Actual source code: rosw.c

      1: /*
      2:   Code for timestepping with Rosenbrock W methods

      4:   Notes:
      5:   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.
     10:   This method is designed to be linearly implicit on F and can use an approximate and lagged Jacobian.

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

     16: #include <petsc/private/kernels/blockinvert.h>

     18: static TSRosWType TSRosWDefault = TSROSWRA34PW2;
     19: static PetscBool  TSRosWRegisterAllCalled;
     20: static PetscBool  TSRosWPackageInitialized;

     22: typedef struct _RosWTableau *RosWTableau;
     23: struct _RosWTableau {
     24:   char      *name;
     25:   PetscInt   order;             /* Classical approximation order of the method */
     26:   PetscInt   s;                 /* Number of stages */
     27:   PetscInt   pinterp;           /* Interpolation order */
     28:   PetscReal *A;                 /* Propagation table, strictly lower triangular */
     29:   PetscReal *Gamma;             /* Stage table, lower triangular with nonzero diagonal */
     30:   PetscBool *GammaZeroDiag;     /* Diagonal entries that are zero in stage table Gamma, vector indicating explicit statages */
     31:   PetscReal *GammaExplicitCorr; /* Coefficients for correction terms needed for explicit stages in transformed variables*/
     32:   PetscReal *b;                 /* Step completion table */
     33:   PetscReal *bembed;            /* Step completion table for embedded method of order one less */
     34:   PetscReal *ASum;              /* Row sum of A */
     35:   PetscReal *GammaSum;          /* Row sum of Gamma, only needed for non-autonomous systems */
     36:   PetscReal *At;                /* Propagation table in transformed variables */
     37:   PetscReal *bt;                /* Step completion table in transformed variables */
     38:   PetscReal *bembedt;           /* Step completion table of order one less in transformed variables */
     39:   PetscReal *GammaInv;          /* Inverse of Gamma, used for transformed variables */
     40:   PetscReal  ccfl;              /* Placeholder for CFL coefficient relative to forward Euler */
     41:   PetscReal *binterpt;          /* Dense output formula */
     42: };
     43: typedef struct _RosWTableauLink *RosWTableauLink;
     44: struct _RosWTableauLink {
     45:   struct _RosWTableau tab;
     46:   RosWTableauLink     next;
     47: };
     48: static RosWTableauLink RosWTableauList;

     50: typedef struct {
     51:   RosWTableau  tableau;
     52:   Vec         *Y;            /* States computed during the step, used to complete the step */
     53:   Vec          Ydot;         /* Work vector holding Ydot during residual evaluation */
     54:   Vec          Ystage;       /* Work vector for the state value at each stage */
     55:   Vec          Zdot;         /* Ydot = Zdot + shift*Y */
     56:   Vec          Zstage;       /* Y = Zstage + Y */
     57:   Vec          vec_sol_prev; /* Solution from the previous step (used for interpolation and rollback)*/
     58:   PetscScalar *work;         /* Scalar work space of length number of stages, used to prepare VecMAXPY() */
     59:   PetscReal    scoeff;       /* shift = scoeff/dt */
     60:   PetscReal    stage_time;
     61:   PetscReal    stage_explicit;     /* Flag indicates that the current stage is explicit */
     62:   PetscBool    recompute_jacobian; /* Recompute the Jacobian at each stage, default is to freeze the Jacobian at the start of each step */
     63:   TSStepStatus status;
     64: } TS_RosW;

     66: /*MC
     67:      TSROSWTHETA1 - One stage first order L-stable Rosenbrock-W scheme (aka theta method).

     69:      Only an approximate Jacobian is needed.

     71:      Level: intermediate

     73: .seealso: [](ch_ts), `TSROSW`
     74: M*/

     76: /*MC
     77:      TSROSWTHETA2 - One stage second order A-stable Rosenbrock-W scheme (aka theta method).

     79:      Only an approximate Jacobian is needed.

     81:      Level: intermediate

     83: .seealso: [](ch_ts), `TSROSW`
     84: M*/

     86: /*MC
     87:      TSROSW2M - Two stage second order L-stable Rosenbrock-W scheme.

     89:      Only an approximate Jacobian is needed. By default, it is only recomputed once per step. This method is a reflection of TSROSW2P.

     91:      Level: intermediate

     93: .seealso: [](ch_ts), `TSROSW`
     94: M*/

     96: /*MC
     97:      TSROSW2P - Two stage second order L-stable Rosenbrock-W scheme.

     99:      Only an approximate Jacobian is needed. By default, it is only recomputed once per step. This method is a reflection of TSROSW2M.

    101:      Level: intermediate

    103: .seealso: [](ch_ts), `TSROSW`
    104: M*/

    106: /*MC
    107:      TSROSWRA3PW - Three stage third order Rosenbrock-W scheme for PDAE of index 1 {cite}`rang_2005`

    109:      Only an approximate Jacobian is needed. By default, it is only recomputed once per step.

    111:      This is strongly A-stable with R(infty) = 0.73. The embedded method of order 2 is strongly A-stable with R(infty) = 0.73.

    113:      Level: intermediate

    115: .seealso: [](ch_ts), `TSROSW`
    116: M*/

    118: /*MC
    119:      TSROSWRA34PW2 - Four stage third order L-stable Rosenbrock-W scheme for PDAE of index 1 {cite}`rang_2005`.

    121:      Only an approximate Jacobian is needed. By default, it is only recomputed once per step.

    123:      This is strongly A-stable with R(infty) = 0. The embedded method of order 2 is strongly A-stable with R(infty) = 0.48.

    125:      Level: intermediate

    127: .seealso: [](ch_ts), `TSROSW`
    128: M*/

    130: /*MC
    131:      TSROSWR34PRW - Four stage third order L-stable Rosenbrock-W scheme for PDAE of index 1 {cite}`rang2015improved`.

    133:      Only an approximate Jacobian is needed. By default, it is only recomputed once per step.

    135:      This is strongly A-stable with R(infty) = 0. The embedded method of order 2 is strongly A-stable with R(infty) = 0.25.
    136:      This method is B_{PR} consistent of order 3.
    137:      This method is spelled "ROS34PRw" in the paper, an improvement to an earlier "ROS34PRW" method from the same author with B_{PR} order 2.

    139:      Level: intermediate

    141: .seealso: [](ch_ts), `TSROSW`
    142: M*/

    144: /*MC
    145:      TSROSWR3PRL2 - Four stage third order L-stable Rosenbrock-W scheme for PDAE of index 1 {cite}`rang2015improved`.

    147:      Only an approximate Jacobian is needed. By default, it is only recomputed once per step.

    149:      This is strongly A-stable with R(infty) = 0. The embedded method of order 2 is strongly A-stable with R(infty) = 0.25.
    150:      This method is B_{PR} consistent of order 3.

    152:      Level: intermediate

    154: .seealso: [](ch_ts), `TSROSW`
    155: M*/

    157: /*MC
    158:      TSROSWRODAS3 - Four stage third order L-stable Rosenbrock scheme {cite}`sandu_1997`

    160:      By default, the Jacobian is only recomputed once per step.

    162:      Both the third order and embedded second order methods are stiffly accurate and L-stable.

    164:      Level: intermediate

    166: .seealso: [](ch_ts), `TSROSW`, `TSROSWSANDU3`
    167: M*/

    169: /*MC
    170:      TSROSWRODASPR - Six stage fourth order L-stable Rosenbrock scheme {cite}`rang2015improved`

    172:      By default, the Jacobian is only recomputed once per step.

    174:      Both the fourth order and embedded third order methods are stiffly accurate and L-stable.
    175:      The method is B_{PR} consistent of order 3, which ensures convergence order for non-stiff, medium stiff, and stiff problems.

    177:      Level: intermediate

    179: .seealso: [](ch_ts), `TSROSW`, `TSROSWR34PRW`, `TSROSWR3PRL2`
    180: M*/

    182: /*MC
    183:      TSROSWRODASPR2 - Six stage fourth order L-stable Rosenbrock scheme {cite}`rang2015improved`

    185:      By default, the Jacobian is only recomputed once per step.

    187:      Both the fourth order and embedded third order methods are stiffly accurate and L-stable.
    188:      The method is B_{PR} consistent of order 3, which ensures convergence order for non-stiff, medium stiff, and stiff problems.
    189:      This method is similar to `TSROSWRODASPR`, but satisfies one extra B_{PR} order condition.

    191:      Developer Note:
    192:      In numerical experiments with ts/tutorials/ex22.c, I (Jed) find this to produce surprisingly poor results.
    193:      Although the coefficients pass basic smoke tests, I'm not confident it was tabulated correctly in the paper.
    194:      It would be informative if someone could reproduce tests from the paper and/or reach out to the author to understand why it fails on this test problem.
    195:      If the method is implemented correctly, doing so might shed light on an additional analysis lens (or further conditions) for robustness on such problems.

    197:      Level: intermediate

    199: .seealso: [](ch_ts), `TSROSW`, `TSROSWRODASPR`
    200: M*/

    202: /*MC
    203:      TSROSWSANDU3 - Three stage third order L-stable Rosenbrock scheme {cite}`sandu_1997`

    205:      By default, the Jacobian is only recomputed once per step.

    207:      The third order method is L-stable, but not stiffly accurate.
    208:      The second order embedded method is strongly A-stable with R(infty) = 0.5.
    209:      The internal stages are L-stable.
    210:      This method is called ROS3 in {cite}`sandu_1997`.

    212:      Level: intermediate

    214: .seealso: [](ch_ts), `TSROSW`, `TSROSWRODAS3`
    215: M*/

    217: /*MC
    218:      TSROSWASSP3P3S1C - A-stable Rosenbrock-W method with SSP explicit part, third order, three stages

    220:      By default, the Jacobian is only recomputed once per step.

    222:      A-stable SPP explicit order 3, 3 stages, CFL 1 (eff = 1/3)

    224:      Level: intermediate

    226: .seealso: [](ch_ts), `TSROSW`, `TSROSWLASSP3P4S2C`, `TSROSWLLSSP3P4S2C`, `SSP`
    227: M*/

    229: /*MC
    230:      TSROSWLASSP3P4S2C - L-stable Rosenbrock-W method with SSP explicit part, third order, four stages

    232:      By default, the Jacobian is only recomputed once per step.

    234:      L-stable (A-stable embedded) SPP explicit order 3, 4 stages, CFL 2 (eff = 1/2)

    236:      Level: intermediate

    238: .seealso: [](ch_ts), `TSROSW`, `TSROSWASSP3P3S1C`, `TSROSWLLSSP3P4S2C`, `TSSSP`
    239: M*/

    241: /*MC
    242:      TSROSWLLSSP3P4S2C - L-stable Rosenbrock-W method with SSP explicit part, third order, four stages

    244:      By default, the Jacobian is only recomputed once per step.

    246:      L-stable (L-stable embedded) SPP explicit order 3, 4 stages, CFL 2 (eff = 1/2)

    248:      Level: intermediate

    250: .seealso: [](ch_ts), `TSROSW`, `TSROSWASSP3P3S1C`, `TSROSWLASSP3P4S2C`, `TSSSP`
    251: M*/

    253: /*MC
    254:      TSROSWGRK4T - four stage, fourth order Rosenbrock (not W) method from Kaps and Rentrop {cite}`kaps1979generalized`

    256:      By default, the Jacobian is only recomputed once per step.

    258:      A(89.3 degrees)-stable, |R(infty)| = 0.454.

    260:      This method does not provide a dense output formula.

    262:      Level: intermediate

    264:      Note:
    265:      See Section 4 Table 7.2 in {cite}`wanner1996solving`

    267: .seealso: [](ch_ts), `TSROSW`, `TSROSWSHAMP4`, `TSROSWVELDD4`, `TSROSW4L`
    268: M*/

    270: /*MC
    271:      TSROSWSHAMP4 - four stage, fourth order Rosenbrock (not W) method from Shampine {cite}`shampine1982implementation`

    273:      By default, the Jacobian is only recomputed once per step.

    275:      A-stable, |R(infty)| = 1/3.

    277:      This method does not provide a dense output formula.

    279:      Level: intermediate

    281:      Note:
    282:      See Section 4 Table 7.2 in {cite}`wanner1996solving`

    284: .seealso: [](ch_ts), `TSROSW`, `TSROSWGRK4T`, `TSROSWVELDD4`, `TSROSW4L`
    285: M*/

    287: /*MC
    288:      TSROSWVELDD4 - four stage, fourth order Rosenbrock (not W) method from van Veldhuizen {cite}`veldhuizen1984d`

    290:      By default, the Jacobian is only recomputed once per step.

    292:      A(89.5 degrees)-stable, |R(infty)| = 0.24.

    294:      This method does not provide a dense output formula.

