Actual source code: arkimex.c

  1: /*
  2:   Code for timestepping with additive Runge-Kutta IMEX method or Diagonally Implicit Runge-Kutta methods.

  4:   Notes:
  5:   For ARK, the general system is written as

  7:   F(t,U,Udot) = G(t,U)

  9:   where F represents the stiff part of the physics and G represents the non-stiff part.

 11: */
 12: #include <petsc/private/tsimpl.h>
 13: #include <petscdm.h>

 15: static TSARKIMEXType  TSARKIMEXDefault = TSARKIMEX3;
 16: static TSDIRKType     TSDIRKDefault    = TSDIRKES213SAL;
 17: static PetscBool      TSARKIMEXRegisterAllCalled;
 18: static PetscBool      TSARKIMEXPackageInitialized;
 19: static PetscErrorCode TSExtrapolate_ARKIMEX(TS, PetscReal, Vec);

 21: typedef struct _ARKTableau *ARKTableau;
 22: struct _ARKTableau {
 23:   char      *name;
 24:   PetscBool  additive;             /* If False, it is a DIRK method */
 25:   PetscInt   order;                /* Classical approximation order of the method */
 26:   PetscInt   s;                    /* Number of stages */
 27:   PetscBool  stiffly_accurate;     /* The implicit part is stiffly accurate */
 28:   PetscBool  FSAL_implicit;        /* The implicit part is FSAL */
 29:   PetscBool  explicit_first_stage; /* The implicit part has an explicit first stage */
 30:   PetscInt   pinterp;              /* Interpolation order */
 31:   PetscReal *At, *bt, *ct;         /* Stiff tableau */
 32:   PetscReal *A, *b, *c;            /* Non-stiff tableau */
 33:   PetscReal *bembedt, *bembed;     /* Embedded formula of order one less (order-1) */
 34:   PetscReal *binterpt, *binterp;   /* Dense output formula */
 35:   PetscReal  ccfl;                 /* Placeholder for CFL coefficient relative to forward Euler */
 36: };
 37: typedef struct _ARKTableauLink *ARKTableauLink;
 38: struct _ARKTableauLink {
 39:   struct _ARKTableau tab;
 40:   ARKTableauLink     next;
 41: };
 42: static ARKTableauLink ARKTableauList;

 44: typedef struct {
 45:   ARKTableau   tableau;
 46:   Vec         *Y;            /* States computed during the step */
 47:   Vec         *YdotI;        /* Time derivatives for the stiff part */
 48:   Vec         *YdotRHS;      /* Function evaluations for the non-stiff part */
 49:   Vec         *Y_prev;       /* States computed during the previous time step */
 50:   Vec         *YdotI_prev;   /* Time derivatives for the stiff part for the previous time step*/
 51:   Vec         *YdotRHS_prev; /* Function evaluations for the non-stiff part for the previous time step*/
 52:   Vec          Ydot0;        /* Holds the slope from the previous step in FSAL case */
 53:   Vec          Ydot;         /* Work vector holding Ydot during residual evaluation */
 54:   Vec          Z;            /* Ydot = shift(Y-Z) */
 55:   PetscScalar *work;         /* Scalar work */
 56:   PetscReal    scoeff;       /* shift = scoeff/dt */
 57:   PetscReal    stage_time;
 58:   PetscBool    imex;
 59:   PetscBool    extrapolate; /* Extrapolate initial guess from previous time-step stage values */
 60:   TSStepStatus status;

 62:   /* context for sensitivity analysis */
 63:   Vec *VecsDeltaLam;   /* Increment of the adjoint sensitivity w.r.t IC at stage */
 64:   Vec *VecsSensiTemp;  /* Vectors to be multiplied with Jacobian transpose */
 65:   Vec *VecsSensiPTemp; /* Temporary Vectors to store JacobianP-transpose-vector product */
 66: } TS_ARKIMEX;

 68: /*MC
 69:      TSARKIMEXARS122 - Second order ARK IMEX scheme, {cite}`ascher_1997`

 71:      This method has one explicit stage and one implicit stage.

 73:      Options Database Key:
 74: .      -ts_arkimex_type ars122 - set arkimex type to ars122

 76:      Level: advanced

 78: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEXSetType()`
 79: M*/

 81: /*MC
 82:      TSARKIMEXA2 - Second order ARK IMEX scheme with A-stable implicit part.

 84:      This method has an explicit stage and one implicit stage, and has an A-stable implicit scheme. This method was provided by Emil Constantinescu.

 86:      Options Database Key:
 87: .      -ts_arkimex_type a2 - set arkimex type to a2

 89:      Level: advanced

 91: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEXSetType()`
 92: M*/

 94: /*MC
 95:      TSARKIMEXL2 - Second order ARK IMEX scheme with L-stable implicit part, {cite}`pareschi_2005`

 97:      This method has two implicit stages, and L-stable implicit scheme.

 99:      Options Database Key:
100: .      -ts_arkimex_type l2 - set arkimex type to l2

102:      Level: advanced

104: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEXSetType()`
105: M*/

107: /*MC
108:      TSARKIMEX1BEE - First order backward Euler represented as an ARK IMEX scheme with extrapolation as error estimator. This is a 3-stage method.

110:      This method is aimed at starting the integration of implicit DAEs when explicit first-stage ARK methods are used.

112:      Options Database Key:
113: .      -ts_arkimex_type 1bee - set arkimex type to 1bee

115:      Level: advanced

117: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEXSetType()`
118: M*/

120: /*MC
121:      TSARKIMEX2C - Second order ARK IMEX scheme with L-stable implicit part.

123:      This method has one explicit stage and two implicit stages. The implicit part is the same as in TSARKIMEX2D and TSARKIMEX2E, but the explicit part has a larger stability region on the negative real axis. This method was provided by Emil Constantinescu.

125:      Options Database Key:
126: .      -ts_arkimex_type 2c - set arkimex type to 2c

128:      Level: advanced

130: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEXSetType()`
131: M*/

133: /*MC
134:      TSARKIMEX2D - Second order ARK IMEX scheme with L-stable implicit part.

136:      This method has one explicit stage and two implicit stages. The stability function is independent of the explicit part in the infinity limit of the implicit component. This method was provided by Emil Constantinescu.

138:      Options Database Key:
139: .      -ts_arkimex_type 2d - set arkimex type to 2d

141:      Level: advanced

143: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEXSetType()`
144: M*/

146: /*MC
147:      TSARKIMEX2E - Second order ARK IMEX scheme with L-stable implicit part.

149:      This method has one explicit stage and two implicit stages. It is is an optimal method developed by Emil Constantinescu.

151:      Options Database Key:
152: .      -ts_arkimex_type 2e - set arkimex type to 2e

154:     Level: advanced

156: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEXSetType()`
157: M*/

159: /*MC
160:      TSARKIMEXPRSSP2 - Second order SSP ARK IMEX scheme, {cite}`pareschi_2005`

162:      This method has three implicit stages.

164:      This method is referred to as SSP2-(3,3,2) in <https://arxiv.org/abs/1110.4375>

166:      Options Database Key:
167: .      -ts_arkimex_type prssp2 - set arkimex type to prssp2

169:      Level: advanced

171: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEXSetType()`
172: M*/

174: /*MC
175:      TSARKIMEX3 - Third order ARK IMEX scheme with L-stable implicit part, {cite}`kennedy_2003`

177:      This method has one explicit stage and three implicit stages.

179:      Options Database Key:
180: .      -ts_arkimex_type 3 - set arkimex type to 3

182:      Level: advanced

184: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEXSetType()`
185: M*/

187: /*MC
188:      TSARKIMEXARS443 - Third order ARK IMEX scheme, {cite}`ascher_1997`

190:      This method has one explicit stage and four implicit stages.

192:      Options Database Key:
193: .      -ts_arkimex_type ars443 - set arkimex type to ars443

195:      Level: advanced

197:      Notes:
198:      This method is referred to as ARS(4,4,3) in <https://arxiv.org/abs/1110.4375>

200: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEXSetType()`
201: M*/

203: /*MC
204:      TSARKIMEXBPR3 - Third order ARK IMEX scheme. Referred to as ARK3 in <https://arxiv.org/abs/1110.4375>

206:      This method has one explicit stage and four implicit stages.

208:      Options Database Key:
209: .      -ts_arkimex_type bpr3 - set arkimex type to bpr3

211:      Level: advanced

213: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEXSetType()`
214: M*/

216: /*MC
217:      TSARKIMEX4 - Fourth order ARK IMEX scheme with L-stable implicit part, {cite}`kennedy_2003`.

219:      This method has one explicit stage and four implicit stages.

221:      Options Database Key:
222: .      -ts_arkimex_type 4 - set arkimex type to4

224:      Level: advanced

226: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEXSetType()`
227: M*/

229: /*MC
230:      TSARKIMEX5 - Fifth order ARK IMEX scheme with L-stable implicit part, {cite}`kennedy_2003`.

232:      This method has one explicit stage and five implicit stages.

234:      Options Database Key:
235: .      -ts_arkimex_type 5 - set arkimex type to 5

237:      Level: advanced

239: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEXSetType()`
240: M*/

242: /*MC
243:      TSDIRKS212 - Second order DIRK scheme.

245:      This method has two implicit stages with an embedded method of other 1.
246:      See `TSDIRK` for additional details.

248:      Options Database Key:
249: .      -ts_dirk_type s212 - select this method.

251:      Level: advanced

253:      Note:
254:      This is the default DIRK scheme in SUNDIALS.

256: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
257: M*/

259: /*MC
260:      TSDIRKES122SAL - First order DIRK scheme <https://arxiv.org/abs/1803.01613>

262:      Uses backward Euler as advancing method and trapezoidal rule as embedded method. See `TSDIRK` for additional details.

264:      Options Database Key:
265: .      -ts_dirk_type es122sal - select this method.

267:      Level: advanced

269: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
270: M*/

272: /*MC
273:      TSDIRKES213SAL - Second order DIRK scheme {cite}`kennedy2019diagonally`. Also known as TR-BDF2, see{cite}`hosea1996analysis`

275:      See `TSDIRK` for additional details.

277:      Options Database Key:
278: .      -ts_dirk_type es213sal - select this method.

280:      Level: advanced

282:      Note:
283:      This is the default DIRK scheme used in PETSc.

285: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
286: M*/

288: /*MC
289:      TSDIRKES324SAL - Third order DIRK scheme, {cite}`kennedy2019diagonally`

291:      See `TSDIRK` for additional details.

293:      Options Database Key:
294: .      -ts_dirk_type es324sal - select this method.

296:      Level: advanced

298: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
299: M*/

301: /*MC
302:      TSDIRKES325SAL - Third order DIRK scheme {cite}`kennedy2019diagonally`.

304:      See `TSDIRK` for additional details.

306:      Options Database Key:
307: .      -ts_dirk_type es325sal - select this method.

309:      Level: advanced

311: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
312: M*/

314: /*MC
315:      TSDIRK657A - Sixth order DIRK scheme <https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs>

317:      See `TSDIRK` for additional details.

319:      Options Database Key:
320: .      -ts_dirk_type 657a - select this method.

322:      Level: advanced

324: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
325: M*/

327: /*MC
328:      TSDIRKES648SA - Sixth order DIRK scheme <https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs>

330:      See `TSDIRK` for additional details.

332:      Options Database Key:
333: .      -ts_dirk_type es648sa - select this method.

335:      Level: advanced

337: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
338: M*/

340: /*MC
341:      TSDIRK658A - Sixth order DIRK scheme  <https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs>

343:      See `TSDIRK` for additional details.

345:      Options Database Key:
346: .      -ts_dirk_type 658a - select this method.

348:      Level: advanced

350: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
351: M*/

353: /*MC
354:      TSDIRKS659A - Sixth order DIRK scheme  <https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs>

356:      See `TSDIRK` for additional details.

358:      Options Database Key:
359: .      -ts_dirk_type s659a - select this method.

361:      Level: advanced

363: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
364: M*/

366: /*MC
367:      TSDIRK7510SAL - Seventh order DIRK scheme <https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs>

369:      See `TSDIRK` for additional details.

371:      Options Database Key:
372: .      -ts_dirk_type 7510sal - select this method.

374:      Level: advanced

376: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
377: M*/

379: /*MC
380:      TSDIRKES7510SA - Seventh order DIRK scheme <https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs>

382:      See `TSDIRK` for additional details.

384:      Options Database Key:
385: .      -ts_dirk_type es7510sa - select this method.

387:      Level: advanced

389: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
390: M*/

392: /*MC
393:      TSDIRK759A - Seventh order DIRK scheme <https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs>

395:      See `TSDIRK` for additional details.

397:      Options Database Key:
398: .      -ts_dirk_type 759a - select this method.

400:      Level: advanced

402: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
403: M*/

405: /*MC
406:      TSDIRKS7511SAL - Seventh order DIRK scheme <https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs>

408:      See `TSDIRK` for additional details.

410:      Options Database Key:
411: .      -ts_dirk_type s7511sal - select this method.

413:      Level: advanced

415: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
416: M*/

418: /*MC
419:      TSDIRK8614A - Eighth order DIRK scheme <https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs>

421:      See `TSDIRK` for additional details.

423:      Options Database Key:
424: .      -ts_dirk_type 8614a - select this method.

426:      Level: advanced

428: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
429: M*/

431: /*MC
432:      TSDIRK8616SAL - Eighth order DIRK scheme <https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs>

434:      See `TSDIRK` for additional details.

436:      Options Database Key:
437: .      -ts_dirk_type 8616sal - select this method.

439:      Level: advanced

441: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
442: M*/

444: /*MC
445:      TSDIRKES8516SAL - Eighth order DIRK scheme <https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs>

447:      See `TSDIRK` for additional details.

449:      Options Database Key:
450: .      -ts_dirk_type es8516sal - select this method.

452:      Level: advanced

454: .seealso: [](ch_ts), `TSDIRK`, `TSDIRKType`, `TSDIRKSetType()`
455: M*/

457: static PetscErrorCode TSHasRHSFunction(TS ts, PetscBool *has)
458: {
459:   TSRHSFunction func;

461:   PetscFunctionBegin;
462:   *has = PETSC_FALSE;
463:   PetscCall(DMTSGetRHSFunction(ts->dm, &func, NULL));
464:   if (func) *has = PETSC_TRUE;
465:   PetscFunctionReturn(PETSC_SUCCESS);
466: }

468: /*@C
469:   TSARKIMEXRegisterAll - Registers all of the additive Runge-Kutta implicit-explicit methods in `TSARKIMEX`

471:   Not Collective, but should be called by all processes which will need the schemes to be registered

473:   Level: advanced

475: .seealso: [](ch_ts), `TS`, `TSARKIMEX`, `TSARKIMEXRegisterDestroy()`
476: @*/
477: PetscErrorCode TSARKIMEXRegisterAll(void)
478: {
479:   PetscFunctionBegin;
480:   if (TSARKIMEXRegisterAllCalled) PetscFunctionReturn(PETSC_SUCCESS);
481:   TSARKIMEXRegisterAllCalled = PETSC_TRUE;

483: #define RC  PetscRealConstant
484: #define s2  RC(1.414213562373095048802)  /* PetscSqrtReal((PetscReal)2.0) */
485: #define us2 RC(0.2928932188134524755992) /* 1.0-1.0/PetscSqrtReal((PetscReal)2.0); */

487:   /* Diagonally implicit methods */
488:   {
489:     /* DIRK212, default of SUNDIALS */
490:     const PetscReal A[2][2] = {
491:       {RC(1.0),  RC(0.0)},
492:       {RC(-1.0), RC(1.0)}
493:     };
494:     const PetscReal b[2]      = {RC(0.5), RC(0.5)};
495:     const PetscReal bembed[2] = {RC(1.0), RC(0.0)};
496:     PetscCall(TSDIRKRegister(TSDIRKS212, 2, 2, &A[0][0], b, NULL, bembed, 1, b));
497:   }

499:   {
500:     /* ESDIRK12 from https://arxiv.org/pdf/1803.01613.pdf */
501:     const PetscReal A[2][2] = {
502:       {RC(0.0), RC(0.0)},
503:       {RC(0.0), RC(1.0)}
504:     };
505:     const PetscReal b[2]      = {RC(0.0), RC(1.0)};
506:     const PetscReal bembed[2] = {RC(0.5), RC(0.5)};
507:     PetscCall(TSDIRKRegister(TSDIRKES122SAL, 1, 2, &A[0][0], b, NULL, bembed, 1, b));
508:   }

510:   {
511:     /* ESDIRK213L[2]SA from KC16.
512:        TR-BDF2 from Hosea and Shampine
513:        ESDIRK23 in https://arxiv.org/pdf/1803.01613.pdf */
514:     const PetscReal A[3][3] = {
515:       {RC(0.0),      RC(0.0),      RC(0.0)},
516:       {us2,          us2,          RC(0.0)},
517:       {s2 / RC(4.0), s2 / RC(4.0), us2    },
518:     };
519:     const PetscReal b[3]      = {s2 / RC(4.0), s2 / RC(4.0), us2};
520:     const PetscReal bembed[3] = {(RC(1.0) - s2 / RC(4.0)) / RC(3.0), (RC(3.0) * s2 / RC(4.0) + RC(1.0)) / RC(3.0), us2 / RC(3.0)};
521:     PetscCall(TSDIRKRegister(TSDIRKES213SAL, 2, 3, &A[0][0], b, NULL, bembed, 1, b));
522:   }

524:   {
525: #define g   RC(0.43586652150845899941601945)
526: #define g2  PetscSqr(g)
527: #define g3  g *g2
528: #define g4  PetscSqr(g2)
529: #define g5  g *g4
530: #define c3  RC(1.0)
531: #define a32 c3 *(c3 - RC(2.0) * g) / (RC(4.0) * g)
532: #define b2  (-RC(2.0) + RC(3.0) * c3 + RC(6.0) * g * (RC(1.0) - c3)) / (RC(12.0) * g * (c3 - RC(2.0) * g))
533: #define b3  (RC(1.0) - RC(6.0) * g + RC(6.0) * g2) / (RC(3.0) * c3 * (c3 - RC(2.0) * g))
534: #if 0
535: /* This is for c3 = 3/5 */
536:   #define bh2 \
537:     c3 * (-RC(1.0) + RC(6.0) * g - RC(23.0) * g3 + RC(12.0) * g4 - RC(6.0) * g5) / (RC(4.0) * (RC(2.0) * g - c3) * (RC(1.0) - RC(6.0) * g + RC(6.0) * g2)) + (RC(3.0) - RC(27.0) * g + RC(68.0) * g2 - RC(55.0) * g3 + RC(21.0) * g4 - RC(6.0) * g5) / (RC(2.0) * (RC(2.0) * g - c3) * (RC(1.0) - RC(6.0) * g + RC(6.0) * g2))
538:   #define bh3 -g * (-RC(2.0) + RC(21.0) * g - RC(68.0) * g2 + RC(79.0) * g3 - RC(33.0) * g4 + RC(12.0) * g5) / (RC(2.0) * (RC(2.0) * g - c3) * (RC(1.0) - RC(6.0) * g + RC(6.0) * g2))
539:   #define bh4 -RC(3.0) * g2 * (-RC(1.0) + RC(4.0) * g - RC(2.0) * g2 + g3) / (RC(1.0) - RC(6.0) * g + RC(6.0) * g2)
540: #else
541:   /* This is for c3 = 1.0 */
542:   #define bh2 a32
543:   #define bh3 g
544:   #define bh4 RC(0.0)
545: #endif
546:     /* ESDIRK3(2I)4L[2]SA from KC16 with c3 = 1.0 */
547:     /* Given by Kvaerno https://link.springer.com/article/10.1023/b:bitn.0000046811.70614.38 */
548:     const PetscReal A[4][4] = {
549:       {RC(0.0),               RC(0.0), RC(0.0), RC(0.0)},
550:       {g,                     g,       RC(0.0), RC(0.0)},
551:       {c3 - a32 - g,          a32,     g,       RC(0.0)},
552:       {RC(1.0) - b2 - b3 - g, b2,      b3,      g      },
553:     };
554:     const PetscReal b[4]      = {RC(1.0) - b2 - b3 - g, b2, b3, g};
555:     const PetscReal bembed[4] = {RC(1.0) - bh2 - bh3 - bh4, bh2, bh3, bh4};
556:     PetscCall(TSDIRKRegister(TSDIRKES324SAL, 3, 4, &A[0][0], b, NULL, bembed, 1, b));
557: #undef g
558: #undef g2
559: #undef g3
560: #undef c3
561: #undef a32
562: #undef b2
563: #undef b3
564: #undef bh2
565: #undef bh3
566: #undef bh4
567:   }

