Actual source code: eimex.c
petsc-3.10.5 2019-03-28
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      2:  #include <petsc/private/tsimpl.h>
      3:  #include <petscdm.h>

      5: static const PetscInt TSEIMEXDefault = 3;

      7: typedef struct {
      8:   PetscInt     row_ind;         /* Return the term T[row_ind][col_ind] */
      9:   PetscInt     col_ind;         /* Return the term T[row_ind][col_ind] */
     10:   PetscInt     nstages;         /* Numbers of stages in current scheme */
     11:   PetscInt     max_rows;        /* Maximum number of rows */
     12:   PetscInt     *N;              /* Harmonic sequence N[max_rows] */
     13:   Vec          Y;               /* States computed during the step, used to complete the step */
     14:   Vec          Z;               /* For shift*(Y-Z) */
     15:   Vec          *T;              /* Working table, size determined by nstages */
     16:   Vec          YdotRHS;         /* f(x) Work vector holding YdotRHS during residual evaluation */
     17:   Vec          YdotI;           /* xdot-g(x) Work vector holding YdotI = G(t,x,xdot) when xdot =0 */
     18:   Vec          Ydot;            /* f(x)+g(x) Work vector */
     19:   Vec          VecSolPrev;      /* Work vector holding the solution from the previous step (used for interpolation) */
     20:   PetscReal    shift;
     21:   PetscReal    ctime;
     22:   PetscBool    recompute_jacobian; /* Recompute the Jacobian at each stage, default is to freeze the Jacobian at the start of each step */
     23:   PetscBool    ord_adapt;       /* order adapativity */
     24:   TSStepStatus status;
     25: } TS_EIMEX;

     27: /* This function is pure */
     28: static PetscInt Map(PetscInt i, PetscInt j, PetscInt s)
     29: {
     30:   return ((2*s-j+1)*j/2+i-j);
     31: }

     34: static PetscErrorCode TSEvaluateStep_EIMEX(TS ts,PetscInt order,Vec X,PetscBool *done)
     35: {
     36:   TS_EIMEX        *ext = (TS_EIMEX*)ts->data;
     37:   const PetscInt  ns = ext->nstages;
     40:   VecCopy(ext->T[Map(ext->row_ind,ext->col_ind,ns)],X);
     41:   return(0);
     42: }

     45: static PetscErrorCode TSStage_EIMEX(TS ts,PetscInt istage)
     46: {
     47:   TS_EIMEX        *ext = (TS_EIMEX*)ts->data;
     48:   PetscReal       h;
     49:   Vec             Y=ext->Y, Z=ext->Z;
     50:   SNES            snes;
     51:   TSAdapt         adapt;
     52:   PetscInt        i,its,lits;
     53:   PetscBool       accept;
     54:   PetscErrorCode  ierr;

     57:   TSGetSNES(ts,&snes);
     58:   h = ts->time_step/ext->N[istage];/* step size for the istage-th stage */
     59:   ext->shift = 1./h;
     60:   SNESSetLagJacobian(snes,-2); /* Recompute the Jacobian on this solve, but not again */
     61:   VecCopy(ext->VecSolPrev,Y); /* Take the previous solution as intial step */

     63:   for(i=0; i<ext->N[istage]; i++){
     64:     ext->ctime = ts->ptime + h*i;
     65:     VecCopy(Y,Z);/* Save the solution of the previous substep */
     66:     SNESSolve(snes,NULL,Y);
     67:     SNESGetIterationNumber(snes,&its);
     68:     SNESGetLinearSolveIterations(snes,&lits);
     69:     ts->snes_its += its; ts->ksp_its += lits;
     70:     TSGetAdapt(ts,&adapt);
     71:     TSAdaptCheckStage(adapt,ts,ext->ctime,Y,&accept);
     72:   }

     74:   return(0);
     75: }

     78: static PetscErrorCode TSStep_EIMEX(TS ts)
     79: {
     80:   TS_EIMEX        *ext = (TS_EIMEX*)ts->data;
     81:   const PetscInt  ns = ext->nstages;
     82:   Vec             *T=ext->T, Y=ext->Y;