    296:      Level: intermediate

    298:      Note:
    299:      See Section 4 Table 7.2 in {cite}`wanner1996solving`

    301: .seealso: [](ch_ts), `TSROSW`, `TSROSWGRK4T`, `TSROSWSHAMP4`, `TSROSW4L`
    302: M*/

    304: /*MC
    305:      TSROSW4L - four stage, fourth order Rosenbrock (not W) method

    307:      By default, the Jacobian is only recomputed once per step.

    309:      A-stable and L-stable

    311:      This method does not provide a dense output formula.

    313:      Level: intermediate

    315:      Note:
    316:      See Section 4 Table 7.2 in {cite}`wanner1996solving`

    318: .seealso: [](ch_ts), `TSROSW`, `TSROSWGRK4T`, `TSROSWSHAMP4`, `TSROSW4L`
    319: M*/

    321: /*@C
    322:   TSRosWRegisterAll - Registers all of the Rosenbrock-W methods in `TSROSW`

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

    326:   Level: advanced

    328: .seealso: [](ch_ts), `TSROSW`, `TSRosWRegisterDestroy()`
    329: @*/
    330: PetscErrorCode TSRosWRegisterAll(void)
    331: {
    332:   PetscFunctionBegin;
    333:   if (TSRosWRegisterAllCalled) PetscFunctionReturn(PETSC_SUCCESS);
    334:   TSRosWRegisterAllCalled = PETSC_TRUE;

    336:   {
    337:     const PetscReal A        = 0;
    338:     const PetscReal Gamma    = 1;
    339:     const PetscReal b        = 1;
    340:     const PetscReal binterpt = 1;

    342:     PetscCall(TSRosWRegister(TSROSWTHETA1, 1, 1, &A, &Gamma, &b, NULL, 1, &binterpt));
    343:   }

    345:   {
    346:     const PetscReal A        = 0;
    347:     const PetscReal Gamma    = 0.5;
    348:     const PetscReal b        = 1;
    349:     const PetscReal binterpt = 1;

    351:     PetscCall(TSRosWRegister(TSROSWTHETA2, 2, 1, &A, &Gamma, &b, NULL, 1, &binterpt));
    352:   }

    354:   {
    355:     /*const PetscReal g = 1. + 1./PetscSqrtReal(2.0);   Direct evaluation: 1.707106781186547524401. Used for setting up arrays of values known at compile time below. */
    356:     const PetscReal A[2][2] = {
    357:       {0,  0},
    358:       {1., 0}
    359:     };
    360:     const PetscReal Gamma[2][2] = {
    361:       {1.707106781186547524401,       0                      },
    362:       {-2. * 1.707106781186547524401, 1.707106781186547524401}
    363:     };
    364:     const PetscReal b[2]  = {0.5, 0.5};
    365:     const PetscReal b1[2] = {1.0, 0.0};
    366:     PetscReal       binterpt[2][2];
    367:     binterpt[0][0] = 1.707106781186547524401 - 1.0;
    368:     binterpt[1][0] = 2.0 - 1.707106781186547524401;
    369:     binterpt[0][1] = 1.707106781186547524401 - 1.5;
    370:     binterpt[1][1] = 1.5 - 1.707106781186547524401;

    372:     PetscCall(TSRosWRegister(TSROSW2P, 2, 2, &A[0][0], &Gamma[0][0], b, b1, 2, &binterpt[0][0]));
    373:   }
    374:   {
    375:     /*const PetscReal g = 1. - 1./PetscSqrtReal(2.0);   Direct evaluation: 0.2928932188134524755992. Used for setting up arrays of values known at compile time below. */
    376:     const PetscReal A[2][2] = {
    377:       {0,  0},
    378:       {1., 0}
    379:     };
    380:     const PetscReal Gamma[2][2] = {
    381:       {0.2928932188134524755992,       0                       },
    382:       {-2. * 0.2928932188134524755992, 0.2928932188134524755992}
    383:     };
    384:     const PetscReal b[2]  = {0.5, 0.5};
    385:     const PetscReal b1[2] = {1.0, 0.0};
    386:     PetscReal       binterpt[2][2];
    387:     binterpt[0][0] = 0.2928932188134524755992 - 1.0;
    388:     binterpt[1][0] = 2.0 - 0.2928932188134524755992;
    389:     binterpt[0][1] = 0.2928932188134524755992 - 1.5;
    390:     binterpt[1][1] = 1.5 - 0.2928932188134524755992;

    392:     PetscCall(TSRosWRegister(TSROSW2M, 2, 2, &A[0][0], &Gamma[0][0], b, b1, 2, &binterpt[0][0]));
    393:   }
    394:   {
    395:     /*const PetscReal g = 7.8867513459481287e-01; Directly written in-place below */
    396:     PetscReal       binterpt[3][2];
    397:     const PetscReal A[3][3] = {
    398:       {0,                      0, 0},
    399:       {1.5773502691896257e+00, 0, 0},
    400:       {0.5,                    0, 0}
    401:     };
    402:     const PetscReal Gamma[3][3] = {
    403:       {7.8867513459481287e-01,  0,                       0                     },
    404:       {-1.5773502691896257e+00, 7.8867513459481287e-01,  0                     },
    405:       {-6.7075317547305480e-01, -1.7075317547305482e-01, 7.8867513459481287e-01}
    406:     };
    407:     const PetscReal b[3]  = {1.0566243270259355e-01, 4.9038105676657971e-02, 8.4529946162074843e-01};
    408:     const PetscReal b2[3] = {-1.7863279495408180e-01, 1. / 3., 8.4529946162074843e-01};

    410:     binterpt[0][0] = -0.8094010767585034;
    411:     binterpt[1][0] = -0.5;
    412:     binterpt[2][0] = 2.3094010767585034;
    413:     binterpt[0][1] = 0.9641016151377548;
    414:     binterpt[1][1] = 0.5;
    415:     binterpt[2][1] = -1.4641016151377548;

    417:     PetscCall(TSRosWRegister(TSROSWRA3PW, 3, 3, &A[0][0], &Gamma[0][0], b, b2, 2, &binterpt[0][0]));
    418:   }
    419:   {
    420:     PetscReal binterpt[4][3];
    421:     /*const PetscReal g = 4.3586652150845900e-01; Directly written in-place below */
    422:     const PetscReal A[4][4] = {
    423:       {0,                      0,                       0,  0},
    424:       {8.7173304301691801e-01, 0,                       0,  0},
    425:       {8.4457060015369423e-01, -1.1299064236484185e-01, 0,  0},
    426:       {0,                      0,                       1., 0}
    427:     };
    428:     const PetscReal Gamma[4][4] = {
    429:       {4.3586652150845900e-01,  0,                       0,                      0                     },
    430:       {-8.7173304301691801e-01, 4.3586652150845900e-01,  0,                      0                     },
    431:       {-9.0338057013044082e-01, 5.4180672388095326e-02,  4.3586652150845900e-01, 0                     },
    432:       {2.4212380706095346e-01,  -1.2232505839045147e+00, 5.4526025533510214e-01, 4.3586652150845900e-01}
    433:     };
    434:     const PetscReal b[4]  = {2.4212380706095346e-01, -1.2232505839045147e+00, 1.5452602553351020e+00, 4.3586652150845900e-01};
    435:     const PetscReal b2[4] = {3.7810903145819369e-01, -9.6042292212423178e-02, 5.0000000000000000e-01, 2.1793326075422950e-01};

    437:     binterpt[0][0] = 1.0564298455794094;
    438:     binterpt[1][0] = 2.296429974281067;
    439:     binterpt[2][0] = -1.307599564525376;
    440:     binterpt[3][0] = -1.045260255335102;
    441:     binterpt[0][1] = -1.3864882699759573;
    442:     binterpt[1][1] = -8.262611700275677;
    443:     binterpt[2][1] = 7.250979895056055;
    444:     binterpt[3][1] = 2.398120075195581;
    445:     binterpt[0][2] = 0.5721822314575016;
    446:     binterpt[1][2] = 4.742931142090097;
    447:     binterpt[2][2] = -4.398120075195578;
    448:     binterpt[3][2] = -0.9169932983520199;

    450:     PetscCall(TSRosWRegister(TSROSWRA34PW2, 3, 4, &A[0][0], &Gamma[0][0], b, b2, 3, &binterpt[0][0]));
    451:   }
    452:   {
    453:     /* const PetscReal g = 4.3586652150845900e-01;       Directly written in-place below */
    454:     const PetscReal A[4][4] = {
    455:       {0,                      0,                       0,                       0},
    456:       {8.7173304301691801e-01, 0,                       0,                       0},
    457:       {1.4722022879435914e+00, -3.1840250568090289e-01, 0,                       0},
    458:       {8.1505192016694938e-01, 5.0000000000000000e-01,  -3.1505192016694938e-01, 0}
    459:     };
    460:     const PetscReal Gamma[4][4] = {
    461:       {4.3586652150845900e-01,  0,                      0,                       0                     },
    462:       {-8.7173304301691801e-01, 4.3586652150845900e-01, 0,                       0                     },
    463:       {-1.2855347382089872e+00, 5.0507005541550687e-01, 4.3586652150845900e-01,  0                     },
    464:       {-4.8201449182864348e-01, 2.1793326075422950e-01, -1.7178529043404503e-01, 4.3586652150845900e-01}
    465:     };
    466:     const PetscReal b[4]  = {3.3303742833830591e-01, 7.1793326075422947e-01, -4.8683721060099439e-01, 4.3586652150845900e-01};
    467:     const PetscReal b2[4] = {2.5000000000000000e-01, 7.4276119608319180e-01, -3.1472922970066219e-01, 3.2196803361747034e-01};

    469:     PetscCall(TSRosWRegister(TSROSWR34PRW, 3, 4, &A[0][0], &Gamma[0][0], b, b2, 0, NULL));
    470:   }
    471:   {
    472:     /* const PetscReal g = 4.3586652150845900e-01;       Directly written in-place below */
    473:     const PetscReal A[4][4] = {
    474:       {0,                      0,   0, 0},
    475:       {1.3075995645253771e+00, 0,   0, 0},
    476:       {0.5,                    0.5, 0, 0},
    477:       {0.5,                    0.5, 0, 0}
    478:     };
    479:     const PetscReal Gamma[4][4] = {
    480:       {4.3586652150845900e-01,  0,                       0,                      0                     },
    481:       {-1.3075995645253771e+00, 4.3586652150845900e-01,  0,                      0                     },
    482:       {-7.0988575860972170e-01, -5.5996735960277766e-01, 4.3586652150845900e-01, 0                     },
    483:       {-1.5550856807552085e-01, -9.5388516575112225e-01, 6.7352721231818413e-01, 4.3586652150845900e-01}
    484:     };
    485:     const PetscReal b[4]  = {3.4449143192447917e-01, -4.5388516575112231e-01, 6.7352721231818413e-01, 4.3586652150845900e-01};
    486:     const PetscReal b2[4] = {5.0000000000000000e-01, -2.5738812086522078e-01, 4.3542008724775044e-01, 3.2196803361747034e-01};

    488:     PetscCall(TSRosWRegister(TSROSWR3PRL2, 3, 4, &A[0][0], &Gamma[0][0], b, b2, 0, NULL));
    489:   }
    490:   {
    491:     /* const PetscReal g = 0.5;       Directly written in-place below */
    492:     const PetscReal A[4][4] = {
    493:       {0,    0,     0,   0},
    494:       {0,    0,     0,   0},
    495:       {1.,   0,     0,   0},
    496:       {0.75, -0.25, 0.5, 0}
    497:     };
    498:     const PetscReal Gamma[4][4] = {
    499:       {0.5,     0,       0,       0  },
    500:       {1.,      0.5,     0,       0  },
    501:       {-0.25,   -0.25,   0.5,     0  },
    502:       {1. / 12, 1. / 12, -2. / 3, 0.5}
    503:     };
    504:     const PetscReal b[4]  = {5. / 6, -1. / 6, -1. / 6, 0.5};
    505:     const PetscReal b2[4] = {0.75, -0.25, 0.5, 0};

