next_time_step;
467       rk->status   = TS_STEP_INCOMPLETE;
468     }
469 reject_step: continue;
470   }
471   if (rk->status != TS_STEP_COMPLETE && !ts->reason) ts->reason = TS_DIVERGED_STEP_REJECTED;
472   PetscFunctionReturn(0);
473 }
474 
475 #undef __FUNCT__
476 #define __FUNCT__ "TSStepAdj_RK"
477 static PetscErrorCode TSStepAdj_RK(TS ts)
478 {
479   TS_RK           *rk   = (TS_RK*)ts->data;
480   RKTableau        tab  = rk->tableau;
481   const PetscInt   s    = tab->s;
482   const PetscReal *A = tab->A,*b = tab->b,*c = tab->c;
483   PetscScalar     *w    = rk->work;
484   Vec             *Y    = rk->Y,*VecDeltaLam = rk->VecDeltaLam,*VecDeltaMu = rk->VecDeltaMu,*VecSensiTemp = rk->VecSensiTemp;
485   PetscInt         i,j,nadj;
486   PetscReal        t;
487   PetscErrorCode   ierr;
488 
489   PetscFunctionBegin;
490 
491   /*ierr = TSStep_RK(ts);CHKERRQ(ierr); // reuse TSStep */
492 
493   t          = ts->ptime;
494   rk->status = TS_STEP_INCOMPLETE;
495   PetscReal h = ts->time_step;
496   ierr = TSPreStep(ts);CHKERRQ(ierr);
497   for (i=s-1; i>=0; i--) {
498     Mat J,Jp;
499     rk->stage_time = t + h*(1.0-c[i]);
500     for (nadj=0; nadj<ts->numberadjs; nadj++) {
501       ierr = VecCopy(ts->vecs_sensi[nadj],VecSensiTemp[nadj]);CHKERRQ(ierr);
502       ierr = VecScale(VecSensiTemp[nadj],-h*b[i]);
503       for (j=i+1; j<s; j++) {
504         ierr = VecAXPY(VecSensiTemp[nadj],-h*A[j*s+i],VecDeltaLam[nadj*s+j]);
505       }
506     }
507     /* Stage values of lambda */
508     ierr = TSGetRHSJacobian(ts,&J,&Jp,NULL,NULL);CHKERRQ(ierr);
509     ierr = TSComputeRHSJacobian(ts,rk->stage_time,Y[i],J,Jp);CHKERRQ(ierr);
510     for (nadj=0; nadj<ts->numberadjs; nadj++) {
511       ierr = MatMultTranspose(J,VecSensiTemp[nadj],VecDeltaLam[nadj*s+i]);CHKERRQ(ierr);
512     }
513 
514     /* Stage values of mu */
515     if(ts->vecs_sensip) {
516       ierr = TSRHSJacobianP(ts,rk->stage_time,Y[i],ts->Jacp);CHKERRQ(ierr);
517       for (nadj=0; nadj<ts->numberadjs; nadj++) {
518         ierr = MatMultTranspose(ts->Jacp,VecSensiTemp[nadj],VecDeltaMu[nadj*s+i]);CHKERRQ(ierr);
519       }
520     }
521   }
522 
523   for (j=0; j<s; j++) w[j] = 1.0;
524   for (nadj=0; nadj<ts->numberadjs; nadj++) {
525     ierr = VecMAXPY(ts->vecs_sensi[nadj],s,w,&VecDeltaLam[nadj*s]);CHKERRQ(ierr);
526     if(ts->vecs_sensip) {
527       ierr = VecMAXPY(ts->vecs_sensip[nadj],s,w,&VecDeltaMu[nadj*s]);CHKERRQ(ierr);
528     }
529   }
530   ts->ptime += ts->time_step;
531   ts->steps++;
532   rk->status = TS_STEP_COMPLETE;
533   ierr = PetscObjectComposedDataSetReal((PetscObject)ts->vecs_sensi,explicit_stage_time_id,ts->ptime);CHKERRQ(ierr);
534   if(ts->vecs_sensip) {
535     ierr = PetscObjectComposedDataSetReal((PetscObject)ts->vecs_sensip,explicit_stage_time_id,ts->ptime);CHKERRQ(ierr);