569:   {
570:     /* ESDIRK3(2I)5L[2]SA from KC16 */
571:     const PetscReal A[5][5] = {
572:       {RC(0.0),                  RC(0.0),                  RC(0.0),                 RC(0.0),                   RC(0.0)           },
573:       {RC(9.0) / RC(40.0),       RC(9.0) / RC(40.0),       RC(0.0),                 RC(0.0),                   RC(0.0)           },
574:       {RC(19.0) / RC(72.0),      RC(14.0) / RC(45.0),      RC(9.0) / RC(40.0),      RC(0.0),                   RC(0.0)           },
575:       {RC(3337.0) / RC(11520.0), RC(233.0) / RC(720.0),    RC(207.0) / RC(1280.0),  RC(9.0) / RC(40.0),        RC(0.0)           },
576:       {RC(7415.0) / RC(34776.0), RC(9920.0) / RC(30429.0), RC(4845.0) / RC(9016.0), -RC(5827.0) / RC(19320.0), RC(9.0) / RC(40.0)},
577:     };
578:     const PetscReal b[5]      = {RC(7415.0) / RC(34776.0), RC(9920.0) / RC(30429.0), RC(4845.0) / RC(9016.0), -RC(5827.0) / RC(19320.0), RC(9.0) / RC(40.0)};
579:     const PetscReal bembed[5] = {RC(23705.0) / RC(104328.0), RC(29720.0) / RC(91287.0), RC(4225.0) / RC(9016.0), -RC(69304987.0) / RC(337732920.0), RC(42843.0) / RC(233080.0)};
580:     PetscCall(TSDIRKRegister(TSDIRKES325SAL, 3, 5, &A[0][0], b, NULL, bembed, 1, b));
581:   }

583:   {
584:     // DIRK(6,6)[1]A[(7,5)A] from https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs
585:     const PetscReal A[7][7] = {
586:       {RC(0.303487844706747),    RC(0.0),                RC(0.0),                   RC(0.0),                   RC(0.0),                 RC(0.0),                RC(0.0)              },
587:       {RC(-0.279756492709814),   RC(0.500032236020747),  RC(0.0),                   RC(0.0),                   RC(0.0),                 RC(0.0),                RC(0.0)              },
588:       {RC(0.280583215743895),    RC(-0.438560061586751), RC(0.217250734515736),     RC(0.0),                   RC(0.0),                 RC(0.0),                RC(0.0)              },
589:       {RC(-0.0677678738539846),  RC(0.984312781232293),  RC(-0.266720192540149),    RC(0.2476680834526),       RC(0.0),                 RC(0.0),                RC(0.0)              },
590:       {RC(0.125671616147993),    RC(-0.995401751002415), RC(0.761333109549059),     RC(-0.210281837202208),    RC(0.866743712636936),   RC(0.0),                RC(0.0)              },
591:       {RC(-0.368056238801488),   RC(-0.999928082701516), RC(0.534734253232519),     RC(-0.174856916279082),    RC(0.615007160285509),   RC(0.696549912132029),  RC(0.0)              },
592:       {RC(-0.00570546839653984), RC(-0.113110431835656), RC(-0.000965563207671587), RC(-0.000130490084629567), RC(0.00111737736895673), RC(-0.279385587378871), RC(0.618455906845342)}
593:     };
594:     const PetscReal b[7]      = {RC(0.257561510484877), RC(0.234281287047716), RC(0.126658904241469), RC(0.252363215441784), RC(0.396701083526306), RC(-0.267566000742152), RC(0.0)};
595:     const PetscReal bembed[7] = {RC(0.257561510484945), RC(0.387312822934391), RC(0.126658904241468), RC(0.252363215441784), RC(0.396701083526306), RC(-0.267566000742225), RC(-0.153031535886669)};
596:     PetscCall(TSDIRKRegister(TSDIRK657A, 6, 7, &A[0][0], b, NULL, bembed, 1, b));
597:   }
598:   {
599:     // ESDIRK(8,6)[2]SA[(8,4)] from https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs
600:     const PetscReal A[8][8] = {
601:       {RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),               RC(0.0),                RC(0.0),                RC(0.0),               RC(0.0)              },
602:       {RC(0.333222149217725),  RC(0.333222149217725),   RC(0.0),                 RC(0.0),               RC(0.0),                RC(0.0),                RC(0.0),               RC(0.0)              },
603:       {RC(0.0639743773182214), RC(-0.0830330224410214), RC(0.333222149217725),   RC(0.0),               RC(0.0),                RC(0.0),                RC(0.0),               RC(0.0)              },
604:       {RC(-0.728522201369326), RC(-0.210414479522485),  RC(0.532519916559342),   RC(0.333222149217725), RC(0.0),                RC(0.0),                RC(0.0),               RC(0.0)              },
605:       {RC(-0.175135269272067), RC(0.666675582067552),   RC(-0.304400907370867),  RC(0.656797712445756), RC(0.333222149217725),  RC(0.0),                RC(0.0),               RC(0.0)              },
606:       {RC(0.222695802705462),  RC(-0.0948971794681061), RC(-0.0234336346686545), RC(-0.45385925012042), RC(0.0283910313826958), RC(0.333222149217725),  RC(0.0),               RC(0.0)              },
607:       {RC(-0.132534078051299), RC(0.702597935004879),   RC(-0.433316453128078),  RC(0.893717488547587), RC(0.057381454791406),  RC(-0.207798411552402), RC(0.333222149217725), RC(0.0)              },
608:       {RC(0.0802253121418085), RC(0.281196044671022),   RC(0.406758926172157),   RC(-0.01945708512416), RC(-0.41785600088526),  RC(0.0545342658870322), RC(0.281376387919675), RC(0.333222149217725)}
609:     };
610:     const PetscReal b[8]      = {RC(0.0802253121418085), RC(0.281196044671022), RC(0.406758926172157), RC(-0.01945708512416), RC(-0.41785600088526), RC(0.0545342658870322), RC(0.281376387919675), RC(0.333222149217725)};
611:     const PetscReal bembed[8] = {RC(0.0), RC(0.292331064554014), RC(0.409676102283681), RC(-0.002094718084982), RC(-0.282771520835975), RC(0.113862336644901), RC(0.181973572260693), RC(0.287023163177669)};
612:     PetscCall(TSDIRKRegister(TSDIRKES648SA, 6, 8, &A[0][0], b, NULL, bembed, 1, b));
613:   }
614:   {
615:     // DIRK(8,6)[1]SAL[(8,5)A] from https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs
616:     const PetscReal A[8][8] = {
617:       {RC(0.477264457385826),    RC(0.0),                RC(0.0),                   RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                 RC(0.0)              },
618:       {RC(-0.197052588415002),   RC(0.476363428459584),  RC(0.0),                   RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                 RC(0.0)              },
619:       {RC(-0.0347674430372966),  RC(0.633051807335483),  RC(0.193634310075028),     RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                 RC(0.0)              },
620:       {RC(0.0967797668578702),   RC(-0.193533526466535), RC(-0.000207622945800473), RC(0.159572204849431),   RC(0.0),                RC(0.0),                RC(0.0),                 RC(0.0)              },
621:       {RC(0.162527231819875),    RC(-0.249672513547382), RC(-0.0459079972041795),   RC(0.36579476400859),    RC(0.255752838307699),  RC(0.0),                RC(0.0),                 RC(0.0)              },
622:       {RC(-0.00707603197171262), RC(0.846299854860295),  RC(0.344020016925018),     RC(-0.0720926054548865), RC(-0.215492331980875), RC(0.104341097622161),  RC(0.0),                 RC(0.0)              },
623:       {RC(0.00176857935179744),  RC(0.0779960013127515), RC(0.303333277564557),     RC(0.213160806732836),   RC(0.351769320319038),  RC(-0.381545894386538), RC(0.433517909105558),   RC(0.0)              },
624:       {RC(0.0),                  RC(0.22732353410559),   RC(0.308415837980118),     RC(0.157263419573007),   RC(0.243551137152275),  RC(-0.120953626732831), RC(-0.0802678473399899), RC(0.264667545261832)}
625:     };
626:     const PetscReal b[8]      = {RC(0.0), RC(0.22732353410559), RC(0.308415837980118), RC(0.157263419573007), RC(0.243551137152275), RC(-0.120953626732831), RC(-0.0802678473399899), RC(0.264667545261832)};
627:     const PetscReal bembed[8] = {RC(0.0), RC(0.22732353410559), RC(0.308415837980118), RC(0.157263419573007), RC(0.243551137152275), RC(-0.103483943222765), RC(-0.0103721771642262), RC(0.177302191576001)};
628:     PetscCall(TSDIRKRegister(TSDIRK658A, 6, 8, &A[0][0], b, NULL, bembed, 1, b));
629:   }
630:   {
631:     // SDIRK(9,6)[1]SAL[(9,5)A] from https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs
632:     const PetscReal A[9][9] = {
633:       {RC(0.218127781944908),   RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)              },
634:       {RC(-0.0903514856119419), RC(0.218127781944908),  RC(0.0),                 RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)              },
635:       {RC(0.172952039138937),   RC(-0.35365501036282),  RC(0.218127781944908),   RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)              },
636:       {RC(0.511999875919193),   RC(0.0289640332201925), RC(-0.0144030945657094), RC(0.218127781944908),   RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)              },
637:       {RC(0.00465303495506782), RC(-0.075635818766597), RC(0.217273030786712),   RC(-0.0206519428725472), RC(0.218127781944908),  RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)              },
638:       {RC(0.896145501762472),   RC(0.139267327700498),  RC(-0.186920979752805),  RC(0.0672971012371723),  RC(-0.350891963442176), RC(0.218127781944908),  RC(0.0),                RC(0.0),                RC(0.0)              },
639:       {RC(0.552959701885751),   RC(-0.439360579793662), RC(0.333704002325091),   RC(-0.0339426520778416), RC(-0.151947445912595), RC(0.0213825661026943), RC(0.218127781944908),  RC(0.0),                RC(0.0)              },
640:       {RC(0.631360374036476),   RC(0.724733619641466),  RC(-0.432170625425258),  RC(0.598611382182477),   RC(-0.709087197034345), RC(-0.483986685696934), RC(0.378391562905131),  RC(0.218127781944908),  RC(0.0)              },
641:       {RC(0.0),                 RC(-0.15504452530869),  RC(0.194518478660789),   RC(0.63515640279203),    RC(0.81172278664173),   RC(0.110736108691585),  RC(-0.495304692414479), RC(-0.319912341007872), RC(0.218127781944908)}
642:     };
643:     const PetscReal b[9]      = {RC(0.0), RC(-0.15504452530869), RC(0.194518478660789), RC(0.63515640279203), RC(0.81172278664173), RC(0.110736108691585), RC(-0.495304692414479), RC(-0.319912341007872), RC(0.218127781944908)};
644:     const PetscReal bembed[9] = {RC(3.62671059311602e-16), RC(0.0736615558278942), RC(0.103527397262229), RC(1.00247481935499), RC(0.361377289250057), RC(-0.785425929961365), RC(-0.0170499047960784), RC(0.296321252214769), RC(-0.0348864791524953)};
645:     PetscCall(TSDIRKRegister(TSDIRKS659A, 6, 9, &A[0][0], b, NULL, bembed, 1, b));
646:   }
647:   {
648:     // DIRK(10,7)[1]SAL[(10,5)A] from https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs
649:     const PetscReal A[10][10] = {
650:       {RC(0.233704632125264),   RC(0.0),                RC(0.0),                  RC(0.0),                  RC(0.0),                   RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),               RC(0.0)              },
651:       {RC(-0.0739324813149407), RC(0.200056838146104),  RC(0.0),                  RC(0.0),                  RC(0.0),                   RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),               RC(0.0)              },
652:       {RC(0.0943790344044812),  RC(0.264056067701605),  RC(0.133245202456465),    RC(0.0),                  RC(0.0),                   RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),               RC(0.0)              },
653:       {RC(0.269084810601201),   RC(-0.503479002548384), RC(-0.00486736469695022), RC(0.251518716213569),    RC(0.0),                   RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),               RC(0.0)              },
654:       {RC(0.145665801918423),   RC(0.204983170463176),  RC(0.407154634069484),    RC(-0.0121039135200389),  RC(0.190243622486334),     RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),               RC(0.0)              },
655:       {RC(0.985450198547345),   RC(0.806942652811456),  RC(-0.808130934167263),   RC(-0.669035819439391),   RC(0.0269384406756128),    RC(0.462144080607327),    RC(0.0),                RC(0.0),                 RC(0.0),               RC(0.0)              },
656:       {RC(0.163902957809563),   RC(0.228315094960095),  RC(0.0745971021260249),   RC(0.000509793400156559), RC(0.0166533681378294),    RC(-0.0229383879045797),  RC(0.103505486637336),  RC(0.0),                 RC(0.0),               RC(0.0)              },
657:       {RC(-0.162694156858437),  RC(0.0453478837428434), RC(0.997443481211424),    RC(0.200251514941093),    RC(-0.000161755458839048), RC(-0.0848134335980281),  RC(-0.36438666566666),  RC(0.158604420136055),   RC(0.0),               RC(0.0)              },
658:       {RC(0.200733156477425),   RC(0.239686443444433),  RC(0.303837014418929),    RC(-0.0534390596279896),  RC(0.0314067599640569),    RC(-0.00764032790448536), RC(0.0609191260198661), RC(-0.0736319201590642), RC(0.204602530607021), RC(0.0)              },
659:       {RC(0.0),                 RC(0.235563761744267),  RC(0.658651488684319),    RC(0.0308877804992098),   RC(-0.906514945595336),    RC(-0.0248488551739974),  RC(-0.309967582365257), RC(0.191663316925525),   RC(0.923933712199542), RC(0.200631323081727)}
660:     };
661:     const PetscReal b[10] = {RC(0.0), RC(0.235563761744267), RC(0.658651488684319), RC(0.0308877804992098), RC(-0.906514945595336), RC(-0.0248488551739974), RC(-0.309967582365257), RC(0.191663316925525), RC(0.923933712199542), RC(0.200631323081727)};
662:     const PetscReal bembed[10] =
663:       {RC(0.0), RC(0.222929376486581), RC(0.950668440138169), RC(0.0342694607044032), RC(0.362875840545746), RC(0.223572979288581), RC(-0.764361723526727), RC(0.563476909230026), RC(-0.690896961894185), RC(0.0974656790270323)};
664:     PetscCall(TSDIRKRegister(TSDIRK7510SAL, 7, 10, &A[0][0], b, NULL, bembed, 1, b));
665:   }
666:   {
667:     // ESDIRK(10,7)[2]SA[(10,5)] from https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs
668:     const PetscReal A[10][10] = {
669:       {RC(0.0),                 RC(0.0),                 RC(0.0),                 RC(0.0),                  RC(0.0),                  RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),               RC(0.0)              },
670:       {RC(0.210055790203419),   RC(0.210055790203419),   RC(0.0),                 RC(0.0),                  RC(0.0),                  RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),               RC(0.0)              },
671:       {RC(0.255781739921086),   RC(0.239850916980976),   RC(0.210055790203419),   RC(0.0),                  RC(0.0),                  RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),               RC(0.0)              },
672:       {RC(0.286789624880437),   RC(0.230494748834778),   RC(0.263925149885491),   RC(0.210055790203419),    RC(0.0),                  RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),               RC(0.0)              },
673:       {RC(-0.0219118128774335), RC(0.897684380345907),   RC(-0.657954605498907),  RC(0.124962304722633),    RC(0.210055790203419),    RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),               RC(0.0)              },
674:       {RC(-0.065614879584776),  RC(-0.0565630711859497), RC(0.0254881105065311),  RC(-0.00368981790650006), RC(-0.0115178258446329),  RC(0.210055790203419),    RC(0.0),                RC(0.0),                 RC(0.0),               RC(0.0)              },
675:       {RC(0.399860851232098),   RC(0.915588469718705),   RC(-0.0758429094934412), RC(-0.263369154872759),   RC(0.719687583564526),    RC(-0.787410407015369),   RC(0.210055790203419),  RC(0.0),                 RC(0.0),               RC(0.0)              },
676:       {RC(0.51693616104628),    RC(1.00000540846973),    RC(-0.0485110663289207), RC(-0.315208041581942),   RC(0.749742806451587),    RC(-0.990975090921248),   RC(0.0159279583407308), RC(0.210055790203419),   RC(0.0),               RC(0.0)              },
677:       {RC(-0.0303062129076945), RC(-0.297035174659034),  RC(0.184724697462164),   RC(-0.0351876079516183),  RC(-0.00324668230690761), RC(0.216151004053531),    RC(-0.126676252098317), RC(0.114040254365262),   RC(0.210055790203419), RC(0.0)              },
678:       {RC(0.0705997961586714),  RC(-0.0281516061956374), RC(0.314600470734633),   RC(-0.0907057557963371),  RC(0.168078953957742),    RC(-0.00655694984590575), RC(0.0505384497804303), RC(-0.0569572058725042), RC(0.368498056875488), RC(0.210055790203419)}
679:     };
680:     const PetscReal b[10]      = {RC(0.0705997961586714),   RC(-0.0281516061956374), RC(0.314600470734633),   RC(-0.0907057557963371), RC(0.168078953957742),
681:                                   RC(-0.00655694984590575), RC(0.0505384497804303),  RC(-0.0569572058725042), RC(0.368498056875488),   RC(0.210055790203419)};
682:     const PetscReal bembed[10] = {RC(-0.015494246543626), RC(0.167657963820093), RC(0.269858958144236),  RC(-0.0443258997755156), RC(0.150049236875266),
683:                                   RC(0.259452082755846),  RC(0.244624573502521), RC(-0.215528446920284), RC(0.0487601760292619),  RC(0.134945602112201)};
684:     PetscCall(TSDIRKRegister(TSDIRKES7510SA, 7, 10, &A[0][0], b, NULL, bembed, 1, b));
685:   }
686:   {
687:     // DIRK(9,7)[1]A[(9,5)A] from https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs
688:     const PetscReal A[9][9] = {
689:       {RC(0.179877789855839),   RC(0.0),                 RC(0.0),                RC(0.0),                  RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
690:       {RC(-0.100405844885157),  RC(0.214948590644819),   RC(0.0),                RC(0.0),                  RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
691:       {RC(0.112251360198995),   RC(-0.206162139150298),  RC(0.125159642941958),  RC(0.0),                  RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
692:       {RC(-0.0335164000768257), RC(0.999942349946143),   RC(-0.491470853833294), RC(0.19820086325566),     RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
693:       {RC(-0.0417345265478321), RC(0.187864510308215),   RC(0.0533789224305102), RC(-0.00822060284862916), RC(0.127670843671646),  RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
694:       {RC(-0.0278257925239257), RC(0.600979340683382),   RC(-0.242632273241134), RC(-0.11318753652081),    RC(0.164326917632931),  RC(0.284116597781395),  RC(0.0),                RC(0.0),                RC(0.0)               },
695:       {RC(0.041465583858922),   RC(0.429657872601836),   RC(-0.381323410582524), RC(0.391934277498434),    RC(-0.245918275501241), RC(-0.35960669741231),  RC(0.184000022289158),  RC(0.0),                RC(0.0)               },
696:       {RC(-0.105565651574538),  RC(-0.0557833155018609), RC(0.358967568942643),  RC(-0.13489263413921),    RC(0.129553247260677),  RC(0.0992493795371489), RC(-0.15716610563461),  RC(0.17918862279814),   RC(0.0)               },
697:       {RC(0.00439696079965225), RC(0.960250486570491),   RC(0.143558372286706),  RC(0.0819015241056593),   RC(0.999562318563625),  RC(0.325203439314358),  RC(-0.679013149331228), RC(-0.990589559837246), RC(0.0773648037639896)}
698:     };