    507:     PetscCall(TSRosWRegister(TSROSWRODAS3, 3, 4, &A[0][0], &Gamma[0][0], b, b2, 0, NULL));
    508:   }
    509:   {
    510:     /* const PetscReal g = 0.25;       Directly written in-place below */
    511:     const PetscReal A[6][6] = {
    512:       {0,                       0,                       0,                       0,                       0,    0},
    513:       {0.75,                    0,                       0,                       0,                       0,    0},
    514:       {7.5162877593868457e-02,  2.4837122406131545e-02,  0,                       0,                       0,    0},
    515:       {1.6532708886396510e+00,  2.1545706385445562e-01,  -1.3157488872766792e+00, 0,                       0,    0},
    516:       {1.9385003738039885e+01,  1.2007117225835324e+00,  -1.9337924059522791e+01, -2.4779140110062559e-01, 0,    0},
    517:       {-7.3844531665375115e+00, -3.0593419030174646e-01, 7.8622074209377981e+00,  5.7817993590145966e-01,  0.25, 0}
    518:     };
    519:     const PetscReal Gamma[6][6] = {
    520:       {0.25,                    0,                       0,                       0,                       0,                       0   },
    521:       {-7.5000000000000000e-01, 0.25,                    0,                       0,                       0,                       0   },
    522:       {-8.8644359075349941e-02, -2.8688974257983398e-02, 0.25,                    0,                       0,                       0   },
    523:       {-4.8470034585330284e+00, -3.1583244269672095e-01, 4.9536568360123221e+00,  0.25,                    0,                       0   },
    524:       {-2.6769456904577400e+01, -1.5066459128852787e+00, 2.7200131480460591e+01,  8.2597133700208525e-01,  0.25,                    0   },
    525:       {6.5876206496361416e+00,  3.6807059172993878e-01,  -6.7423520694658121e+00, -1.0619631475741095e-01, -3.5714285714285715e-01, 0.25}
    526:     };
    527:     const PetscReal b[6]  = {-7.9683251690137014e-01, 6.2136401428192344e-02, 1.1198553514719862e+00, 4.7198362114404874e-01, -1.0714285714285714e-01, 0.25};
    528:     const PetscReal b2[6] = {-7.3844531665375115e+00, -3.0593419030174646e-01, 7.8622074209377981e+00, 5.7817993590145966e-01, 0.25, 0.0};

    530:     PetscCall(TSRosWRegister(TSROSWRODASPR, 4, 6, &A[0][0], &Gamma[0][0], b, b2, 0, NULL));
    531:   }
    532:   {
    533:     /* const PetscReal g = 0.3125;       Directly written in-place below */
    534:     const PetscReal A[6][6] = {
    535:       {0,                       0,                       0,                       0,                      0,                      0},
    536:       {9.3750000000000000e-01,  0,                       0,                       0,                      0,                      0},
    537:       {-4.7145892646261345e-02, 5.4531286650471122e-01,  0,                       0,                      0,                      0},
    538:       {4.6915543899742240e-01,  4.4490537602383673e-01,  -2.2498239334061121e-01, 0,                      0,                      0},
    539:       {1.0950372887345903e+00,  6.3223023457294381e-01,  -8.9232966090485821e-01, 1.6506213759732410e-01, 0,                      0},
    540:       {-1.7746585073632790e-01, -5.8241418952602364e-01, 6.8180612588238165e-01,  7.6557391437996980e-01, 3.1250000000000000e-01, 0}
    541:     };
    542:     const PetscReal Gamma[6][6] = {
    543:       {0.3125,                  0,                       0,                       0,                      0,                       0     },
    544:       {-9.3750000000000000e-01, 0.3125,                  0,                       0,                      0,                       0     },
    545:       {-9.7580572085994507e-02, -5.8666328499964138e-01, 0.3125,                  0,                      0,                       0     },
    546:       {-4.9407065013256957e-01, -5.6819726428975503e-01, 5.0318949274167679e-01,  0.3125,                 0,                       0     },
    547:       {-1.2725031394709183e+00, -1.2146444240989676e+00, 1.5741357867872399e+00,  6.0051177678264578e-01, 0.3125,                  0     },
    548:       {6.9690744901421153e-01,  6.2237005730756434e-01,  -1.1553701989197045e+00, 1.8350029013386296e-01, -6.5990759753593431e-01, 0.3125}
    549:     };
    550:     const PetscReal b[6]  = {5.1944159827788361e-01, 3.9955867781540699e-02, -4.7356407303732290e-01, 9.4907420451383284e-01, -3.4740759753593431e-01, 0.3125};
    551:     const PetscReal b2[6] = {-1.7746585073632790e-01, -5.8241418952602364e-01, 6.8180612588238165e-01, 7.6557391437996980e-01, 0.3125, 0};

    553:     PetscCall(TSRosWRegister(TSROSWRODASPR2, 4, 6, &A[0][0], &Gamma[0][0], b, b2, 0, NULL));
    554:   }
    555:   {
    556:     /*const PetscReal g = 0.43586652150845899941601945119356;       Directly written in-place below */
    557:     const PetscReal A[3][3] = {
    558:       {0,                                  0, 0},
    559:       {0.43586652150845899941601945119356, 0, 0},
    560:       {0.43586652150845899941601945119356, 0, 0}
    561:     };
    562:     const PetscReal Gamma[3][3] = {
    563:       {0.43586652150845899941601945119356,  0,                                  0                                 },
    564:       {-0.19294655696029095575009695436041, 0.43586652150845899941601945119356, 0                                 },
    565:       {0,                                   1.74927148125794685173529749738960, 0.43586652150845899941601945119356}
    566:     };
    567:     const PetscReal b[3]  = {-0.75457412385404315829818998646589, 1.94100407061964420292840123379419, -0.18642994676560104463021124732829};
    568:     const PetscReal b2[3] = {-1.53358745784149585370766523913002, 2.81745131148625772213931745457622, -0.28386385364476186843165221544619};

    570:     PetscReal binterpt[3][2];
    571:     binterpt[0][0] = 3.793692883777660870425141387941;
    572:     binterpt[1][0] = -2.918692883777660870425141387941;
    573:     binterpt[2][0] = 0.125;
    574:     binterpt[0][1] = -0.725741064379812106687651020584;
    575:     binterpt[1][1] = 0.559074397713145440020984353917;
    576:     binterpt[2][1] = 0.16666666666666666666666666666667;

    578:     PetscCall(TSRosWRegister(TSROSWSANDU3, 3, 3, &A[0][0], &Gamma[0][0], b, b2, 2, &binterpt[0][0]));
    579:   }
    580:   {
    581:     /*const PetscReal s3 = PetscSqrtReal(3.),g = (3.0+s3)/6.0;
    582:      * Direct evaluation: s3 = 1.732050807568877293527;
    583:      *                     g = 0.7886751345948128822546;
    584:      * Values are directly inserted below to ensure availability at compile time (compiler warnings otherwise...) */
    585:     const PetscReal A[3][3] = {
    586:       {0,    0,    0},
    587:       {1,    0,    0},
    588:       {0.25, 0.25, 0}
    589:     };
    590:     const PetscReal Gamma[3][3] = {
    591:       {0,                                       0,                                      0                       },
    592:       {(-3.0 - 1.732050807568877293527) / 6.0,  0.7886751345948128822546,               0                       },
    593:       {(-3.0 - 1.732050807568877293527) / 24.0, (-3.0 - 1.732050807568877293527) / 8.0, 0.7886751345948128822546}
    594:     };
    595:     const PetscReal b[3]  = {1. / 6., 1. / 6., 2. / 3.};
    596:     const PetscReal b2[3] = {1. / 4., 1. / 4., 1. / 2.};
    597:     PetscReal       binterpt[3][2];

    599:     binterpt[0][0] = 0.089316397477040902157517886164709;
    600:     binterpt[1][0] = -0.91068360252295909784248211383529;
    601:     binterpt[2][0] = 1.8213672050459181956849642276706;
    602:     binterpt[0][1] = 0.077350269189625764509148780501957;
    603:     binterpt[1][1] = 1.077350269189625764509148780502;
    604:     binterpt[2][1] = -1.1547005383792515290182975610039;

    606:     PetscCall(TSRosWRegister(TSROSWASSP3P3S1C, 3, 3, &A[0][0], &Gamma[0][0], b, b2, 2, &binterpt[0][0]));
    607:   }

    609:   {
    610:     const PetscReal A[4][4] = {
    611:       {0,       0,       0,       0},
    612:       {1. / 2., 0,       0,       0},
    613:       {1. / 2., 1. / 2., 0,       0},
    614:       {1. / 6., 1. / 6., 1. / 6., 0}
    615:     };
    616:     const PetscReal Gamma[4][4] = {
    617:       {1. / 2., 0,        0,       0},
    618:       {0.0,     1. / 4.,  0,       0},
    619:       {-2.,     -2. / 3., 2. / 3., 0},
    620:       {1. / 2., 5. / 36., -2. / 9, 0}
    621:     };
    622:     const PetscReal b[4]  = {1. / 6., 1. / 6., 1. / 6., 1. / 2.};
    623:     const PetscReal b2[4] = {1. / 8., 3. / 4., 1. / 8., 0};
    624:     PetscReal       binterpt[4][3];

    626:     binterpt[0][0] = 6.25;
    627:     binterpt[1][0] = -30.25;
    628:     binterpt[2][0] = 1.75;
    629:     binterpt[3][0] = 23.25;
    630:     binterpt[0][1] = -9.75;
    631:     binterpt[1][1] = 58.75;
    632:     binterpt[2][1] = -3.25;
    633:     binterpt[3][1] = -45.75;
    634:     binterpt[0][2] = 3.6666666666666666666666666666667;
    635:     binterpt[1][2] = -28.333333333333333333333333333333;
    636:     binterpt[2][2] = 1.6666666666666666666666666666667;
    637:     binterpt[3][2] = 23.;

    639:     PetscCall(TSRosWRegister(TSROSWLASSP3P4S2C, 3, 4, &A[0][0], &Gamma[0][0], b, b2, 3, &binterpt[0][0]));
    640:   }

    642:   {
    643:     const PetscReal A[4][4] = {
    644:       {0,       0,       0,       0},
    645:       {1. / 2., 0,       0,       0},
    646:       {1. / 2., 1. / 2., 0,       0},
    647:       {1. / 6., 1. / 6., 1. / 6., 0}
    648:     };
    649:     const PetscReal Gamma[4][4] = {
    650:       {1. / 2.,  0,          0,        0},
    651:       {0.0,      3. / 4.,    0,        0},
    652:       {-2. / 3., -23. / 9.,  2. / 9.,  0},
    653:       {1. / 18., 65. / 108., -2. / 27, 0}
    654:     };
    655:     const PetscReal b[4]  = {1. / 6., 1. / 6., 1. / 6., 1. / 2.};
    656:     const PetscReal b2[4] = {3. / 16., 10. / 16., 3. / 16., 0};
    657:     PetscReal       binterpt[4][3];

    659:     binterpt[0][0] = 1.6911764705882352941176470588235;
    660:     binterpt[1][0] = 3.6813725490196078431372549019608;
    661:     binterpt[2][0] = 0.23039215686274509803921568627451;
    662:     binterpt[3][0] = -4.6029411764705882352941176470588;
    663:     binterpt[0][1] = -0.95588235294117647058823529411765;
    664:     binterpt[1][1] = -6.2401960784313725490196078431373;
    665:     binterpt[2][1] = -0.31862745098039215686274509803922;
    666:     binterpt[3][1] = 7.5147058823529411764705882352941;
    667:     binterpt[0][2] = -0.56862745098039215686274509803922;
    668:     binterpt[1][2] = 2.7254901960784313725490196078431;
    669:     binterpt[2][2] = 0.25490196078431372549019607843137;
    670:     binterpt[3][2] = -2.4117647058823529411764705882353;

    672:     PetscCall(TSRosWRegister(TSROSWLLSSP3P4S2C, 3, 4, &A[0][0], &Gamma[0][0], b, b2, 3, &binterpt[0][0]));
    673:   }

    675:   {
    676:     PetscReal A[4][4], Gamma[4][4], b[4], b2[4];
    677:     PetscReal binterpt[4][3];

    679:     Gamma[0][0] = 0.4358665215084589994160194475295062513822671686978816;
    680:     Gamma[0][1] = 0;
    681:     Gamma[0][2] = 0;
    682:     Gamma[0][3] = 0;
    683:     Gamma[1][0] = -1.997527830934941248426324674704153457289527280554476;
    684:     Gamma[1][1] = 0.4358665215084589994160194475295062513822671686978816;
    685:     Gamma[1][2] = 0;
    686:     Gamma[1][3] = 0;
    687:     Gamma[2][0] = -1.007948511795029620852002345345404191008352770119903;
    688:     Gamma[2][1] = -0.004648958462629345562774289390054679806993396798458131;
    689:     Gamma[2][2] = 0.4358665215084589994160194475295062513822671686978816;
    690:     Gamma[2][3] = 0;
    691:     Gamma[3][0] = -0.6685429734233467180451604600279552604364311322650783;
    692:     Gamma[3][1] = 0.6056625986449338476089525334450053439525178740492984;
    693:     Gamma[3][2] = -0.9717899277217721234705114616271378792182450260943198;
    694:     Gamma[3][3] = 0;

    696:     A[0][0] = 0;
    697:     A[0][1] = 0;
    698:     A[0][2] = 0;
    699:     A[0][3] = 0;
    700:     A[1][0] = 0.8717330430169179988320388950590125027645343373957631;
    701:     A[1][1] = 0;
    702:     A[1][2] = 0;
    703:     A[1][3] = 0;
    704:     A[2][0] = 0.5275890119763004115618079766722914408876108660811028;
    705:     A[2][1] = 0.07241098802369958843819203208518599088698057726988732;
    706:     A[2][2] = 0;
    707:     A[2][3] = 0;
    708:     A[3][0] = 0.3990960076760701320627260685975778145384666450351314;
    709:     A[3][1] = -0.4375576546135194437228463747348862825846903771419953;
    710:     A[3][2] = 1.038461646937449311660120300601880176655352737312713;
    711:     A[3][3] = 0;

