Actual source code: alpha2.c

      1: /*
      2:   Code for timestepping with implicit generalized-\alpha method
      3:   for second order systems.
      4: */
      5: #include <petsc/private/tsimpl.h>

      7: static PetscBool  cited = PETSC_FALSE;
      8: static const char citation[] =
      9:   "@article{Chung1993,\n"
     10:   "  title   = {A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation: The Generalized-$\\alpha$ Method},\n"
     11:   "  author  = {J. Chung, G. M. Hubert},\n"
     12:   "  journal = {ASME Journal of Applied Mechanics},\n"
     13:   "  volume  = {60},\n"
     14:   "  number  = {2},\n"
     15:   "  pages   = {371--375},\n"
     16:   "  year    = {1993},\n"
     17:   "  issn    = {0021-8936},\n"
     18:   "  doi     = {http://dx.doi.org/10.1115/1.2900803}\n}\n";

     20: typedef struct {
     21:   PetscReal stage_time;
     22:   PetscReal shift_V;
     23:   PetscReal shift_A;
     24:   PetscReal scale_F;
     25:   Vec       X0,Xa,X1;
     26:   Vec       V0,Va,V1;
     27:   Vec       A0,Aa,A1;

     29:   Vec       vec_dot;

     31:   PetscReal Alpha_m;
     32:   PetscReal Alpha_f;
     33:   PetscReal Gamma;
     34:   PetscReal Beta;
     35:   PetscInt  order;

     37:   Vec       vec_sol_prev;
     38:   Vec       vec_dot_prev;
     39:   Vec       vec_lte_work[2];

     41:   TSStepStatus status;
     42: } TS_Alpha;

     44: static PetscErrorCode TSAlpha_StageTime(TS ts)
     45: {
     46:   TS_Alpha  *th = (TS_Alpha*)ts->data;
     47:   PetscReal t  = ts->ptime;
     48:   PetscReal dt = ts->time_step;
     49:   PetscReal Alpha_m = th->Alpha_m;
     50:   PetscReal Alpha_f = th->Alpha_f;
     51:   PetscReal Gamma   = th->Gamma;
     52:   PetscReal Beta    = th->Beta;

     54:   th->stage_time = t + Alpha_f*dt;
     55:   th->shift_V = Gamma/(dt*Beta);
     56:   th->shift_A = Alpha_m/(Alpha_f*dt*dt*Beta);
     57:   th->scale_F = 1/Alpha_f;
     58:   return 0;
     59: }

     61: static PetscErrorCode TSAlpha_StageVecs(TS ts,Vec X)
     62: {
     63:   TS_Alpha       *th = (TS_Alpha*)ts->data;
     64:   Vec            X1 = X,      V1 = th->V1, A1 = th->A1;
     65:   Vec            Xa = th->Xa, Va = th->Va, Aa = th->Aa;
     66:   Vec            X0 = th->X0, V0 = th->V0, A0 = th->A0;
     67:   PetscReal      dt = ts->time_step;
     68:   PetscReal      Alpha_m = th->Alpha_m;
     69:   PetscReal      Alpha_f = th->Alpha_f;
     70:   PetscReal      Gamma   = th->Gamma;
     71:   PetscReal      Beta    = th->Beta;

     73:   /* A1 = ... */
     74:   VecWAXPY(A1,-1.0,X0,X1);
     75:   VecAXPY (A1,-dt,V0);
     76:   VecAXPBY(A1,-(1-2*Beta)/(2*Beta),1/(dt*dt*Beta),A0);
     77:   /* V1 = ... */
     78:   VecWAXPY(V1,(1.0-Gamma)/Gamma,A0,A1);
     79:   VecAYPX (V1,dt*Gamma,V0);
     80:   /* Xa = X0 + Alpha_f*(X1-X0) */
     81:   VecWAXPY(Xa,-1.0,X0,X1);
     82:   VecAYPX (Xa,Alpha_f,X0);
     83:   /* Va = V0 + Alpha_f*(V1-V0) */
     84:   VecWAXPY(Va,-1.0,V0,V1);
     85:   VecAYPX (Va,Alpha_f,V0);
     86:   /* Aa = A0 + Alpha_m*(A1-A0) */
     87:   VecWAXPY(Aa,-1.0,A0,A1);
     88:   VecAYPX (Aa,Alpha_m,A0);
     89:   return 0;
     90: }

