Actual source code: eimex.c

      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: }

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

     43: static PetscErrorCode TSStage_EIMEX(TS ts,PetscInt istage)
     44: {
     45:   TS_EIMEX        *ext = (TS_EIMEX*)ts->data;
     46:   PetscReal       h;
     47:   Vec             Y=ext->Y, Z=ext->Z;
     48:   SNES            snes;
     49:   TSAdapt         adapt;
     50:   PetscInt        i,its,lits;
     51:   PetscBool       accept;
     52:   PetscErrorCode  ierr;

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

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

     74: static PetscErrorCode TSStep_EIMEX(TS ts)
     75: {
     76:   TS_EIMEX        *ext = (TS_EIMEX*)ts->data;
     77:   const PetscInt  ns = ext->nstages;
     78:   Vec             *T=ext->T, Y=ext->Y;

     80:   SNES            snes;
     81:   PetscInt        i,j;
     82:   PetscBool       accept = PETSC_FALSE;
     83:   PetscErrorCode  ierr;
     84:   PetscReal       alpha,local_error,local_error_a,local_error_r;

     87:   TSGetSNES(ts,&snes);
     88:   SNESSetType(snes,"ksponly");
     89:   ext->status = TS_STEP_INCOMPLETE;

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

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

     99:   for (i=1;i<ns;i++){
    100:     for (j=i;j<ns;j++){
    101:       alpha = -(PetscReal)ext->N[j]/ext->N[j-i];
    102:       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] */
    103:       alpha = 1.0/(1.0 + alpha);
    104:       VecScale(T[Map(j,i,ns)],alpha);
    105:     }
    106:   }

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

    110:   if (ext->ord_adapt && ext->nstages < ext->max_rows){
    111:         accept = PETSC_FALSE;
    112:         while (!accept && ext->nstages < ext->max_rows){
    113:           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);
    114:           accept = (local_error < 1.0)? PETSC_TRUE : PETSC_FALSE;

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

    142:         if (ext->nstages == ext->max_rows){
    143:           PetscInfo(ts,"Max number of rows has been used\n");
    144:         }
    145:   }/*end if ext->ord_adapt*/
    146:   ts->ptime += ts->time_step;
    147:   ext->status = TS_STEP_COMPLETE;

    149:   if (ext->status != TS_STEP_COMPLETE && !ts->reason) ts->reason = TS_DIVERGED_STEP_REJECTED;
    150:   return(0);
    151: }

    153: /* cubic Hermit spline */
    154: static PetscErrorCode TSInterpolate_EIMEX(TS ts,PetscReal itime,Vec X)
    155: {
    156:   TS_EIMEX       *ext = (TS_EIMEX*)ts->data;
    157:   PetscReal      t,a,b;
    158:   Vec            Y0=ext->VecSolPrev,Y1=ext->Y,Ydot=ext->Ydot,YdotI=ext->YdotI;
    159:   const PetscReal h = ts->ptime - ts->ptime_prev;
    162:   t = (itime -ts->ptime + h)/h;
    163:   /* YdotI = -f(x)-g(x) */

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

    168:   a    = 2.0*t*t*t - 3.0*t*t + 1.0;
    169:   b    = -(t*t*t - 2.0*t*t + t)*h;
    170:   VecAXPBYPCZ(X,a,b,0.0,Y0,YdotI);

    172:   TSComputeIFunction(ts,ts->ptime,Y1,Ydot,YdotI,PETSC_FALSE);
    173:   a    = -2.0*t*t*t+3.0*t*t;
    174:   b    = -(t*t*t - t*t)*h;
    175:   VecAXPBYPCZ(X,a,b,1.0,Y1,YdotI);

    177:   return(0);
    178: }

    180: static PetscErrorCode TSReset_EIMEX(TS ts)
    181: {
    182:   TS_EIMEX        *ext = (TS_EIMEX*)ts->data;
    183:   PetscInt        ns;
    184:   PetscErrorCode  ierr;