700:     const PetscReal b[9]      = {RC(0.0), RC(0.179291520437966), RC(0.115310295273026), RC(-0.857943261453138), RC(0.654911318641998), RC(1.18713633508094), RC(-0.0949482361570542), RC(-0.37661430946407), RC(0.19285633764033)};
701:     const PetscReal bembed[9] = {RC(0.0), RC(0.1897135479408), RC(0.127461414808862), RC(-0.835810807663404), RC(0.665114177777166), RC(1.16481046518346), RC(-0.11661858889792), RC(-0.387303251022099), RC(0.192633041873135)};
702:     PetscCall(TSDIRKRegister(TSDIRK759A, 7, 9, &A[0][0], b, NULL, bembed, 1, b));
703:   }
704:   {
705:     // SDIRK(11,7)[1]SAL[(11,5)A] from https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs
706:     const PetscReal A[11][11] = {
707:       {RC(0.200252661187742),  RC(0.0),                 RC(0.0),                  RC(0.0),                 RC(0.0),                 RC(0.0),                 RC(0.0),                 RC(0.0),                 RC(0.0),               RC(0.0),               RC(0.0)              },
708:       {RC(-0.082947368165267), RC(0.200252661187742),   RC(0.0),                  RC(0.0),                 RC(0.0),                 RC(0.0),                 RC(0.0),                 RC(0.0),                 RC(0.0),               RC(0.0),               RC(0.0)              },
709:       {RC(0.483452690540751),  RC(0.0),                 RC(0.200252661187742),    RC(0.0),                 RC(0.0),                 RC(0.0),                 RC(0.0),                 RC(0.0),                 RC(0.0),               RC(0.0),               RC(0.0)              },
710:       {RC(0.771076453481321),  RC(-0.22936926341842),   RC(0.289733373208823),    RC(0.200252661187742),   RC(0.0),                 RC(0.0),                 RC(0.0),                 RC(0.0),                 RC(0.0),               RC(0.0),               RC(0.0)              },
711:       {RC(0.0329683054968892), RC(-0.162397421903366),  RC(0.000951777538562805), RC(0.0),                 RC(0.200252661187742),   RC(0.0),                 RC(0.0),                 RC(0.0),                 RC(0.0),               RC(0.0),               RC(0.0)              },
712:       {RC(0.265888743485945),  RC(0.606743151103931),   RC(0.173443800537369),    RC(-0.0433968261546912), RC(-0.385211017224481),  RC(0.200252661187742),   RC(0.0),                 RC(0.0),                 RC(0.0),               RC(0.0),               RC(0.0)              },
713:       {RC(0.220662294551146),  RC(-0.0465078507657608), RC(-0.0333111995282464),  RC(0.011801580836998),   RC(0.169480801030105),   RC(-0.0167974432139385), RC(0.200252661187742),   RC(0.0),                 RC(0.0),               RC(0.0),               RC(0.0)              },
714:       {RC(0.323099728365267),  RC(0.0288371831672575),  RC(-0.0543404318773196),  RC(0.0137765831431662),  RC(0.0516799019060702),  RC(-0.0421359763835713), RC(0.181297932037826),   RC(0.200252661187742),   RC(0.0),               RC(0.0),               RC(0.0)              },
715:       {RC(-0.164226696476538), RC(0.187552004946792),   RC(0.0628674420973025),   RC(-0.0108886582703428), RC(-0.0117628641717889), RC(0.0432176880867965),  RC(-0.0315206836275473), RC(-0.0846007021638797), RC(0.200252661187742), RC(0.0),               RC(0.0)              },
716:       {RC(0.651428598623771),  RC(-0.10208078475356),   RC(0.198305701801888),    RC(-0.0117354096673789), RC(-0.0440385966743686), RC(-0.0358364455795087), RC(-0.0075408087654097), RC(0.160320941654639),   RC(0.017940248694499), RC(0.200252661187742), RC(0.0)              },
717:       {RC(0.0),                RC(-0.266259448580236),  RC(-0.615982357748271),   RC(0.561474126687165),   RC(0.266911112787025),   RC(0.219775952207137),   RC(0.387847665451514),   RC(0.612483137773236),   RC(0.330027015806089), RC(-0.6965298655714),  RC(0.200252661187742)}
718:     };
719:     const PetscReal b[11] =
720:       {RC(0.0), RC(-0.266259448580236), RC(-0.615982357748271), RC(0.561474126687165), RC(0.266911112787025), RC(0.219775952207137), RC(0.387847665451514), RC(0.612483137773236), RC(0.330027015806089), RC(-0.6965298655714), RC(0.200252661187742)};
721:     const PetscReal bembed[11] =
722:       {RC(0.0), RC(0.180185524442613), RC(-0.628869710835338), RC(0.186185675988647), RC(0.0484716652630425), RC(0.203927720607141), RC(0.44041662512573), RC(0.615710527731245), RC(0.0689648839032607), RC(-0.253599870605903), RC(0.138606958379488)};
723:     PetscCall(TSDIRKRegister(TSDIRKS7511SAL, 7, 11, &A[0][0], b, NULL, bembed, 1, b));
724:   }
725:   {
726:     // DIRK(13,8)[1]A[(14,6)A] from https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs
727:     const PetscReal A[14][14] = {
728:       {RC(0.421050745442291),   RC(0.0),                RC(0.0),                 RC(0.0),                  RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
729:       {RC(-0.0761079419591268), RC(0.264353986580857),  RC(0.0),                 RC(0.0),                  RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
730:       {RC(0.0727106904170694),  RC(-0.204265976977285), RC(0.181608196544136),   RC(0.0),                  RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
731:       {RC(0.55763054816611),    RC(-0.409773579543499), RC(0.510926516886944),   RC(0.259892204518476),    RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
732:       {RC(0.0228083864844437),  RC(-0.445569051836454), RC(-0.0915242778636248), RC(0.00450055909321655),  RC(0.6397807199983),      RC(0.0),                RC(0.0),                 RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
733:       {RC(-0.135945849505152),  RC(0.0946509646963754), RC(-0.236110197279175),  RC(0.00318944206456517),  RC(0.255453021028118),    RC(0.174805219173446),  RC(0.0),                 RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
734:       {RC(-0.147960260670772),  RC(-0.402188192230535), RC(-0.703014530043888),  RC(0.00941974677418186),  RC(0.885747111289207),    RC(0.261314066449028),  RC(0.16307697503668),    RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
735:       {RC(0.165597241042244),   RC(0.824182962188923),  RC(-0.0280136160783609), RC(0.282372386631758),    RC(-0.957721354131182),   RC(0.489439550159977),  RC(0.170094415598103),   RC(0.0522519785718563),   RC(0.0),                RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
736:       {RC(0.0335292011495618),  RC(0.575750388029166),  RC(0.223289855356637),   RC(-0.00317458833242804), RC(-0.112890382135193),   RC(-0.419809267954284), RC(0.0466136902102104),  RC(-0.00115413813041085), RC(0.109685363692383),  RC(0.0),                 RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
737:       {RC(-0.0512616878252355), RC(0.699261265830807),  RC(-0.117939611738769),  RC(0.0021745241931243),   RC(-0.00932826702640947), RC(-0.267575057469428), RC(0.126949139814065),   RC(0.00330353204502163),  RC(0.185949445053766),  RC(0.0938215615963721),  RC(0.0),                RC(0.0),                RC(0.0),                RC(0.0)               },
738:       {RC(-0.106521517960343),  RC(0.41835889096168),   RC(0.353585905881916),   RC(-0.0746474161579599),  RC(-0.015450626460289),   RC(-0.46224659192275),  RC(-0.0576406327329181), RC(-0.00712066942504018), RC(0.377776558014452),  RC(0.36890054338294),    RC(0.0618488746331837), RC(0.0),                RC(0.0),                RC(0.0)               },
739:       {RC(-0.163079104890997),  RC(0.644561721693806),  RC(0.636968661639572),   RC(-0.122346720085377),   RC(-0.333062564990312),   RC(-0.3054226490478),   RC(-0.357820712828352),  RC(-0.0125510510334706),  RC(0.371263681186311),  RC(0.371979640363694),   RC(0.0531090658708968), RC(0.0518279459132049), RC(0.0),                RC(0.0)               },
740:       {RC(0.579993784455521),   RC(-0.188833728676494), RC(0.999975696843775),   RC(0.0572810855901161),   RC(-0.264374735003671),   RC(0.165091739976854),  RC(-0.546675809010452),  RC(-0.0283821822291982),  RC(-0.102639860418374), RC(-0.0343251040446405), RC(0.4762598462591),    RC(-0.304153104931261), RC(0.0953911855943621), RC(0.0)               },
741:       {RC(0.0848552694007844),  RC(0.287193912340074),  RC(0.543683503004232),   RC(-0.081311059300692),   RC(-0.0328661289388557),  RC(-0.323456834372922), RC(-0.240378871658975),  RC(-0.0189913019930369),  RC(0.220663114082036),  RC(0.253029984360864),   RC(0.252011799370563),  RC(-0.154882222605423), RC(0.0315202264687415), RC(0.0514095812104714)}
742:     };
743:     const PetscReal b[14] = {RC(0.0), RC(0.516650324205117), RC(0.0773227217357826), RC(-0.12474204666975), RC(-0.0241052115180679), RC(-0.325821145180359), RC(0.0907237460123951), RC(0.0459271880596652), RC(0.221012259404702), RC(0.235510906761942), RC(0.491109674204385), RC(-0.323506525837343), RC(0.119918108821531), RC(0.0)};
744:     const PetscReal bembed[14] = {RC(2.32345691433618e-16), RC(0.499150900944401), RC(0.080991997189243), RC(-0.0359440417166322), RC(-0.0258910397441454), RC(-0.304540350278636),  RC(0.0836627473632563),
745:                                   RC(0.0417664613347638),   RC(0.223636394275293), RC(0.231569156867596), RC(0.240526201277663),   RC(-0.222933582911926),  RC(-0.0111479879597561), RC(0.19915314335888)};
746:     PetscCall(TSDIRKRegister(TSDIRK8614A, 8, 14, &A[0][0], b, NULL, bembed, 1, b));
747:   }
748:   {
749:     // DIRK(15,8)[1]SAL[(16,6)A] from https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs
750:     const PetscReal A[16][16] = {
751:       {RC(0.498904981271193),   RC(0.0),                  RC(0.0),                 RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                   RC(0.0),                 RC(0.0),                  RC(0.0),                 RC(0.0),               RC(0.0),                 RC(0.0),                 RC(0.0)              },
752:       {RC(-0.303806037341816),  RC(0.886299445992379),    RC(0.0),                 RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                   RC(0.0),                 RC(0.0),                  RC(0.0),                 RC(0.0),               RC(0.0),                 RC(0.0),                 RC(0.0)              },
753:       {RC(-0.581440223471476),  RC(0.371003719460259),    RC(0.43844717752802),    RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                   RC(0.0),                 RC(0.0),                  RC(0.0),                 RC(0.0),               RC(0.0),                 RC(0.0),                 RC(0.0)              },
754:       {RC(0.531852638870051),   RC(-0.339363014907108),   RC(0.422373239795441),   RC(0.223854203543397),    RC(0.0),                RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                   RC(0.0),                 RC(0.0),                  RC(0.0),                 RC(0.0),               RC(0.0),                 RC(0.0),                 RC(0.0)              },
755:       {RC(0.118517891868867),   RC(-0.0756235584174296),  RC(-0.0864284870668712), RC(0.000536692838658312), RC(0.10101418329932),   RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                   RC(0.0),                 RC(0.0),                  RC(0.0),                 RC(0.0),               RC(0.0),                 RC(0.0),                 RC(0.0)              },
756:       {RC(0.218733626116401),   RC(-0.139568928299635),   RC(0.30473612813488),    RC(0.00354038623073564),  RC(0.0932085751160559), RC(0.140161806097591),   RC(0.0),                RC(0.0),                 RC(0.0),                   RC(0.0),                 RC(0.0),                  RC(0.0),                 RC(0.0),               RC(0.0),                 RC(0.0),                 RC(0.0)              },
757:       {RC(0.0692944686081835),  RC(-0.0442152168939502),  RC(-0.0903375348855603), RC(0.00259030241156141),  RC(0.204514233679515),  RC(-0.0245383758960002), RC(0.199289437094059),  RC(0.0),                 RC(0.0),                   RC(0.0),                 RC(0.0),                  RC(0.0),                 RC(0.0),               RC(0.0),                 RC(0.0),                 RC(0.0)              },
758:       {RC(0.990640016505571),   RC(-0.632104756315967),   RC(0.856971425234221),   RC(0.174494099232246),    RC(-0.113715829680145), RC(-0.151494045307366),  RC(-0.438268629569005), RC(0.120578398912139),   RC(0.0),                   RC(0.0),                 RC(0.0),                  RC(0.0),                 RC(0.0),               RC(0.0),                 RC(0.0),                 RC(0.0)              },
759:       {RC(-0.099415677713136),  RC(0.211832014309207),    RC(-0.245998265866888),  RC(-0.182249672235861),   RC(0.167897635713799),  RC(0.212850335030069),   RC(-0.391739299440123), RC(-0.0118718506876767), RC(0.526293701659093),     RC(0.0),                 RC(0.0),                  RC(0.0),                 RC(0.0),               RC(0.0),                 RC(0.0),                 RC(0.0)              },
760:       {RC(0.383983914845461),   RC(-0.245011361219604),   RC(0.46717278554955),    RC(-0.0361272447593202),  RC(0.0742234660511333), RC(-0.0474816271948766), RC(-0.229859978525756), RC(0.0516283729206322),  RC(0.0),                   RC(0.193823890777594),   RC(0.0),                  RC(0.0),                 RC(0.0),               RC(0.0),                 RC(0.0),                 RC(0.0)              },
761:       {RC(0.0967855003180134),  RC(-0.0481037037916184),  RC(0.191268138832434),   RC(0.234977164564126),    RC(0.0620265921753097), RC(0.403432826534738),   RC(0.152403846687238),  RC(-0.118420429237746),  RC(0.0582141598685892),    RC(-0.13924540906863),   RC(0.106661313117545),    RC(0.0),                 RC(0.0),               RC(0.0),                 RC(0.0),                 RC(0.0)              },
762:       {RC(0.133941307432055),   RC(-0.0722076602896254),  RC(0.217086297689275),   RC(0.00495499602192887),  RC(0.0306090174933995), RC(0.26483526755746),    RC(0.204442440745605),  RC(0.196883395136708),   RC(0.056527012583996),     RC(-0.150216381356784),  RC(-0.217209415757333),   RC(0.330353722743315),   RC(0.0),               RC(0.0),                 RC(0.0),                 RC(0.0)              },
763:       {RC(0.157014274561299),   RC(-0.0883810256381874),  RC(0.117193033885034),   RC(-0.0362304243769466),  RC(0.0169030211466111), RC(-0.169835753576141),  RC(0.399749979234113),  RC(0.31806704093008),    RC(0.050340008347693),     RC(0.120284837472214),   RC(-0.235313193645423),   RC(0.232488522208926),   RC(0.117719679450729), RC(0.0),                 RC(0.0),                 RC(0.0)              },
764:       {RC(0.00276453816875833), RC(-0.00366028255231782), RC(-0.331078914515559),  RC(0.623377549031949),    RC(0.167618142989491),  RC(0.0748467945312516),  RC(0.797629286699677),  RC(-0.390714256799583),  RC(-0.00808553925131555),  RC(0.014840324980952),   RC(-0.0856180410248133),  RC(0.602943304937827),   RC(-0.5771359338496),  RC(0.112273026653282),   RC(0.0),                 RC(0.0)              },
765:       {RC(0.0),                 RC(0.0),                  RC(0.085283971980307),   RC(0.51334393454179),     RC(0.144355978013514),  RC(0.255379109487853),   RC(0.225075750790524),  RC(-0.343241323394982),  RC(0.0),                   RC(0.0798250392218852),  RC(0.0528824734082655),   RC(-0.0830350888900362), RC(0.022567388707279), RC(-0.0592631119040204), RC(0.106825878037621),   RC(0.0)              },
766:       {RC(0.173784481207652),   RC(-0.110887906116241),   RC(0.190052513365204),   RC(-0.0688345422674029),  RC(0.10326505079603),   RC(0.267127097115219),   RC(0.141703423176897),  RC(0.0117966866651728),  RC(-6.65650091812762e-15), RC(-0.0213725083662519), RC(-0.00931148598712566), RC(-0.10007679077114),   RC(0.123471797451553), RC(0.00203684241073055), RC(-0.0294320891781173), RC(0.195746619921528)}
767:     };
768:     const PetscReal b[16] = {RC(0.0), RC(0.0), RC(0.085283971980307), RC(0.51334393454179), RC(0.144355978013514), RC(0.255379109487853), RC(0.225075750790524), RC(-0.343241323394982), RC(0.0), RC(0.0798250392218852), RC(0.0528824734082655), RC(-0.0830350888900362), RC(0.022567388707279), RC(-0.0592631119040204), RC(0.106825878037621), RC(0.0)};
769:     const PetscReal bembed[16] = {RC(-1.31988512519898e-15), RC(7.53606601764004e-16), RC(0.0886789133915965),   RC(0.0968726531622137),  RC(0.143815375874267),     RC(0.335214773313601),  RC(0.221862366978063),  RC(-0.147408947987273),
770:                                   RC(4.16297599203445e-16),  RC(0.000727276166520566), RC(-0.00284892677941246), RC(0.00512492274297611), RC(-0.000275595071215218), RC(0.0136014719350733), RC(0.0165190013607726), RC(0.228116714912817)};
771:     PetscCall(TSDIRKRegister(TSDIRK8616SAL, 8, 16, &A[0][0], b, NULL, bembed, 1, b));
772:   }
773:   {
774:     // ESDIRK(16,8)[2]SAL[(16,5)] from https://github.com/yousefalamri55/High_Order_DIRK_Methods_Coeffs
775:     const PetscReal A[16][16] = {
776:       {RC(0.0),                  RC(0.0),                 RC(0.0),                  RC(0.0),                   RC(0.0),                  RC(0.0),                  RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                  RC(0.0),               RC(0.0)              },
777:       {RC(0.117318819358521),    RC(0.117318819358521),   RC(0.0),                  RC(0.0),                   RC(0.0),                  RC(0.0),                  RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                  RC(0.0),               RC(0.0)              },
778:       {RC(0.0557014605974616),   RC(0.385525646638742),   RC(0.117318819358521),    RC(0.0),                   RC(0.0),                  RC(0.0),                  RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                  RC(0.0),               RC(0.0)              },
779:       {RC(0.063493276428895),    RC(0.373556126263681),   RC(0.0082994166438953),   RC(0.117318819358521),     RC(0.0),                  RC(0.0),                  RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                  RC(0.0),               RC(0.0)              },
780:       {RC(0.0961351856230088),   RC(0.335558324517178),   RC(0.207077765910132),    RC(-0.0581917140797146),   RC(0.117318819358521),    RC(0.0),                  RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                  RC(0.0),               RC(0.0)              },
781:       {RC(0.0497669214238319),   RC(0.384288616546039),   RC(0.0821728117583936),   RC(0.120337007107103),     RC(0.202262782645888),    RC(0.117318819358521),    RC(0.0),                  RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                  RC(0.0),               RC(0.0)              },
782:       {RC(0.00626710666809847),  RC(0.496491452640725),   RC(-0.111303249827358),   RC(0.170478821683603),     RC(0.166517073971103),    RC(-0.0328669811542241),  RC(0.117318819358521),    RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                  RC(0.0),               RC(0.0)              },
783:       {RC(0.0463439767281591),   RC(0.00306724391019652), RC(-0.00816305222386205), RC(-0.0353302599538294),   RC(0.0139313601702569),   RC(-0.00992014507967429), RC(0.0210087909090165),   RC(0.117318819358521),  RC(0.0),                 RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                  RC(0.0),               RC(0.0)              },
784:       {RC(0.111574049232048),    RC(0.467639166482209),   RC(0.237773114804619),    RC(0.0798895699267508),    RC(0.109580615914593),    RC(0.0307353103825936),   RC(-0.0404391509541147),  RC(-0.16942110744293),  RC(0.117318819358521),   RC(0.0),                 RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                  RC(0.0),               RC(0.0)              },
785:       {RC(-0.0107072484863877),  RC(-0.231376703354252),  RC(0.017541113036611),    RC(0.144871527682418),     RC(-0.041855459769806),   RC(0.0841832168332261),   RC(-0.0850020937282192),  RC(0.486170343825899),  RC(-0.0526717116822739), RC(0.117318819358521),   RC(0.0),                RC(0.0),                 RC(0.0),                 RC(0.0),                  RC(0.0),               RC(0.0)              },
786:       {RC(-0.0142238262314935),  RC(0.14752923682514),    RC(0.238235830732566),    RC(0.037950291904103),     RC(0.252075123381518),    RC(0.0474266904224567),   RC(-0.00363139069342027), RC(0.274081442388563),  RC(-0.0599166970745255), RC(-0.0527138812389185), RC(0.117318819358521),  RC(0.0),                 RC(0.0),                 RC(0.0),                  RC(0.0),               RC(0.0)              },
787:       {RC(-0.11837020183211),    RC(-0.635712481821264),  RC(0.239738832602538),    RC(0.330058936651707),     RC(-0.325784087988237),   RC(-0.0506514314589253),  RC(-0.281914404487009),   RC(0.852596345144291),  RC(0.651444614298805),   RC(-0.103476387303591),  RC(-0.354835880209975), RC(0.117318819358521),   RC(0.0),                 RC(0.0),                  RC(0.0),               RC(0.0)              },
788:       {RC(-0.00458164025442349), RC(0.296219694015248),   RC(0.322146049419995),    RC(0.15917778285238),      RC(0.284864871688843),    RC(0.185509526463076),    RC(-0.0784621067883274),  RC(0.166312223692047),  RC(-0.284152486083397),  RC(-0.357125104338944),  RC(0.078437074055306),  RC(0.0884129667114481),  RC(0.117318819358521),   RC(0.0),                  RC(0.0),               RC(0.0)              },
789:       {RC(-0.0545561913848106),  RC(0.675785423442753),   RC(0.423066443201941),    RC(-0.000165300126841193), RC(0.104252994793763),    RC(-0.105763019303021),   RC(-0.15988308809318),    RC(0.0515050001032011), RC(0.56013979290924),    RC(-0.45781539708603),   RC(-0.255870699752664), RC(0.026960254296416),   RC(-0.0721245985053681), RC(0.117318819358521),    RC(0.0),               RC(0.0)              },
790:       {RC(0.0649253995775223),   RC(-0.0216056457922249), RC(-0.073738139377975),   RC(0.0931033310077225),    RC(-0.0194339577299149),  RC(-0.0879623837313009),  RC(0.057125517179467),    RC(0.205120850488097),  RC(0.132576503537441),   RC(0.489416890627328),   RC(-0.1106765720501),   RC(-0.081038793996096),  RC(0.0606031613503788),  RC(-0.00241467937442272), RC(0.117318819358521), RC(0.0)              },
791:       {RC(0.0459979286336779),   RC(0.0780075394482806),  RC(0.015021874148058),    RC(0.195180277284195),     RC(-0.00246643310153235), RC(0.0473977117068314),   RC(-0.0682773558610363),  RC(0.19568019123878),   RC(-0.0876765449323747), RC(0.177874852409192),   RC(-0.337519251582222), RC(-0.0123255553640736), RC(0.311573291192553),   RC(0.0458604327754991),   RC(0.278352222645651), RC(0.117318819358521)}
792:     };
793:     const PetscReal b[16]      = {RC(0.0459979286336779),  RC(0.0780075394482806), RC(0.015021874148058),  RC(0.195180277284195),   RC(-0.00246643310153235), RC(0.0473977117068314), RC(-0.0682773558610363), RC(0.19568019123878),
794:                                   RC(-0.0876765449323747), RC(0.177874852409192),  RC(-0.337519251582222), RC(-0.0123255553640736), RC(0.311573291192553),    RC(0.0458604327754991), RC(0.278352222645651),   RC(0.117318819358521)};
795:     const PetscReal bembed[16] = {RC(0.0603373529853206),   RC(0.175453809423998),  RC(0.0537707777611352), RC(0.195309248607308),  RC(0.0135893741970232), RC(-0.0221160259296707), RC(-0.00726526156430691), RC(0.102961059369124),
796:                                   RC(0.000900215457460583), RC(0.0547959465692338), RC(-0.334995726863153), RC(0.0464409662093384), RC(0.301388101652194),  RC(0.00524851570622031), RC(0.229538601845236),    RC(0.124643044573514)};
797:     PetscCall(TSDIRKRegister(TSDIRKES8516SAL, 8, 16, &A[0][0], b, NULL, bembed, 1, b));
798:   }