     84:   SNES            snes;
     85:   PetscInt        i,j;
     86:   PetscBool       accept = PETSC_FALSE;
     87:   PetscErrorCode  ierr;
     88:   PetscReal       alpha,local_error,local_error_a,local_error_r;

     91:   TSGetSNES(ts,&snes);
     92:   SNESSetType(snes,"ksponly");
     93:   ext->status = TS_STEP_INCOMPLETE;

     95:   VecCopy(ts->vec_sol,ext->VecSolPrev);

     97:   /* Apply n_j steps of the base method to obtain solutions of T(j,1),1<=j<=s */
     98:   for(j=0; j<ns; j++){
     99:         TSStage_EIMEX(ts,j);
    100:         VecCopy(Y,T[j]);
    101:   }

    103:   for(i=1;i<ns;i++){
    104:     for(j=i;j<ns;j++){
    105:       alpha = -(PetscReal)ext->N[j]/ext->N[j-i];
    106:       VecAXPBYPCZ(T[Map(j,i,ns)],alpha,1.0,0,T[Map(j,i-1,ns)],T[Map(j-1,i-1,ns)]);/* T[j][i]=alpha*T[j][i-1]+T[j-1][i-1] */
    107:       alpha = 1.0/(1.0 + alpha);
    108:       VecScale(T[Map(j,i,ns)],alpha);
    109:     }
    110:   }

    112:   TSEvaluateStep(ts,ns,ts->vec_sol,NULL);/*update ts solution */

    114:   if(ext->ord_adapt && ext->nstages < ext->max_rows){
    115:         accept = PETSC_FALSE;
    116:         while(!accept && ext->nstages < ext->max_rows){
    117:           TSErrorWeightedNorm(ts,ts->vec_sol,T[Map(ext->nstages-1,ext->nstages-2,ext->nstages)],ts->adapt->wnormtype,&local_error,&local_error_a,&local_error_r);
    118:           accept = (local_error < 1.0)? PETSC_TRUE : PETSC_FALSE;

    120:           if(!accept){/* add one more stage*/
    121:             TSStage_EIMEX(ts,ext->nstages);
    122:             ext->nstages++; ext->row_ind++; ext->col_ind++;
    123:             /*T table need to be recycled*/
    124:             VecDuplicateVecs(ts->vec_sol,(1+ext->nstages)*ext->nstages/2,&ext->T);
    125:             for(i=0; i<ext->nstages-1; i++){
    126:               for(j=0; j<=i; j++){
    127:                 VecCopy(T[Map(i,j,ext->nstages-1)],ext->T[Map(i,j,ext->nstages)]);
    128:               }
    129:             }
    130:             VecDestroyVecs(ext->nstages*(ext->nstages-1)/2,&T);
    131:             T = ext->T; /*reset the pointer*/
    132:             /*recycling finished, store the new solution*/
    133:             VecCopy(Y,T[ext->nstages-1]);
    134:             /*extrapolation for the newly added stage*/
    135:             for(i=1;i<ext->nstages;i++){
    136:               alpha = -(PetscReal)ext->N[ext->nstages-1]/ext->N[ext->nstages-1-i];
    137:               VecAXPBYPCZ(T[Map(ext->nstages-1,i,ext->nstages)],alpha,1.0,0,T[Map(ext->nstages-1,i-1,ext->nstages)],T[Map(ext->nstages-1-1,i-1,ext->nstages)]);/*T[ext->nstages-1][i]=alpha*T[ext->nstages-1][i-1]+T[ext->nstages-1-1][i-1]*/
    138:               alpha = 1.0/(1.0 + alpha);
    139:               VecScale(T[Map(ext->nstages-1,i,ext->nstages)],alpha);
    140:             }
    141:             /*update ts solution */
    142:             TSEvaluateStep(ts,ext->nstages,ts->vec_sol,NULL);
    143:           }/*end if !accept*/
    144:         }/*end while*/

    146:         if(ext->nstages == ext->max_rows){
    147:                 PetscInfo(ts,"Max number of rows has been used\n");
    148:         }
    149:   }/*end if ext->ord_adapt*/
    150:   ts->ptime += ts->time_step;
    151:   ext->status = TS_STEP_COMPLETE;