536   }
537   PetscFunctionReturn(0);
538 }
539 
540 #undef __FUNCT__
541 #define __FUNCT__ "TSInterpolate_RK"
542 static PetscErrorCode TSInterpolate_RK(TS ts,PetscReal itime,Vec X)
543 {
544   TS_RK           *rk = (TS_RK*)ts->data;
545   PetscInt         s  = rk->tableau->s,pinterp = rk->tableau->pinterp,i,j;
546   PetscReal        h;
547   PetscReal        tt,t;
548   PetscScalar     *b;
549   const PetscReal *B = rk->tableau->binterp;
550   PetscErrorCode   ierr;
551 
552   PetscFunctionBegin;
553   if (!B) SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_SUP,"TSRK %s does not have an interpolation formula",rk->tableau->name);
554   switch (rk->status) {
555   case TS_STEP_INCOMPLETE:
556   case TS_STEP_PENDING:
557     h = ts->time_step;
558     t = (itime - ts->ptime)/h;
559     break;
560   case TS_STEP_COMPLETE:
561     h = ts->time_step_prev;
562     t = (itime - ts->ptime)/h + 1; /* In the interval [0,1] */
563     break;
564   default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
565   }
566   ierr = PetscMalloc1(s,&b);CHKERRQ(ierr);
567   for (i=0; i<s; i++) b[i] = 0;
568   for (j=0,tt=t; j<pinterp; j++,tt*=t) {
569     for (i=0; i<s; i++) {
570       b[i]  += h * B[i*pinterp+j] * tt;
571     }
572   }
573   ierr = VecCopy(rk->Y[0],X);CHKERRQ(ierr);
574   ierr = VecMAXPY(X,s,b,rk->YdotRHS);CHKERRQ(ierr);
575   ierr = PetscFree(b);CHKERRQ(ierr);
576   PetscFunctionReturn(0);
577 }
578 
579 /*------------------------------------------------------------*/
580 #undef __FUNCT__
581 #define __FUNCT__ "TSReset_RK"
582 static PetscErrorCode TSReset_RK(TS ts)
583 {
584   TS_RK         *rk = (TS_RK*)ts->data;
585   PetscInt       s;
586   PetscErrorCode ierr;
587 
588   PetscFunctionBegin;
589   if (!rk->tableau) PetscFunctionReturn(0);
590   s    = rk->tableau->s;
591   ierr = VecDestroyVecs(s,&rk->Y);CHKERRQ(ierr);
592   ierr = VecDestroyVecs(s,&rk->YdotRHS);CHKERRQ(ierr);
593   if(ts->reverse_mode) {
594     ierr = VecDestroyVecs(s*ts->numberadjs,&rk->VecDeltaLam);CHKERRQ(ierr);
595     if(rk->VecDeltaMu) {
596       ierr = VecDestroyVecs(s*ts->numberadjs,&rk->VecDeltaMu);CHKERRQ(ierr);
597     }
598     ierr = VecDestroyVecs(ts->numberadjs,&rk->VecSensiTemp);CHKERRQ(ierr);
599   }
600   ierr = PetscFree(rk->work);CHKERRQ(ierr);
601   PetscFunctionReturn(0);
602 }
603 
604 #undef __FUNCT__
605 #define __FUNCT__ "TSDestroy_RK"
606 static PetscErrorCode TSDestroy_RK(TS ts)
607 {
608   PetscErrorCode ierr;
609 
610   PetscFunctionBegin;
611   ierr = TSReset_RK(ts);CHKERRQ(ierr);
612   ierr = PetscFree(ts->data);CHKERRQ(ierr);
613   ierr = PetscObjectComposeFunction((PetscObject)ts,"TSRKGetType_C",NULL);CHKERRQ(ierr);
614   ierr = PetscObjectComposeFunction((PetscObject)ts,"TSRKSetType_C",NULL);CHKERRQ(ierr);