     92: static PetscErrorCode TSAlpha_SNESSolve(TS ts,Vec b,Vec x)
     93: {
     94:   PetscInt       nits,lits;

     96:   SNESSolve(ts->snes,b,x);
     97:   SNESGetIterationNumber(ts->snes,&nits);
     98:   SNESGetLinearSolveIterations(ts->snes,&lits);
     99:   ts->snes_its += nits; ts->ksp_its += lits;
    100:   return 0;
    101: }

    103: /*
    104:   Compute a consistent initial state for the generalized-alpha method.
    105:   - Solve two successive backward Euler steps with halved time step.
    106:   - Compute the initial second time derivative using backward differences.
    107:   - If using adaptivity, estimate the LTE of the initial step.
    108: */
    109: static PetscErrorCode TSAlpha_Restart(TS ts,PetscBool *initok)
    110: {
    111:   TS_Alpha       *th = (TS_Alpha*)ts->data;
    112:   PetscReal      time_step;
    113:   PetscReal      alpha_m,alpha_f,gamma,beta;
    114:   Vec            X0 = ts->vec_sol, X1, X2 = th->X1;
    115:   Vec            V0 = ts->vec_dot, V1, V2 = th->V1;
    116:   PetscBool      stageok;

    118:   VecDuplicate(X0,&X1);
    119:   VecDuplicate(V0,&V1);

    121:   /* Setup backward Euler with halved time step */
    122:   TSAlpha2GetParams(ts,&alpha_m,&alpha_f,&gamma,&beta);
    123:   TSAlpha2SetParams(ts,1,1,1,0.5);
    124:   TSGetTimeStep(ts,&time_step);
    125:   ts->time_step = time_step/2;
    126:   TSAlpha_StageTime(ts);
    127:   th->stage_time = ts->ptime;
    128:   VecZeroEntries(th->A0);

    130:   /* First BE step, (t0,X0,V0) -> (t1,X1,V1) */
    131:   th->stage_time += ts->time_step;
    132:   VecCopy(X0,th->X0);
    133:   VecCopy(V0,th->V0);
    134:   TSPreStage(ts,th->stage_time);
    135:   VecCopy(th->X0,X1);
    136:   TSAlpha_SNESSolve(ts,NULL,X1);
    137:   VecCopy(th->V1,V1);
    138:   TSPostStage(ts,th->stage_time,0,&X1);
    139:   TSAdaptCheckStage(ts->adapt,ts,th->stage_time,X1,&stageok);
    140:   if (!stageok) goto finally;

    142:   /* Second BE step, (t1,X1,V1) -> (t2,X2,V2) */
    143:   th->stage_time += ts->time_step;
    144:   VecCopy(X1,th->X0);
    145:   VecCopy(V1,th->V0);
    146:   TSPreStage(ts,th->stage_time);
    147:   VecCopy(th->X0,X2);
    148:   TSAlpha_SNESSolve(ts,NULL,X2);
    149:   VecCopy(th->V1,V2);
    150:   TSPostStage(ts,th->stage_time,0,&X2);
    151:   TSAdaptCheckStage(ts->adapt,ts,th->stage_time,X1,&stageok);
    152:   if (!stageok) goto finally;

    154:   /* Compute A0 ~ dV/dt at t0 with backward differences */
    155:   VecZeroEntries(th->A0);
    156:   VecAXPY(th->A0,-3/ts->time_step,V0);
    157:   VecAXPY(th->A0,+4/ts->time_step,V1);
    158:   VecAXPY(th->A0,-1/ts->time_step,V2);