    153:   if (ext->status != TS_STEP_COMPLETE && !ts->reason) ts->reason = TS_DIVERGED_STEP_REJECTED;
    154:   return(0);
    155: }

    157: /* cubic Hermit spline */
    158: static PetscErrorCode TSInterpolate_EIMEX(TS ts,PetscReal itime,Vec X)
    159: {
    160:   TS_EIMEX       *ext = (TS_EIMEX*)ts->data;
    161:   PetscReal      t,a,b;
    162:   Vec            Y0=ext->VecSolPrev,Y1=ext->Y,Ydot=ext->Ydot,YdotI=ext->YdotI;
    163:   const PetscReal h = ts->ptime - ts->ptime_prev;
    166:   t = (itime -ts->ptime + h)/h;
    167:   /* YdotI = -f(x)-g(x) */

    169:   VecZeroEntries(Ydot);
    170:   TSComputeIFunction(ts,ts->ptime-h,Y0,Ydot,YdotI,PETSC_FALSE);

    172:   a    = 2.0*t*t*t - 3.0*t*t + 1.0;
    173:   b    = -(t*t*t - 2.0*t*t + t)*h;
    174:   VecAXPBYPCZ(X,a,b,0.0,Y0,YdotI);

    176:   TSComputeIFunction(ts,ts->ptime,Y1,Ydot,YdotI,PETSC_FALSE);
    177:   a    = -2.0*t*t*t+3.0*t*t;
    178:   b    = -(t*t*t - t*t)*h;
    179:   VecAXPBYPCZ(X,a,b,1.0,Y1,YdotI);

    181:   return(0);
    182: }

    185: static PetscErrorCode TSReset_EIMEX(TS ts)
    186: {
    187:   TS_EIMEX        *ext = (TS_EIMEX*)ts->data;
    188:   PetscInt        ns;
    189:   PetscErrorCode  ierr;

    192:   ns = ext->nstages;
    193:   VecDestroyVecs((1+ns)*ns/2,&ext->T);
    194:   VecDestroy(&ext->Y);
    195:   VecDestroy(&ext->Z);
    196:   VecDestroy(&ext->YdotRHS);
    197:   VecDestroy(&ext->YdotI);
    198:   VecDestroy(&ext->Ydot);
    199:   VecDestroy(&ext->VecSolPrev);
    200:   PetscFree(ext->N);
    201:   return(0);
    202: }

    204: static PetscErrorCode TSDestroy_EIMEX(TS ts)
    205: {
    206:   PetscErrorCode  ierr;

    209:   TSReset_EIMEX(ts);
    210:   PetscFree(ts->data);
    211:   PetscObjectComposeFunction((PetscObject)ts,"TSEIMEXSetMaxRows_C",NULL);
    212:   PetscObjectComposeFunction((PetscObject)ts,"TSEIMEXSetRowCol_C",NULL);
    213:   PetscObjectComposeFunction((PetscObject)ts,"TSEIMEXSetOrdAdapt_C",NULL);

    215:   return(0);
    216: }

    219: static PetscErrorCode TSEIMEXGetVecs(TS ts,DM dm,Vec *Z,Vec *Ydot,Vec *YdotI, Vec *YdotRHS)
    220: {
    221:   TS_EIMEX       *ext = (TS_EIMEX*)ts->data;

    225:   if (Z) {
    226:     if (dm && dm != ts->dm) {
    227:       DMGetNamedGlobalVector(dm,"TSEIMEX_Z",Z);
    228:     } else *Z = ext->Z;
    229:   }
    230:   if (Ydot) {
    231:     if (dm && dm != ts->dm) {
    232:       DMGetNamedGlobalVector(dm,"TSEIMEX_Ydot",Ydot);
    233:     } else *Ydot = ext->Ydot;
    234:   }
    235:   if (YdotI) {
    236:     if (dm && dm != ts->dm) {
    237:       DMGetNamedGlobalVector(dm,"TSEIMEX_YdotI",YdotI);
    238:     } else *YdotI = ext->YdotI;
    239:   }
    240:   if (YdotRHS) {
    241:     if (dm && dm != ts->dm) {
    242:       DMGetNamedGlobalVector(dm,"TSEIMEX_YdotRHS",YdotRHS);
    243:     } else *YdotRHS = ext->YdotRHS;
    244:   }
    245:   return(0);
    246: }