615   PetscFunctionReturn(0);
616 }
617 
618 
619 #undef __FUNCT__
620 #define __FUNCT__ "DMCoarsenHook_TSRK"
621 static PetscErrorCode DMCoarsenHook_TSRK(DM fine,DM coarse,void *ctx)
622 {
623   PetscFunctionBegin;
624   PetscFunctionReturn(0);
625 }
626 
627 #undef __FUNCT__
628 #define __FUNCT__ "DMRestrictHook_TSRK"
629 static PetscErrorCode DMRestrictHook_TSRK(DM fine,Mat restrct,Vec rscale,Mat inject,DM coarse,void *ctx)
630 {
631   PetscFunctionBegin;
632   PetscFunctionReturn(0);
633 }
634 
635 
636 #undef __FUNCT__
637 #define __FUNCT__ "DMSubDomainHook_TSRK"
638 static PetscErrorCode DMSubDomainHook_TSRK(DM dm,DM subdm,void *ctx)
639 {
640   PetscFunctionBegin;
641   PetscFunctionReturn(0);
642 }
643 
644 #undef __FUNCT__
645 #define __FUNCT__ "DMSubDomainRestrictHook_TSRK"
646 static PetscErrorCode DMSubDomainRestrictHook_TSRK(DM dm,VecScatter gscat,VecScatter lscat,DM subdm,void *ctx)
647 {
648 
649   PetscFunctionBegin;
650   PetscFunctionReturn(0);
651 }
652 /*
653 #undef __FUNCT__
654 #define __FUNCT__ "RKSetAdjCoe"
655 static PetscErrorCode RKSetAdjCoe(RKTableau tab)
656 {
657   PetscReal *A,*b;
658   PetscInt        s,i,j;
659   PetscErrorCode  ierr;
660 
661   PetscFunctionBegin;
662   s    = tab->s;
663   ierr = PetscMalloc2(s*s,&A,s,&b);CHKERRQ(ierr);
664 
665   for (i=0; i<s; i++)
666     for (j=0; j<s; j++) {
667       A[i*s+j] = (tab->b[s-1-i]==0) ? 0: -tab->A[s-1-i+(s-1-j)*s] * tab->b[s-1-j] / tab->b[s-1-i];
668       ierr = PetscPrintf(PETSC_COMM_WORLD,"Coefficients: A[%D][%D]=%.6f\n",i,j,A[i*s+j]);CHKERRQ(ierr);
669     }
670 
671   for (i=0; i<s; i++) b[i] = (tab->b[s-1-i]==0)? 0: -tab->b[s-1-i];
672 
673   ierr  = PetscMemcpy(tab->A,A,s*s*sizeof(A[0]));CHKERRQ(ierr);
674   ierr  = PetscMemcpy(tab->b,b,s*sizeof(b[0]));CHKERRQ(ierr);
675   ierr  = PetscFree2(A,b);CHKERRQ(ierr);
676   PetscFunctionReturn(0);
677 }
678 */
679 #undef __FUNCT__
680 #define __FUNCT__ "TSSetUp_RK"
681 static PetscErrorCode TSSetUp_RK(TS ts)
682 {
683   TS_RK         *rk = (TS_RK*)ts->data;
684   RKTableau      tab;
685   PetscInt       s;
686   PetscErrorCode ierr;
687   DM             dm;
688 
689   PetscFunctionBegin;
690   if (!rk->tableau) {
691     ierr = TSRKSetType(ts,TSRKDefault);CHKERRQ(ierr);
692   }
693   tab  = rk->tableau;
694   s    = tab->s;
695   /* old */
696   /*if (ts->reverse_mode) {
697     ierr = RKSetAdjCoe(tab);CHKERRQ(ierr);
698   }*/
699   ierr = VecDuplicateVecs(ts->vec_sol,s,&rk->Y);CHKERRQ(ierr);
700   ierr = VecDuplicateVecs(ts->vec_sol,s,&rk->YdotRHS);CHKERRQ(ierr);
701   if (ts->reverse_mode) {
702     ierr = VecDuplicateVecs(ts->vecs_sensi[0],s*ts->numberadjs,&rk->VecDeltaLam);CHKERRQ(ierr);
703     if(ts->vecs_sensip) {
704       ierr = VecDuplicateVecs(ts->vecs_sensip[0],s*ts->numberadjs,&rk->VecDeltaMu);CHKERRQ(ierr);
705     }