800:   /* Additive methods */
801:   {
802:     const PetscReal A[3][3] = {
803:       {0.0, 0.0, 0.0},
804:       {0.0, 0.0, 0.0},
805:       {0.0, 0.5, 0.0}
806:     };
807:     const PetscReal At[3][3] = {
808:       {1.0, 0.0, 0.0},
809:       {0.0, 0.5, 0.0},
810:       {0.0, 0.5, 0.5}
811:     };
812:     const PetscReal b[3]       = {0.0, 0.5, 0.5};
813:     const PetscReal bembedt[3] = {1.0, 0.0, 0.0};
814:     PetscCall(TSARKIMEXRegister(TSARKIMEX1BEE, 2, 3, &At[0][0], b, NULL, &A[0][0], b, NULL, bembedt, bembedt, 1, b, NULL));
815:   }
816:   {
817:     const PetscReal A[2][2] = {
818:       {0.0, 0.0},
819:       {0.5, 0.0}
820:     };
821:     const PetscReal At[2][2] = {
822:       {0.0, 0.0},
823:       {0.0, 0.5}
824:     };
825:     const PetscReal b[2]       = {0.0, 1.0};
826:     const PetscReal bembedt[2] = {0.5, 0.5};
827:     /* binterpt[2][2] = {{1.0,-1.0},{0.0,1.0}};  second order dense output has poor stability properties and hence it is not currently in use */
828:     PetscCall(TSARKIMEXRegister(TSARKIMEXARS122, 2, 2, &At[0][0], b, NULL, &A[0][0], b, NULL, bembedt, bembedt, 1, b, NULL));
829:   }
830:   {
831:     const PetscReal A[2][2] = {
832:       {0.0, 0.0},
833:       {1.0, 0.0}
834:     };
835:     const PetscReal At[2][2] = {
836:       {0.0, 0.0},
837:       {0.5, 0.5}
838:     };
839:     const PetscReal b[2]       = {0.5, 0.5};
840:     const PetscReal bembedt[2] = {0.0, 1.0};
841:     /* binterpt[2][2] = {{1.0,-0.5},{0.0,0.5}}  second order dense output has poor stability properties and hence it is not currently in use */
842:     PetscCall(TSARKIMEXRegister(TSARKIMEXA2, 2, 2, &At[0][0], b, NULL, &A[0][0], b, NULL, bembedt, bembedt, 1, b, NULL));
843:   }
844:   {
845:     const PetscReal A[2][2] = {
846:       {0.0, 0.0},
847:       {1.0, 0.0}
848:     };
849:     const PetscReal At[2][2] = {
850:       {us2,             0.0},
851:       {1.0 - 2.0 * us2, us2}
852:     };
853:     const PetscReal b[2]           = {0.5, 0.5};
854:     const PetscReal bembedt[2]     = {0.0, 1.0};
855:     const PetscReal binterpt[2][2] = {
856:       {(us2 - 1.0) / (2.0 * us2 - 1.0),     -1 / (2.0 * (1.0 - 2.0 * us2))},
857:       {1 - (us2 - 1.0) / (2.0 * us2 - 1.0), -1 / (2.0 * (1.0 - 2.0 * us2))}
858:     };
859:     const PetscReal binterp[2][2] = {
860:       {1.0, -0.5},
861:       {0.0, 0.5 }
862:     };
863:     PetscCall(TSARKIMEXRegister(TSARKIMEXL2, 2, 2, &At[0][0], b, NULL, &A[0][0], b, NULL, bembedt, bembedt, 2, binterpt[0], binterp[0]));
864:   }
865:   {
866:     const PetscReal A[3][3] = {
867:       {0,      0,   0},
868:       {2 - s2, 0,   0},
869:       {0.5,    0.5, 0}
870:     };
871:     const PetscReal At[3][3] = {
872:       {0,            0,            0         },
873:       {1 - 1 / s2,   1 - 1 / s2,   0         },
874:       {1 / (2 * s2), 1 / (2 * s2), 1 - 1 / s2}
875:     };
876:     const PetscReal bembedt[3]     = {(4. - s2) / 8., (4. - s2) / 8., 1 / (2. * s2)};
877:     const PetscReal binterpt[3][2] = {
878:       {1.0 / s2, -1.0 / (2.0 * s2)},
879:       {1.0 / s2, -1.0 / (2.0 * s2)},
880:       {1.0 - s2, 1.0 / s2         }
881:     };
882:     PetscCall(TSARKIMEXRegister(TSARKIMEX2C, 2, 3, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembedt, 2, binterpt[0], NULL));
883:   }
884:   {
885:     const PetscReal A[3][3] = {
886:       {0,      0,    0},
887:       {2 - s2, 0,    0},
888:       {0.75,   0.25, 0}
889:     };
890:     const PetscReal At[3][3] = {
891:       {0,            0,            0         },
892:       {1 - 1 / s2,   1 - 1 / s2,   0         },
893:       {1 / (2 * s2), 1 / (2 * s2), 1 - 1 / s2}
894:     };
895:     const PetscReal bembedt[3]     = {(4. - s2) / 8., (4. - s2) / 8., 1 / (2. * s2)};
896:     const PetscReal binterpt[3][2] = {
897:       {1.0 / s2, -1.0 / (2.0 * s2)},
898:       {1.0 / s2, -1.0 / (2.0 * s2)},
899:       {1.0 - s2, 1.0 / s2         }
900:     };
901:     PetscCall(TSARKIMEXRegister(TSARKIMEX2D, 2, 3, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembedt, 2, binterpt[0], NULL));
902:   }
903:   { /* Optimal for linear implicit part */
904:     const PetscReal A[3][3] = {
905:       {0,                0,                0},
906:       {2 - s2,           0,                0},
907:       {(3 - 2 * s2) / 6, (3 + 2 * s2) / 6, 0}
908:     };
909:     const PetscReal At[3][3] = {
910:       {0,            0,            0         },
911:       {1 - 1 / s2,   1 - 1 / s2,   0         },
912:       {1 / (2 * s2), 1 / (2 * s2), 1 - 1 / s2}
913:     };
914:     const PetscReal bembedt[3]     = {(4. - s2) / 8., (4. - s2) / 8., 1 / (2. * s2)};
915:     const PetscReal binterpt[3][2] = {
916:       {1.0 / s2, -1.0 / (2.0 * s2)},
917:       {1.0 / s2, -1.0 / (2.0 * s2)},
918:       {1.0 - s2, 1.0 / s2         }
919:     };
920:     PetscCall(TSARKIMEXRegister(TSARKIMEX2E, 2, 3, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembedt, 2, binterpt[0], NULL));
921:   }
922:   { /* Optimal for linear implicit part */
923:     const PetscReal A[3][3] = {
924:       {0,   0,   0},
925:       {0.5, 0,   0},
926:       {0.5, 0.5, 0}
927:     };
928:     const PetscReal At[3][3] = {
929:       {0.25,   0,      0     },
930:       {0,      0.25,   0     },
931:       {1. / 3, 1. / 3, 1. / 3}
932:     };
933:     PetscCall(TSARKIMEXRegister(TSARKIMEXPRSSP2, 2, 3, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, NULL, NULL, 0, NULL, NULL));
934:   }
935:   {
936:     const PetscReal A[4][4] = {
937:       {0,                                0,                                0,                                 0},
938:       {1767732205903. / 2027836641118.,  0,                                0,                                 0},
939:       {5535828885825. / 10492691773637., 788022342437. / 10882634858940.,  0,                                 0},
940:       {6485989280629. / 16251701735622., -4246266847089. / 9704473918619., 10755448449292. / 10357097424841., 0}
941:     };
942:     const PetscReal At[4][4] = {
943:       {0,                                0,                                0,                                 0                              },
944:       {1767732205903. / 4055673282236.,  1767732205903. / 4055673282236.,  0,                                 0                              },
945:       {2746238789719. / 10658868560708., -640167445237. / 6845629431997.,  1767732205903. / 4055673282236.,   0                              },
946:       {1471266399579. / 7840856788654.,  -4482444167858. / 7529755066697., 11266239266428. / 11593286722821., 1767732205903. / 4055673282236.}
947:     };
948:     const PetscReal bembedt[4]     = {2756255671327. / 12835298489170., -10771552573575. / 22201958757719., 9247589265047. / 10645013368117., 2193209047091. / 5459859503100.};
949:     const PetscReal binterpt[4][2] = {
950:       {4655552711362. / 22874653954995.,  -215264564351. / 13552729205753.  },
951:       {-18682724506714. / 9892148508045., 17870216137069. / 13817060693119. },
952:       {34259539580243. / 13192909600954., -28141676662227. / 17317692491321.},
953:       {584795268549. / 6622622206610.,    2508943948391. / 7218656332882.   }
954:     };
955:     PetscCall(TSARKIMEXRegister(TSARKIMEX3, 3, 4, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembedt, 2, binterpt[0], NULL));
956:   }
957:   {
958:     const PetscReal A[5][5] = {
959:       {0,        0,       0,      0,       0},
960:       {1. / 2,   0,       0,      0,       0},
961:       {11. / 18, 1. / 18, 0,      0,       0},
962:       {5. / 6,   -5. / 6, .5,     0,       0},
963:       {1. / 4,   7. / 4,  3. / 4, -7. / 4, 0}
964:     };
965:     const PetscReal At[5][5] = {
966:       {0, 0,       0,       0,      0     },
967:       {0, 1. / 2,  0,       0,      0     },
968:       {0, 1. / 6,  1. / 2,  0,      0     },
969:       {0, -1. / 2, 1. / 2,  1. / 2, 0     },
970:       {0, 3. / 2,  -3. / 2, 1. / 2, 1. / 2}
971:     };
972:     PetscCall(TSARKIMEXRegister(TSARKIMEXARS443, 3, 5, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, NULL, NULL, 0, NULL, NULL));
973:   }
974:   {
975:     const PetscReal A[5][5] = {
976:       {0,      0,      0,      0, 0},
977:       {1,      0,      0,      0, 0},
978:       {4. / 9, 2. / 9, 0,      0, 0},
979:       {1. / 4, 0,      3. / 4, 0, 0},
980:       {1. / 4, 0,      3. / 5, 0, 0}
981:     };
982:     const PetscReal At[5][5] = {
983:       {0,       0,       0,   0,   0 },
984:       {.5,      .5,      0,   0,   0 },
985:       {5. / 18, -1. / 9, .5,  0,   0 },
986:       {.5,      0,       0,   .5,  0 },
987:       {.25,     0,       .75, -.5, .5}
988:     };
989:     PetscCall(TSARKIMEXRegister(TSARKIMEXBPR3, 3, 5, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, NULL, NULL, 0, NULL, NULL));
990:   }
991:   {
992:     const PetscReal A[6][6] = {
993:       {0,                               0,                                 0,                                 0,                                0,              0},
994:       {1. / 2,                          0,                                 0,                                 0,                                0,              0},
995:       {13861. / 62500.,                 6889. / 62500.,                    0,                                 0,                                0,              0},
996:       {-116923316275. / 2393684061468., -2731218467317. / 15368042101831., 9408046702089. / 11113171139209.,  0,                                0,              0},
997:       {-451086348788. / 2902428689909., -2682348792572. / 7519795681897.,  12662868775082. / 11960479115383., 3355817975965. / 11060851509271., 0,              0},
998:       {647845179188. / 3216320057751.,  73281519250. / 8382639484533.,     552539513391. / 3454668386233.,    3354512671639. / 8306763924573.,  4040. / 17871., 0}
999:     };
1000:     const PetscReal At[6][6] = {
1001:       {0,                            0,                       0,                       0,                   0,             0     },
1002:       {1. / 4,                       1. / 4,                  0,                       0,                   0,             0     },
1003:       {8611. / 62500.,               -1743. / 31250.,         1. / 4,                  0,                   0,             0     },
1004:       {5012029. / 34652500.,         -654441. / 2922500.,     174375. / 388108.,       1. / 4,              0,             0     },
1005:       {15267082809. / 155376265600., -71443401. / 120774400., 730878875. / 902184768., 2285395. / 8070912., 1. / 4,        0     },
1006:       {82889. / 524892.,             0,                       15625. / 83664.,         69875. / 102672.,    -2260. / 8211, 1. / 4}
1007:     };
1008:     const PetscReal bembedt[6]     = {4586570599. / 29645900160., 0, 178811875. / 945068544., 814220225. / 1159782912., -3700637. / 11593932., 61727. / 225920.};
1009:     const PetscReal binterpt[6][3] = {
1010:       {6943876665148. / 7220017795957.,   -54480133. / 30881146., 6818779379841. / 7100303317025.  },
1011:       {0,                                 0,                      0                                },
1012:       {7640104374378. / 9702883013639.,   -11436875. / 14766696., 2173542590792. / 12501825683035. },
1013:       {-20649996744609. / 7521556579894., 174696575. / 18121608., -31592104683404. / 5083833661969.},
1014:       {8854892464581. / 2390941311638.,   -12120380. / 966161.,   61146701046299. / 7138195549469. },
1015:       {-11397109935349. / 6675773540249., 3843. / 706.,           -17219254887155. / 4939391667607.}
1016:     };
1017:     PetscCall(TSARKIMEXRegister(TSARKIMEX4, 4, 6, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembedt, 3, binterpt[0], NULL));
1018:   }
1019:   {
1020:     const PetscReal A[8][8] = {
1021:       {0,                                  0,                              0,                                 0,                                  0,                               0,                                 0,                               0},
1022:       {41. / 100,                          0,                              0,                                 0,                                  0,                               0,                                 0,                               0},
1023:       {367902744464. / 2072280473677.,     677623207551. / 8224143866563., 0,                                 0,                                  0,                               0,                                 0,                               0},
1024:       {1268023523408. / 10340822734521.,   0,                              1029933939417. / 13636558850479.,  0,                                  0,                               0,                                 0,                               0},
1025:       {14463281900351. / 6315353703477.,   0,                              66114435211212. / 5879490589093.,  -54053170152839. / 4284798021562.,  0,                               0,                                 0,                               0},
1026:       {14090043504691. / 34967701212078.,  0,                              15191511035443. / 11219624916014., -18461159152457. / 12425892160975., -281667163811. / 9011619295870., 0,                                 0,                               0},
1027:       {19230459214898. / 13134317526959.,  0,                              21275331358303. / 2942455364971.,  -38145345988419. / 4862620318723.,  -1. / 8,                         -1. / 8,                           0,                               0},
1028:       {-19977161125411. / 11928030595625., 0,                              -40795976796054. / 6384907823539., 177454434618887. / 12078138498510., 782672205425. / 8267701900261.,  -69563011059811. / 9646580694205., 7356628210526. / 4942186776405., 0}
1029:     };
1030:     const PetscReal At[8][8] = {
1031:       {0,                                0,                                0,                                 0,                                  0,                                0,                                  0,                                 0         },
1032:       {41. / 200.,                       41. / 200.,                       0,                                 0,                                  0,                                0,                                  0,                                 0         },
1033:       {41. / 400.,                       -567603406766. / 11931857230679., 41. / 200.,                        0,                                  0,                                0,                                  0,                                 0         },
1034:       {683785636431. / 9252920307686.,   0,                                -110385047103. / 1367015193373.,   41. / 200.,                         0,                                0,                                  0,                                 0         },
1035:       {3016520224154. / 10081342136671., 0,                                30586259806659. / 12414158314087., -22760509404356. / 11113319521817., 41. / 200.,                       0,                                  0,                                 0         },
1036:       {218866479029. / 1489978393911.,   0,                                638256894668. / 5436446318841.,    -1179710474555. / 5321154724896.,   -60928119172. / 8023461067671.,   41. / 200.,                         0,                                 0         },
1037:       {1020004230633. / 5715676835656.,  0,                                25762820946817. / 25263940353407., -2161375909145. / 9755907335909.,   -211217309593. / 5846859502534.,  -4269925059573. / 7827059040749.,   41. / 200,                         0         },
1038:       {-872700587467. / 9133579230613.,  0,                                0,                                 22348218063261. / 9555858737531.,   -1143369518992. / 8141816002931., -39379526789629. / 19018526304540., 32727382324388. / 42900044865799., 41. / 200.}
1039:     };
1040:     const PetscReal bembedt[8]     = {-975461918565. / 9796059967033., 0, 0, 78070527104295. / 32432590147079., -548382580838. / 3424219808633., -33438840321285. / 15594753105479., 3629800801594. / 4656183773603., 4035322873751. / 18575991585200.};
1041:     const PetscReal binterpt[8][3] = {
1042:       {-17674230611817. / 10670229744614., 43486358583215. / 12773830924787.,  -9257016797708. / 5021505065439. },
1043:       {0,                                  0,                                  0                                },
1044:       {0,                                  0,                                  0                                },
1045:       {65168852399939. / 7868540260826.,   -91478233927265. / 11067650958493., 26096422576131. / 11239449250142.},
1046:       {15494834004392. / 5936557850923.,   -79368583304911. / 10890268929626., 92396832856987. / 20362823103730.},
1047:       {-99329723586156. / 26959484932159., -12239297817655. / 9152339842473.,  30029262896817. / 10175596800299.},
1048:       {-19024464361622. / 5461577185407.,  115839755401235. / 10719374521269., -26136350496073. / 3983972220547.},
1049:       {-6511271360970. / 6095937251113.,   5843115559534. / 2180450260947.,    -5289405421727. / 3760307252460. }
1050:     };
1051:     PetscCall(TSARKIMEXRegister(TSARKIMEX5, 5, 8, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembedt, 3, binterpt[0], NULL));
1052:   }
1053: #undef RC
1054: #undef us2
1055: #undef s2
1056:   PetscFunctionReturn(PETSC_SUCCESS);
1057: }

1059: /*@C
1060:   TSARKIMEXRegisterDestroy - Frees the list of schemes that were registered by `TSARKIMEXRegister()`.