    713:     b[0] = 0.1876410243467238251612921333138006734899663569186926;
    714:     b[1] = -0.5952974735769549480478230473706443582188442040780541;
    715:     b[2] = 0.9717899277217721234705114616271378792182450260943198;
    716:     b[3] = 0.4358665215084589994160194475295062513822671686978816;

    718:     b2[0] = 0.2147402862233891404862383521089097657790734483804460;
    719:     b2[1] = -0.4851622638849390928209050538171743017757490232519684;
    720:     b2[2] = 0.8687250025203875511662123688667549217531982787600080;
    721:     b2[3] = 0.4016969751411624011684543450940068201770721128357014;

    723:     binterpt[0][0] = 2.2565812720167954547104627844105;
    724:     binterpt[1][0] = 1.349166413351089573796243820819;
    725:     binterpt[2][0] = -2.4695174540533503758652847586647;
    726:     binterpt[3][0] = -0.13623023131453465264142184656474;
    727:     binterpt[0][1] = -3.0826699111559187902922463354557;
    728:     binterpt[1][1] = -2.4689115685996042534544925650515;
    729:     binterpt[2][1] = 5.7428279814696677152129332773553;
    730:     binterpt[3][1] = -0.19124650171414467146619437684812;
    731:     binterpt[0][2] = 1.0137296634858471607430756831148;
    732:     binterpt[1][2] = 0.52444768167155973161042570784064;
    733:     binterpt[2][2] = -2.3015205996945452158771370439586;
    734:     binterpt[3][2] = 0.76334325453713832352363565300308;

    736:     PetscCall(TSRosWRegister(TSROSWARK3, 3, 4, &A[0][0], &Gamma[0][0], b, b2, 3, &binterpt[0][0]));
    737:   }
    738:   PetscCall(TSRosWRegisterRos4(TSROSWGRK4T, 0.231, PETSC_CURRENT, PETSC_CURRENT, 0, -0.1282612945269037e+01));
    739:   PetscCall(TSRosWRegisterRos4(TSROSWSHAMP4, 0.5, PETSC_CURRENT, PETSC_CURRENT, 0, 125. / 108.));
    740:   PetscCall(TSRosWRegisterRos4(TSROSWVELDD4, 0.22570811482256823492, PETSC_CURRENT, PETSC_CURRENT, 0, -1.355958941201148));
    741:   PetscCall(TSRosWRegisterRos4(TSROSW4L, 0.57282, PETSC_CURRENT, PETSC_CURRENT, 0, -1.093502252409163));
    742:   PetscFunctionReturn(PETSC_SUCCESS);
    743: }

    745: /*@C
    746:   TSRosWRegisterDestroy - Frees the list of schemes that were registered by `TSRosWRegister()`.

    748:   Not Collective

    750:   Level: advanced

    752: .seealso: [](ch_ts), `TSRosWRegister()`, `TSRosWRegisterAll()`
    753: @*/
    754: PetscErrorCode TSRosWRegisterDestroy(void)
    755: {
    756:   RosWTableauLink link;

    758:   PetscFunctionBegin;
    759:   while ((link = RosWTableauList)) {
    760:     RosWTableau t   = &link->tab;
    761:     RosWTableauList = link->next;
    762:     PetscCall(PetscFree5(t->A, t->Gamma, t->b, t->ASum, t->GammaSum));
    763:     PetscCall(PetscFree5(t->At, t->bt, t->GammaInv, t->GammaZeroDiag, t->GammaExplicitCorr));
    764:     PetscCall(PetscFree2(t->bembed, t->bembedt));
    765:     PetscCall(PetscFree(t->binterpt));
    766:     PetscCall(PetscFree(t->name));
    767:     PetscCall(PetscFree(link));
    768:   }
    769:   TSRosWRegisterAllCalled = PETSC_FALSE;
    770:   PetscFunctionReturn(PETSC_SUCCESS);
    771: }

    773: /*@C
    774:   TSRosWInitializePackage - This function initializes everything in the `TSROSW` package. It is called
    775:   from `TSInitializePackage()`.

    777:   Level: developer

    779: .seealso: [](ch_ts), `TSROSW`, `PetscInitialize()`, `TSRosWFinalizePackage()`
    780: @*/
    781: PetscErrorCode TSRosWInitializePackage(void)
    782: {
    783:   PetscFunctionBegin;
    784:   if (TSRosWPackageInitialized) PetscFunctionReturn(PETSC_SUCCESS);
    785:   TSRosWPackageInitialized = PETSC_TRUE;
    786:   PetscCall(TSRosWRegisterAll());
    787:   PetscCall(PetscRegisterFinalize(TSRosWFinalizePackage));
    788:   PetscFunctionReturn(PETSC_SUCCESS);
    789: }

    791: /*@C
    792:   TSRosWFinalizePackage - This function destroys everything in the `TSROSW` package. It is
    793:   called from `PetscFinalize()`.

    795:   Level: developer

    797: .seealso: [](ch_ts), `TSROSW`, `PetscFinalize()`, `TSRosWInitializePackage()`
    798: @*/
    799: PetscErrorCode TSRosWFinalizePackage(void)
    800: {
    801:   PetscFunctionBegin;
    802:   TSRosWPackageInitialized = PETSC_FALSE;
    803:   PetscCall(TSRosWRegisterDestroy());
    804:   PetscFunctionReturn(PETSC_SUCCESS);
    805: }

    807: /*@C
    808:   TSRosWRegister - register a `TSROSW`, Rosenbrock W scheme by providing the entries in the Butcher tableau and optionally embedded approximations and interpolation

    810:   Not Collective, but the same schemes should be registered on all processes on which they will be used

    812:   Input Parameters:
    813: + name     - identifier for method
    814: . order    - approximation order of method
    815: . s        - number of stages, this is the dimension of the matrices below
    816: . A        - Table of propagated stage coefficients (dimension s*s, row-major), strictly lower triangular
    817: . Gamma    - Table of coefficients in implicit stage equations (dimension s*s, row-major), lower triangular with nonzero diagonal
    818: . b        - Step completion table (dimension s)
    819: . bembed   - Step completion table for a scheme of order one less (dimension s, NULL if no embedded scheme is available)
    820: . pinterp  - Order of the interpolation scheme, equal to the number of columns of binterpt
    821: - binterpt - Coefficients of the interpolation formula (dimension s*pinterp)

    823:   Level: advanced

    825:   Note:
    826:   Several Rosenbrock W methods are provided, this function is only needed to create new methods.

    828: .seealso: [](ch_ts), `TSROSW`
    829: @*/
    830: PetscErrorCode TSRosWRegister(TSRosWType name, PetscInt order, PetscInt s, const PetscReal A[], const PetscReal Gamma[], const PetscReal b[], const PetscReal bembed[], PetscInt pinterp, const PetscReal binterpt[])
    831: {
    832:   RosWTableauLink link;
    833:   RosWTableau     t;
    834:   PetscInt        i, j, k;
    835:   PetscScalar    *GammaInv;

    837:   PetscFunctionBegin;
    838:   PetscAssertPointer(name, 1);
    839:   PetscAssertPointer(A, 4);
    840:   PetscAssertPointer(Gamma, 5);
    841:   PetscAssertPointer(b, 6);
    842:   if (bembed) PetscAssertPointer(bembed, 7);

    844:   PetscCall(TSRosWInitializePackage());
    845:   PetscCall(PetscNew(&link));
    846:   t = &link->tab;
    847:   PetscCall(PetscStrallocpy(name, &t->name));
    848:   t->order = order;
    849:   t->s     = s;
    850:   PetscCall(PetscMalloc5(s * s, &t->A, s * s, &t->Gamma, s, &t->b, s, &t->ASum, s, &t->GammaSum));
    851:   PetscCall(PetscMalloc5(s * s, &t->At, s, &t->bt, s * s, &t->GammaInv, s, &t->GammaZeroDiag, s * s, &t->GammaExplicitCorr));
    852:   PetscCall(PetscArraycpy(t->A, A, s * s));
    853:   PetscCall(PetscArraycpy(t->Gamma, Gamma, s * s));
    854:   PetscCall(PetscArraycpy(t->GammaExplicitCorr, Gamma, s * s));
    855:   PetscCall(PetscArraycpy(t->b, b, s));
    856:   if (bembed) {
    857:     PetscCall(PetscMalloc2(s, &t->bembed, s, &t->bembedt));
    858:     PetscCall(PetscArraycpy(t->bembed, bembed, s));
    859:   }
    860:   for (i = 0; i < s; i++) {
    861:     t->ASum[i]     = 0;
    862:     t->GammaSum[i] = 0;
    863:     for (j = 0; j < s; j++) {
    864:       t->ASum[i] += A[i * s + j];
    865:       t->GammaSum[i] += Gamma[i * s + j];
    866:     }
    867:   }
    868:   PetscCall(PetscMalloc1(s * s, &GammaInv)); /* Need to use Scalar for inverse, then convert back to Real */
    869:   for (i = 0; i < s * s; i++) GammaInv[i] = Gamma[i];
    870:   for (i = 0; i < s; i++) {
    871:     if (Gamma[i * s + i] == 0.0) {
    872:       GammaInv[i * s + i] = 1.0;
    873:       t->GammaZeroDiag[i] = PETSC_TRUE;
    874:     } else {
    875:       t->GammaZeroDiag[i] = PETSC_FALSE;
    876:     }
    877:   }

    879:   switch (s) {
    880:   case 1:
    881:     GammaInv[0] = 1. / GammaInv[0];
    882:     break;
    883:   case 2:
    884:     PetscCall(PetscKernel_A_gets_inverse_A_2(GammaInv, 0, PETSC_FALSE, NULL));
    885:     break;
    886:   case 3:
    887:     PetscCall(PetscKernel_A_gets_inverse_A_3(GammaInv, 0, PETSC_FALSE, NULL));
    888:     break;
    889:   case 4:
    890:     PetscCall(PetscKernel_A_gets_inverse_A_4(GammaInv, 0, PETSC_FALSE, NULL));
    891:     break;
    892:   case 5: {
    893:     PetscInt  ipvt5[5];
    894:     MatScalar work5[5 * 5];
    895:     PetscCall(PetscKernel_A_gets_inverse_A_5(GammaInv, ipvt5, work5, 0, PETSC_FALSE, NULL));
    896:     break;
    897:   }
    898:   case 6:
    899:     PetscCall(PetscKernel_A_gets_inverse_A_6(GammaInv, 0, PETSC_FALSE, NULL));
    900:     break;
    901:   case 7:
    902:     PetscCall(PetscKernel_A_gets_inverse_A_7(GammaInv, 0, PETSC_FALSE, NULL));
    903:     break;
    904:   default:
    905:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Not implemented for %" PetscInt_FMT " stages", s);
    906:   }
    907:   for (i = 0; i < s * s; i++) t->GammaInv[i] = PetscRealPart(GammaInv[i]);
    908:   PetscCall(PetscFree(GammaInv));

    910:   for (i = 0; i < s; i++) {
    911:     for (k = 0; k < i + 1; k++) {
    912:       t->GammaExplicitCorr[i * s + k] = (t->GammaExplicitCorr[i * s + k]) * (t->GammaInv[k * s + k]);
    913:       for (j = k + 1; j < i + 1; j++) t->GammaExplicitCorr[i * s + k] += (t->GammaExplicitCorr[i * s + j]) * (t->GammaInv[j * s + k]);
    914:     }
    915:   }

    917:   for (i = 0; i < s; i++) {
    918:     for (j = 0; j < s; j++) {
    919:       t->At[i * s + j] = 0;
    920:       for (k = 0; k < s; k++) t->At[i * s + j] += t->A[i * s + k] * t->GammaInv[k * s + j];
    921:     }
    922:     t->bt[i] = 0;
    923:     for (j = 0; j < s; j++) t->bt[i] += t->b[j] * t->GammaInv[j * s + i];
    924:     if (bembed) {
    925:       t->bembedt[i] = 0;
    926:       for (j = 0; j < s; j++) t->bembedt[i] += t->bembed[j] * t->GammaInv[j * s + i];
    927:     }
    928:   }
    929:   t->ccfl = 1.0; /* Fix this */

    931:   t->pinterp = pinterp;
    932:   PetscCall(PetscMalloc1(s * pinterp, &t->binterpt));
    933:   PetscCall(PetscArraycpy(t->binterpt, binterpt, s * pinterp));
    934:   link->next      = RosWTableauList;
    935:   RosWTableauList = link;
    936:   PetscFunctionReturn(PETSC_SUCCESS);
    937: }

    939: /*@C
    940:   TSRosWRegisterRos4 - register a fourth order Rosenbrock scheme by providing parameter choices

    942:   Not Collective, but the same schemes should be registered on all processes on which they will be used

    944:   Input Parameters:
    945: + name  - identifier for method
    946: . gamma - leading coefficient (diagonal entry)
    947: . a2    - design parameter, see Table 7.2 of {cite}`wanner1996solving`
    948: . a3    - design parameter or `PETSC_DETERMINE` to satisfy one of the order five conditions (Eq 7.22)
    949: . b3    - design parameter, see Table 7.2 of {cite}`wanner1996solving`
    950: - e4    - design parameter for embedded method, see coefficient E4 in ros4.f code from Hairer

    952:   Level: developer

    954:   Notes:
    955:   This routine encodes the design of fourth order Rosenbrock methods as described in {cite}`wanner1996solving`
    956:   It is used here to implement several methods from the book and can be used to experiment with new methods.
    957:   It was written this way instead of by copying coefficients in order to provide better than double precision satisfaction of the order conditions.