    249: static PetscErrorCode TSEIMEXRestoreVecs(TS ts,DM dm,Vec *Z,Vec *Ydot,Vec *YdotI,Vec *YdotRHS)
    250: {

    254:   if (Z) {
    255:     if (dm && dm != ts->dm) {
    256:       DMRestoreNamedGlobalVector(dm,"TSEIMEX_Z",Z);
    257:     }
    258:   }
    259:   if (Ydot) {
    260:     if (dm && dm != ts->dm) {
    261:       DMRestoreNamedGlobalVector(dm,"TSEIMEX_Ydot",Ydot);
    262:     }
    263:   }
    264:   if (YdotI) {
    265:     if (dm && dm != ts->dm) {
    266:       DMRestoreNamedGlobalVector(dm,"TSEIMEX_YdotI",YdotI);
    267:     }
    268:   }
    269:   if (YdotRHS) {
    270:     if (dm && dm != ts->dm) {
    271:       DMRestoreNamedGlobalVector(dm,"TSEIMEX_YdotRHS",YdotRHS);
    272:     }
    273:   }
    274:   return(0);
    275: }

    278: /*
    279:   This defines the nonlinear equation that is to be solved with SNES
    280:   Fn[t0+Theta*dt, U, (U-U0)*shift] = 0
    281:   In the case of Backward Euler, Fn = (U-U0)/h-g(t1,U))
    282:   Since FormIFunction calculates G = ydot - g(t,y), ydot will be set to (U-U0)/h
    283: */
    284: static PetscErrorCode SNESTSFormFunction_EIMEX(SNES snes,Vec X,Vec G,TS ts)
    285: {
    286:   TS_EIMEX        *ext = (TS_EIMEX*)ts->data;
    287:   PetscErrorCode  ierr;
    288:   Vec             Ydot,Z;
    289:   DM              dm,dmsave;

    292:   VecZeroEntries(G);

    294:   SNESGetDM(snes,&dm);
    295:   TSEIMEXGetVecs(ts,dm,&Z,&Ydot,NULL,NULL);
    296:   VecZeroEntries(Ydot);
    297:   dmsave = ts->dm;
    298:   ts->dm = dm;
    299:   TSComputeIFunction(ts,ext->ctime,X,Ydot,G,PETSC_FALSE);
    300:   /* PETSC_FALSE indicates non-imex, adding explicit RHS to the implicit I function.  */
    301:   VecCopy(G,Ydot);
    302:   ts->dm = dmsave;
    303:   TSEIMEXRestoreVecs(ts,dm,&Z,&Ydot,NULL,NULL);

    305:   return(0);
    306: }

    308: /*
    309:  This defined the Jacobian matrix for SNES. Jn = (I/h-g'(t,y))
    310:  */
    311: static PetscErrorCode SNESTSFormJacobian_EIMEX(SNES snes,Vec X,Mat A,Mat B,TS ts)
    312: {
    313:   TS_EIMEX        *ext = (TS_EIMEX*)ts->data;
    314:   Vec             Ydot;
    315:   PetscErrorCode  ierr;
    316:   DM              dm,dmsave;
    318:   SNESGetDM(snes,&dm);
    319:   TSEIMEXGetVecs(ts,dm,NULL,&Ydot,NULL,NULL);
    320:   /*  VecZeroEntries(Ydot); */
    321:   /* ext->Ydot have already been computed in SNESTSFormFunction_EIMEX (SNES guarantees this) */
    322:   dmsave = ts->dm;
    323:   ts->dm = dm;
    324:   TSComputeIJacobian(ts,ts->ptime,X,Ydot,ext->shift,A,B,PETSC_TRUE);
    325:   ts->dm = dmsave;
    326:   TSEIMEXRestoreVecs(ts,dm,NULL,&Ydot,NULL,NULL);
    327:   return(0);
    328: }

    330: static PetscErrorCode DMCoarsenHook_TSEIMEX(DM fine,DM coarse,void *ctx)
    331: {