1062:   Not Collective

1064:   Level: advanced

1066: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXRegister()`, `TSARKIMEXRegisterAll()`
1067: @*/
1068: PetscErrorCode TSARKIMEXRegisterDestroy(void)
1069: {
1070:   ARKTableauLink link;

1072:   PetscFunctionBegin;
1073:   while ((link = ARKTableauList)) {
1074:     ARKTableau t   = &link->tab;
1075:     ARKTableauList = link->next;
1076:     PetscCall(PetscFree6(t->At, t->bt, t->ct, t->A, t->b, t->c));
1077:     PetscCall(PetscFree2(t->bembedt, t->bembed));
1078:     PetscCall(PetscFree2(t->binterpt, t->binterp));
1079:     PetscCall(PetscFree(t->name));
1080:     PetscCall(PetscFree(link));
1081:   }
1082:   TSARKIMEXRegisterAllCalled = PETSC_FALSE;
1083:   PetscFunctionReturn(PETSC_SUCCESS);
1084: }

1086: /*@C
1087:   TSARKIMEXInitializePackage - This function initializes everything in the `TSARKIMEX` package. It is called
1088:   from `TSInitializePackage()`.

1090:   Level: developer

1092: .seealso: [](ch_ts), `PetscInitialize()`, `TSARKIMEXFinalizePackage()`
1093: @*/
1094: PetscErrorCode TSARKIMEXInitializePackage(void)
1095: {
1096:   PetscFunctionBegin;
1097:   if (TSARKIMEXPackageInitialized) PetscFunctionReturn(PETSC_SUCCESS);
1098:   TSARKIMEXPackageInitialized = PETSC_TRUE;
1099:   PetscCall(TSARKIMEXRegisterAll());
1100:   PetscCall(PetscRegisterFinalize(TSARKIMEXFinalizePackage));
1101:   PetscFunctionReturn(PETSC_SUCCESS);
1102: }

1104: /*@C
1105:   TSARKIMEXFinalizePackage - This function destroys everything in the `TSARKIMEX` package. It is
1106:   called from `PetscFinalize()`.

1108:   Level: developer

1110: .seealso: [](ch_ts), `PetscFinalize()`, `TSARKIMEXInitializePackage()`
1111: @*/
1112: PetscErrorCode TSARKIMEXFinalizePackage(void)
1113: {
1114:   PetscFunctionBegin;
1115:   TSARKIMEXPackageInitialized = PETSC_FALSE;
1116:   PetscCall(TSARKIMEXRegisterDestroy());
1117:   PetscFunctionReturn(PETSC_SUCCESS);
1118: }

1120: /*@C
1121:   TSARKIMEXRegister - register a `TSARKIMEX` scheme by providing the entries in the Butcher tableau and optionally embedded approximations and interpolation

1123:   Logically Collective.

1125:   Input Parameters:
1126: + name     - identifier for method
1127: . order    - approximation order of method
1128: . s        - number of stages, this is the dimension of the matrices below
1129: . At       - Butcher table of stage coefficients for stiff part (dimension s*s, row-major)
1130: . bt       - Butcher table for completing the stiff part of the step (dimension s; NULL to use the last row of At)
1131: . ct       - Abscissa of each stiff stage (dimension s, NULL to use row sums of At)
1132: . A        - Non-stiff stage coefficients (dimension s*s, row-major)
1133: . b        - Non-stiff step completion table (dimension s; NULL to use last row of At)
1134: . c        - Non-stiff abscissa (dimension s; NULL to use row sums of A)
1135: . bembedt  - Stiff part of completion table for embedded method (dimension s; NULL if not available)
1136: . bembed   - Non-stiff part of completion table for embedded method (dimension s; NULL to use bembedt if provided)
1137: . pinterp  - Order of the interpolation scheme, equal to the number of columns of binterpt and binterp
1138: . binterpt - Coefficients of the interpolation formula for the stiff part (dimension s*pinterp)
1139: - binterp  - Coefficients of the interpolation formula for the non-stiff part (dimension s*pinterp; NULL to reuse binterpt)

1141:   Level: advanced

1143:   Note:
1144:   Several `TSARKIMEX` methods are provided, this function is only needed to create new methods.

1146: .seealso: [](ch_ts), `TSARKIMEX`, `TSType`, `TS`
1147: @*/
1148: PetscErrorCode TSARKIMEXRegister(TSARKIMEXType name, PetscInt order, PetscInt s, const PetscReal At[], const PetscReal bt[], const PetscReal ct[], const PetscReal A[], const PetscReal b[], const PetscReal c[], const PetscReal bembedt[], const PetscReal bembed[], PetscInt pinterp, const PetscReal binterpt[], const PetscReal binterp[])
1149: {
1150:   ARKTableauLink link;
1151:   ARKTableau     t;
1152:   PetscInt       i, j;

1154:   PetscFunctionBegin;
1155:   PetscCall(TSARKIMEXInitializePackage());
1156:   for (link = ARKTableauList; link; link = link->next) {
1157:     PetscBool match;

1159:     PetscCall(PetscStrcmp(link->tab.name, name, &match));
1160:     PetscCheck(!match, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Method with name \"%s\" already registered", name);
1161:   }
1162:   PetscCall(PetscNew(&link));
1163:   t = &link->tab;
1164:   PetscCall(PetscStrallocpy(name, &t->name));
1165:   t->order = order;
1166:   t->s     = s;
1167:   PetscCall(PetscMalloc6(s * s, &t->At, s, &t->bt, s, &t->ct, s * s, &t->A, s, &t->b, s, &t->c));
1168:   PetscCall(PetscArraycpy(t->At, At, s * s));
1169:   if (A) {
1170:     PetscCall(PetscArraycpy(t->A, A, s * s));
1171:     t->additive = PETSC_TRUE;
1172:   }

1174:   if (bt) PetscCall(PetscArraycpy(t->bt, bt, s));
1175:   else
1176:     for (i = 0; i < s; i++) t->bt[i] = At[(s - 1) * s + i];

1178:   if (t->additive) {
1179:     if (b) PetscCall(PetscArraycpy(t->b, b, s));
1180:     else
1181:       for (i = 0; i < s; i++) t->b[i] = t->bt[i];
1182:   } else PetscCall(PetscArrayzero(t->b, s));

1184:   if (ct) PetscCall(PetscArraycpy(t->ct, ct, s));
1185:   else
1186:     for (i = 0; i < s; i++)
1187:       for (j = 0, t->ct[i] = 0; j < s; j++) t->ct[i] += At[i * s + j];

1189:   if (t->additive) {
1190:     if (c) PetscCall(PetscArraycpy(t->c, c, s));
1191:     else
1192:       for (i = 0; i < s; i++)
1193:         for (j = 0, t->c[i] = 0; j < s; j++) t->c[i] += A[i * s + j];
1194:   } else PetscCall(PetscArrayzero(t->c, s));

1196:   t->stiffly_accurate = PETSC_TRUE;
1197:   for (i = 0; i < s; i++)
1198:     if (t->At[(s - 1) * s + i] != t->bt[i]) t->stiffly_accurate = PETSC_FALSE;

1200:   t->explicit_first_stage = PETSC_TRUE;
1201:   for (i = 0; i < s; i++)
1202:     if (t->At[i] != 0.0) t->explicit_first_stage = PETSC_FALSE;

1204:   /* def of FSAL can be made more precise */
1205:   t->FSAL_implicit = (PetscBool)(t->explicit_first_stage && t->stiffly_accurate);

1207:   if (bembedt) {
1208:     PetscCall(PetscMalloc2(s, &t->bembedt, s, &t->bembed));
1209:     PetscCall(PetscArraycpy(t->bembedt, bembedt, s));
1210:     PetscCall(PetscArraycpy(t->bembed, bembed ? bembed : bembedt, s));
1211:   }

1213:   t->pinterp = pinterp;
1214:   PetscCall(PetscMalloc2(s * pinterp, &t->binterpt, s * pinterp, &t->binterp));
1215:   PetscCall(PetscArraycpy(t->binterpt, binterpt, s * pinterp));
1216:   PetscCall(PetscArraycpy(t->binterp, binterp ? binterp : binterpt, s * pinterp));

1218:   link->next     = ARKTableauList;
1219:   ARKTableauList = link;
1220:   PetscFunctionReturn(PETSC_SUCCESS);
1221: }

1223: /*@C
1224:   TSDIRKRegister - register a `TSDIRK` scheme by providing the entries in its Butcher tableau and, optionally, embedded approximations and interpolation

1226:   Logically Collective.

1228:   Input Parameters:
1229: + name     - identifier for method
1230: . order    - approximation order of method
1231: . s        - number of stages, this is the dimension of the matrices below
1232: . At       - Butcher table of stage coefficients (dimension `s`*`s`, row-major order)
1233: . bt       - Butcher table for completing the step (dimension `s`; pass `NULL` to use the last row of `At`)
1234: . ct       - Abscissa of each stage (dimension s, NULL to use row sums of At)
1235: . bembedt  - Stiff part of completion table for embedded method (dimension s; `NULL` if not available)
1236: . pinterp  - Order of the interpolation scheme, equal to the number of columns of `binterpt` and `binterp`
1237: - binterpt - Coefficients of the interpolation formula (dimension s*pinterp)

1239:   Level: advanced

1241:   Note:
1242:   Several `TSDIRK` methods are provided, the use of this function is only needed to create new methods.

1244: .seealso: [](ch_ts), `TSDIRK`, `TSType`, `TS`
1245: @*/
1246: PetscErrorCode TSDIRKRegister(TSDIRKType name, PetscInt order, PetscInt s, const PetscReal At[], const PetscReal bt[], const PetscReal ct[], const PetscReal bembedt[], PetscInt pinterp, const PetscReal binterpt[])
1247: {
1248:   PetscFunctionBegin;
1249:   PetscCall(TSARKIMEXRegister(name, order, s, At, bt, ct, NULL, NULL, NULL, bembedt, NULL, pinterp, binterpt, NULL));
1250:   PetscFunctionReturn(PETSC_SUCCESS);
1251: }

1253: /*
1254:  The step completion formula is

1256:  x1 = x0 - h bt^T YdotI + h b^T YdotRHS

1258:  This function can be called before or after ts->vec_sol has been updated.
1259:  Suppose we have a completion formula (bt,b) and an embedded formula (bet,be) of different order.
1260:  We can write

1262:  x1e = x0 - h bet^T YdotI + h be^T YdotRHS
1263:      = x1 + h bt^T YdotI - h b^T YdotRHS - h bet^T YdotI + h be^T YdotRHS
1264:      = x1 - h (bet - bt)^T YdotI + h (be - b)^T YdotRHS

1266:  so we can evaluate the method with different order even after the step has been optimistically completed.
1267: */
1268: static PetscErrorCode TSEvaluateStep_ARKIMEX(TS ts, PetscInt order, Vec X, PetscBool *done)
1269: {
1270:   TS_ARKIMEX  *ark = (TS_ARKIMEX *)ts->data;
1271:   ARKTableau   tab = ark->tableau;
1272:   PetscScalar *w   = ark->work;
1273:   PetscReal    h;
1274:   PetscInt     s = tab->s, j;
1275:   PetscBool    hasE;

1277:   PetscFunctionBegin;
1278:   switch (ark->status) {
1279:   case TS_STEP_INCOMPLETE:
1280:   case TS_STEP_PENDING:
1281:     h = ts->time_step;
1282:     break;
1283:   case TS_STEP_COMPLETE:
1284:     h = ts->ptime - ts->ptime_prev;
1285:     break;
1286:   default:
1287:     SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_PLIB, "Invalid TSStepStatus");
1288:   }
1289:   if (order == tab->order) {
1290:     if (ark->status == TS_STEP_INCOMPLETE) {
1291:       if (!ark->imex && tab->stiffly_accurate) { /* Only the stiffly accurate implicit formula is used */
1292:         PetscCall(VecCopy(ark->Y[s - 1], X));
1293:       } else { /* Use the standard completion formula (bt,b) */
1294:         PetscCall(VecCopy(ts->vec_sol, X));
1295:         for (j = 0; j < s; j++) w[j] = h * tab->bt[j];
1296:         PetscCall(VecMAXPY(X, s, w, ark->YdotI));
1297:         if (tab->additive && ark->imex) { /* Method is IMEX, complete the explicit formula */
1298:           PetscCall(TSHasRHSFunction(ts, &hasE));
1299:           if (hasE) {
1300:             for (j = 0; j < s; j++) w[j] = h * tab->b[j];
1301:             PetscCall(VecMAXPY(X, s, w, ark->YdotRHS));
1302:           }
1303:         }
1304:       }
1305:     } else PetscCall(VecCopy(ts->vec_sol, X));
1306:     if (done) *done = PETSC_TRUE;
1307:     PetscFunctionReturn(PETSC_SUCCESS);
1308:   } else if (order == tab->order - 1) {
1309:     if (!tab->bembedt) goto unavailable;
1310:     if (ark->status == TS_STEP_INCOMPLETE) { /* Complete with the embedded method (bet,be) */
1311:       PetscCall(VecCopy(ts->vec_sol, X));
1312:       for (j = 0; j < s; j++) w[j] = h * tab->bembedt[j];
1313:       PetscCall(VecMAXPY(X, s, w, ark->YdotI));
1314:       if (tab->additive) {
1315:         PetscCall(TSHasRHSFunction(ts, &hasE));
1316:         if (hasE) {
1317:           for (j = 0; j < s; j++) w[j] = h * tab->bembed[j];
1318:           PetscCall(VecMAXPY(X, s, w, ark->YdotRHS));
1319:         }
1320:       }
1321:     } else { /* Rollback and re-complete using (bet-be,be-b) */
1322:       PetscCall(VecCopy(ts->vec_sol, X));
1323:       for (j = 0; j < s; j++) w[j] = h * (tab->bembedt[j] - tab->bt[j]);
1324:       PetscCall(VecMAXPY(X, tab->s, w, ark->YdotI));
1325:       if (tab->additive) {
1326:         PetscCall(TSHasRHSFunction(ts, &hasE));
1327:         if (hasE) {
1328:           for (j = 0; j < s; j++) w[j] = h * (tab->bembed[j] - tab->b[j]);
1329:           PetscCall(VecMAXPY(X, s, w, ark->YdotRHS));
1330:         }
1331:       }
1332:     }
1333:     if (done) *done = PETSC_TRUE;
1334:     PetscFunctionReturn(PETSC_SUCCESS);
1335:   }
1336: unavailable:
1337:   if (done) *done = PETSC_FALSE;
1338:   else
1339:     SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "ARKIMEX '%s' of order %" PetscInt_FMT " cannot evaluate step at order %" PetscInt_FMT ". Consider using -ts_adapt_type none or a different method that has an embedded estimate.", tab->name,
1340:             tab->order, order);
1341:   PetscFunctionReturn(PETSC_SUCCESS);
1342: }

1344: static PetscErrorCode TSARKIMEXTestMassIdentity(TS ts, PetscBool *id)
1345: {
1346:   Vec         Udot, Y1, Y2;
1347:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;
1348:   PetscReal   norm;

1350:   PetscFunctionBegin;
1351:   PetscCall(VecDuplicate(ts->vec_sol, &Udot));
1352:   PetscCall(VecDuplicate(ts->vec_sol, &Y1));
1353:   PetscCall(VecDuplicate(ts->vec_sol, &Y2));
1354:   PetscCall(TSComputeIFunction(ts, ts->ptime, ts->vec_sol, Udot, Y1, ark->imex));
1355:   PetscCall(VecSetRandom(Udot, NULL));
1356:   PetscCall(TSComputeIFunction(ts, ts->ptime, ts->vec_sol, Udot, Y2, ark->imex));
1357:   PetscCall(VecAXPY(Y2, -1.0, Y1));
1358:   PetscCall(VecAXPY(Y2, -1.0, Udot));
1359:   PetscCall(VecNorm(Y2, NORM_2, &norm));
1360:   if (norm < 100.0 * PETSC_MACHINE_EPSILON) {
1361:     *id = PETSC_TRUE;
1362:   } else {
1363:     *id = PETSC_FALSE;
1364:     PetscCall(PetscInfo((PetscObject)ts, "IFunction(Udot = random) - IFunction(Udot = 0) is not near Udot, %g, suspect mass matrix implied in IFunction() is not the identity as required\n", (double)norm));
1365:   }
1366:   PetscCall(VecDestroy(&Udot));
1367:   PetscCall(VecDestroy(&Y1));
1368:   PetscCall(VecDestroy(&Y2));
1369:   PetscFunctionReturn(PETSC_SUCCESS);
1370: }

1372: static PetscErrorCode TSRollBack_ARKIMEX(TS ts)
1373: {
1374:   TS_ARKIMEX      *ark = (TS_ARKIMEX *)ts->data;
1375:   ARKTableau       tab = ark->tableau;
1376:   const PetscInt   s   = tab->s;
1377:   const PetscReal *bt = tab->bt, *b = tab->b;
1378:   PetscScalar     *w     = ark->work;
1379:   Vec             *YdotI = ark->YdotI, *YdotRHS = ark->YdotRHS;
1380:   PetscInt         j;
1381:   PetscReal        h;

1383:   PetscFunctionBegin;
1384:   switch (ark->status) {
1385:   case TS_STEP_INCOMPLETE:
1386:   case TS_STEP_PENDING:
1387:     h = ts->time_step;
1388:     break;
1389:   case TS_STEP_COMPLETE:
1390:     h = ts->ptime - ts->ptime_prev;
1391:     break;
1392:   default:
1393:     SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_PLIB, "Invalid TSStepStatus");
1394:   }
1395:   for (j = 0; j < s; j++) w[j] = -h * bt[j];
1396:   PetscCall(VecMAXPY(ts->vec_sol, s, w, YdotI));
1397:   if (tab->additive) {
1398:     PetscBool hasE;

1400:     PetscCall(TSHasRHSFunction(ts, &hasE));
1401:     if (hasE) {
1402:       for (j = 0; j < s; j++) w[j] = -h * b[j];
1403:       PetscCall(VecMAXPY(ts->vec_sol, s, w, YdotRHS));
1404:     }
1405:   }
1406:   PetscFunctionReturn(PETSC_SUCCESS);
1407: }

1409: static PetscErrorCode TSStep_ARKIMEX(TS ts)
1410: {
1411:   TS_ARKIMEX      *ark = (TS_ARKIMEX *)ts->data;
1412:   ARKTableau       tab = ark->tableau;
1413:   const PetscInt   s   = tab->s;
1414:   const PetscReal *At = tab->At, *A = tab->A, *ct = tab->ct, *c = tab->c;
1415:   PetscScalar     *w = ark->work;
1416:   Vec             *Y = ark->Y, *YdotI = ark->YdotI, *YdotRHS = ark->YdotRHS, Ydot = ark->Ydot, Ydot0 = ark->Ydot0, Z = ark->Z;
1417:   PetscBool        extrapolate = ark->extrapolate;
1418:   TSAdapt          adapt;
1419:   SNES             snes;
1420:   PetscInt         i, j, its, lits;
1421:   PetscInt         rejections = 0;
1422:   PetscBool        hasE = PETSC_FALSE, dirk = (PetscBool)(!tab->additive), stageok, accept = PETSC_TRUE;
1423:   PetscReal        next_time_step = ts->time_step;

1425:   PetscFunctionBegin;
1426:   if (ark->extrapolate && !ark->Y_prev) {
1427:     PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &ark->Y_prev));
1428:     PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &ark->YdotI_prev));
1429:     if (tab->additive) PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &ark->YdotRHS_prev));
1430:   }

1432:   if (!dirk) PetscCall(TSHasRHSFunction(ts, &hasE));
1433:   if (!hasE) dirk = PETSC_TRUE;

1435:   if (!ts->steprollback) {
1436:     if (dirk || ts->equation_type >= TS_EQ_IMPLICIT) { /* Save the initial slope for the next step */
1437:       PetscCall(VecCopy(YdotI[s - 1], Ydot0));
1438:     }
1439:     if (ark->extrapolate && !ts->steprestart) { /* Save the Y, YdotI, YdotRHS for extrapolation initial guess */
1440:       for (i = 0; i < s; i++) {
1441:         PetscCall(VecCopy(Y[i], ark->Y_prev[i]));
1442:         PetscCall(VecCopy(YdotI[i], ark->YdotI_prev[i]));
1443:         if (tab->additive && hasE) PetscCall(VecCopy(YdotRHS[i], ark->YdotRHS_prev[i]));
1444:       }
1445:     }
1446:   }