    959: .seealso: [](ch_ts), `TSRosW`, `TSRosWRegister()`
    960: @*/
    961: PetscErrorCode TSRosWRegisterRos4(TSRosWType name, PetscReal gamma, PetscReal a2, PetscReal a3, PetscReal b3, PetscReal e4)
    962: {
    963:   /* Declare numeric constants so they can be quad precision without being truncated at double */
    964:   const PetscReal one = 1, two = 2, three = 3, four = 4, five = 5, six = 6, eight = 8, twelve = 12, twenty = 20, twentyfour = 24, p32 = one / six - gamma + gamma * gamma, p42 = one / eight - gamma / three, p43 = one / twelve - gamma / three, p44 = one / twentyfour - gamma / two + three / two * gamma * gamma - gamma * gamma * gamma, p56 = one / twenty - gamma / four;
    965:   PetscReal   a4, a32, a42, a43, b1, b2, b4, beta2p, beta3p, beta4p, beta32, beta42, beta43, beta32beta2p, beta4jbetajp;
    966:   PetscReal   A[4][4], Gamma[4][4], b[4], bm[4];
    967:   PetscScalar M[3][3], rhs[3];

    969:   PetscFunctionBegin;
    970:   /* Step 1: choose Gamma (input) */
    971:   /* Step 2: choose a2,a3,a4; b1,b2,b3,b4 to satisfy order conditions */
    972:   if (a3 == (PetscReal)PETSC_DEFAULT || a3 == (PetscReal)PETSC_DETERMINE) a3 = (one / five - a2 / four) / (one / four - a2 / three); /* Eq 7.22 */
    973:   a4 = a3;                                                                                                                           /* consequence of 7.20 */

    975:   /* Solve order conditions 7.15a, 7.15c, 7.15e */
    976:   M[0][0] = one;
    977:   M[0][1] = one;
    978:   M[0][2] = one; /* 7.15a */
    979:   M[1][0] = 0.0;
    980:   M[1][1] = a2 * a2;
    981:   M[1][2] = a4 * a4; /* 7.15c */
    982:   M[2][0] = 0.0;
    983:   M[2][1] = a2 * a2 * a2;
    984:   M[2][2] = a4 * a4 * a4; /* 7.15e */
    985:   rhs[0]  = one - b3;
    986:   rhs[1]  = one / three - a3 * a3 * b3;
    987:   rhs[2]  = one / four - a3 * a3 * a3 * b3;
    988:   PetscCall(PetscKernel_A_gets_inverse_A_3(&M[0][0], 0, PETSC_FALSE, NULL));
    989:   b1 = PetscRealPart(M[0][0] * rhs[0] + M[0][1] * rhs[1] + M[0][2] * rhs[2]);
    990:   b2 = PetscRealPart(M[1][0] * rhs[0] + M[1][1] * rhs[1] + M[1][2] * rhs[2]);
    991:   b4 = PetscRealPart(M[2][0] * rhs[0] + M[2][1] * rhs[1] + M[2][2] * rhs[2]);

    993:   /* Step 3 */
    994:   beta43       = (p56 - a2 * p43) / (b4 * a3 * a3 * (a3 - a2)); /* 7.21 */
    995:   beta32beta2p = p44 / (b4 * beta43);                           /* 7.15h */
    996:   beta4jbetajp = (p32 - b3 * beta32beta2p) / b4;
    997:   M[0][0]      = b2;
    998:   M[0][1]      = b3;
    999:   M[0][2]      = b4;
    1000:   M[1][0]      = a4 * a4 * beta32beta2p - a3 * a3 * beta4jbetajp;
    1001:   M[1][1]      = a2 * a2 * beta4jbetajp;
    1002:   M[1][2]      = -a2 * a2 * beta32beta2p;
    1003:   M[2][0]      = b4 * beta43 * a3 * a3 - p43;
    1004:   M[2][1]      = -b4 * beta43 * a2 * a2;
    1005:   M[2][2]      = 0;
    1006:   rhs[0]       = one / two - gamma;
    1007:   rhs[1]       = 0;
    1008:   rhs[2]       = -a2 * a2 * p32;
    1009:   PetscCall(PetscKernel_A_gets_inverse_A_3(&M[0][0], 0, PETSC_FALSE, NULL));
    1010:   beta2p = PetscRealPart(M[0][0] * rhs[0] + M[0][1] * rhs[1] + M[0][2] * rhs[2]);
    1011:   beta3p = PetscRealPart(M[1][0] * rhs[0] + M[1][1] * rhs[1] + M[1][2] * rhs[2]);
    1012:   beta4p = PetscRealPart(M[2][0] * rhs[0] + M[2][1] * rhs[1] + M[2][2] * rhs[2]);

    1014:   /* Step 4: back-substitute */
    1015:   beta32 = beta32beta2p / beta2p;
    1016:   beta42 = (beta4jbetajp - beta43 * beta3p) / beta2p;

    1018:   /* Step 5: 7.15f and 7.20, then 7.16 */
    1019:   a43 = 0;
    1020:   a32 = p42 / (b3 * a3 * beta2p + b4 * a4 * beta2p);
    1021:   a42 = a32;

    1023:   A[0][0]     = 0;
    1024:   A[0][1]     = 0;
    1025:   A[0][2]     = 0;
    1026:   A[0][3]     = 0;
    1027:   A[1][0]     = a2;
    1028:   A[1][1]     = 0;
    1029:   A[1][2]     = 0;
    1030:   A[1][3]     = 0;
    1031:   A[2][0]     = a3 - a32;
    1032:   A[2][1]     = a32;
    1033:   A[2][2]     = 0;
    1034:   A[2][3]     = 0;
    1035:   A[3][0]     = a4 - a43 - a42;
    1036:   A[3][1]     = a42;
    1037:   A[3][2]     = a43;
    1038:   A[3][3]     = 0;
    1039:   Gamma[0][0] = gamma;
    1040:   Gamma[0][1] = 0;
    1041:   Gamma[0][2] = 0;
    1042:   Gamma[0][3] = 0;
    1043:   Gamma[1][0] = beta2p - A[1][0];
    1044:   Gamma[1][1] = gamma;
    1045:   Gamma[1][2] = 0;
    1046:   Gamma[1][3] = 0;
    1047:   Gamma[2][0] = beta3p - beta32 - A[2][0];
    1048:   Gamma[2][1] = beta32 - A[2][1];
    1049:   Gamma[2][2] = gamma;
    1050:   Gamma[2][3] = 0;
    1051:   Gamma[3][0] = beta4p - beta42 - beta43 - A[3][0];
    1052:   Gamma[3][1] = beta42 - A[3][1];
    1053:   Gamma[3][2] = beta43 - A[3][2];
    1054:   Gamma[3][3] = gamma;
    1055:   b[0]        = b1;
    1056:   b[1]        = b2;
    1057:   b[2]        = b3;
    1058:   b[3]        = b4;

    1060:   /* Construct embedded formula using given e4. We are solving Equation 7.18. */
    1061:   bm[3] = b[3] - e4 * gamma;                                              /* using definition of E4 */
    1062:   bm[2] = (p32 - beta4jbetajp * bm[3]) / (beta32 * beta2p);               /* fourth row of 7.18 */
    1063:   bm[1] = (one / two - gamma - beta3p * bm[2] - beta4p * bm[3]) / beta2p; /* second row */
    1064:   bm[0] = one - bm[1] - bm[2] - bm[3];                                    /* first row */

    1066:   {
    1067:     const PetscReal misfit = a2 * a2 * bm[1] + a3 * a3 * bm[2] + a4 * a4 * bm[3] - one / three;
    1068:     PetscCheck(PetscAbs(misfit) <= PETSC_SMALL, PETSC_COMM_SELF, PETSC_ERR_SUP, "Assumptions violated, could not construct a third order embedded method");
    1069:   }
    1070:   PetscCall(TSRosWRegister(name, 4, 4, &A[0][0], &Gamma[0][0], b, bm, 0, NULL));
    1071:   PetscFunctionReturn(PETSC_SUCCESS);
    1072: }

    1074: /*
    1075:  The step completion formula is

    1077:  x1 = x0 + b^T Y

    1079:  where Y is the multi-vector of stages corrections. This function can be called before or after ts->vec_sol has been
    1080:  updated. Suppose we have a completion formula b and an embedded formula be of different order. We can write

    1082:  x1e = x0 + be^T Y
    1083:      = x1 - b^T Y + be^T Y
    1084:      = x1 + (be - b)^T Y

    1086:  so we can evaluate the method of different order even after the step has been optimistically completed.
    1087: */
    1088: static PetscErrorCode TSEvaluateStep_RosW(TS ts, PetscInt order, Vec U, PetscBool *done)
    1089: {
    1090:   TS_RosW     *ros = (TS_RosW *)ts->data;
    1091:   RosWTableau  tab = ros->tableau;
    1092:   PetscScalar *w   = ros->work;
    1093:   PetscInt     i;

    1095:   PetscFunctionBegin;
    1096:   if (order == tab->order) {
    1097:     if (ros->status == TS_STEP_INCOMPLETE) { /* Use standard completion formula */
    1098:       PetscCall(VecCopy(ts->vec_sol, U));
    1099:       for (i = 0; i < tab->s; i++) w[i] = tab->bt[i];
    1100:       PetscCall(VecMAXPY(U, tab->s, w, ros->Y));
    1101:     } else PetscCall(VecCopy(ts->vec_sol, U));
    1102:     if (done) *done = PETSC_TRUE;
    1103:     PetscFunctionReturn(PETSC_SUCCESS);
    1104:   } else if (order == tab->order - 1) {
    1105:     if (!tab->bembedt) goto unavailable;
    1106:     if (ros->status == TS_STEP_INCOMPLETE) { /* Use embedded completion formula */
    1107:       PetscCall(VecCopy(ts->vec_sol, U));
    1108:       for (i = 0; i < tab->s; i++) w[i] = tab->bembedt[i];
    1109:       PetscCall(VecMAXPY(U, tab->s, w, ros->Y));
    1110:     } else { /* Use rollback-and-recomplete formula (bembedt - bt) */
    1111:       for (i = 0; i < tab->s; i++) w[i] = tab->bembedt[i] - tab->bt[i];
    1112:       PetscCall(VecCopy(ts->vec_sol, U));
    1113:       PetscCall(VecMAXPY(U, tab->s, w, ros->Y));
    1114:     }
    1115:     if (done) *done = PETSC_TRUE;
    1116:     PetscFunctionReturn(PETSC_SUCCESS);
    1117:   }
    1118: unavailable:
    1119:   if (done) *done = PETSC_FALSE;
    1120:   else
    1121:     SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "Rosenbrock-W '%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,
    1122:             tab->order, order);
    1123:   PetscFunctionReturn(PETSC_SUCCESS);
    1124: }

    1126: static PetscErrorCode TSStep_RosW(TS ts)
    1127: {
    1128:   TS_RosW         *ros = (TS_RosW *)ts->data;
    1129:   RosWTableau      tab = ros->tableau;
    1130:   const PetscInt   s   = tab->s;
    1131:   const PetscReal *At = tab->At, *Gamma = tab->Gamma, *ASum = tab->ASum, *GammaInv = tab->GammaInv;
    1132:   const PetscReal *GammaExplicitCorr = tab->GammaExplicitCorr;
    1133:   const PetscBool *GammaZeroDiag     = tab->GammaZeroDiag;
    1134:   PetscScalar     *w                 = ros->work;
    1135:   Vec             *Y = ros->Y, Ydot = ros->Ydot, Zdot = ros->Zdot, Zstage = ros->Zstage;
    1136:   SNES             snes;
    1137:   TSAdapt          adapt;
    1138:   PetscInt         i, j, its, lits;
    1139:   PetscInt         rejections = 0;
    1140:   PetscBool        stageok, accept = PETSC_TRUE;
    1141:   PetscReal        next_time_step = ts->time_step;
    1142:   PetscInt         lag;