706     ierr = VecDuplicateVecs(ts->vecs_sensi[0],ts->numberadjs,&rk->VecSensiTemp);CHKERRQ(ierr);
707   }
708   ierr = PetscMalloc1(s,&rk->work);CHKERRQ(ierr);
709   ierr = TSGetDM(ts,&dm);CHKERRQ(ierr);
710   if (dm) {
711     ierr = DMCoarsenHookAdd(dm,DMCoarsenHook_TSRK,DMRestrictHook_TSRK,ts);CHKERRQ(ierr);
712     ierr = DMSubDomainHookAdd(dm,DMSubDomainHook_TSRK,DMSubDomainRestrictHook_TSRK,ts);CHKERRQ(ierr);
713   }
714   PetscFunctionReturn(0);
715 }
716 
717 /*------------------------------------------------------------*/
718 
719 #undef __FUNCT__
720 #define __FUNCT__ "TSSetFromOptions_RK"
721 static PetscErrorCode TSSetFromOptions_RK(PetscOptions *PetscOptionsObject,TS ts)
722 {
723   PetscErrorCode ierr;
724   char           rktype[256];
725 
726   PetscFunctionBegin;
727   ierr = PetscOptionsHead(PetscOptionsObject,"RK ODE solver options");CHKERRQ(ierr);
728   {
729     RKTableauLink  link;
730     PetscInt       count,choice;
731     PetscBool      flg;
732     const char   **namelist;
733     ierr = PetscStrncpy(rktype,TSRKDefault,sizeof(rktype));CHKERRQ(ierr);
734     for (link=RKTableauList,count=0; link; link=link->next,count++) ;
735     ierr = PetscMalloc1(count,&namelist);CHKERRQ(ierr);
736     for (link=RKTableauList,count=0; link; link=link->next,count++) namelist[count] = link->tab.name;
737     ierr      = PetscOptionsEList("-ts_rk_type","Family of RK method","TSRKSetType",(const char*const*)namelist,count,rktype,&choice,&flg);CHKERRQ(ierr);
738     ierr      = TSRKSetType(ts,flg ? namelist[choice] : rktype);CHKERRQ(ierr);
739     ierr      = PetscFree(namelist);CHKERRQ(ierr);
740   }
741   ierr = PetscOptionsTail();CHKERRQ(ierr);
742   PetscFunctionReturn(0);
743 }
744 
745 #undef __FUNCT__
746 #define __FUNCT__ "PetscFormatRealArray"
747 static PetscErrorCode PetscFormatRealArray(char buf[],size_t len,const char *fmt,PetscInt n,const PetscReal x[])
748 {
749   PetscErrorCode ierr;
750   PetscInt       i;
751   size_t         left,count;
752   char           *p;
753 
754   PetscFunctionBegin;
755   for (i=0,p=buf,left=len; i<n; i++) {
756     ierr = PetscSNPrintfCount(p,left,fmt,&count,x[i]);CHKERRQ(ierr);
757     if (count >= left) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Insufficient space in buffer");
758     left -= count;
759     p    += count;
760     *p++  = ' ';
761   }
762   p[i ? 0 : -1] = 0;
763   PetscFunctionReturn(0);
764 }
765 
766 #undef __FUNCT__
767 #define __FUNCT__ "TSView_RK"
768 static PetscErrorCode TSView_RK(TS ts,PetscViewer viewer)
769 {
770   TS_RK         *rk   = (TS_RK*)ts->data;
771   RKTableau      tab  = rk->tableau;
772   PetscBool      iascii;
773   PetscErrorCode ierr;
774   TSAdapt        adapt;
775 
776   PetscFunctionBegin;
777   ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);CHKERRQ(ierr);