1448:   /*
1449:      For fully implicit formulations we must solve the equations

1451:        F(t_n,x_n,xdot) = 0

1453:      for the explicit first stage.
1454:      Here we call SNESSolve using PETSC_MAX_REAL as shift to flag it.
1455:      Special handling is inside SNESTSFormFunction_ARKIMEX and SNESTSFormJacobian_ARKIMEX
1456:   */
1457:   if (dirk && tab->explicit_first_stage && ts->steprestart) {
1458:     ark->scoeff = PETSC_MAX_REAL;
1459:     PetscCall(VecCopy(ts->vec_sol, Z));
1460:     PetscCall(TSGetSNES(ts, &snes));
1461:     PetscCall(SNESSolve(snes, NULL, Ydot0));
1462:   }

1464:   /* For IMEX we compute a step */
1465:   if (!dirk && ts->equation_type >= TS_EQ_IMPLICIT && tab->explicit_first_stage && ts->steprestart) {
1466:     TS ts_start;
1467:     if (PetscDefined(USE_DEBUG) && hasE) {
1468:       PetscBool id = PETSC_FALSE;
1469:       PetscCall(TSARKIMEXTestMassIdentity(ts, &id));
1470:       PetscCheck(id, PetscObjectComm((PetscObject)ts), PETSC_ERR_ARG_INCOMP, "This scheme requires an identity mass matrix, however the TSIFunction you provided does not utilize an identity mass matrix");
1471:     }
1472:     PetscCall(TSClone(ts, &ts_start));
1473:     PetscCall(TSSetSolution(ts_start, ts->vec_sol));
1474:     PetscCall(TSSetTime(ts_start, ts->ptime));
1475:     PetscCall(TSSetMaxSteps(ts_start, ts->steps + 1));
1476:     PetscCall(TSSetMaxTime(ts_start, ts->ptime + ts->time_step));
1477:     PetscCall(TSSetExactFinalTime(ts_start, TS_EXACTFINALTIME_STEPOVER));
1478:     PetscCall(TSSetTimeStep(ts_start, ts->time_step));
1479:     PetscCall(TSSetType(ts_start, TSARKIMEX));
1480:     PetscCall(TSARKIMEXSetFullyImplicit(ts_start, PETSC_TRUE));
1481:     PetscCall(TSARKIMEXSetType(ts_start, TSARKIMEX1BEE));

1483:     PetscCall(TSRestartStep(ts_start));
1484:     PetscCall(TSSolve(ts_start, ts->vec_sol));
1485:     PetscCall(TSGetTime(ts_start, &ts->ptime));
1486:     PetscCall(TSGetTimeStep(ts_start, &ts->time_step));

1488:     { /* Save the initial slope for the next step */
1489:       TS_ARKIMEX *ark_start = (TS_ARKIMEX *)ts_start->data;
1490:       PetscCall(VecCopy(ark_start->YdotI[ark_start->tableau->s - 1], Ydot0));
1491:     }
1492:     ts->steps++;

1494:     /* Set the correct TS in SNES */
1495:     /* We'll try to bypass this by changing the method on the fly */
1496:     {
1497:       PetscCall(TSGetSNES(ts, &snes));
1498:       PetscCall(TSSetSNES(ts, snes));
1499:     }
1500:     PetscCall(TSDestroy(&ts_start));
1501:   }

1503:   ark->status = TS_STEP_INCOMPLETE;
1504:   while (!ts->reason && ark->status != TS_STEP_COMPLETE) {
1505:     PetscReal t = ts->ptime;
1506:     PetscReal h = ts->time_step;
1507:     for (i = 0; i < s; i++) {
1508:       ark->stage_time = t + h * ct[i];
1509:       PetscCall(TSPreStage(ts, ark->stage_time));
1510:       if (At[i * s + i] == 0) { /* This stage is explicit */
1511:         PetscCheck(i == 0 || ts->equation_type < TS_EQ_IMPLICIT, PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "Explicit stages other than the first one are not supported for implicit problems");
1512:         PetscCall(VecCopy(ts->vec_sol, Y[i]));
1513:         for (j = 0; j < i; j++) w[j] = h * At[i * s + j];
1514:         PetscCall(VecMAXPY(Y[i], i, w, YdotI));
1515:         if (tab->additive && hasE) {
1516:           for (j = 0; j < i; j++) w[j] = h * A[i * s + j];
1517:           PetscCall(VecMAXPY(Y[i], i, w, YdotRHS));
1518:         }
1519:       } else {
1520:         ark->scoeff = 1. / At[i * s + i];
1521:         /* Ydot = shift*(Y-Z) */
1522:         PetscCall(VecCopy(ts->vec_sol, Z));
1523:         for (j = 0; j < i; j++) w[j] = h * At[i * s + j];
1524:         PetscCall(VecMAXPY(Z, i, w, YdotI));
1525:         if (tab->additive && hasE) {
1526:           for (j = 0; j < i; j++) w[j] = h * A[i * s + j];
1527:           PetscCall(VecMAXPY(Z, i, w, YdotRHS));
1528:         }
1529:         if (extrapolate && !ts->steprestart) {
1530:           /* Initial guess extrapolated from previous time step stage values */
1531:           PetscCall(TSExtrapolate_ARKIMEX(ts, c[i], Y[i]));
1532:         } else {
1533:           /* Initial guess taken from last stage */
1534:           PetscCall(VecCopy(i > 0 ? Y[i - 1] : ts->vec_sol, Y[i]));
1535:         }
1536:         PetscCall(TSGetSNES(ts, &snes));
1537:         PetscCall(SNESSolve(snes, NULL, Y[i]));
1538:         PetscCall(SNESGetIterationNumber(snes, &its));
1539:         PetscCall(SNESGetLinearSolveIterations(snes, &lits));
1540:         ts->snes_its += its;
1541:         ts->ksp_its += lits;
1542:         PetscCall(TSGetAdapt(ts, &adapt));
1543:         PetscCall(TSAdaptCheckStage(adapt, ts, ark->stage_time, Y[i], &stageok));
1544:         if (!stageok) {
1545:           /* We are likely rejecting the step because of solver or function domain problems so we should not attempt to
1546:            * use extrapolation to initialize the solves on the next attempt. */
1547:           extrapolate = PETSC_FALSE;
1548:           goto reject_step;
1549:         }
1550:       }
1551:       if (dirk || ts->equation_type >= TS_EQ_IMPLICIT) {
1552:         if (i == 0 && tab->explicit_first_stage) {
1553:           PetscCheck(tab->stiffly_accurate, PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "%s %s is not stiffly accurate and therefore explicit-first stage methods cannot be used if the equation is implicit because the slope cannot be evaluated",
1554:                      ((PetscObject)ts)->type_name, ark->tableau->name);
1555:           PetscCall(VecCopy(Ydot0, YdotI[0])); /* YdotI = YdotI(tn-1) */
1556:         } else {
1557:           PetscCall(VecAXPBYPCZ(YdotI[i], -ark->scoeff / h, ark->scoeff / h, 0, Z, Y[i])); /* YdotI = shift*(X-Z) */
1558:         }
1559:       } else {
1560:         if (i == 0 && tab->explicit_first_stage) {
1561:           PetscCall(VecZeroEntries(Ydot));
1562:           PetscCall(TSComputeIFunction(ts, t + h * ct[i], Y[i], Ydot, YdotI[i], ark->imex)); /* YdotI = -G(t,Y,0)   */
1563:           PetscCall(VecScale(YdotI[i], -1.0));
1564:         } else {
1565:           PetscCall(VecAXPBYPCZ(YdotI[i], -ark->scoeff / h, ark->scoeff / h, 0, Z, Y[i])); /* YdotI = shift*(X-Z) */
1566:         }
1567:         if (hasE) {
1568:           if (ark->imex) {
1569:             PetscCall(TSComputeRHSFunction(ts, t + h * c[i], Y[i], YdotRHS[i]));
1570:           } else {
1571:             PetscCall(VecZeroEntries(YdotRHS[i]));
1572:           }
1573:         }
1574:       }
1575:       PetscCall(TSPostStage(ts, ark->stage_time, i, Y));
1576:     }

1578:     ark->status = TS_STEP_INCOMPLETE;
1579:     PetscCall(TSEvaluateStep_ARKIMEX(ts, tab->order, ts->vec_sol, NULL));
1580:     ark->status = TS_STEP_PENDING;
1581:     PetscCall(TSGetAdapt(ts, &adapt));
1582:     PetscCall(TSAdaptCandidatesClear(adapt));
1583:     PetscCall(TSAdaptCandidateAdd(adapt, tab->name, tab->order, 1, tab->ccfl, (PetscReal)tab->s, PETSC_TRUE));
1584:     PetscCall(TSAdaptChoose(adapt, ts, ts->time_step, NULL, &next_time_step, &accept));
1585:     ark->status = accept ? TS_STEP_COMPLETE : TS_STEP_INCOMPLETE;
1586:     if (!accept) { /* Roll back the current step */
1587:       PetscCall(TSRollBack_ARKIMEX(ts));
1588:       ts->time_step = next_time_step;
1589:       goto reject_step;
1590:     }

1592:     ts->ptime += ts->time_step;
1593:     ts->time_step = next_time_step;
1594:     break;

1596:   reject_step:
1597:     ts->reject++;
1598:     accept = PETSC_FALSE;
1599:     if (!ts->reason && ++rejections > ts->max_reject && ts->max_reject >= 0) {
1600:       ts->reason = TS_DIVERGED_STEP_REJECTED;
1601:       PetscCall(PetscInfo(ts, "Step=%" PetscInt_FMT ", step rejections %" PetscInt_FMT " greater than current TS allowed, stopping solve\n", ts->steps, rejections));
1602:     }
1603:   }
1604:   PetscFunctionReturn(PETSC_SUCCESS);
1605: }

1607: /*
1608:   This adjoint step function assumes the partitioned ODE system has an identity mass matrix and thus can be represented in the form
1609:   Udot = H(t,U) + G(t,U)
1610:   This corresponds to F(t,U,Udot) = Udot-H(t,U).

1612:   The complete adjoint equations are
1613:   (shift*I - dHdu) lambda_s[i]   = 1/at[i][i] (
1614:     dGdU (b_i lambda_{n+1} + \sum_{j=i+1}^s a[j][i] lambda_s[j])
1615:     + dHdU (bt[i] lambda_{n+1} +  \sum_{j=i+1}^s at[j][i] lambda_s[j])), i = s-1,...,0
1616:   lambda_n = lambda_{n+1} + \sum_{j=1}^s lambda_s[j]
1617:   mu_{n+1}[i]   = h (at[i][i] dHdP lambda_s[i]
1618:     + dGdP (b_i lambda_{n+1} + \sum_{j=i+1}^s a[j][i] lambda_s[j])
1619:     + dHdP (bt[i] lambda_{n+1} + \sum_{j=i+1}^s at[j][i] lambda_s[j])), i = s-1,...,0
1620:   mu_n = mu_{n+1} + \sum_{j=1}^s mu_{n+1}[j]
1621:   where shift = 1/(at[i][i]*h)

1623:   If at[i][i] is 0, the first equation falls back to
1624:   lambda_s[i] = h (
1625:     (b_i dGdU + bt[i] dHdU) lambda_{n+1} + dGdU \sum_{j=i+1}^s a[j][i] lambda_s[j]
1626:     + dHdU \sum_{j=i+1}^s at[j][i] lambda_s[j])

1628: */
1629: static PetscErrorCode TSAdjointStep_ARKIMEX(TS ts)
1630: {
1631:   TS_ARKIMEX      *ark = (TS_ARKIMEX *)ts->data;
1632:   ARKTableau       tab = ark->tableau;
1633:   const PetscInt   s   = tab->s;
1634:   const PetscReal *At = tab->At, *A = tab->A, *ct = tab->ct, *c = tab->c, *b = tab->b, *bt = tab->bt;
1635:   PetscScalar     *w = ark->work;
1636:   Vec             *Y = ark->Y, Ydot = ark->Ydot, *VecsDeltaLam = ark->VecsDeltaLam, *VecsSensiTemp = ark->VecsSensiTemp, *VecsSensiPTemp = ark->VecsSensiPTemp;
1637:   Mat              Jex, Jim, Jimpre;
1638:   PetscInt         i, j, nadj;
1639:   PetscReal        t                 = ts->ptime, stage_time_ex;
1640:   PetscReal        adjoint_time_step = -ts->time_step; /* always positive since ts->time_step is negative */
1641:   KSP              ksp;

1643:   PetscFunctionBegin;
1644:   ark->status = TS_STEP_INCOMPLETE;
1645:   PetscCall(SNESGetKSP(ts->snes, &ksp));
1646:   PetscCall(TSGetRHSMats_Private(ts, &Jex, NULL));
1647:   PetscCall(TSGetIJacobian(ts, &Jim, &Jimpre, NULL, NULL));

1649:   for (i = s - 1; i >= 0; i--) {
1650:     ark->stage_time = t - adjoint_time_step * (1.0 - ct[i]);
1651:     stage_time_ex   = t - adjoint_time_step * (1.0 - c[i]);
1652:     if (At[i * s + i] == 0) { // This stage is explicit
1653:       ark->scoeff = 0.;
1654:     } else {
1655:       ark->scoeff = -1. / At[i * s + i]; // this makes shift=ark->scoeff/ts->time_step positive since ts->time_step is negative
1656:     }
1657:     PetscCall(TSComputeSNESJacobian(ts, Y[i], Jim, Jimpre));
1658:     PetscCall(TSComputeRHSJacobian(ts, stage_time_ex, Y[i], Jex, Jex));
1659:     if (ts->vecs_sensip) {
1660:       PetscCall(TSComputeIJacobianP(ts, ark->stage_time, Y[i], Ydot, ark->scoeff / adjoint_time_step, ts->Jacp, PETSC_TRUE)); // get dFdP (-dHdP), Ydot not really used since mass matrix is identity
1661:       PetscCall(TSComputeRHSJacobianP(ts, stage_time_ex, Y[i], ts->Jacprhs));                                                 // get dGdP
1662:     }
1663:     /* Build RHS (stored in VecsDeltaLam) for first-order adjoint */
1664:     for (nadj = 0; nadj < ts->numcost; nadj++) {
1665:       /* build implicit part */
1666:       PetscCall(VecSet(VecsSensiTemp[nadj], 0));
1667:       if (s - i - 1 > 0) {
1668:         /* Temp = -\sum_{j=i+1}^s at[j][i] lambda_s[j] */
1669:         for (j = i + 1; j < s; j++) w[j - i - 1] = -At[j * s + i];
1670:         PetscCall(VecMAXPY(VecsSensiTemp[nadj], s - i - 1, w, &VecsDeltaLam[nadj * s + i + 1]));
1671:       }
1672:       /* Temp = Temp - bt[i] lambda_{n+1} */
1673:       PetscCall(VecAXPY(VecsSensiTemp[nadj], -bt[i], ts->vecs_sensi[nadj]));
1674:       if (bt[i] || s - i - 1 > 0) {
1675:         /* (shift I - dHdU) Temp */
1676:         PetscCall(MatMultTranspose(Jim, VecsSensiTemp[nadj], VecsDeltaLam[nadj * s + i]));
1677:         /* cancel out shift Temp where shift=-scoeff/h */
1678:         PetscCall(VecAXPY(VecsDeltaLam[nadj * s + i], ark->scoeff / adjoint_time_step, VecsSensiTemp[nadj]));
1679:         if (ts->vecs_sensip) {
1680:           /* - dHdP Temp */
1681:           PetscCall(MatMultTranspose(ts->Jacp, VecsSensiTemp[nadj], VecsSensiPTemp[nadj]));
1682:           /* mu_n += -h dHdP Temp */
1683:           PetscCall(VecAXPY(ts->vecs_sensip[nadj], adjoint_time_step, VecsSensiPTemp[nadj]));
1684:         }
1685:       } else {
1686:         PetscCall(VecSet(VecsDeltaLam[nadj * s + i], 0)); // make sure it is initialized
1687:       }
1688:       /* build explicit part */
1689:       PetscCall(VecSet(VecsSensiTemp[nadj], 0));
1690:       if (s - i - 1 > 0) {
1691:         /* Temp = \sum_{j=i+1}^s a[j][i] lambda_s[j] */
1692:         for (j = i + 1; j < s; j++) w[j - i - 1] = A[j * s + i];
1693:         PetscCall(VecMAXPY(VecsSensiTemp[nadj], s - i - 1, w, &VecsDeltaLam[nadj * s + i + 1]));
1694:       }
1695:       /* Temp = Temp + b[i] lambda_{n+1} */
1696:       PetscCall(VecAXPY(VecsSensiTemp[nadj], b[i], ts->vecs_sensi[nadj]));
1697:       if (b[i] || s - i - 1 > 0) {
1698:         /* dGdU Temp */
1699:         PetscCall(MatMultTransposeAdd(Jex, VecsSensiTemp[nadj], VecsDeltaLam[nadj * s + i], VecsDeltaLam[nadj * s + i]));
1700:         if (ts->vecs_sensip) {
1701:           /* dGdP Temp */
1702:           PetscCall(MatMultTranspose(ts->Jacprhs, VecsSensiTemp[nadj], VecsSensiPTemp[nadj]));
1703:           /* mu_n += h dGdP Temp */
1704:           PetscCall(VecAXPY(ts->vecs_sensip[nadj], adjoint_time_step, VecsSensiPTemp[nadj]));
1705:         }
1706:       }
1707:       /* Build LHS for first-order adjoint */
1708:       if (At[i * s + i] == 0) { // This stage is explicit
1709:         PetscCall(VecScale(VecsDeltaLam[nadj * s + i], adjoint_time_step));
1710:       } else {
1711:         KSP                ksp;
1712:         KSPConvergedReason kspreason;
1713:         PetscCall(SNESGetKSP(ts->snes, &ksp));
1714:         PetscCall(KSPSetOperators(ksp, Jim, Jimpre));
1715:         PetscCall(VecScale(VecsDeltaLam[nadj * s + i], 1. / At[i * s + i]));
1716:         PetscCall(KSPSolveTranspose(ksp, VecsDeltaLam[nadj * s + i], VecsDeltaLam[nadj * s + i]));
1717:         PetscCall(KSPGetConvergedReason(ksp, &kspreason));
1718:         if (kspreason < 0) {
1719:           ts->reason = TSADJOINT_DIVERGED_LINEAR_SOLVE;
1720:           PetscCall(PetscInfo(ts, "Step=%" PetscInt_FMT ", %" PetscInt_FMT "th cost function, transposed linear solve fails, stopping 1st-order adjoint solve\n", ts->steps, nadj));
1721:         }
1722:         if (ts->vecs_sensip) {
1723:           /* -dHdP lambda_s[i] */
1724:           PetscCall(MatMultTranspose(ts->Jacp, VecsDeltaLam[nadj * s + i], VecsSensiPTemp[nadj]));
1725:           /* mu_n += h at[i][i] dHdP lambda_s[i] */
1726:           PetscCall(VecAXPY(ts->vecs_sensip[nadj], -At[i * s + i] * adjoint_time_step, VecsSensiPTemp[nadj]));
1727:         }
1728:       }
1729:     }
1730:   }
1731:   for (j = 0; j < s; j++) w[j] = 1.0;
1732:   for (nadj = 0; nadj < ts->numcost; nadj++) // no need to do this for mu's
1733:     PetscCall(VecMAXPY(ts->vecs_sensi[nadj], s, w, &VecsDeltaLam[nadj * s]));
1734:   ark->status = TS_STEP_COMPLETE;
1735:   PetscFunctionReturn(PETSC_SUCCESS);
1736: }

1738: static PetscErrorCode TSInterpolate_ARKIMEX(TS ts, PetscReal itime, Vec X)
1739: {
1740:   TS_ARKIMEX      *ark = (TS_ARKIMEX *)ts->data;
1741:   ARKTableau       tab = ark->tableau;
1742:   PetscInt         s = tab->s, pinterp = tab->pinterp, i, j;
1743:   PetscReal        h;
1744:   PetscReal        tt, t;
1745:   PetscScalar     *bt = ark->work, *b = ark->work + s;
1746:   const PetscReal *Bt = tab->binterpt, *B = tab->binterp;