    1144:   PetscFunctionBegin;
    1145:   if (!ts->steprollback) PetscCall(VecCopy(ts->vec_sol, ros->vec_sol_prev));

    1147:   ros->status = TS_STEP_INCOMPLETE;
    1148:   while (!ts->reason && ros->status != TS_STEP_COMPLETE) {
    1149:     const PetscReal h = ts->time_step;
    1150:     for (i = 0; i < s; i++) {
    1151:       ros->stage_time = ts->ptime + h * ASum[i];
    1152:       PetscCall(TSPreStage(ts, ros->stage_time));
    1153:       if (GammaZeroDiag[i]) {
    1154:         ros->stage_explicit = PETSC_TRUE;
    1155:         ros->scoeff         = 1.;
    1156:       } else {
    1157:         ros->stage_explicit = PETSC_FALSE;
    1158:         ros->scoeff         = 1. / Gamma[i * s + i];
    1159:       }

    1161:       PetscCall(VecCopy(ts->vec_sol, Zstage));
    1162:       for (j = 0; j < i; j++) w[j] = At[i * s + j];
    1163:       PetscCall(VecMAXPY(Zstage, i, w, Y));

    1165:       for (j = 0; j < i; j++) w[j] = 1. / h * GammaInv[i * s + j];
    1166:       PetscCall(VecZeroEntries(Zdot));
    1167:       PetscCall(VecMAXPY(Zdot, i, w, Y));

    1169:       /* Initial guess taken from last stage */
    1170:       PetscCall(VecZeroEntries(Y[i]));

    1172:       if (!ros->stage_explicit) {
    1173:         PetscCall(TSGetSNES(ts, &snes));
    1174:         if (!ros->recompute_jacobian && !i) {
    1175:           PetscCall(SNESGetLagJacobian(snes, &lag));
    1176:           if (lag == 1) {                            /* use did not set a nontrivial lag, so lag over all stages */
    1177:             PetscCall(SNESSetLagJacobian(snes, -2)); /* Recompute the Jacobian on this solve, but not again for the rest of the stages */
    1178:           }
    1179:         }
    1180:         PetscCall(SNESSolve(snes, NULL, Y[i]));
    1181:         if (!ros->recompute_jacobian && i == s - 1 && lag == 1) { PetscCall(SNESSetLagJacobian(snes, lag)); /* Set lag back to 1 so we know user did not set it */ }
    1182:         PetscCall(SNESGetIterationNumber(snes, &its));
    1183:         PetscCall(SNESGetLinearSolveIterations(snes, &lits));
    1184:         ts->snes_its += its;
    1185:         ts->ksp_its += lits;
    1186:       } else {
    1187:         Mat J, Jp;
    1188:         PetscCall(VecZeroEntries(Ydot)); /* Evaluate Y[i]=G(t,Ydot=0,Zstage) */
    1189:         PetscCall(TSComputeIFunction(ts, ros->stage_time, Zstage, Ydot, Y[i], PETSC_FALSE));
    1190:         PetscCall(VecScale(Y[i], -1.0));
    1191:         PetscCall(VecAXPY(Y[i], -1.0, Zdot)); /*Y[i] = F(Zstage)-Zdot[=GammaInv*Y]*/

    1193:         PetscCall(VecZeroEntries(Zstage)); /* Zstage = GammaExplicitCorr[i,j] * Y[j] */
    1194:         for (j = 0; j < i; j++) w[j] = GammaExplicitCorr[i * s + j];
    1195:         PetscCall(VecMAXPY(Zstage, i, w, Y));

    1197:         /* Y[i] = Y[i] + Jac*Zstage[=Jac*GammaExplicitCorr[i,j] * Y[j]] */
    1198:         PetscCall(TSGetIJacobian(ts, &J, &Jp, NULL, NULL));
    1199:         PetscCall(TSComputeIJacobian(ts, ros->stage_time, ts->vec_sol, Ydot, 0, J, Jp, PETSC_FALSE));
    1200:         PetscCall(MatMult(J, Zstage, Zdot));
    1201:         PetscCall(VecAXPY(Y[i], -1.0, Zdot));
    1202:         ts->ksp_its += 1;

    1204:         PetscCall(VecScale(Y[i], h));
    1205:       }
    1206:       PetscCall(TSPostStage(ts, ros->stage_time, i, Y));
    1207:       PetscCall(TSGetAdapt(ts, &adapt));
    1208:       PetscCall(TSAdaptCheckStage(adapt, ts, ros->stage_time, Y[i], &stageok));
    1209:       if (!stageok) goto reject_step;
    1210:     }

    1212:     ros->status = TS_STEP_INCOMPLETE;
    1213:     PetscCall(TSEvaluateStep_RosW(ts, tab->order, ts->vec_sol, NULL));
    1214:     ros->status = TS_STEP_PENDING;
    1215:     PetscCall(TSGetAdapt(ts, &adapt));
    1216:     PetscCall(TSAdaptCandidatesClear(adapt));
    1217:     PetscCall(TSAdaptCandidateAdd(adapt, tab->name, tab->order, 1, tab->ccfl, (PetscReal)tab->s, PETSC_TRUE));
    1218:     PetscCall(TSAdaptChoose(adapt, ts, ts->time_step, NULL, &next_time_step, &accept));
    1219:     ros->status = accept ? TS_STEP_COMPLETE : TS_STEP_INCOMPLETE;
    1220:     if (!accept) { /* Roll back the current step */
    1221:       PetscCall(VecCopy(ts->vec_sol0, ts->vec_sol));
    1222:       ts->time_step = next_time_step;
    1223:       goto reject_step;
    1224:     }

    1226:     ts->ptime += ts->time_step;
    1227:     ts->time_step = next_time_step;
    1228:     break;

    1230:   reject_step:
    1231:     ts->reject++;
    1232:     accept = PETSC_FALSE;
    1233:     if (!ts->reason && ++rejections > ts->max_reject && ts->max_reject >= 0) {
    1234:       ts->reason = TS_DIVERGED_STEP_REJECTED;
    1235:       PetscCall(PetscInfo(ts, "Step=%" PetscInt_FMT ", step rejections %" PetscInt_FMT " greater than current TS allowed, stopping solve\n", ts->steps, rejections));
    1236:     }
    1237:   }
    1238:   PetscFunctionReturn(PETSC_SUCCESS);
    1239: }

    1241: static PetscErrorCode TSInterpolate_RosW(TS ts, PetscReal itime, Vec U)
    1242: {
    1243:   TS_RosW         *ros = (TS_RosW *)ts->data;
    1244:   PetscInt         s = ros->tableau->s, pinterp = ros->tableau->pinterp, i, j;
    1245:   PetscReal        h;
    1246:   PetscReal        tt, t;
    1247:   PetscScalar     *bt;
    1248:   const PetscReal *Bt       = ros->tableau->binterpt;
    1249:   const PetscReal *GammaInv = ros->tableau->GammaInv;
    1250:   PetscScalar     *w        = ros->work;
    1251:   Vec             *Y        = ros->Y;

    1253:   PetscFunctionBegin;
    1254:   PetscCheck(Bt, PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "TSRosW %s does not have an interpolation formula", ros->tableau->name);

    1256:   switch (ros->status) {
    1257:   case TS_STEP_INCOMPLETE:
    1258:   case TS_STEP_PENDING:
    1259:     h = ts->time_step;
    1260:     t = (itime - ts->ptime) / h;
    1261:     break;
    1262:   case TS_STEP_COMPLETE:
    1263:     h = ts->ptime - ts->ptime_prev;
    1264:     t = (itime - ts->ptime) / h + 1; /* In the interval [0,1] */
    1265:     break;
    1266:   default:
    1267:     SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_PLIB, "Invalid TSStepStatus");
    1268:   }
    1269:   PetscCall(PetscMalloc1(s, &bt));
    1270:   for (i = 0; i < s; i++) bt[i] = 0;
    1271:   for (j = 0, tt = t; j < pinterp; j++, tt *= t) {
    1272:     for (i = 0; i < s; i++) bt[i] += Bt[i * pinterp + j] * tt;
    1273:   }

    1275:   /* y(t+tt*h) = y(t) + Sum bt(tt) * GammaInv * Ydot */
    1276:   /* U <- 0*/
    1277:   PetscCall(VecZeroEntries(U));
    1278:   /* U <- Sum bt_i * GammaInv(i,1:i) * Y(1:i) */
    1279:   for (j = 0; j < s; j++) w[j] = 0;
    1280:   for (j = 0; j < s; j++) {
    1281:     for (i = j; i < s; i++) w[j] += bt[i] * GammaInv[i * s + j];
    1282:   }
    1283:   PetscCall(VecMAXPY(U, i, w, Y));
    1284:   /* U <- y(t) + U */
    1285:   PetscCall(VecAXPY(U, 1, ros->vec_sol_prev));

    1287:   PetscCall(PetscFree(bt));
    1288:   PetscFunctionReturn(PETSC_SUCCESS);
    1289: }

    1291: /*------------------------------------------------------------*/

    1293: static PetscErrorCode TSRosWTableauReset(TS ts)
    1294: {
    1295:   TS_RosW    *ros = (TS_RosW *)ts->data;
    1296:   RosWTableau tab = ros->tableau;

    1298:   PetscFunctionBegin;
    1299:   if (!tab) PetscFunctionReturn(PETSC_SUCCESS);
    1300:   PetscCall(VecDestroyVecs(tab->s, &ros->Y));
    1301:   PetscCall(PetscFree(ros->work));
    1302:   PetscFunctionReturn(PETSC_SUCCESS);
    1303: }

    1305: static PetscErrorCode TSReset_RosW(TS ts)
    1306: {
    1307:   TS_RosW *ros = (TS_RosW *)ts->data;

    1309:   PetscFunctionBegin;
    1310:   PetscCall(TSRosWTableauReset(ts));
    1311:   PetscCall(VecDestroy(&ros->Ydot));
    1312:   PetscCall(VecDestroy(&ros->Ystage));
    1313:   PetscCall(VecDestroy(&ros->Zdot));
    1314:   PetscCall(VecDestroy(&ros->Zstage));
    1315:   PetscCall(VecDestroy(&ros->vec_sol_prev));
    1316:   PetscFunctionReturn(PETSC_SUCCESS);
    1317: }

    1319: static PetscErrorCode TSRosWGetVecs(TS ts, DM dm, Vec *Ydot, Vec *Zdot, Vec *Ystage, Vec *Zstage)
    1320: {
    1321:   TS_RosW *rw = (TS_RosW *)ts->data;

    1323:   PetscFunctionBegin;
    1324:   if (Ydot) {
    1325:     if (dm && dm != ts->dm) {
    1326:       PetscCall(DMGetNamedGlobalVector(dm, "TSRosW_Ydot", Ydot));
    1327:     } else *Ydot = rw->Ydot;
    1328:   }
    1329:   if (Zdot) {
    1330:     if (dm && dm != ts->dm) {
    1331:       PetscCall(DMGetNamedGlobalVector(dm, "TSRosW_Zdot", Zdot));
    1332:     } else *Zdot = rw->Zdot;
    1333:   }
    1334:   if (Ystage) {
    1335:     if (dm && dm != ts->dm) {
    1336:       PetscCall(DMGetNamedGlobalVector(dm, "TSRosW_Ystage", Ystage));
    1337:     } else *Ystage = rw->Ystage;
    1338:   }
    1339:   if (Zstage) {
    1340:     if (dm && dm != ts->dm) {
    1341:       PetscCall(DMGetNamedGlobalVector(dm, "TSRosW_Zstage", Zstage));
    1342:     } else *Zstage = rw->Zstage;
    1343:   }
    1344:   PetscFunctionReturn(PETSC_SUCCESS);
    1345: }

    1347: static PetscErrorCode TSRosWRestoreVecs(TS ts, DM dm, Vec *Ydot, Vec *Zdot, Vec *Ystage, Vec *Zstage)
    1348: {
    1349:   PetscFunctionBegin;
    1350:   if (Ydot) {
    1351:     if (dm && dm != ts->dm) PetscCall(DMRestoreNamedGlobalVector(dm, "TSRosW_Ydot", Ydot));
    1352:   }
    1353:   if (Zdot) {
    1354:     if (dm && dm != ts->dm) PetscCall(DMRestoreNamedGlobalVector(dm, "TSRosW_Zdot", Zdot));
    1355:   }
    1356:   if (Ystage) {
    1357:     if (dm && dm != ts->dm) PetscCall(DMRestoreNamedGlobalVector(dm, "TSRosW_Ystage", Ystage));
    1358:   }
    1359:   if (Zstage) {
    1360:     if (dm && dm != ts->dm) PetscCall(DMRestoreNamedGlobalVector(dm, "TSRosW_Zstage", Zstage));
    1361:   }
    1362:   PetscFunctionReturn(PETSC_SUCCESS);
    1363: }