    160:   /* Rough, lower-order estimate LTE of the initial step */
    161:   if (th->vec_lte_work[0]) {
    162:     VecZeroEntries(th->vec_lte_work[0]);
    163:     VecAXPY(th->vec_lte_work[0],+2,X2);
    164:     VecAXPY(th->vec_lte_work[0],-4,X1);
    165:     VecAXPY(th->vec_lte_work[0],+2,X0);
    166:   }
    167:   if (th->vec_lte_work[1]) {
    168:     VecZeroEntries(th->vec_lte_work[1]);
    169:     VecAXPY(th->vec_lte_work[1],+2,V2);
    170:     VecAXPY(th->vec_lte_work[1],-4,V1);
    171:     VecAXPY(th->vec_lte_work[1],+2,V0);
    172:   }

    174:  finally:
    175:   /* Revert TSAlpha to the initial state (t0,X0,V0) */
    176:   if (initok) *initok = stageok;
    177:   TSSetTimeStep(ts,time_step);
    178:   TSAlpha2SetParams(ts,alpha_m,alpha_f,gamma,beta);
    179:   VecCopy(ts->vec_sol,th->X0);
    180:   VecCopy(ts->vec_dot,th->V0);

    182:   VecDestroy(&X1);
    183:   VecDestroy(&V1);
    184:   return 0;
    185: }

    187: static PetscErrorCode TSStep_Alpha(TS ts)
    188: {
    189:   TS_Alpha       *th = (TS_Alpha*)ts->data;
    190:   PetscInt       rejections = 0;
    191:   PetscBool      stageok,accept = PETSC_TRUE;
    192:   PetscReal      next_time_step = ts->time_step;

    194:   PetscCitationsRegister(citation,&cited);

    196:   if (!ts->steprollback) {
    197:     if (th->vec_sol_prev) VecCopy(th->X0,th->vec_sol_prev);
    198:     if (th->vec_dot_prev) VecCopy(th->V0,th->vec_dot_prev);
    199:     VecCopy(ts->vec_sol,th->X0);
    200:     VecCopy(ts->vec_dot,th->V0);
    201:     VecCopy(th->A1,th->A0);
    202:   }

    204:   th->status = TS_STEP_INCOMPLETE;
    205:   while (!ts->reason && th->status != TS_STEP_COMPLETE) {

    207:     if (ts->steprestart) {
    208:       TSAlpha_Restart(ts,&stageok);
    209:       if (!stageok) goto reject_step;
    210:     }

    212:     TSAlpha_StageTime(ts);
    213:     VecCopy(th->X0,th->X1);
    214:     TSPreStage(ts,th->stage_time);
    215:     TSAlpha_SNESSolve(ts,NULL,th->X1);
    216:     TSPostStage(ts,th->stage_time,0,&th->Xa);
    217:     TSAdaptCheckStage(ts->adapt,ts,th->stage_time,th->Xa,&stageok);
    218:     if (!stageok) goto reject_step;

    220:     th->status = TS_STEP_PENDING;
    221:     VecCopy(th->X1,ts->vec_sol);
    222:     VecCopy(th->V1,ts->vec_dot);
    223:     TSAdaptChoose(ts->adapt,ts,ts->time_step,NULL,&next_time_step,&accept);
    224:     th->status = accept ? TS_STEP_COMPLETE : TS_STEP_INCOMPLETE;
    225:     if (!accept) {
    226:       VecCopy(th->X0,ts->vec_sol);
    227:       VecCopy(th->V0,ts->vec_dot);
    228:       ts->time_step = next_time_step;
    229:       goto reject_step;
    230:     }

    232:     ts->ptime += ts->time_step;
    233:     ts->time_step = next_time_step;
    234:     break;