1748:   PetscFunctionBegin;
1749:   PetscCheck(Bt && B, PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "%s %s does not have an interpolation formula", ((PetscObject)ts)->type_name, ark->tableau->name);
1750:   switch (ark->status) {
1751:   case TS_STEP_INCOMPLETE:
1752:   case TS_STEP_PENDING:
1753:     h = ts->time_step;
1754:     t = (itime - ts->ptime) / h;
1755:     break;
1756:   case TS_STEP_COMPLETE:
1757:     h = ts->ptime - ts->ptime_prev;
1758:     t = (itime - ts->ptime) / h + 1; /* In the interval [0,1] */
1759:     break;
1760:   default:
1761:     SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_PLIB, "Invalid TSStepStatus");
1762:   }
1763:   for (i = 0; i < s; i++) bt[i] = b[i] = 0;
1764:   for (j = 0, tt = t; j < pinterp; j++, tt *= t) {
1765:     for (i = 0; i < s; i++) {
1766:       bt[i] += h * Bt[i * pinterp + j] * tt;
1767:       b[i] += h * B[i * pinterp + j] * tt;
1768:     }
1769:   }
1770:   PetscCall(VecCopy(ark->Y[0], X));
1771:   PetscCall(VecMAXPY(X, s, bt, ark->YdotI));
1772:   if (tab->additive) {
1773:     PetscBool hasE;
1774:     PetscCall(TSHasRHSFunction(ts, &hasE));
1775:     if (hasE) PetscCall(VecMAXPY(X, s, b, ark->YdotRHS));
1776:   }
1777:   PetscFunctionReturn(PETSC_SUCCESS);
1778: }

1780: static PetscErrorCode TSExtrapolate_ARKIMEX(TS ts, PetscReal c, Vec X)
1781: {
1782:   TS_ARKIMEX      *ark = (TS_ARKIMEX *)ts->data;
1783:   ARKTableau       tab = ark->tableau;
1784:   PetscInt         s = tab->s, pinterp = tab->pinterp, i, j;
1785:   PetscReal        h, h_prev, t, tt;
1786:   PetscScalar     *bt = ark->work, *b = ark->work + s;
1787:   const PetscReal *Bt = tab->binterpt, *B = tab->binterp;

1789:   PetscFunctionBegin;
1790:   PetscCheck(Bt && B, PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "TSARKIMEX %s does not have an interpolation formula", ark->tableau->name);
1791:   h      = ts->time_step;
1792:   h_prev = ts->ptime - ts->ptime_prev;
1793:   t      = 1 + h / h_prev * c;
1794:   for (i = 0; i < s; i++) bt[i] = b[i] = 0;
1795:   for (j = 0, tt = t; j < pinterp; j++, tt *= t) {
1796:     for (i = 0; i < s; i++) {
1797:       bt[i] += h * Bt[i * pinterp + j] * tt;
1798:       b[i] += h * B[i * pinterp + j] * tt;
1799:     }
1800:   }
1801:   PetscCheck(ark->Y_prev, PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "Stages from previous step have not been stored");
1802:   PetscCall(VecCopy(ark->Y_prev[0], X));
1803:   PetscCall(VecMAXPY(X, s, bt, ark->YdotI_prev));
1804:   if (tab->additive) {
1805:     PetscBool hasE;
1806:     PetscCall(TSHasRHSFunction(ts, &hasE));
1807:     if (hasE) PetscCall(VecMAXPY(X, s, b, ark->YdotRHS_prev));
1808:   }
1809:   PetscFunctionReturn(PETSC_SUCCESS);
1810: }

1812: static PetscErrorCode TSARKIMEXTableauReset(TS ts)
1813: {
1814:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;
1815:   ARKTableau  tab = ark->tableau;

1817:   PetscFunctionBegin;
1818:   if (!tab) PetscFunctionReturn(PETSC_SUCCESS);
1819:   PetscCall(PetscFree(ark->work));
1820:   PetscCall(VecDestroyVecs(tab->s, &ark->Y));
1821:   PetscCall(VecDestroyVecs(tab->s, &ark->YdotI));
1822:   PetscCall(VecDestroyVecs(tab->s, &ark->YdotRHS));
1823:   PetscCall(VecDestroyVecs(tab->s, &ark->Y_prev));
1824:   PetscCall(VecDestroyVecs(tab->s, &ark->YdotI_prev));
1825:   PetscCall(VecDestroyVecs(tab->s, &ark->YdotRHS_prev));
1826:   PetscFunctionReturn(PETSC_SUCCESS);
1827: }

1829: static PetscErrorCode TSReset_ARKIMEX(TS ts)
1830: {
1831:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;

1833:   PetscFunctionBegin;
1834:   PetscCall(TSARKIMEXTableauReset(ts));
1835:   PetscCall(VecDestroy(&ark->Ydot));
1836:   PetscCall(VecDestroy(&ark->Ydot0));
1837:   PetscCall(VecDestroy(&ark->Z));
1838:   PetscFunctionReturn(PETSC_SUCCESS);
1839: }

1841: static PetscErrorCode TSAdjointReset_ARKIMEX(TS ts)
1842: {
1843:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;
1844:   ARKTableau  tab = ark->tableau;

1846:   PetscFunctionBegin;
1847:   PetscCall(VecDestroyVecs(tab->s * ts->numcost, &ark->VecsDeltaLam));
1848:   PetscCall(VecDestroyVecs(ts->numcost, &ark->VecsSensiTemp));
1849:   PetscCall(VecDestroyVecs(ts->numcost, &ark->VecsSensiPTemp));
1850:   PetscFunctionReturn(PETSC_SUCCESS);
1851: }

1853: static PetscErrorCode TSARKIMEXGetVecs(TS ts, DM dm, Vec *Z, Vec *Ydot)
1854: {
1855:   TS_ARKIMEX *ax = (TS_ARKIMEX *)ts->data;

1857:   PetscFunctionBegin;
1858:   if (Z) {
1859:     if (dm && dm != ts->dm) {
1860:       PetscCall(DMGetNamedGlobalVector(dm, "TSARKIMEX_Z", Z));
1861:     } else *Z = ax->Z;
1862:   }
1863:   if (Ydot) {
1864:     if (dm && dm != ts->dm) {
1865:       PetscCall(DMGetNamedGlobalVector(dm, "TSARKIMEX_Ydot", Ydot));
1866:     } else *Ydot = ax->Ydot;
1867:   }
1868:   PetscFunctionReturn(PETSC_SUCCESS);
1869: }

1871: static PetscErrorCode TSARKIMEXRestoreVecs(TS ts, DM dm, Vec *Z, Vec *Ydot)
1872: {
1873:   PetscFunctionBegin;
1874:   if (Z) {
1875:     if (dm && dm != ts->dm) PetscCall(DMRestoreNamedGlobalVector(dm, "TSARKIMEX_Z", Z));
1876:   }
1877:   if (Ydot) {
1878:     if (dm && dm != ts->dm) PetscCall(DMRestoreNamedGlobalVector(dm, "TSARKIMEX_Ydot", Ydot));
1879:   }
1880:   PetscFunctionReturn(PETSC_SUCCESS);
1881: }

1883: PETSC_SINGLE_LIBRARY_INTERN PetscErrorCode MatFindNonzeroRowsOrCols_Basic(Mat, PetscBool, PetscReal, IS *);

1885: /* This defines the nonlinear equation that is to be solved with SNES */
1886: static PetscErrorCode SNESTSFormFunction_ARKIMEX(SNES snes, Vec X, Vec F, TS ts)
1887: {
1888:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;
1889:   DM          dm, dmsave;
1890:   Vec         Z, Ydot;

1892:   PetscFunctionBegin;
1893:   PetscCall(SNESGetDM(snes, &dm));
1894:   PetscCall(TSARKIMEXGetVecs(ts, dm, &Z, &Ydot));
1895:   dmsave = ts->dm;
1896:   ts->dm = dm;

1898:   if (ark->scoeff == PETSC_MAX_REAL) {
1899:     /* We are solving F(t_n,x_n,xdot) = 0 to start the method */
1900:     PetscCall(TSComputeIFunction(ts, ark->stage_time, Z, X, F, ark->imex));
1901:   } else {
1902:     PetscReal shift = ark->scoeff / ts->time_step;
1903:     PetscCall(VecAXPBYPCZ(Ydot, -shift, shift, 0, Z, X)); /* Ydot = shift*(X-Z) */
1904:     PetscCall(TSComputeIFunction(ts, ark->stage_time, X, Ydot, F, ark->imex));
1905:   }

1907:   ts->dm = dmsave;
1908:   PetscCall(TSARKIMEXRestoreVecs(ts, dm, &Z, &Ydot));
1909:   PetscFunctionReturn(PETSC_SUCCESS);
1910: }

1912: static PetscErrorCode SNESTSFormJacobian_ARKIMEX(SNES snes, Vec X, Mat A, Mat B, TS ts)
1913: {
1914:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;
1915:   DM          dm, dmsave;
1916:   Vec         Ydot, Z;
1917:   PetscReal   shift;

1919:   PetscFunctionBegin;
1920:   PetscCall(SNESGetDM(snes, &dm));
1921:   PetscCall(TSARKIMEXGetVecs(ts, dm, &Z, &Ydot));
1922:   /* ark->Ydot has already been computed in SNESTSFormFunction_ARKIMEX (SNES guarantees this) */
1923:   dmsave = ts->dm;
1924:   ts->dm = dm;

1926:   if (ark->scoeff == PETSC_MAX_REAL) {
1927:     PetscBool hasZeroRows;
1928:     IS        alg_is;

1930:     /* We are solving F(t_n,x_n,xdot) = 0 to start the method
1931:        Jed's proposal is to compute with a very large shift and then scale back the matrix */
1932:     shift = 1.0 / PETSC_MACHINE_EPSILON;
1933:     PetscCall(TSComputeIJacobian(ts, ark->stage_time, Z, X, shift, A, B, ark->imex));
1934:     PetscCall(MatScale(B, PETSC_MACHINE_EPSILON));
1935:     /* DAEs need special handling for preconditioning purposes only.
1936:        We need to locate the algebraic variables and modify the preconditioning matrix by
1937:        calling MatZeroRows with identity on these variables.
1938:        We must store the IS in the DM since this function can be called by multilevel solvers.
1939:     */
1940:     PetscCall(PetscObjectQuery((PetscObject)dm, "TSARKIMEX_ALG_IS", (PetscObject *)&alg_is));
1941:     if (!alg_is) {
1942:       PetscInt m, n;
1943:       IS       nonzeroRows;

1945:       PetscCall(MatViewFromOptions(B, (PetscObject)snes, "-ts_arkimex_alg_mat_view_pre"));
1946:       PetscCall(MatFindNonzeroRowsOrCols_Basic(B, PETSC_FALSE, 100 * PETSC_MACHINE_EPSILON, &nonzeroRows));
1947:       if (nonzeroRows) PetscCall(ISViewFromOptions(nonzeroRows, (PetscObject)snes, "-ts_arkimex_alg_is_view_pre"));
1948:       PetscCall(MatGetOwnershipRange(B, &m, &n));
1949:       if (nonzeroRows) PetscCall(ISComplement(nonzeroRows, m, n, &alg_is));
1950:       else PetscCall(ISCreateStride(PetscObjectComm((PetscObject)snes), 0, m, 1, &alg_is));
1951:       PetscCall(ISDestroy(&nonzeroRows));
1952:       PetscCall(PetscObjectCompose((PetscObject)dm, "TSARKIMEX_ALG_IS", (PetscObject)alg_is));
1953:       PetscCall(ISDestroy(&alg_is));
1954:     }
1955:     PetscCall(PetscObjectQuery((PetscObject)dm, "TSARKIMEX_ALG_IS", (PetscObject *)&alg_is));
1956:     PetscCall(ISViewFromOptions(alg_is, (PetscObject)snes, "-ts_arkimex_alg_is_view"));
1957:     PetscCall(MatHasOperation(B, MATOP_ZERO_ROWS, &hasZeroRows));
1958:     if (hasZeroRows) {
1959:       /* the default of AIJ is to not keep the pattern! We should probably change it someday */
1960:       PetscCall(MatSetOption(B, MAT_KEEP_NONZERO_PATTERN, PETSC_TRUE));
1961:       PetscCall(MatZeroRowsIS(B, alg_is, 1.0, NULL, NULL));
1962:     }
1963:     PetscCall(MatViewFromOptions(B, (PetscObject)snes, "-ts_arkimex_alg_mat_view"));
1964:     if (A != B) PetscCall(MatScale(A, PETSC_MACHINE_EPSILON));
1965:   } else {
1966:     shift = ark->scoeff / ts->time_step;
1967:     PetscCall(TSComputeIJacobian(ts, ark->stage_time, X, Ydot, shift, A, B, ark->imex));
1968:   }
1969:   ts->dm = dmsave;
1970:   PetscCall(TSARKIMEXRestoreVecs(ts, dm, &Z, &Ydot));
1971:   PetscFunctionReturn(PETSC_SUCCESS);
1972: }

1974: static PetscErrorCode TSGetStages_ARKIMEX(TS ts, PetscInt *ns, Vec *Y[])
1975: {
1976:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;

1978:   PetscFunctionBegin;
1979:   if (ns) *ns = ark->tableau->s;
1980:   if (Y) *Y = ark->Y;
1981:   PetscFunctionReturn(PETSC_SUCCESS);
1982: }

1984: static PetscErrorCode DMCoarsenHook_TSARKIMEX(DM fine, DM coarse, void *ctx)
1985: {
1986:   PetscFunctionBegin;
1987:   PetscFunctionReturn(PETSC_SUCCESS);
1988: }

1990: static PetscErrorCode DMRestrictHook_TSARKIMEX(DM fine, Mat restrct, Vec rscale, Mat inject, DM coarse, void *ctx)
1991: {
1992:   TS  ts = (TS)ctx;
1993:   Vec Z, Z_c;

1995:   PetscFunctionBegin;
1996:   PetscCall(TSARKIMEXGetVecs(ts, fine, &Z, NULL));
1997:   PetscCall(TSARKIMEXGetVecs(ts, coarse, &Z_c, NULL));
1998:   PetscCall(MatRestrict(restrct, Z, Z_c));
1999:   PetscCall(VecPointwiseMult(Z_c, rscale, Z_c));
2000:   PetscCall(TSARKIMEXRestoreVecs(ts, fine, &Z, NULL));
2001:   PetscCall(TSARKIMEXRestoreVecs(ts, coarse, &Z_c, NULL));
2002:   PetscFunctionReturn(PETSC_SUCCESS);
2003: }

2005: static PetscErrorCode DMSubDomainHook_TSARKIMEX(DM dm, DM subdm, void *ctx)
2006: {
2007:   PetscFunctionBegin;
2008:   PetscFunctionReturn(PETSC_SUCCESS);
2009: }

2011: static PetscErrorCode DMSubDomainRestrictHook_TSARKIMEX(DM dm, VecScatter gscat, VecScatter lscat, DM subdm, void *ctx)
2012: {
2013:   TS  ts = (TS)ctx;
2014:   Vec Z, Z_c;

2016:   PetscFunctionBegin;
2017:   PetscCall(TSARKIMEXGetVecs(ts, dm, &Z, NULL));
2018:   PetscCall(TSARKIMEXGetVecs(ts, subdm, &Z_c, NULL));

2020:   PetscCall(VecScatterBegin(gscat, Z, Z_c, INSERT_VALUES, SCATTER_FORWARD));
2021:   PetscCall(VecScatterEnd(gscat, Z, Z_c, INSERT_VALUES, SCATTER_FORWARD));

2023:   PetscCall(TSARKIMEXRestoreVecs(ts, dm, &Z, NULL));
2024:   PetscCall(TSARKIMEXRestoreVecs(ts, subdm, &Z_c, NULL));
2025:   PetscFunctionReturn(PETSC_SUCCESS);
2026: }

2028: static PetscErrorCode TSARKIMEXTableauSetUp(TS ts)
2029: {
2030:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;
2031:   ARKTableau  tab = ark->tableau;

2033:   PetscFunctionBegin;
2034:   PetscCall(PetscMalloc1(2 * tab->s, &ark->work));
2035:   PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &ark->Y));
2036:   PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &ark->YdotI));
2037:   if (tab->additive) PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &ark->YdotRHS));
2038:   if (ark->extrapolate) {
2039:     PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &ark->Y_prev));
2040:     PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &ark->YdotI_prev));
2041:     if (tab->additive) PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &ark->YdotRHS_prev));
2042:   }
2043:   PetscFunctionReturn(PETSC_SUCCESS);
2044: }

2046: static PetscErrorCode TSSetUp_ARKIMEX(TS ts)
2047: {
2048:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;
2049:   DM          dm;
2050:   SNES        snes;

2052:   PetscFunctionBegin;
2053:   PetscCall(TSARKIMEXTableauSetUp(ts));
2054:   PetscCall(VecDuplicate(ts->vec_sol, &ark->Ydot));
2055:   PetscCall(VecDuplicate(ts->vec_sol, &ark->Ydot0));
2056:   PetscCall(VecDuplicate(ts->vec_sol, &ark->Z));
2057:   PetscCall(TSGetDM(ts, &dm));
2058:   PetscCall(DMCoarsenHookAdd(dm, DMCoarsenHook_TSARKIMEX, DMRestrictHook_TSARKIMEX, ts));
2059:   PetscCall(DMSubDomainHookAdd(dm, DMSubDomainHook_TSARKIMEX, DMSubDomainRestrictHook_TSARKIMEX, ts));
2060:   PetscCall(TSGetSNES(ts, &snes));
2061:   PetscFunctionReturn(PETSC_SUCCESS);
2062: }

2064: static PetscErrorCode TSAdjointSetUp_ARKIMEX(TS ts)
2065: {
2066:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;
2067:   ARKTableau  tab = ark->tableau;

2069:   PetscFunctionBegin;
2070:   PetscCall(VecDuplicateVecs(ts->vecs_sensi[0], tab->s * ts->numcost, &ark->VecsDeltaLam));
2071:   PetscCall(VecDuplicateVecs(ts->vecs_sensi[0], ts->numcost, &ark->VecsSensiTemp));
2072:   if (ts->vecs_sensip) { PetscCall(VecDuplicateVecs(ts->vecs_sensip[0], ts->numcost, &ark->VecsSensiPTemp)); }
2073:   if (PetscDefined(USE_DEBUG)) {
2074:     PetscBool id = PETSC_FALSE;
2075:     PetscCall(TSARKIMEXTestMassIdentity(ts, &id));
2076:     PetscCheck(id, PetscObjectComm((PetscObject)ts), PETSC_ERR_ARG_INCOMP, "Adjoint ARKIMEX requires an identity mass matrix, however the TSIFunction you provided does not utilize an identity mass matrix");
2077:   }
2078:   PetscFunctionReturn(PETSC_SUCCESS);
2079: }

2081: static PetscErrorCode TSSetFromOptions_ARKIMEX(TS ts, PetscOptionItems *PetscOptionsObject)
2082: {
2083:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;
2084:   PetscBool   dirk;

2086:   PetscFunctionBegin;
2087:   PetscCall(PetscObjectTypeCompare((PetscObject)ts, TSDIRK, &dirk));
2088:   PetscOptionsHeadBegin(PetscOptionsObject, dirk ? "DIRK ODE solver options" : "ARKIMEX ODE solver options");
2089:   {
2090:     ARKTableauLink link;
2091:     PetscInt       count, choice;
2092:     PetscBool      flg;
2093:     const char   **namelist;
2094:     for (link = ARKTableauList, count = 0; link; link = link->next) {
2095:       if (!dirk && link->tab.additive) count++;
2096:       if (dirk && !link->tab.additive) count++;
2097:     }
2098:     PetscCall(PetscMalloc1(count, (char ***)&namelist));
2099:     for (link = ARKTableauList, count = 0; link; link = link->next) {
2100:       if (!dirk && link->tab.additive) namelist[count++] = link->tab.name;
2101:       if (dirk && !link->tab.additive) namelist[count++] = link->tab.name;
2102:     }
2103:     if (dirk) {
2104:       PetscCall(PetscOptionsEList("-ts_dirk_type", "Family of DIRK method", "TSDIRKSetType", (const char *const *)namelist, count, ark->tableau->name, &choice, &flg));
2105:       if (flg) PetscCall(TSDIRKSetType(ts, namelist[choice]));
2106:     } else {
2107:       PetscCall(PetscOptionsEList("-ts_arkimex_type", "Family of ARK IMEX method", "TSARKIMEXSetType", (const char *const *)namelist, count, ark->tableau->name, &choice, &flg));
2108:       if (flg) PetscCall(TSARKIMEXSetType(ts, namelist[choice]));
2109:       flg = (PetscBool)!ark->imex;
2110:       PetscCall(PetscOptionsBool("-ts_arkimex_fully_implicit", "Solve the problem fully implicitly", "TSARKIMEXSetFullyImplicit", flg, &flg, NULL));
2111:       ark->imex = (PetscBool)!flg;
2112:     }
2113:     PetscCall(PetscFree(namelist));
2114:     PetscCall(PetscOptionsBool("-ts_arkimex_initial_guess_extrapolate", "Extrapolate the initial guess for the stage solution from stage values of the previous time step", "", ark->extrapolate, &ark->extrapolate, NULL));
2115:   }
2116:   PetscOptionsHeadEnd();
2117:   PetscFunctionReturn(PETSC_SUCCESS);
2118: }