    334:   return(0);
    335: }

    337: static PetscErrorCode DMRestrictHook_TSEIMEX(DM fine,Mat restrct,Vec rscale,Mat inject,DM coarse,void *ctx)
    338: {
    339:   TS ts = (TS)ctx;
    341:   Vec Z,Z_c;

    344:   TSEIMEXGetVecs(ts,fine,&Z,NULL,NULL,NULL);
    345:   TSEIMEXGetVecs(ts,coarse,&Z_c,NULL,NULL,NULL);
    346:   MatRestrict(restrct,Z,Z_c);
    347:   VecPointwiseMult(Z_c,rscale,Z_c);
    348:   TSEIMEXRestoreVecs(ts,fine,&Z,NULL,NULL,NULL);
    349:   TSEIMEXRestoreVecs(ts,coarse,&Z_c,NULL,NULL,NULL);
    350:   return(0);
    351: }

    354: static PetscErrorCode TSSetUp_EIMEX(TS ts)
    355: {
    356:   TS_EIMEX       *ext = (TS_EIMEX*)ts->data;
    358:   DM             dm;

    361:   if (!ext->N){ /* ext->max_rows not set */
    362:     TSEIMEXSetMaxRows(ts,TSEIMEXDefault);
    363:   }
    364:   if(-1 == ext->row_ind && -1 == ext->col_ind){
    365:         TSEIMEXSetRowCol(ts,ext->max_rows,ext->max_rows);
    366:   } else{/* ext->row_ind and col_ind already set */
    367:     if (ext->ord_adapt){
    368:       PetscInfo(ts,"Order adaptivity is enabled and TSEIMEXSetRowCol or -ts_eimex_row_col option will take no effect\n");
    369:     }
    370:   }

    372:   if(ext->ord_adapt){
    373:     ext->nstages = 2; /* Start with the 2-stage scheme */
    374:     TSEIMEXSetRowCol(ts,ext->nstages,ext->nstages);
    375:   } else{
    376:     ext->nstages = ext->max_rows; /* by default nstages is the same as max_rows, this can be changed by setting order adaptivity */
    377:   }

    379:   TSGetAdapt(ts,&ts->adapt);

    381:   VecDuplicateVecs(ts->vec_sol,(1+ext->nstages)*ext->nstages/2,&ext->T);/* full T table */
    382:   VecDuplicate(ts->vec_sol,&ext->YdotI);
    383:   VecDuplicate(ts->vec_sol,&ext->YdotRHS);
    384:   VecDuplicate(ts->vec_sol,&ext->Ydot);
    385:   VecDuplicate(ts->vec_sol,&ext->VecSolPrev);
    386:   VecDuplicate(ts->vec_sol,&ext->Y);
    387:   VecDuplicate(ts->vec_sol,&ext->Z);
    388:   TSGetDM(ts,&dm);
    389:   if (dm) {
    390:     DMCoarsenHookAdd(dm,DMCoarsenHook_TSEIMEX,DMRestrictHook_TSEIMEX,ts);
    391:   }
    392:   return(0);
    393: }

    395: static PetscErrorCode TSSetFromOptions_EIMEX(PetscOptionItems *PetscOptionsObject,TS ts)
    396: {
    397:   TS_EIMEX       *ext = (TS_EIMEX*)ts->data;
    399:   PetscInt       tindex[2];
    400:   PetscInt       np = 2, nrows=TSEIMEXDefault;

    403:   tindex[0] = TSEIMEXDefault;
    404:   tindex[1] = TSEIMEXDefault;
    405:   PetscOptionsHead(PetscOptionsObject,"EIMEX ODE solver options");
    406:   {
    407:     PetscBool flg;
    408:     PetscOptionsInt("-ts_eimex_max_rows","Define the maximum number of rows used","TSEIMEXSetMaxRows",nrows,&nrows,&flg); /* default value 3 */
    409:     if(flg){
    410:       TSEIMEXSetMaxRows(ts,nrows);
    411:     }
    412:     PetscOptionsIntArray("-ts_eimex_row_col","Return the specific term in the T table","TSEIMEXSetRowCol",tindex,&np,&flg);
    413:     if(flg){
    414:       TSEIMEXSetRowCol(ts,tindex[0],tindex[1]);
    415:     }
    416:     PetscOptionsBool("-ts_eimex_order_adapt","Solve the problem with adaptive order","TSEIMEXSetOrdAdapt",ext->ord_adapt,&ext->ord_adapt,NULL);
    417:   }
    418:   PetscOptionsTail();
    419:   return(0);
    420: }