    236:   reject_step:
    237:     ts->reject++; accept = PETSC_FALSE;
    238:     if (!ts->reason && ++rejections > ts->max_reject && ts->max_reject >= 0) {
    239:       ts->reason = TS_DIVERGED_STEP_REJECTED;
    240:       PetscInfo(ts,"Step=%D, step rejections %D greater than current TS allowed, stopping solve\n",ts->steps,rejections);
    241:     }

    243:   }
    244:   return 0;
    245: }

    247: static PetscErrorCode TSEvaluateWLTE_Alpha(TS ts,NormType wnormtype,PetscInt *order,PetscReal *wlte)
    248: {
    249:   TS_Alpha       *th = (TS_Alpha*)ts->data;
    250:   Vec            X = th->X1;              /* X = solution */
    251:   Vec            V = th->V1;              /* V = solution */
    252:   Vec            Y = th->vec_lte_work[0]; /* Y = X + LTE  */
    253:   Vec            Z = th->vec_lte_work[1]; /* Z = V + LTE  */
    254:   PetscReal      enormX,enormV,enormXa,enormVa,enormXr,enormVr;

    256:   if (!th->vec_sol_prev) {*wlte = -1; return 0;}
    257:   if (!th->vec_dot_prev) {*wlte = -1; return 0;}
    258:   if (!th->vec_lte_work[0]) {*wlte = -1; return 0;}
    259:   if (!th->vec_lte_work[1]) {*wlte = -1; return 0;}
    260:   if (ts->steprestart) {
    261:     /* th->vec_lte_prev is set to the LTE in TSAlpha_Restart() */
    262:     VecAXPY(Y,1,X);
    263:     VecAXPY(Z,1,V);
    264:   } else {
    265:     /* Compute LTE using backward differences with non-constant time step */
    266:     PetscReal   h = ts->time_step, h_prev = ts->ptime - ts->ptime_prev;
    267:     PetscReal   a = 1 + h_prev/h;
    268:     PetscScalar scal[3]; Vec vecX[3],vecV[3];
    269:     scal[0] = +1/a;   scal[1] = -1/(a-1); scal[2] = +1/(a*(a-1));
    270:     vecX[0] = th->X1; vecX[1] = th->X0;   vecX[2] = th->vec_sol_prev;
    271:     vecV[0] = th->V1; vecV[1] = th->V0;   vecV[2] = th->vec_dot_prev;
    272:     VecCopy(X,Y);
    273:     VecMAXPY(Y,3,scal,vecX);
    274:     VecCopy(V,Z);
    275:     VecMAXPY(Z,3,scal,vecV);
    276:   }
    277:   /* XXX ts->atol and ts->vatol are not appropriate for computing enormV */
    278:   TSErrorWeightedNorm(ts,X,Y,wnormtype,&enormX,&enormXa,&enormXr);
    279:   TSErrorWeightedNorm(ts,V,Z,wnormtype,&enormV,&enormVa,&enormVr);
    280:   if (wnormtype == NORM_2)
    281:     *wlte = PetscSqrtReal(PetscSqr(enormX)/2 + PetscSqr(enormV)/2);
    282:   else
    283:     *wlte = PetscMax(enormX,enormV);
    284:   if (order) *order = 2;
    285:   return 0;
    286: }

    288: static PetscErrorCode TSRollBack_Alpha(TS ts)
    289: {
    290:   TS_Alpha       *th = (TS_Alpha*)ts->data;

    292:   VecCopy(th->X0,ts->vec_sol);
    293:   VecCopy(th->V0,ts->vec_dot);
    294:   return 0;
    295: }

    297: /*
    298: static PetscErrorCode TSInterpolate_Alpha(TS ts,PetscReal t,Vec X,Vec V)
    299: {
    300:   TS_Alpha       *th = (TS_Alpha*)ts->data;
    301:   PetscReal      dt  = t - ts->ptime;