2120: static PetscErrorCode TSView_ARKIMEX(TS ts, PetscViewer viewer)
2121: {
2122:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;
2123:   PetscBool   iascii, dirk;

2125:   PetscFunctionBegin;
2126:   PetscCall(PetscObjectTypeCompare((PetscObject)ts, TSDIRK, &dirk));
2127:   PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &iascii));
2128:   if (iascii) {
2129:     PetscViewerFormat format;
2130:     ARKTableau        tab = ark->tableau;
2131:     TSARKIMEXType     arktype;
2132:     char              buf[2048];
2133:     PetscBool         flg;

2135:     PetscCall(TSARKIMEXGetType(ts, &arktype));
2136:     PetscCall(TSARKIMEXGetFullyImplicit(ts, &flg));
2137:     PetscCall(PetscViewerASCIIPrintf(viewer, "  %s %s\n", dirk ? "DIRK" : "ARK IMEX", arktype));
2138:     PetscCall(PetscFormatRealArray(buf, sizeof(buf), "% 8.6f", tab->s, tab->ct));
2139:     PetscCall(PetscViewerASCIIPrintf(viewer, "  %sabscissa       ct = %s\n", dirk ? "" : "Stiff ", buf));
2140:     PetscCall(PetscViewerGetFormat(viewer, &format));
2141:     if (format == PETSC_VIEWER_ASCII_INFO_DETAIL) {
2142:       PetscCall(PetscViewerASCIIPrintf(viewer, "  %sAt =\n", dirk ? "" : "Stiff "));
2143:       for (PetscInt i = 0; i < tab->s; i++) {
2144:         PetscCall(PetscFormatRealArray(buf, sizeof(buf), "% 8.6f", tab->s, tab->At + i * tab->s));
2145:         PetscCall(PetscViewerASCIIPrintf(viewer, "    %s\n", buf));
2146:       }
2147:       PetscCall(PetscFormatRealArray(buf, sizeof(buf), "% 8.6f", tab->s, tab->bt));
2148:       PetscCall(PetscViewerASCIIPrintf(viewer, "  %sbt = %s\n", dirk ? "" : "Stiff ", buf));
2149:       PetscCall(PetscFormatRealArray(buf, sizeof(buf), "% 8.6f", tab->s, tab->bembedt));
2150:       PetscCall(PetscViewerASCIIPrintf(viewer, "  %sbet = %s\n", dirk ? "" : "Stiff ", buf));
2151:     }
2152:     PetscCall(PetscViewerASCIIPrintf(viewer, "Fully implicit: %s\n", flg ? "yes" : "no"));
2153:     PetscCall(PetscViewerASCIIPrintf(viewer, "Stiffly accurate: %s\n", tab->stiffly_accurate ? "yes" : "no"));
2154:     PetscCall(PetscViewerASCIIPrintf(viewer, "Explicit first stage: %s\n", tab->explicit_first_stage ? "yes" : "no"));
2155:     PetscCall(PetscViewerASCIIPrintf(viewer, "FSAL property: %s\n", tab->FSAL_implicit ? "yes" : "no"));
2156:     if (!dirk) {
2157:       PetscCall(PetscFormatRealArray(buf, sizeof(buf), "% 8.6f", tab->s, tab->c));
2158:       PetscCall(PetscViewerASCIIPrintf(viewer, "  Nonstiff abscissa     c = %s\n", buf));
2159:     }
2160:   }
2161:   PetscFunctionReturn(PETSC_SUCCESS);
2162: }

2164: static PetscErrorCode TSLoad_ARKIMEX(TS ts, PetscViewer viewer)
2165: {
2166:   SNES    snes;
2167:   TSAdapt adapt;

2169:   PetscFunctionBegin;
2170:   PetscCall(TSGetAdapt(ts, &adapt));
2171:   PetscCall(TSAdaptLoad(adapt, viewer));
2172:   PetscCall(TSGetSNES(ts, &snes));
2173:   PetscCall(SNESLoad(snes, viewer));
2174:   /* function and Jacobian context for SNES when used with TS is always ts object */
2175:   PetscCall(SNESSetFunction(snes, NULL, NULL, ts));
2176:   PetscCall(SNESSetJacobian(snes, NULL, NULL, NULL, ts));
2177:   PetscFunctionReturn(PETSC_SUCCESS);
2178: }

2180: /*@C
2181:   TSARKIMEXSetType - Set the type of `TSARKIMEX` scheme

2183:   Logically Collective

2185:   Input Parameters:
2186: + ts      - timestepping context
2187: - arktype - type of `TSARKIMEX` scheme

2189:   Options Database Key:
2190: . -ts_arkimex_type <1bee,a2,l2,ars122,2c,2d,2e,prssp2,3,bpr3,ars443,4,5> - set `TSARKIMEX` scheme type

2192:   Level: intermediate

2194: .seealso: [](ch_ts), `TSARKIMEXGetType()`, `TSARKIMEX`, `TSARKIMEXType`, `TSARKIMEX1BEE`, `TSARKIMEXA2`, `TSARKIMEXL2`, `TSARKIMEXARS122`, `TSARKIMEX2C`, `TSARKIMEX2D`, `TSARKIMEX2E`, `TSARKIMEXPRSSP2`,
2195:           `TSARKIMEX3`, `TSARKIMEXBPR3`, `TSARKIMEXARS443`, `TSARKIMEX4`, `TSARKIMEX5`
2196: @*/
2197: PetscErrorCode TSARKIMEXSetType(TS ts, TSARKIMEXType arktype)
2198: {
2199:   PetscFunctionBegin;
2201:   PetscAssertPointer(arktype, 2);
2202:   PetscTryMethod(ts, "TSARKIMEXSetType_C", (TS, TSARKIMEXType), (ts, arktype));
2203:   PetscFunctionReturn(PETSC_SUCCESS);
2204: }

2206: /*@C
2207:   TSARKIMEXGetType - Get the type of `TSARKIMEX` scheme

2209:   Logically Collective

2211:   Input Parameter:
2212: . ts - timestepping context

2214:   Output Parameter:
2215: . arktype - type of `TSARKIMEX` scheme

2217:   Level: intermediate

2219: .seealso: [](ch_ts), `TSARKIMEXc`
2220: @*/
2221: PetscErrorCode TSARKIMEXGetType(TS ts, TSARKIMEXType *arktype)
2222: {
2223:   PetscFunctionBegin;
2225:   PetscUseMethod(ts, "TSARKIMEXGetType_C", (TS, TSARKIMEXType *), (ts, arktype));
2226:   PetscFunctionReturn(PETSC_SUCCESS);
2227: }

2229: /*@
2230:   TSARKIMEXSetFullyImplicit - Solve both parts of the equation implicitly, including the part that is normally solved explicitly

2232:   Logically Collective

2234:   Input Parameters:
2235: + ts  - timestepping context
2236: - flg - `PETSC_TRUE` for fully implicit

2238:   Level: intermediate

2240: .seealso: [](ch_ts), `TSARKIMEX`, `TSARKIMEXGetType()`, `TSARKIMEXGetFullyImplicit()`
2241: @*/
2242: PetscErrorCode TSARKIMEXSetFullyImplicit(TS ts, PetscBool flg)
2243: {
2244:   PetscFunctionBegin;
2247:   PetscTryMethod(ts, "TSARKIMEXSetFullyImplicit_C", (TS, PetscBool), (ts, flg));
2248:   PetscFunctionReturn(PETSC_SUCCESS);
2249: }

2251: /*@
2252:   TSARKIMEXGetFullyImplicit - Inquires if both parts of the equation are solved implicitly

2254:   Logically Collective

2256:   Input Parameter:
2257: . ts - timestepping context

2259:   Output Parameter:
2260: . flg - `PETSC_TRUE` for fully implicit

2262:   Level: intermediate

2264: .seealso: [](ch_ts), `TSARKIMEXGetType()`, `TSARKIMEXSetFullyImplicit()`
2265: @*/
2266: PetscErrorCode TSARKIMEXGetFullyImplicit(TS ts, PetscBool *flg)
2267: {
2268:   PetscFunctionBegin;
2270:   PetscAssertPointer(flg, 2);
2271:   PetscUseMethod(ts, "TSARKIMEXGetFullyImplicit_C", (TS, PetscBool *), (ts, flg));
2272:   PetscFunctionReturn(PETSC_SUCCESS);
2273: }

2275: static PetscErrorCode TSARKIMEXGetType_ARKIMEX(TS ts, TSARKIMEXType *arktype)
2276: {
2277:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;

2279:   PetscFunctionBegin;
2280:   *arktype = ark->tableau->name;
2281:   PetscFunctionReturn(PETSC_SUCCESS);
2282: }

2284: static PetscErrorCode TSARKIMEXSetType_ARKIMEX(TS ts, TSARKIMEXType arktype)
2285: {
2286:   TS_ARKIMEX    *ark = (TS_ARKIMEX *)ts->data;
2287:   PetscBool      match;
2288:   ARKTableauLink link;

2290:   PetscFunctionBegin;
2291:   if (ark->tableau) {
2292:     PetscCall(PetscStrcmp(ark->tableau->name, arktype, &match));
2293:     if (match) PetscFunctionReturn(PETSC_SUCCESS);
2294:   }
2295:   for (link = ARKTableauList; link; link = link->next) {
2296:     PetscCall(PetscStrcmp(link->tab.name, arktype, &match));
2297:     if (match) {
2298:       if (ts->setupcalled) PetscCall(TSARKIMEXTableauReset(ts));
2299:       ark->tableau = &link->tab;
2300:       if (ts->setupcalled) PetscCall(TSARKIMEXTableauSetUp(ts));
2301:       ts->default_adapt_type = ark->tableau->bembed ? TSADAPTBASIC : TSADAPTNONE;
2302:       PetscFunctionReturn(PETSC_SUCCESS);
2303:     }
2304:   }
2305:   SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_ARG_UNKNOWN_TYPE, "Could not find '%s'", arktype);
2306: }

2308: static PetscErrorCode TSARKIMEXSetFullyImplicit_ARKIMEX(TS ts, PetscBool flg)
2309: {
2310:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;

2312:   PetscFunctionBegin;
2313:   ark->imex = (PetscBool)!flg;
2314:   PetscFunctionReturn(PETSC_SUCCESS);
2315: }

2317: static PetscErrorCode TSARKIMEXGetFullyImplicit_ARKIMEX(TS ts, PetscBool *flg)
2318: {
2319:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;

2321:   PetscFunctionBegin;
2322:   *flg = (PetscBool)!ark->imex;
2323:   PetscFunctionReturn(PETSC_SUCCESS);
2324: }

2326: static PetscErrorCode TSDestroy_ARKIMEX(TS ts)
2327: {
2328:   PetscFunctionBegin;
2329:   PetscCall(TSReset_ARKIMEX(ts));
2330:   if (ts->dm) {
2331:     PetscCall(DMCoarsenHookRemove(ts->dm, DMCoarsenHook_TSARKIMEX, DMRestrictHook_TSARKIMEX, ts));
2332:     PetscCall(DMSubDomainHookRemove(ts->dm, DMSubDomainHook_TSARKIMEX, DMSubDomainRestrictHook_TSARKIMEX, ts));
2333:   }
2334:   PetscCall(PetscFree(ts->data));
2335:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSDIRKGetType_C", NULL));
2336:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSDIRKSetType_C", NULL));
2337:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSARKIMEXGetType_C", NULL));
2338:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSARKIMEXSetType_C", NULL));
2339:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSARKIMEXSetFullyImplicit_C", NULL));
2340:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSARKIMEXGetFullyImplicit_C", NULL));
2341:   PetscFunctionReturn(PETSC_SUCCESS);
2342: }

2344: /* ------------------------------------------------------------ */
2345: /*MC
2346:       TSARKIMEX - ODE and DAE solver using additive Runge-Kutta IMEX schemes

2348:   These methods are intended for problems with well-separated time scales, especially when a slow scale is strongly
2349:   nonlinear such that it is expensive to solve with a fully implicit method. The user should provide the stiff part
2350:   of the equation using `TSSetIFunction()` and the non-stiff part with `TSSetRHSFunction()`.

2352:   Level: beginner

2354:   Notes:
2355:   The default is `TSARKIMEX3`, it can be changed with `TSARKIMEXSetType()` or -ts_arkimex_type

2357:   If the equation is implicit or a DAE, then `TSSetEquationType()` needs to be set accordingly. Refer to the manual for further information.

2359:   Methods with an explicit stage can only be used with ODE in which the stiff part G(t,X,Xdot) has the form Xdot + Ghat(t,X).

2361:   Consider trying `TSROSW` if the stiff part is linear or weakly nonlinear.

2363: .seealso: [](ch_ts), `TSCreate()`, `TS`, `TSSetType()`, `TSARKIMEXSetType()`, `TSARKIMEXGetType()`, `TSARKIMEXSetFullyImplicit()`, `TSARKIMEXGetFullyImplicit()`,
2364:           `TSARKIMEX1BEE`, `TSARKIMEX2C`, `TSARKIMEX2D`, `TSARKIMEX2E`, `TSARKIMEX3`, `TSARKIMEXL2`, `TSARKIMEXA2`, `TSARKIMEXARS122`,
2365:           `TSARKIMEX4`, `TSARKIMEX5`, `TSARKIMEXPRSSP2`, `TSARKIMEXARS443`, `TSARKIMEXBPR3`, `TSARKIMEXType`, `TSARKIMEXRegister()`, `TSType`
2366: M*/
2367: PETSC_EXTERN PetscErrorCode TSCreate_ARKIMEX(TS ts)
2368: {
2369:   TS_ARKIMEX *ark;
2370:   PetscBool   dirk;

2372:   PetscFunctionBegin;
2373:   PetscCall(TSARKIMEXInitializePackage());
2374:   PetscCall(PetscObjectTypeCompare((PetscObject)ts, TSDIRK, &dirk));

2376:   ts->ops->reset          = TSReset_ARKIMEX;
2377:   ts->ops->adjointreset   = TSAdjointReset_ARKIMEX;
2378:   ts->ops->destroy        = TSDestroy_ARKIMEX;
2379:   ts->ops->view           = TSView_ARKIMEX;
2380:   ts->ops->load           = TSLoad_ARKIMEX;
2381:   ts->ops->setup          = TSSetUp_ARKIMEX;
2382:   ts->ops->adjointsetup   = TSAdjointSetUp_ARKIMEX;
2383:   ts->ops->step           = TSStep_ARKIMEX;
2384:   ts->ops->interpolate    = TSInterpolate_ARKIMEX;
2385:   ts->ops->evaluatestep   = TSEvaluateStep_ARKIMEX;
2386:   ts->ops->rollback       = TSRollBack_ARKIMEX;
2387:   ts->ops->setfromoptions = TSSetFromOptions_ARKIMEX;
2388:   ts->ops->snesfunction   = SNESTSFormFunction_ARKIMEX;
2389:   ts->ops->snesjacobian   = SNESTSFormJacobian_ARKIMEX;
2390:   ts->ops->getstages      = TSGetStages_ARKIMEX;
2391:   ts->ops->adjointstep    = TSAdjointStep_ARKIMEX;

2393:   ts->usessnes = PETSC_TRUE;

2395:   PetscCall(PetscNew(&ark));
2396:   ts->data  = (void *)ark;
2397:   ark->imex = dirk ? PETSC_FALSE : PETSC_TRUE;

2399:   ark->VecsDeltaLam   = NULL;
2400:   ark->VecsSensiTemp  = NULL;
2401:   ark->VecsSensiPTemp = NULL;

2403:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSARKIMEXGetType_C", TSARKIMEXGetType_ARKIMEX));
2404:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSARKIMEXGetFullyImplicit_C", TSARKIMEXGetFullyImplicit_ARKIMEX));
2405:   if (!dirk) {
2406:     PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSARKIMEXSetType_C", TSARKIMEXSetType_ARKIMEX));
2407:     PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSARKIMEXSetFullyImplicit_C", TSARKIMEXSetFullyImplicit_ARKIMEX));
2408:     PetscCall(TSARKIMEXSetType(ts, TSARKIMEXDefault));
2409:   }
2410:   PetscFunctionReturn(PETSC_SUCCESS);
2411: }

2413: /* ------------------------------------------------------------ */

2415: static PetscErrorCode TSDIRKSetType_DIRK(TS ts, TSDIRKType dirktype)
2416: {
2417:   TS_ARKIMEX *ark = (TS_ARKIMEX *)ts->data;

2419:   PetscFunctionBegin;
2420:   PetscCall(TSARKIMEXSetType_ARKIMEX(ts, dirktype));
2421:   PetscCheck(!ark->tableau->additive, PetscObjectComm((PetscObject)ts), PETSC_ERR_ARG_WRONG, "Method \"%s\" is not DIRK", dirktype);
2422:   PetscFunctionReturn(PETSC_SUCCESS);
2423: }

2425: /*@C
2426:   TSDIRKSetType - Set the type of `TSDIRK` scheme

2428:   Logically Collective

2430:   Input Parameters:
2431: + ts       - timestepping context
2432: - dirktype - type of `TSDIRK` scheme

2434:   Options Database Key:
2435: . -ts_dirkimex_type - set `TSDIRK` scheme type

2437:   Level: intermediate

2439: .seealso: [](ch_ts), `TSDIRKGetType()`, `TSDIRK`, `TSDIRKType`
2440: @*/
2441: PetscErrorCode TSDIRKSetType(TS ts, TSDIRKType dirktype)
2442: {
2443:   PetscFunctionBegin;
2445:   PetscAssertPointer(dirktype, 2);
2446:   PetscTryMethod(ts, "TSDIRKSetType_C", (TS, TSDIRKType), (ts, dirktype));
2447:   PetscFunctionReturn(PETSC_SUCCESS);
2448: }

2450: /*@C
2451:   TSDIRKGetType - Get the type of `TSDIRK` scheme

2453:   Logically Collective

2455:   Input Parameter:
2456: . ts - timestepping context

2458:   Output Parameter:
2459: . dirktype - type of `TSDIRK` scheme

2461:   Level: intermediate

2463: .seealso: [](ch_ts), `TSDIRKSetType()`
2464: @*/
2465: PetscErrorCode TSDIRKGetType(TS ts, TSDIRKType *dirktype)
2466: {
2467:   PetscFunctionBegin;
2469:   PetscUseMethod(ts, "TSDIRKGetType_C", (TS, TSDIRKType *), (ts, dirktype));
2470:   PetscFunctionReturn(PETSC_SUCCESS);
2471: }

2473: /*MC
2474:       TSDIRK - ODE and DAE solver using Diagonally implicit Runge-Kutta schemes.

2476:   Level: beginner

2478:   Notes:
2479:   The default is `TSDIRKES213SAL`, it can be changed with `TSDIRKSetType()` or -ts_dirk_type.
2480:   The convention used in PETSc to name the DIRK methods is TSDIRK[E][S]PQS[SA][L][A] with:
2481: + E - whether the method has an explicit first stage
2482: . S - whether the method is single diagonal
2483: . P - order of the advancing method
2484: . Q - order of the embedded method
2485: . S - number of stages
2486: . SA - whether the method is stiffly accurate
2487: . L - whether the method is L-stable
2488: - A - whether the method is A-stable

2490: .seealso: [](ch_ts), `TSCreate()`, `TS`, `TSSetType()`, `TSDIRKSetType()`, `TSDIRKGetType()`, `TSDIRKRegister()`.
2491: M*/
2492: PETSC_EXTERN PetscErrorCode TSCreate_DIRK(TS ts)
2493: {
2494:   PetscFunctionBegin;
2495:   PetscCall(TSCreate_ARKIMEX(ts));
2496:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSDIRKGetType_C", TSARKIMEXGetType_ARKIMEX));
2497:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSDIRKSetType_C", TSDIRKSetType_DIRK));
2498:   PetscCall(TSDIRKSetType(ts, TSDIRKDefault));
2499:   PetscFunctionReturn(PETSC_SUCCESS);
2500: }