    1365: static PetscErrorCode DMCoarsenHook_TSRosW(DM fine, DM coarse, void *ctx)
    1366: {
    1367:   PetscFunctionBegin;
    1368:   PetscFunctionReturn(PETSC_SUCCESS);
    1369: }

    1371: static PetscErrorCode DMRestrictHook_TSRosW(DM fine, Mat restrct, Vec rscale, Mat inject, DM coarse, void *ctx)
    1372: {
    1373:   TS  ts = (TS)ctx;
    1374:   Vec Ydot, Zdot, Ystage, Zstage;
    1375:   Vec Ydotc, Zdotc, Ystagec, Zstagec;

    1377:   PetscFunctionBegin;
    1378:   PetscCall(TSRosWGetVecs(ts, fine, &Ydot, &Ystage, &Zdot, &Zstage));
    1379:   PetscCall(TSRosWGetVecs(ts, coarse, &Ydotc, &Ystagec, &Zdotc, &Zstagec));
    1380:   PetscCall(MatRestrict(restrct, Ydot, Ydotc));
    1381:   PetscCall(VecPointwiseMult(Ydotc, rscale, Ydotc));
    1382:   PetscCall(MatRestrict(restrct, Ystage, Ystagec));
    1383:   PetscCall(VecPointwiseMult(Ystagec, rscale, Ystagec));
    1384:   PetscCall(MatRestrict(restrct, Zdot, Zdotc));
    1385:   PetscCall(VecPointwiseMult(Zdotc, rscale, Zdotc));
    1386:   PetscCall(MatRestrict(restrct, Zstage, Zstagec));
    1387:   PetscCall(VecPointwiseMult(Zstagec, rscale, Zstagec));
    1388:   PetscCall(TSRosWRestoreVecs(ts, fine, &Ydot, &Ystage, &Zdot, &Zstage));
    1389:   PetscCall(TSRosWRestoreVecs(ts, coarse, &Ydotc, &Ystagec, &Zdotc, &Zstagec));
    1390:   PetscFunctionReturn(PETSC_SUCCESS);
    1391: }

    1393: static PetscErrorCode DMSubDomainHook_TSRosW(DM fine, DM coarse, void *ctx)
    1394: {
    1395:   PetscFunctionBegin;
    1396:   PetscFunctionReturn(PETSC_SUCCESS);
    1397: }

    1399: static PetscErrorCode DMSubDomainRestrictHook_TSRosW(DM dm, VecScatter gscat, VecScatter lscat, DM subdm, void *ctx)
    1400: {
    1401:   TS  ts = (TS)ctx;
    1402:   Vec Ydot, Zdot, Ystage, Zstage;
    1403:   Vec Ydots, Zdots, Ystages, Zstages;

    1405:   PetscFunctionBegin;
    1406:   PetscCall(TSRosWGetVecs(ts, dm, &Ydot, &Ystage, &Zdot, &Zstage));
    1407:   PetscCall(TSRosWGetVecs(ts, subdm, &Ydots, &Ystages, &Zdots, &Zstages));

    1409:   PetscCall(VecScatterBegin(gscat, Ydot, Ydots, INSERT_VALUES, SCATTER_FORWARD));
    1410:   PetscCall(VecScatterEnd(gscat, Ydot, Ydots, INSERT_VALUES, SCATTER_FORWARD));

    1412:   PetscCall(VecScatterBegin(gscat, Ystage, Ystages, INSERT_VALUES, SCATTER_FORWARD));
    1413:   PetscCall(VecScatterEnd(gscat, Ystage, Ystages, INSERT_VALUES, SCATTER_FORWARD));

    1415:   PetscCall(VecScatterBegin(gscat, Zdot, Zdots, INSERT_VALUES, SCATTER_FORWARD));
    1416:   PetscCall(VecScatterEnd(gscat, Zdot, Zdots, INSERT_VALUES, SCATTER_FORWARD));

    1418:   PetscCall(VecScatterBegin(gscat, Zstage, Zstages, INSERT_VALUES, SCATTER_FORWARD));
    1419:   PetscCall(VecScatterEnd(gscat, Zstage, Zstages, INSERT_VALUES, SCATTER_FORWARD));

    1421:   PetscCall(TSRosWRestoreVecs(ts, dm, &Ydot, &Ystage, &Zdot, &Zstage));
    1422:   PetscCall(TSRosWRestoreVecs(ts, subdm, &Ydots, &Ystages, &Zdots, &Zstages));
    1423:   PetscFunctionReturn(PETSC_SUCCESS);
    1424: }

    1426: static PetscErrorCode SNESTSFormFunction_RosW(SNES snes, Vec U, Vec F, TS ts)
    1427: {
    1428:   TS_RosW  *ros = (TS_RosW *)ts->data;
    1429:   Vec       Ydot, Zdot, Ystage, Zstage;
    1430:   PetscReal shift = ros->scoeff / ts->time_step;
    1431:   DM        dm, dmsave;

    1433:   PetscFunctionBegin;
    1434:   PetscCall(SNESGetDM(snes, &dm));
    1435:   PetscCall(TSRosWGetVecs(ts, dm, &Ydot, &Zdot, &Ystage, &Zstage));
    1436:   PetscCall(VecWAXPY(Ydot, shift, U, Zdot));   /* Ydot = shift*U + Zdot */
    1437:   PetscCall(VecWAXPY(Ystage, 1.0, U, Zstage)); /* Ystage = U + Zstage */
    1438:   dmsave = ts->dm;
    1439:   ts->dm = dm;
    1440:   PetscCall(TSComputeIFunction(ts, ros->stage_time, Ystage, Ydot, F, PETSC_FALSE));
    1441:   ts->dm = dmsave;
    1442:   PetscCall(TSRosWRestoreVecs(ts, dm, &Ydot, &Zdot, &Ystage, &Zstage));
    1443:   PetscFunctionReturn(PETSC_SUCCESS);
    1444: }

    1446: static PetscErrorCode SNESTSFormJacobian_RosW(SNES snes, Vec U, Mat A, Mat B, TS ts)
    1447: {
    1448:   TS_RosW  *ros = (TS_RosW *)ts->data;
    1449:   Vec       Ydot, Zdot, Ystage, Zstage;
    1450:   PetscReal shift = ros->scoeff / ts->time_step;
    1451:   DM        dm, dmsave;

    1453:   PetscFunctionBegin;
    1454:   /* ros->Ydot and ros->Ystage have already been computed in SNESTSFormFunction_RosW (SNES guarantees this) */
    1455:   PetscCall(SNESGetDM(snes, &dm));
    1456:   PetscCall(TSRosWGetVecs(ts, dm, &Ydot, &Zdot, &Ystage, &Zstage));
    1457:   dmsave = ts->dm;
    1458:   ts->dm = dm;
    1459:   PetscCall(TSComputeIJacobian(ts, ros->stage_time, Ystage, Ydot, shift, A, B, PETSC_TRUE));
    1460:   ts->dm = dmsave;
    1461:   PetscCall(TSRosWRestoreVecs(ts, dm, &Ydot, &Zdot, &Ystage, &Zstage));
    1462:   PetscFunctionReturn(PETSC_SUCCESS);
    1463: }

    1465: static PetscErrorCode TSRosWTableauSetUp(TS ts)
    1466: {
    1467:   TS_RosW    *ros = (TS_RosW *)ts->data;
    1468:   RosWTableau tab = ros->tableau;

    1470:   PetscFunctionBegin;
    1471:   PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &ros->Y));
    1472:   PetscCall(PetscMalloc1(tab->s, &ros->work));
    1473:   PetscFunctionReturn(PETSC_SUCCESS);
    1474: }

    1476: static PetscErrorCode TSSetUp_RosW(TS ts)
    1477: {
    1478:   TS_RosW         *ros = (TS_RosW *)ts->data;
    1479:   DM               dm;
    1480:   SNES             snes;
    1481:   TSRHSJacobianFn *rhsjacobian;

    1483:   PetscFunctionBegin;
    1484:   PetscCall(TSRosWTableauSetUp(ts));
    1485:   PetscCall(VecDuplicate(ts->vec_sol, &ros->Ydot));
    1486:   PetscCall(VecDuplicate(ts->vec_sol, &ros->Ystage));
    1487:   PetscCall(VecDuplicate(ts->vec_sol, &ros->Zdot));
    1488:   PetscCall(VecDuplicate(ts->vec_sol, &ros->Zstage));
    1489:   PetscCall(VecDuplicate(ts->vec_sol, &ros->vec_sol_prev));
    1490:   PetscCall(TSGetDM(ts, &dm));
    1491:   PetscCall(DMCoarsenHookAdd(dm, DMCoarsenHook_TSRosW, DMRestrictHook_TSRosW, ts));
    1492:   PetscCall(DMSubDomainHookAdd(dm, DMSubDomainHook_TSRosW, DMSubDomainRestrictHook_TSRosW, ts));
    1493:   /* Rosenbrock methods are linearly implicit, so set that unless the user has specifically asked for something else */
    1494:   PetscCall(TSGetSNES(ts, &snes));
    1495:   if (!((PetscObject)snes)->type_name) PetscCall(SNESSetType(snes, SNESKSPONLY));
    1496:   PetscCall(DMTSGetRHSJacobian(dm, &rhsjacobian, NULL));
    1497:   if (rhsjacobian == TSComputeRHSJacobianConstant) {
    1498:     Mat Amat, Pmat;

    1500:     /* Set the SNES matrix to be different from the RHS matrix because there is no way to reconstruct shift*M-J */
    1501:     PetscCall(SNESGetJacobian(snes, &Amat, &Pmat, NULL, NULL));
    1502:     if (Amat && Amat == ts->Arhs) {
    1503:       if (Amat == Pmat) {
    1504:         PetscCall(MatDuplicate(ts->Arhs, MAT_COPY_VALUES, &Amat));
    1505:         PetscCall(SNESSetJacobian(snes, Amat, Amat, NULL, NULL));
    1506:       } else {
    1507:         PetscCall(MatDuplicate(ts->Arhs, MAT_COPY_VALUES, &Amat));
    1508:         PetscCall(SNESSetJacobian(snes, Amat, NULL, NULL, NULL));
    1509:         if (Pmat && Pmat == ts->Brhs) {
    1510:           PetscCall(MatDuplicate(ts->Brhs, MAT_COPY_VALUES, &Pmat));
    1511:           PetscCall(SNESSetJacobian(snes, NULL, Pmat, NULL, NULL));
    1512:           PetscCall(MatDestroy(&Pmat));
    1513:         }
    1514:       }
    1515:       PetscCall(MatDestroy(&Amat));
    1516:     }
    1517:   }
    1518:   PetscFunctionReturn(PETSC_SUCCESS);
    1519: }
    1520: /*------------------------------------------------------------*/

    1522: static PetscErrorCode TSSetFromOptions_RosW(TS ts, PetscOptionItems *PetscOptionsObject)
    1523: {
    1524:   TS_RosW *ros = (TS_RosW *)ts->data;
    1525:   SNES     snes;

    1527:   PetscFunctionBegin;
    1528:   PetscOptionsHeadBegin(PetscOptionsObject, "RosW ODE solver options");
    1529:   {
    1530:     RosWTableauLink link;
    1531:     PetscInt        count, choice;
    1532:     PetscBool       flg;
    1533:     const char    **namelist;

    1535:     for (link = RosWTableauList, count = 0; link; link = link->next, count++);
    1536:     PetscCall(PetscMalloc1(count, (char ***)&namelist));
    1537:     for (link = RosWTableauList, count = 0; link; link = link->next, count++) namelist[count] = link->tab.name;
    1538:     PetscCall(PetscOptionsEList("-ts_rosw_type", "Family of Rosenbrock-W method", "TSRosWSetType", (const char *const *)namelist, count, ros->tableau->name, &choice, &flg));
    1539:     if (flg) PetscCall(TSRosWSetType(ts, namelist[choice]));
    1540:     PetscCall(PetscFree(namelist));

    1542:     PetscCall(PetscOptionsBool("-ts_rosw_recompute_jacobian", "Recompute the Jacobian at each stage", "TSRosWSetRecomputeJacobian", ros->recompute_jacobian, &ros->recompute_jacobian, NULL));
    1543:   }
    1544:   PetscOptionsHeadEnd();
    1545:   /* Rosenbrock methods are linearly implicit, so set that unless the user has specifically asked for something else */
    1546:   PetscCall(TSGetSNES(ts, &snes));
    1547:   if (!((PetscObject)snes)->type_name) PetscCall(SNESSetType(snes, SNESKSPONLY));
    1548:   PetscFunctionReturn(PETSC_SUCCESS);
    1549: }

    1551: static PetscErrorCode TSView_RosW(TS ts, PetscViewer viewer)
    1552: {
    1553:   TS_RosW  *ros = (TS_RosW *)ts->data;
    1554:   PetscBool iascii;