    303:   VecCopy(ts->vec_dot,V);
    304:   VecAXPY(V,dt*(1-th->Gamma),th->A0);
    305:   VecAXPY(V,dt*th->Gamma,th->A1);
    306:   VecCopy(ts->vec_sol,X);
    307:   VecAXPY(X,dt,V);
    308:   VecAXPY(X,dt*dt*((PetscReal)0.5-th->Beta),th->A0);
    309:   VecAXPY(X,dt*dt*th->Beta,th->A1);
    310:   return 0;
    311: }
    312: */

    314: static PetscErrorCode SNESTSFormFunction_Alpha(PETSC_UNUSED SNES snes,Vec X,Vec F,TS ts)
    315: {
    316:   TS_Alpha       *th = (TS_Alpha*)ts->data;
    317:   PetscReal      ta = th->stage_time;
    318:   Vec            Xa = th->Xa, Va = th->Va, Aa = th->Aa;

    320:   TSAlpha_StageVecs(ts,X);
    321:   /* F = Function(ta,Xa,Va,Aa) */
    322:   TSComputeI2Function(ts,ta,Xa,Va,Aa,F);
    323:   VecScale(F,th->scale_F);
    324:   return 0;
    325: }

    327: static PetscErrorCode SNESTSFormJacobian_Alpha(PETSC_UNUSED SNES snes,PETSC_UNUSED Vec X,Mat J,Mat P,TS ts)
    328: {
    329:   TS_Alpha       *th = (TS_Alpha*)ts->data;
    330:   PetscReal      ta = th->stage_time;
    331:   Vec            Xa = th->Xa, Va = th->Va, Aa = th->Aa;
    332:   PetscReal      dVdX = th->shift_V, dAdX = th->shift_A;

    334:   /* J,P = Jacobian(ta,Xa,Va,Aa) */
    335:   TSComputeI2Jacobian(ts,ta,Xa,Va,Aa,dVdX,dAdX,J,P);
    336:   return 0;
    337: }

    339: static PetscErrorCode TSReset_Alpha(TS ts)
    340: {
    341:   TS_Alpha       *th = (TS_Alpha*)ts->data;

    343:   VecDestroy(&th->X0);
    344:   VecDestroy(&th->Xa);
    345:   VecDestroy(&th->X1);
    346:   VecDestroy(&th->V0);
    347:   VecDestroy(&th->Va);
    348:   VecDestroy(&th->V1);
    349:   VecDestroy(&th->A0);
    350:   VecDestroy(&th->Aa);
    351:   VecDestroy(&th->A1);
    352:   VecDestroy(&th->vec_sol_prev);
    353:   VecDestroy(&th->vec_dot_prev);
    354:   VecDestroy(&th->vec_lte_work[0]);
    355:   VecDestroy(&th->vec_lte_work[1]);
    356:   return 0;
    357: }

    359: static PetscErrorCode TSDestroy_Alpha(TS ts)
    360: {
    361:   TSReset_Alpha(ts);
    362:   PetscFree(ts->data);

    364:   PetscObjectComposeFunction((PetscObject)ts,"TSAlpha2SetRadius_C",NULL);
    365:   PetscObjectComposeFunction((PetscObject)ts,"TSAlpha2SetParams_C",NULL);
    366:   PetscObjectComposeFunction((PetscObject)ts,"TSAlpha2GetParams_C",NULL);
    367:   return 0;
    368: }

    370: static PetscErrorCode TSSetUp_Alpha(TS ts)
    371: {
    372:   TS_Alpha       *th = (TS_Alpha*)ts->data;
    373:   PetscBool      match;

    375:   VecDuplicate(ts->vec_sol,&th->X0);
    376:   VecDuplicate(ts->vec_sol,&th->Xa);
    377:   VecDuplicate(ts->vec_sol,&th->X1);
    378:   VecDuplicate(ts->vec_sol,&th->V0);
    379:   VecDuplicate(ts->vec_sol,&th->Va);
    380:   VecDuplicate(ts->vec_sol,&th->V1);
    381:   VecDuplicate(ts->vec_sol,&th->A0);
    382:   VecDuplicate(ts->vec_sol,&th->Aa);
    383:   VecDuplicate(ts->vec_sol,&th->A1);