    187:   ns = ext->nstages;
    188:   VecDestroyVecs((1+ns)*ns/2,&ext->T);
    189:   VecDestroy(&ext->Y);
    190:   VecDestroy(&ext->Z);
    191:   VecDestroy(&ext->YdotRHS);
    192:   VecDestroy(&ext->YdotI);
    193:   VecDestroy(&ext->Ydot);
    194:   VecDestroy(&ext->VecSolPrev);
    195:   PetscFree(ext->N);
    196:   return(0);
    197: }

    199: static PetscErrorCode TSDestroy_EIMEX(TS ts)
    200: {
    201:   PetscErrorCode  ierr;

    204:   TSReset_EIMEX(ts);
    205:   PetscFree(ts->data);
    206:   PetscObjectComposeFunction((PetscObject)ts,"TSEIMEXSetMaxRows_C",NULL);
    207:   PetscObjectComposeFunction((PetscObject)ts,"TSEIMEXSetRowCol_C",NULL);
    208:   PetscObjectComposeFunction((PetscObject)ts,"TSEIMEXSetOrdAdapt_C",NULL);
    209:   return(0);
    210: }

    212: static PetscErrorCode TSEIMEXGetVecs(TS ts,DM dm,Vec *Z,Vec *Ydot,Vec *YdotI, Vec *YdotRHS)
    213: {
    214:   TS_EIMEX       *ext = (TS_EIMEX*)ts->data;

    218:   if (Z) {
    219:     if (dm && dm != ts->dm) {
    220:       DMGetNamedGlobalVector(dm,"TSEIMEX_Z",Z);
    221:     } else *Z = ext->Z;
    222:   }
    223:   if (Ydot) {
    224:     if (dm && dm != ts->dm) {
    225:       DMGetNamedGlobalVector(dm,"TSEIMEX_Ydot",Ydot);
    226:     } else *Ydot = ext->Ydot;
    227:   }
    228:   if (YdotI) {
    229:     if (dm && dm != ts->dm) {
    230:       DMGetNamedGlobalVector(dm,"TSEIMEX_YdotI",YdotI);
    231:     } else *YdotI = ext->YdotI;
    232:   }
    233:   if (YdotRHS) {
    234:     if (dm && dm != ts->dm) {
    235:       DMGetNamedGlobalVector(dm,"TSEIMEX_YdotRHS",YdotRHS);
    236:     } else *YdotRHS = ext->YdotRHS;
    237:   }
    238:   return(0);
    239: }

    241: static PetscErrorCode TSEIMEXRestoreVecs(TS ts,DM dm,Vec *Z,Vec *Ydot,Vec *YdotI,Vec *YdotRHS)
    242: {

    246:   if (Z) {
    247:     if (dm && dm != ts->dm) {
    248:       DMRestoreNamedGlobalVector(dm,"TSEIMEX_Z",Z);
    249:     }
    250:   }
    251:   if (Ydot) {
    252:     if (dm && dm != ts->dm) {
    253:       DMRestoreNamedGlobalVector(dm,"TSEIMEX_Ydot",Ydot);
    254:     }
    255:   }
    256:   if (YdotI) {
    257:     if (dm && dm != ts->dm) {
    258:       DMRestoreNamedGlobalVector(dm,"TSEIMEX_YdotI",YdotI);
    259:     }
    260:   }
    261:   if (YdotRHS) {
    262:     if (dm && dm != ts->dm) {
    263:       DMRestoreNamedGlobalVector(dm,"TSEIMEX_YdotRHS",YdotRHS);
    264:     }
    265:   }
    266:   return(0);
    267: }

    269: /*
    270:   This defines the nonlinear equation that is to be solved with SNES
    271:   Fn[t0+Theta*dt, U, (U-U0)*shift] = 0
    272:   In the case of Backward Euler, Fn = (U-U0)/h-g(t1,U))
    273:   Since FormIFunction calculates G = ydot - g(t,y), ydot will be set to (U-U0)/h
    274: */
    275: static PetscErrorCode SNESTSFormFunction_EIMEX(SNES snes,Vec X,Vec G,TS ts)
    276: {
    277:   TS_EIMEX        *ext = (TS_EIMEX*)ts->data;
    278:   PetscErrorCode  ierr;
    279:   Vec             Ydot,Z;
    280:   DM              dm,dmsave;