    422: static PetscErrorCode TSView_EIMEX(TS ts,PetscViewer viewer)
    423: {
    425:   return(0);
    426: }

    429: /*@C
    430:   TSEIMEXSetMaxRows - Set the maximum number of rows for EIMEX schemes

    432:   Logically collective

    434:   Input Parameter:
    435: +  ts - timestepping context
    436: -  nrows - maximum number of rows

    438:   Level: intermediate

    440: .seealso: TSEIMEXSetRowCol(), TSEIMEXSetOrdAdapt(), TSEIMEX
    441: @*/
    442: PetscErrorCode TSEIMEXSetMaxRows(TS ts, PetscInt nrows)
    443: {
    447:   PetscTryMethod(ts,"TSEIMEXSetMaxRows_C",(TS,PetscInt),(ts,nrows));
    448:   return(0);
    449: }

    452: /*@C
    453:   TSEIMEXSetRowCol - Set the type index in the T table for the return value

    455:   Logically collective

    457:   Input Parameter:
    458: +  ts - timestepping context
    459: -  tindex - index in the T table

    461:   Level: intermediate

    463: .seealso: TSEIMEXSetMaxRows(), TSEIMEXSetOrdAdapt(), TSEIMEX
    464: @*/
    465: PetscErrorCode TSEIMEXSetRowCol(TS ts, PetscInt row, PetscInt col)
    466: {
    470:   PetscTryMethod(ts,"TSEIMEXSetRowCol_C",(TS,PetscInt, PetscInt),(ts,row,col));
    471:   return(0);
    472: }

    475: /*@C
    476:   TSEIMEXSetOrdAdapt - Set the order adaptativity

    478:   Logically collective

    480:   Input Parameter:
    481: +  ts - timestepping context
    482: -  tindex - index in the T table

    484:   Level: intermediate

    486: .seealso: TSEIMEXSetRowCol(), TSEIMEXSetOrdAdapt(), TSEIMEX
    487: @*/
    488: PetscErrorCode TSEIMEXSetOrdAdapt(TS ts, PetscBool flg)
    489: {
    493:   PetscTryMethod(ts,"TSEIMEXSetOrdAdapt_C",(TS,PetscBool),(ts,flg));
    494:   return(0);
    495: }

    498: static PetscErrorCode TSEIMEXSetMaxRows_EIMEX(TS ts,PetscInt nrows)
    499: {
    500:   TS_EIMEX *ext = (TS_EIMEX*)ts->data;
    502:   PetscInt       i;

    505:   if (nrows < 0 || nrows > 100) SETERRQ1(((PetscObject)ts)->comm,PETSC_ERR_ARG_OUTOFRANGE,"Max number of rows (current value %D) should be an integer number between 1 and 100\n",nrows);
    506:   PetscFree(ext->N);
    507:   ext->max_rows = nrows;
    508:   PetscMalloc1(nrows,&ext->N);
    509:   for(i=0;i<nrows;i++) ext->N[i]=i+1;
    510:   return(0);
    511: }

    513: static PetscErrorCode TSEIMEXSetRowCol_EIMEX(TS ts,PetscInt row,PetscInt col)
    514: {
    515:   TS_EIMEX *ext = (TS_EIMEX*)ts->data;

    518:   if (row < 1 || col < 1) SETERRQ2(((PetscObject)ts)->comm,PETSC_ERR_ARG_OUTOFRANGE,"The row or column index (current value %d,%d) should not be less than 1 \n",row,col);
    519:   if (row > ext->max_rows || col > ext->max_rows) SETERRQ3(((PetscObject)ts)->comm,PETSC_ERR_ARG_OUTOFRANGE,"The row or column index (current value %d,%d) exceeds the maximum number of rows %d\n",row,col,ext->max_rows);
    520:   if (col > row) SETERRQ2(((PetscObject)ts)->comm,PETSC_ERR_ARG_OUTOFRANGE,"The column index (%d) exceeds the row index (%d)\n",col,row);