    385:   TSGetAdapt(ts,&ts->adapt);
    386:   TSAdaptCandidatesClear(ts->adapt);
    387:   PetscObjectTypeCompare((PetscObject)ts->adapt,TSADAPTNONE,&match);
    388:   if (!match) {
    389:     VecDuplicate(ts->vec_sol,&th->vec_sol_prev);
    390:     VecDuplicate(ts->vec_sol,&th->vec_dot_prev);
    391:     VecDuplicate(ts->vec_sol,&th->vec_lte_work[0]);
    392:     VecDuplicate(ts->vec_sol,&th->vec_lte_work[1]);
    393:   }

    395:   TSGetSNES(ts,&ts->snes);
    396:   return 0;
    397: }

    399: static PetscErrorCode TSSetFromOptions_Alpha(PetscOptionItems *PetscOptionsObject,TS ts)
    400: {
    401:   TS_Alpha       *th = (TS_Alpha*)ts->data;

    403:   PetscOptionsHead(PetscOptionsObject,"Generalized-Alpha ODE solver options");
    404:   {
    405:     PetscBool flg;
    406:     PetscReal radius = 1;
    407:     PetscOptionsReal("-ts_alpha_radius","Spectral radius (high-frequency dissipation)","TSAlpha2SetRadius",radius,&radius,&flg);
    408:     if (flg) TSAlpha2SetRadius(ts,radius);
    409:     PetscOptionsReal("-ts_alpha_alpha_m","Algorithmic parameter alpha_m","TSAlpha2SetParams",th->Alpha_m,&th->Alpha_m,NULL);
    410:     PetscOptionsReal("-ts_alpha_alpha_f","Algorithmic parameter alpha_f","TSAlpha2SetParams",th->Alpha_f,&th->Alpha_f,NULL);
    411:     PetscOptionsReal("-ts_alpha_gamma","Algorithmic parameter gamma","TSAlpha2SetParams",th->Gamma,&th->Gamma,NULL);
    412:     PetscOptionsReal("-ts_alpha_beta","Algorithmic parameter beta","TSAlpha2SetParams",th->Beta,&th->Beta,NULL);
    413:     TSAlpha2SetParams(ts,th->Alpha_m,th->Alpha_f,th->Gamma,th->Beta);
    414:   }
    415:   PetscOptionsTail();
    416:   return 0;
    417: }

    419: static PetscErrorCode TSView_Alpha(TS ts,PetscViewer viewer)
    420: {
    421:   TS_Alpha       *th = (TS_Alpha*)ts->data;
    422:   PetscBool      iascii;

    424:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
    425:   if (iascii) {
    426:     PetscViewerASCIIPrintf(viewer,"  Alpha_m=%g, Alpha_f=%g, Gamma=%g, Beta=%g\n",(double)th->Alpha_m,(double)th->Alpha_f,(double)th->Gamma,(double)th->Beta);
    427:   }
    428:   return 0;
    429: }

    431: static PetscErrorCode TSAlpha2SetRadius_Alpha(TS ts,PetscReal radius)
    432: {
    433:   PetscReal      alpha_m,alpha_f,gamma,beta;

    436:   alpha_m = (2-radius)/(1+radius);
    437:   alpha_f = 1/(1+radius);
    438:   gamma   = (PetscReal)0.5 + alpha_m - alpha_f;
    439:   beta    = (PetscReal)0.5 * (1 + alpha_m - alpha_f); beta *= beta;
    440:   TSAlpha2SetParams(ts,alpha_m,alpha_f,gamma,beta);
    441:   return 0;
    442: }

    444: static PetscErrorCode TSAlpha2SetParams_Alpha(TS ts,PetscReal alpha_m,PetscReal alpha_f,PetscReal gamma,PetscReal beta)
    445: {
    446:   TS_Alpha  *th = (TS_Alpha*)ts->data;
    447:   PetscReal tol = 100*PETSC_MACHINE_EPSILON;
    448:   PetscReal res = ((PetscReal)0.5 + alpha_m - alpha_f) - gamma;