    283:   VecZeroEntries(G);

    285:   SNESGetDM(snes,&dm);
    286:   TSEIMEXGetVecs(ts,dm,&Z,&Ydot,NULL,NULL);
    287:   VecZeroEntries(Ydot);
    288:   dmsave = ts->dm;
    289:   ts->dm = dm;
    290:   TSComputeIFunction(ts,ext->ctime,X,Ydot,G,PETSC_FALSE);
    291:   /* PETSC_FALSE indicates non-imex, adding explicit RHS to the implicit I function.  */
    292:   VecCopy(G,Ydot);
    293:   ts->dm = dmsave;
    294:   TSEIMEXRestoreVecs(ts,dm,&Z,&Ydot,NULL,NULL);

    296:   return(0);
    297: }

    299: /*
    300:  This defined the Jacobian matrix for SNES. Jn = (I/h-g'(t,y))
    301:  */
    302: static PetscErrorCode SNESTSFormJacobian_EIMEX(SNES snes,Vec X,Mat A,Mat B,TS ts)
    303: {
    304:   TS_EIMEX        *ext = (TS_EIMEX*)ts->data;
    305:   Vec             Ydot;
    306:   PetscErrorCode  ierr;
    307:   DM              dm,dmsave;
    309:   SNESGetDM(snes,&dm);
    310:   TSEIMEXGetVecs(ts,dm,NULL,&Ydot,NULL,NULL);
    311:   /*  VecZeroEntries(Ydot); */
    312:   /* ext->Ydot have already been computed in SNESTSFormFunction_EIMEX (SNES guarantees this) */
    313:   dmsave = ts->dm;
    314:   ts->dm = dm;
    315:   TSComputeIJacobian(ts,ts->ptime,X,Ydot,ext->shift,A,B,PETSC_TRUE);
    316:   ts->dm = dmsave;
    317:   TSEIMEXRestoreVecs(ts,dm,NULL,&Ydot,NULL,NULL);
    318:   return(0);
    319: }

    321: static PetscErrorCode DMCoarsenHook_TSEIMEX(DM fine,DM coarse,void *ctx)
    322: {

    325:   return(0);
    326: }

    328: static PetscErrorCode DMRestrictHook_TSEIMEX(DM fine,Mat restrct,Vec rscale,Mat inject,DM coarse,void *ctx)
    329: {
    330:   TS ts = (TS)ctx;
    332:   Vec Z,Z_c;

    335:   TSEIMEXGetVecs(ts,fine,&Z,NULL,NULL,NULL);
    336:   TSEIMEXGetVecs(ts,coarse,&Z_c,NULL,NULL,NULL);
    337:   MatRestrict(restrct,Z,Z_c);
    338:   VecPointwiseMult(Z_c,rscale,Z_c);
    339:   TSEIMEXRestoreVecs(ts,fine,&Z,NULL,NULL,NULL);
    340:   TSEIMEXRestoreVecs(ts,coarse,&Z_c,NULL,NULL,NULL);
    341:   return(0);
    342: }

    344: static PetscErrorCode TSSetUp_EIMEX(TS ts)
    345: {
    346:   TS_EIMEX       *ext = (TS_EIMEX*)ts->data;
    348:   DM             dm;

    351:   if (!ext->N){ /* ext->max_rows not set */
    352:     TSEIMEXSetMaxRows(ts,TSEIMEXDefault);
    353:   }
    354:   if (-1 == ext->row_ind && -1 == ext->col_ind){
    355:         TSEIMEXSetRowCol(ts,ext->max_rows,ext->max_rows);
    356:   } else{/* ext->row_ind and col_ind already set */
    357:     if (ext->ord_adapt){
    358:       PetscInfo(ts,"Order adaptivity is enabled and TSEIMEXSetRowCol or -ts_eimex_row_col option will take no effect\n");
    359:     }
    360:   }

    362:   if (ext->ord_adapt){
    363:     ext->nstages = 2; /* Start with the 2-stage scheme */
    364:     TSEIMEXSetRowCol(ts,ext->nstages,ext->nstages);
    365:   } else{
    366:     ext->nstages = ext->max_rows; /* by default nstages is the same as max_rows, this can be changed by setting order adaptivity */
    367:   }