778   if (iascii) {
779     TSRKType rktype;
780     char     buf[512];
781     ierr = TSRKGetType(ts,&rktype);CHKERRQ(ierr);
782     ierr = PetscViewerASCIIPrintf(viewer,"  RK %s\n",rktype);CHKERRQ(ierr);
783     ierr = PetscFormatRealArray(buf,sizeof(buf),"% 8.6f",tab->s,tab->c);CHKERRQ(ierr);
784     ierr = PetscViewerASCIIPrintf(viewer,"  Abscissa     c = %s\n",buf);CHKERRQ(ierr);
785     ierr = PetscViewerASCIIPrintf(viewer,"FSAL: %s\n",tab->FSAL ? "yes" : "no");CHKERRQ(ierr);
786   }
787   ierr = TSGetAdapt(ts,&adapt);CHKERRQ(ierr);
788   ierr = TSAdaptView(adapt,viewer);CHKERRQ(ierr);
789   PetscFunctionReturn(0);
790 }
791 
792 #undef __FUNCT__
793 #define __FUNCT__ "TSLoad_RK"
794 static PetscErrorCode TSLoad_RK(TS ts,PetscViewer viewer)
795 {
796   PetscErrorCode ierr;
797   TSAdapt        tsadapt;
798 
799   PetscFunctionBegin;
800   ierr = TSGetAdapt(ts,&tsadapt);CHKERRQ(ierr);
801   ierr = TSAdaptLoad(tsadapt,viewer);CHKERRQ(ierr);
802   PetscFunctionReturn(0);
803 }
804 
805 #undef __FUNCT__
806 #define __FUNCT__ "TSRKSetType"
807 /*@C
808   TSRKSetType - Set the type of RK scheme
809 
810   Logically collective
811 
812   Input Parameter:
813 +  ts - timestepping context
814 -  rktype - type of RK-scheme
815 
816   Level: intermediate
817 
818 .seealso: TSRKGetType(), TSRK, TSRK2, TSRK3, TSRKPRSSP2, TSRK4, TSRK5
819 @*/
820 PetscErrorCode TSRKSetType(TS ts,TSRKType rktype)
821 {
822   PetscErrorCode ierr;
823 
824   PetscFunctionBegin;
825   PetscValidHeaderSpecific(ts,TS_CLASSID,1);
826   ierr = PetscTryMethod(ts,"TSRKSetType_C",(TS,TSRKType),(ts,rktype));CHKERRQ(ierr);
827   PetscFunctionReturn(0);
828 }
829 
830 #undef __FUNCT__
831 #define __FUNCT__ "TSRKGetType"
832 /*@C
833   TSRKGetType - Get the type of RK scheme
834 
835   Logically collective
836 
837   Input Parameter:
838 .  ts - timestepping context
839 
840   Output Parameter:
841 .  rktype - type of RK-scheme
842 
843   Level: intermediate
844 
845 .seealso: TSRKGetType()
846 @*/
847 PetscErrorCode TSRKGetType(TS ts,TSRKType *rktype)
848 {
849   PetscErrorCode ierr;
850 
851   PetscFunctionBegin;
852   PetscValidHeaderSpecific(ts,TS_CLASSID,1);
853   ierr = PetscUseMethod(ts,"TSRKGetType_C",(TS,TSRKType*),(ts,rktype));CHKERRQ(ierr);
854   PetscFunctionReturn(0);
855 }
856 
857 #undef __FUNCT__
858 #define __FUNCT__ "TSRKGetType_RK"
859 PetscErrorCode  TSRKGetType_RK(TS ts,TSRKType *rktype)
860 {
861   TS_RK     *rk = (TS_RK*)ts->data;
862   PetscErrorCode ierr;
863 
864   PetscFunctionBegin;
865   if (!rk->tableau) {
866     ierr = TSRKSetType(ts,TSRKDefault);CHKERRQ(ierr);
867   }
868   *rktype = rk->tableau->name;
869   PetscFunctionReturn(0);
870 }
871 #undef __FUNCT__
872 #define __FUNCT__ "TSRKSetType_RK"
873 PetscErrorCode  TSRKSetType_RK(TS ts,TSRKType rktype)