    450:   th->Alpha_m = alpha_m;
    451:   th->Alpha_f = alpha_f;
    452:   th->Gamma   = gamma;
    453:   th->Beta    = beta;
    454:   th->order   = (PetscAbsReal(res) < tol) ? 2 : 1;
    455:   return 0;
    456: }

    458: static PetscErrorCode TSAlpha2GetParams_Alpha(TS ts,PetscReal *alpha_m,PetscReal *alpha_f,PetscReal *gamma,PetscReal *beta)
    459: {
    460:   TS_Alpha *th = (TS_Alpha*)ts->data;

    462:   if (alpha_m) *alpha_m = th->Alpha_m;
    463:   if (alpha_f) *alpha_f = th->Alpha_f;
    464:   if (gamma)   *gamma   = th->Gamma;
    465:   if (beta)    *beta    = th->Beta;
    466:   return 0;
    467: }

    469: /*MC
    470:       TSALPHA2 - ODE/DAE solver using the implicit Generalized-Alpha method
    471:                  for second-order systems

    473:   Level: beginner

    475:   References:
    476: . * - J. Chung, G.M.Hubert. "A Time Integration Algorithm for Structural
    477:   Dynamics with Improved Numerical Dissipation: The Generalized-alpha
    478:   Method" ASME Journal of Applied Mechanics, 60, 371:375, 1993.

    480: .seealso:  TS, TSCreate(), TSSetType(), TSAlpha2SetRadius(), TSAlpha2SetParams()
    481: M*/
    482: PETSC_EXTERN PetscErrorCode TSCreate_Alpha2(TS ts)
    483: {
    484:   TS_Alpha       *th;

    486:   ts->ops->reset          = TSReset_Alpha;
    487:   ts->ops->destroy        = TSDestroy_Alpha;
    488:   ts->ops->view           = TSView_Alpha;
    489:   ts->ops->setup          = TSSetUp_Alpha;
    490:   ts->ops->setfromoptions = TSSetFromOptions_Alpha;
    491:   ts->ops->step           = TSStep_Alpha;
    492:   ts->ops->evaluatewlte   = TSEvaluateWLTE_Alpha;
    493:   ts->ops->rollback       = TSRollBack_Alpha;
    494:   /*ts->ops->interpolate  = TSInterpolate_Alpha;*/
    495:   ts->ops->snesfunction   = SNESTSFormFunction_Alpha;
    496:   ts->ops->snesjacobian   = SNESTSFormJacobian_Alpha;
    497:   ts->default_adapt_type  = TSADAPTNONE;

    499:   ts->usessnes = PETSC_TRUE;

    501:   PetscNewLog(ts,&th);
    502:   ts->data = (void*)th;

    504:   th->Alpha_m = 0.5;
    505:   th->Alpha_f = 0.5;
    506:   th->Gamma   = 0.5;
    507:   th->Beta    = 0.25;
    508:   th->order   = 2;

    510:   PetscObjectComposeFunction((PetscObject)ts,"TSAlpha2SetRadius_C",TSAlpha2SetRadius_Alpha);
    511:   PetscObjectComposeFunction((PetscObject)ts,"TSAlpha2SetParams_C",TSAlpha2SetParams_Alpha);
    512:   PetscObjectComposeFunction((PetscObject)ts,"TSAlpha2GetParams_C",TSAlpha2GetParams_Alpha);
    513:   return 0;
    514: }

    516: /*@
    517:   TSAlpha2SetRadius - sets the desired spectral radius of the method
    518:                       (i.e. high-frequency numerical damping)

    520:   Logically Collective on TS

    522:   The algorithmic parameters \alpha_m and \alpha_f of the
    523:   generalized-\alpha method can be computed in terms of a specified
    524:   spectral radius \rho in [0,1] for infinite time step in order to
    525:   control high-frequency numerical damping:
    526:     \alpha_m = (2-\rho)/(1+\rho)
    527:     \alpha_f = 1/(1+\rho)