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

    371:   VecDuplicateVecs(ts->vec_sol,(1+ext->nstages)*ext->nstages/2,&ext->T);/* full T table */
    372:   VecDuplicate(ts->vec_sol,&ext->YdotI);
    373:   VecDuplicate(ts->vec_sol,&ext->YdotRHS);
    374:   VecDuplicate(ts->vec_sol,&ext->Ydot);
    375:   VecDuplicate(ts->vec_sol,&ext->VecSolPrev);
    376:   VecDuplicate(ts->vec_sol,&ext->Y);
    377:   VecDuplicate(ts->vec_sol,&ext->Z);
    378:   TSGetDM(ts,&dm);
    379:   if (dm) {
    380:     DMCoarsenHookAdd(dm,DMCoarsenHook_TSEIMEX,DMRestrictHook_TSEIMEX,ts);
    381:   }
    382:   return(0);
    383: }

    385: static PetscErrorCode TSSetFromOptions_EIMEX(PetscOptionItems *PetscOptionsObject,TS ts)
    386: {
    387:   TS_EIMEX       *ext = (TS_EIMEX*)ts->data;
    389:   PetscInt       tindex[2];
    390:   PetscInt       np = 2, nrows=TSEIMEXDefault;

    393:   tindex[0] = TSEIMEXDefault;
    394:   tindex[1] = TSEIMEXDefault;
    395:   PetscOptionsHead(PetscOptionsObject,"EIMEX ODE solver options");
    396:   {
    397:     PetscBool flg;
    398:     PetscOptionsInt("-ts_eimex_max_rows","Define the maximum number of rows used","TSEIMEXSetMaxRows",nrows,&nrows,&flg); /* default value 3 */
    399:     if (flg){
    400:       TSEIMEXSetMaxRows(ts,nrows);
    401:     }
    402:     PetscOptionsIntArray("-ts_eimex_row_col","Return the specific term in the T table","TSEIMEXSetRowCol",tindex,&np,&flg);
    403:     if (flg){
    404:       TSEIMEXSetRowCol(ts,tindex[0],tindex[1]);
    405:     }
    406:     PetscOptionsBool("-ts_eimex_order_adapt","Solve the problem with adaptive order","TSEIMEXSetOrdAdapt",ext->ord_adapt,&ext->ord_adapt,NULL);
    407:   }
    408:   PetscOptionsTail();
    409:   return(0);
    410: }

    412: static PetscErrorCode TSView_EIMEX(TS ts,PetscViewer viewer)
    413: {
    415:   return(0);
    416: }

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

    421:   Logically collective

    423:   Input Parameter:
    424: +  ts - timestepping context
    425: -  nrows - maximum number of rows

    427:   Level: intermediate

    429: .seealso: TSEIMEXSetRowCol(), TSEIMEXSetOrdAdapt(), TSEIMEX
    430: @*/
    431: PetscErrorCode TSEIMEXSetMaxRows(TS ts, PetscInt nrows)
    432: {
    436:   PetscTryMethod(ts,"TSEIMEXSetMaxRows_C",(TS,PetscInt),(ts,nrows));
    437:   return(0);
    438: }

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

    443:   Logically collective

    445:   Input Parameter:
    446: +  ts - timestepping context
    447: -  tindex - index in the T table

    449:   Level: intermediate

    451: .seealso: TSEIMEXSetMaxRows(), TSEIMEXSetOrdAdapt(), TSEIMEX
    452: @*/
    453: PetscErrorCode TSEIMEXSetRowCol(TS ts, PetscInt row, PetscInt col)
    454: {
    458:   PetscTryMethod(ts,"TSEIMEXSetRowCol_C",(TS,PetscInt, PetscInt),(ts,row,col));
    459:   return(0);
    460: }

    462: /*@C
    463:   TSEIMEXSetOrdAdapt - Set the order adaptativity

    465:   Logically collective

    467:   Input Parameter:
    468: +  ts - timestepping context
    469: -  tindex - index in the T table