874 {
875   TS_RK     *rk = (TS_RK*)ts->data;
876   PetscErrorCode ierr;
877   PetscBool      match;
878   RKTableauLink link;
879 
880   PetscFunctionBegin;
881   if (rk->tableau) {
882     ierr = PetscStrcmp(rk->tableau->name,rktype,&match);CHKERRQ(ierr);
883     if (match) PetscFunctionReturn(0);
884   }
885   for (link = RKTableauList; link; link=link->next) {
886     ierr = PetscStrcmp(link->tab.name,rktype,&match);CHKERRQ(ierr);
887     if (match) {
888       ierr = TSReset_RK(ts);CHKERRQ(ierr);
889       rk->tableau = &link->tab;
890       PetscFunctionReturn(0);
891     }
892   }
893   SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_ARG_UNKNOWN_TYPE,"Could not find '%s'",rktype);
894   PetscFunctionReturn(0);
895 }
896 
897 #undef __FUNCT__
898 #define __FUNCT__ "TSGetStages_RK"
899 static PetscErrorCode  TSGetStages_RK(TS ts,PetscInt *ns,Vec **Y)
900 {
901   TS_RK          *rk = (TS_RK*)ts->data;
902 
903   PetscFunctionBegin;
904   *ns = rk->tableau->s;
905   if(Y) *Y  = rk->Y;
906   PetscFunctionReturn(0);
907 }
908 
909 
910 /* ------------------------------------------------------------ */
911 /*MC
912       TSRK - ODE and DAE solver using Runge-Kutta schemes
913 
914   The user should provide the right hand side of the equation
915   using TSSetRHSFunction().
916 
917   Notes:
918   The default is TSRK3, it can be changed with TSRKSetType() or -ts_rk_type
919 
920   Level: beginner
921 
922 .seealso:  TSCreate(), TS, TSSetType(), TSRKSetType(), TSRKGetType(), TSRKSetFullyImplicit(), TSRK2D, TTSRK2E, TSRK3,
923            TSRK4, TSRK5, TSRKPRSSP2, TSRKBPR3, TSRKType, TSRKRegister()
924 
925 M*/
926 #undef __FUNCT__
927 #define __FUNCT__ "TSCreate_RK"
928 PETSC_EXTERN PetscErrorCode TSCreate_RK(TS ts)
929 {
930   TS_RK     *th;
931   PetscErrorCode ierr;
932 
933   PetscFunctionBegin;
934 #if !defined(PETSC_USE_DYNAMIC_LIBRARIES)
935   ierr = TSRKInitializePackage();CHKERRQ(ierr);
936 #endif
937 
938   ts->ops->reset          = TSReset_RK;
939   ts->ops->destroy        = TSDestroy_RK;
940   ts->ops->view           = TSView_RK;
941   ts->ops->load           = TSLoad_RK;
942   ts->ops->setup          = TSSetUp_RK;
943   ts->ops->step           = TSStep_RK;
944   ts->ops->interpolate    = TSInterpolate_RK;
945   ts->ops->evaluatestep   = TSEvaluateStep_RK;
946   ts->ops->setfromoptions = TSSetFromOptions_RK;
947   ts->ops->getstages      = TSGetStages_RK;
948   ts->ops->stepadj        = TSStepAdj_RK;
949 
950   ierr = PetscNewLog(ts,&th);CHKERRQ(ierr);
951   ts->data = (void*)th;
952 
953   ierr = PetscObjectComposeFunction((PetscObject)ts,"TSRKGetType_C",TSRKGetType_RK);CHKERRQ(ierr);
954   ierr = PetscObjectComposeFunction((PetscObject)ts,"TSRKSetType_C",TSRKSetType_RK);CHKERRQ(ierr);
955   PetscFunctionReturn(0);
956 }
957