    529:   Input Parameters:
    530: +  ts - timestepping context
    531: -  radius - the desired spectral radius

    533:   Options Database:
    534: .  -ts_alpha_radius <radius> - set the desired spectral radius

    536:   Level: intermediate

    538: .seealso: TSAlpha2SetParams(), TSAlpha2GetParams()
    539: @*/
    540: PetscErrorCode TSAlpha2SetRadius(TS ts,PetscReal radius)
    541: {
    545:   PetscTryMethod(ts,"TSAlpha2SetRadius_C",(TS,PetscReal),(ts,radius));
    546:   return 0;
    547: }

    549: /*@
    550:   TSAlpha2SetParams - sets the algorithmic parameters for TSALPHA2

    552:   Logically Collective on TS

    554:   Second-order accuracy can be obtained so long as:
    555:     \gamma = 1/2 + alpha_m - alpha_f
    556:     \beta  = 1/4 (1 + alpha_m - alpha_f)^2

    558:   Unconditional stability requires:
    559:     \alpha_m >= \alpha_f >= 1/2

    561:   Input Parameters:
    562: + ts - timestepping context
    563: . \alpha_m - algorithmic parameter
    564: . \alpha_f - algorithmic parameter
    565: . \gamma   - algorithmic parameter
    566: - \beta    - algorithmic parameter

    568:   Options Database:
    569: + -ts_alpha_alpha_m <alpha_m> - set alpha_m
    570: . -ts_alpha_alpha_f <alpha_f> - set alpha_f
    571: . -ts_alpha_gamma   <gamma> - set gamma
    572: - -ts_alpha_beta    <beta> - set beta

    574:   Note:
    575:   Use of this function is normally only required to hack TSALPHA2 to
    576:   use a modified integration scheme. Users should call
    577:   TSAlpha2SetRadius() to set the desired spectral radius of the methods
    578:   (i.e. high-frequency damping) in order so select optimal values for
    579:   these parameters.

    581:   Level: advanced

    583: .seealso: TSAlpha2SetRadius(), TSAlpha2GetParams()
    584: @*/
    585: PetscErrorCode TSAlpha2SetParams(TS ts,PetscReal alpha_m,PetscReal alpha_f,PetscReal gamma,PetscReal beta)
    586: {
    592:   PetscTryMethod(ts,"TSAlpha2SetParams_C",(TS,PetscReal,PetscReal,PetscReal,PetscReal),(ts,alpha_m,alpha_f,gamma,beta));
    593:   return 0;
    594: }

    596: /*@
    597:   TSAlpha2GetParams - gets the algorithmic parameters for TSALPHA2

    599:   Not Collective

    601:   Input Parameter:
    602: . ts - timestepping context

    604:   Output Parameters:
    605: + \alpha_m - algorithmic parameter
    606: . \alpha_f - algorithmic parameter
    607: . \gamma   - algorithmic parameter
    608: - \beta    - algorithmic parameter

    610:   Note:
    611:   Use of this function is normally only required to hack TSALPHA2 to
    612:   use a modified integration scheme. Users should call
    613:   TSAlpha2SetRadius() to set the high-frequency damping (i.e. spectral
    614:   radius of the method) in order so select optimal values for these
    615:   parameters.

    617:   Level: advanced

    619: .seealso: TSAlpha2SetRadius(), TSAlpha2SetParams()
    620: @*/
    621: PetscErrorCode TSAlpha2GetParams(TS ts,PetscReal *alpha_m,PetscReal *alpha_f,PetscReal *gamma,PetscReal *beta)
    622: {
    628:   PetscUseMethod(ts,"TSAlpha2GetParams_C",(TS,PetscReal*,PetscReal*,PetscReal*,PetscReal*),(ts,alpha_m,alpha_f,gamma,beta));
    629:   return 0;
    630: }