    471:   Level: intermediate

    473: .seealso: TSEIMEXSetRowCol(), TSEIMEXSetOrdAdapt(), TSEIMEX
    474: @*/
    475: PetscErrorCode TSEIMEXSetOrdAdapt(TS ts, PetscBool flg)
    476: {
    480:   PetscTryMethod(ts,"TSEIMEXSetOrdAdapt_C",(TS,PetscBool),(ts,flg));
    481:   return(0);
    482: }

    484: static PetscErrorCode TSEIMEXSetMaxRows_EIMEX(TS ts,PetscInt nrows)
    485: {
    486:   TS_EIMEX *ext = (TS_EIMEX*)ts->data;
    488:   PetscInt       i;

    491:   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);
    492:   PetscFree(ext->N);
    493:   ext->max_rows = nrows;
    494:   PetscMalloc1(nrows,&ext->N);
    495:   for (i=0;i<nrows;i++) ext->N[i]=i+1;
    496:   return(0);
    497: }

    499: static PetscErrorCode TSEIMEXSetRowCol_EIMEX(TS ts,PetscInt row,PetscInt col)
    500: {
    501:   TS_EIMEX *ext = (TS_EIMEX*)ts->data;

    504:   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);
    505:   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);
    506:   if (col > row) SETERRQ2(((PetscObject)ts)->comm,PETSC_ERR_ARG_OUTOFRANGE,"The column index (%d) exceeds the row index (%d)\n",col,row);

    508:   ext->row_ind = row - 1;
    509:   ext->col_ind = col - 1; /* Array index in C starts from 0 */
    510:   return(0);
    511: }

    513: static PetscErrorCode TSEIMEXSetOrdAdapt_EIMEX(TS ts,PetscBool flg)
    514: {
    515:   TS_EIMEX *ext = (TS_EIMEX*)ts->data;
    517:   ext->ord_adapt = flg;
    518:   return(0);
    519: }

    521: /*MC
    522:       TSEIMEX - Time stepping with Extrapolated IMEX methods.

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

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

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

    533:   The general system is written as

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

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

    541:   Another common form for the system is

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

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

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

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

    553:       Level: beginner

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

    557:  M*/
    558: PETSC_EXTERN PetscErrorCode TSCreate_EIMEX(TS ts)
    559: {
    560:   TS_EIMEX       *ext;

    565:   ts->ops->reset          = TSReset_EIMEX;
    566:   ts->ops->destroy        = TSDestroy_EIMEX;
    567:   ts->ops->view           = TSView_EIMEX;
    568:   ts->ops->setup          = TSSetUp_EIMEX;
    569:   ts->ops->step           = TSStep_EIMEX;
    570:   ts->ops->interpolate    = TSInterpolate_EIMEX;
    571:   ts->ops->evaluatestep   = TSEvaluateStep_EIMEX;
    572:   ts->ops->setfromoptions = TSSetFromOptions_EIMEX;
    573:   ts->ops->snesfunction   = SNESTSFormFunction_EIMEX;
    574:   ts->ops->snesjacobian   = SNESTSFormJacobian_EIMEX;
    575:   ts->default_adapt_type  = TSADAPTNONE;

    577:   ts->usessnes = PETSC_TRUE;

    579:   PetscNewLog(ts,&ext);
    580:   ts->data = (void*)ext;

    582:   ext->ord_adapt = PETSC_FALSE; /* By default, no order adapativity */
    583:   ext->row_ind   = -1;
    584:   ext->col_ind   = -1;
    585:   ext->max_rows  = TSEIMEXDefault;
    586:   ext->nstages   = TSEIMEXDefault;

    588:   PetscObjectComposeFunction((PetscObject)ts,"TSEIMEXSetMaxRows_C", TSEIMEXSetMaxRows_EIMEX);
    589:   PetscObjectComposeFunction((PetscObject)ts,"TSEIMEXSetRowCol_C",  TSEIMEXSetRowCol_EIMEX);
    590:   PetscObjectComposeFunction((PetscObject)ts,"TSEIMEXSetOrdAdapt_C",TSEIMEXSetOrdAdapt_EIMEX);
    591:   return(0);
    592: }