    522:   ext->row_ind = row - 1;
    523:   ext->col_ind = col - 1; /* Array index in C starts from 0 */
    524:   return(0);
    525: }

    527: static PetscErrorCode TSEIMEXSetOrdAdapt_EIMEX(TS ts,PetscBool flg)
    528: {
    529:   TS_EIMEX *ext = (TS_EIMEX*)ts->data;
    531:   ext->ord_adapt = flg;
    532:   return(0);
    533: }

    535: /*MC
    536:       TSEIMEX - Time stepping with Extrapolated IMEX methods.

    538:    These methods are intended for problems with well-separated time scales, especially when a slow scale is strongly nonlinear such that it
    539:    is expensive to solve with a fully implicit method. The user should provide the stiff part of the equation using TSSetIFunction() and the
    540:    non-stiff part with TSSetRHSFunction().

    542:    Notes:
    543:   The default is a 3-stage scheme, it can be changed with TSEIMEXSetMaxRows() or -ts_eimex_max_rows

    545:   This method currently only works with ODE, for which the stiff part G(t,X,Xdot) has the form Xdot + Ghat(t,X).

    547:   The general system is written as

    549:   G(t,X,Xdot) = F(t,X)

    551:   where G represents the stiff part and F represents the non-stiff part. The user should provide the stiff part
    552:   of the equation using TSSetIFunction() and the non-stiff part with TSSetRHSFunction().
    553:   This method is designed to be linearly implicit on G and can use an approximate and lagged Jacobian.

    555:   Another common form for the system is

    557:   y'=f(x)+g(x)

    559:   The relationship between F,G and f,g is

    561:   G = y'-g(x), F = f(x)

    563:  References
    564:   E. Constantinescu and A. Sandu, Extrapolated implicit-explicit time stepping, SIAM Journal on Scientific
    565: Computing, 31 (2010), pp. 4452-4477.

    567:       Level: beginner

    569: .seealso:  TSCreate(), TS, TSSetType(), TSEIMEXSetMaxRows(), TSEIMEXSetRowCol(), TSEIMEXSetOrdAdapt()

    571:  M*/
    572: PETSC_EXTERN PetscErrorCode TSCreate_EIMEX(TS ts)
    573: {
    574:   TS_EIMEX       *ext;

    579:   ts->ops->reset          = TSReset_EIMEX;
    580:   ts->ops->destroy        = TSDestroy_EIMEX;
    581:   ts->ops->view           = TSView_EIMEX;
    582:   ts->ops->setup          = TSSetUp_EIMEX;
    583:   ts->ops->step           = TSStep_EIMEX;
    584:   ts->ops->interpolate    = TSInterpolate_EIMEX;
    585:   ts->ops->evaluatestep   = TSEvaluateStep_EIMEX;
    586:   ts->ops->setfromoptions = TSSetFromOptions_EIMEX;
    587:   ts->ops->snesfunction   = SNESTSFormFunction_EIMEX;
    588:   ts->ops->snesjacobian   = SNESTSFormJacobian_EIMEX;
    589:   ts->default_adapt_type  = TSADAPTNONE;

    591:   ts->usessnes = PETSC_TRUE;

    593:   PetscNewLog(ts,&ext);
    594:   ts->data = (void*)ext;

    596:   ext->ord_adapt = PETSC_FALSE; /* By default, no order adapativity */
    597:   ext->row_ind   = -1;
    598:   ext->col_ind   = -1;
    599:   ext->max_rows  = TSEIMEXDefault;
    600:   ext->nstages   = TSEIMEXDefault;

    602:   PetscObjectComposeFunction((PetscObject)ts,"TSEIMEXSetMaxRows_C", TSEIMEXSetMaxRows_EIMEX);
    603:   PetscObjectComposeFunction((PetscObject)ts,"TSEIMEXSetRowCol_C",  TSEIMEXSetRowCol_EIMEX);
    604:   PetscObjectComposeFunction((PetscObject)ts,"TSEIMEXSetOrdAdapt_C",TSEIMEXSetOrdAdapt_EIMEX);
    605:   return(0);
    606: }