    1556:   PetscFunctionBegin;
    1557:   PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &iascii));
    1558:   if (iascii) {
    1559:     RosWTableau tab = ros->tableau;
    1560:     TSRosWType  rostype;
    1561:     char        buf[512];
    1562:     PetscInt    i;
    1563:     PetscReal   abscissa[512];
    1564:     PetscCall(TSRosWGetType(ts, &rostype));
    1565:     PetscCall(PetscViewerASCIIPrintf(viewer, "  Rosenbrock-W %s\n", rostype));
    1566:     PetscCall(PetscFormatRealArray(buf, sizeof(buf), "% 8.6f", tab->s, tab->ASum));
    1567:     PetscCall(PetscViewerASCIIPrintf(viewer, "  Abscissa of A       = %s\n", buf));
    1568:     for (i = 0; i < tab->s; i++) abscissa[i] = tab->ASum[i] + tab->Gamma[i];
    1569:     PetscCall(PetscFormatRealArray(buf, sizeof(buf), "% 8.6f", tab->s, abscissa));
    1570:     PetscCall(PetscViewerASCIIPrintf(viewer, "  Abscissa of A+Gamma = %s\n", buf));
    1571:   }
    1572:   PetscFunctionReturn(PETSC_SUCCESS);
    1573: }

    1575: static PetscErrorCode TSLoad_RosW(TS ts, PetscViewer viewer)
    1576: {
    1577:   SNES    snes;
    1578:   TSAdapt adapt;

    1580:   PetscFunctionBegin;
    1581:   PetscCall(TSGetAdapt(ts, &adapt));
    1582:   PetscCall(TSAdaptLoad(adapt, viewer));
    1583:   PetscCall(TSGetSNES(ts, &snes));
    1584:   PetscCall(SNESLoad(snes, viewer));
    1585:   /* function and Jacobian context for SNES when used with TS is always ts object */
    1586:   PetscCall(SNESSetFunction(snes, NULL, NULL, ts));
    1587:   PetscCall(SNESSetJacobian(snes, NULL, NULL, NULL, ts));
    1588:   PetscFunctionReturn(PETSC_SUCCESS);
    1589: }

    1591: /*@
    1592:   TSRosWSetType - Set the type of Rosenbrock-W, `TSROSW`, scheme

    1594:   Logically Collective

    1596:   Input Parameters:
    1597: + ts       - timestepping context
    1598: - roswtype - type of Rosenbrock-W scheme

    1600:   Level: beginner

    1602: .seealso: [](ch_ts), `TSRosWGetType()`, `TSROSW`, `TSROSW2M`, `TSROSW2P`, `TSROSWRA3PW`, `TSROSWRA34PW2`, `TSROSWRODAS3`, `TSROSWSANDU3`, `TSROSWASSP3P3S1C`, `TSROSWLASSP3P4S2C`, `TSROSWLLSSP3P4S2C`, `TSROSWARK3`
    1603: @*/
    1604: PetscErrorCode TSRosWSetType(TS ts, TSRosWType roswtype)
    1605: {
    1606:   PetscFunctionBegin;
    1608:   PetscAssertPointer(roswtype, 2);
    1609:   PetscTryMethod(ts, "TSRosWSetType_C", (TS, TSRosWType), (ts, roswtype));
    1610:   PetscFunctionReturn(PETSC_SUCCESS);
    1611: }

    1613: /*@
    1614:   TSRosWGetType - Get the type of Rosenbrock-W scheme

    1616:   Logically Collective

    1618:   Input Parameter:
    1619: . ts - timestepping context

    1621:   Output Parameter:
    1622: . rostype - type of Rosenbrock-W scheme

    1624:   Level: intermediate

    1626: .seealso: [](ch_ts), `TSRosWType`, `TSRosWSetType()`
    1627: @*/
    1628: PetscErrorCode TSRosWGetType(TS ts, TSRosWType *rostype)
    1629: {
    1630:   PetscFunctionBegin;
    1632:   PetscUseMethod(ts, "TSRosWGetType_C", (TS, TSRosWType *), (ts, rostype));
    1633:   PetscFunctionReturn(PETSC_SUCCESS);
    1634: }

    1636: /*@
    1637:   TSRosWSetRecomputeJacobian - Set whether to recompute the Jacobian at each stage. The default is to update the Jacobian once per step.

    1639:   Logically Collective

    1641:   Input Parameters:
    1642: + ts  - timestepping context
    1643: - flg - `PETSC_TRUE` to recompute the Jacobian at each stage

    1645:   Level: intermediate

    1647: .seealso: [](ch_ts), `TSRosWType`, `TSRosWGetType()`
    1648: @*/
    1649: PetscErrorCode TSRosWSetRecomputeJacobian(TS ts, PetscBool flg)
    1650: {
    1651:   PetscFunctionBegin;
    1653:   PetscTryMethod(ts, "TSRosWSetRecomputeJacobian_C", (TS, PetscBool), (ts, flg));
    1654:   PetscFunctionReturn(PETSC_SUCCESS);
    1655: }

    1657: static PetscErrorCode TSRosWGetType_RosW(TS ts, TSRosWType *rostype)
    1658: {
    1659:   TS_RosW *ros = (TS_RosW *)ts->data;

    1661:   PetscFunctionBegin;
    1662:   *rostype = ros->tableau->name;
    1663:   PetscFunctionReturn(PETSC_SUCCESS);
    1664: }

    1666: static PetscErrorCode TSRosWSetType_RosW(TS ts, TSRosWType rostype)
    1667: {
    1668:   TS_RosW        *ros = (TS_RosW *)ts->data;
    1669:   PetscBool       match;
    1670:   RosWTableauLink link;

    1672:   PetscFunctionBegin;
    1673:   if (ros->tableau) {
    1674:     PetscCall(PetscStrcmp(ros->tableau->name, rostype, &match));
    1675:     if (match) PetscFunctionReturn(PETSC_SUCCESS);
    1676:   }
    1677:   for (link = RosWTableauList; link; link = link->next) {
    1678:     PetscCall(PetscStrcmp(link->tab.name, rostype, &match));
    1679:     if (match) {
    1680:       if (ts->setupcalled) PetscCall(TSRosWTableauReset(ts));
    1681:       ros->tableau = &link->tab;
    1682:       if (ts->setupcalled) PetscCall(TSRosWTableauSetUp(ts));
    1683:       ts->default_adapt_type = ros->tableau->bembed ? TSADAPTBASIC : TSADAPTNONE;
    1684:       PetscFunctionReturn(PETSC_SUCCESS);
    1685:     }
    1686:   }
    1687:   SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_ARG_UNKNOWN_TYPE, "Could not find '%s'", rostype);
    1688: }

    1690: static PetscErrorCode TSRosWSetRecomputeJacobian_RosW(TS ts, PetscBool flg)
    1691: {
    1692:   TS_RosW *ros = (TS_RosW *)ts->data;

    1694:   PetscFunctionBegin;
    1695:   ros->recompute_jacobian = flg;
    1696:   PetscFunctionReturn(PETSC_SUCCESS);
    1697: }

    1699: static PetscErrorCode TSDestroy_RosW(TS ts)
    1700: {
    1701:   PetscFunctionBegin;
    1702:   PetscCall(TSReset_RosW(ts));
    1703:   if (ts->dm) {
    1704:     PetscCall(DMCoarsenHookRemove(ts->dm, DMCoarsenHook_TSRosW, DMRestrictHook_TSRosW, ts));
    1705:     PetscCall(DMSubDomainHookRemove(ts->dm, DMSubDomainHook_TSRosW, DMSubDomainRestrictHook_TSRosW, ts));
    1706:   }
    1707:   PetscCall(PetscFree(ts->data));
    1708:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRosWGetType_C", NULL));
    1709:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRosWSetType_C", NULL));
    1710:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRosWSetRecomputeJacobian_C", NULL));
    1711:   PetscFunctionReturn(PETSC_SUCCESS);
    1712: }

    1714: /* ------------------------------------------------------------ */
    1715: /*MC
    1716:       TSROSW - ODE solver using Rosenbrock-W schemes

    1718:   These methods are intended for problems with well-separated time scales, especially when a slow scale is strongly
    1719:   nonlinear such that it is expensive to solve with a fully implicit method. The user should provide the stiff part
    1720:   of the equation using `TSSetIFunction()` and the non-stiff part with `TSSetRHSFunction()`.

    1722:   Level: beginner

    1724:   Notes:
    1725:   This method currently only works with autonomous ODE and DAE.

    1727:   Consider trying `TSARKIMEX` if the stiff part is strongly nonlinear.

    1729:   Since this uses a single linear solve per time-step if you wish to lag the jacobian or preconditioner computation you must use also -snes_lag_jacobian_persists true or -snes_lag_jacobian_preconditioner true

    1731:   Developer Notes:
    1732:   Rosenbrock-W methods are typically specified for autonomous ODE

    1734: $  udot = f(u)

    1736:   by the stage equations

    1738: $  k_i = h f(u_0 + sum_j alpha_ij k_j) + h J sum_j gamma_ij k_j

    1740:   and step completion formula

    1742: $  u_1 = u_0 + sum_j b_j k_j

    1744:   with step size h and coefficients alpha_ij, gamma_ij, and b_i. Implementing the method in this form would require f(u)
    1745:   and the Jacobian J to be available, in addition to the shifted matrix I - h gamma_ii J. Following Hairer and Wanner,
    1746:   we define new variables for the stage equations

    1748: $  y_i = gamma_ij k_j

    1750:   The k_j can be recovered because Gamma is invertible. Let C be the lower triangular part of Gamma^{-1} and define

    1752: $  A = Alpha Gamma^{-1}, bt^T = b^T Gamma^{-1}

    1754:   to rewrite the method as

    1756: .vb
    1757:   [M/(h gamma_ii) - J] y_i = f(u_0 + sum_j a_ij y_j) + M sum_j (c_ij/h) y_j
    1758:   u_1 = u_0 + sum_j bt_j y_j
    1759: .ve

    1761:    where we have introduced the mass matrix M. Continue by defining

    1763: $  ydot_i = 1/(h gamma_ii) y_i - sum_j (c_ij/h) y_j

    1765:    or, more compactly in tensor notation

    1767: $  Ydot = 1/h (Gamma^{-1} \otimes I) Y .

    1769:    Note that Gamma^{-1} is lower triangular. With this definition of Ydot in terms of known quantities and the current
    1770:    stage y_i, the stage equations reduce to performing one Newton step (typically with a lagged Jacobian) on the
    1771:    equation

    1773: $  g(u_0 + sum_j a_ij y_j + y_i, ydot_i) = 0

    1775:    with initial guess y_i = 0.

    1777: .seealso: [](ch_ts), `TSCreate()`, `TS`, `TSSetType()`, `TSRosWSetType()`, `TSRosWRegister()`, `TSROSWTHETA1`, `TSROSWTHETA2`, `TSROSW2M`, `TSROSW2P`, `TSROSWRA3PW`, `TSROSWRA34PW2`, `TSROSWRODAS3`,
    1778:           `TSROSWSANDU3`, `TSROSWASSP3P3S1C`, `TSROSWLASSP3P4S2C`, `TSROSWLLSSP3P4S2C`, `TSROSWGRK4T`, `TSROSWSHAMP4`, `TSROSWVELDD4`, `TSROSW4L`, `TSType`
    1779: M*/
    1780: PETSC_EXTERN PetscErrorCode TSCreate_RosW(TS ts)
    1781: {
    1782:   TS_RosW *ros;

    1784:   PetscFunctionBegin;
    1785:   PetscCall(TSRosWInitializePackage());

    1787:   ts->ops->reset          = TSReset_RosW;
    1788:   ts->ops->destroy        = TSDestroy_RosW;
    1789:   ts->ops->view           = TSView_RosW;
    1790:   ts->ops->load           = TSLoad_RosW;
    1791:   ts->ops->setup          = TSSetUp_RosW;
    1792:   ts->ops->step           = TSStep_RosW;
    1793:   ts->ops->interpolate    = TSInterpolate_RosW;
    1794:   ts->ops->evaluatestep   = TSEvaluateStep_RosW;
    1795:   ts->ops->setfromoptions = TSSetFromOptions_RosW;
    1796:   ts->ops->snesfunction   = SNESTSFormFunction_RosW;
    1797:   ts->ops->snesjacobian   = SNESTSFormJacobian_RosW;

    1799:   ts->usessnes = PETSC_TRUE;

    1801:   PetscCall(PetscNew(&ros));
    1802:   ts->data = (void *)ros;

    1804:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRosWGetType_C", TSRosWGetType_RosW));
    1805:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRosWSetType_C", TSRosWSetType_RosW));
    1806:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRosWSetRecomputeJacobian_C", TSRosWSetRecomputeJacobian_RosW));

    1808:   PetscCall(TSRosWSetType(ts, TSRosWDefault));
    1809:   PetscFunctionReturn(PETSC_SUCCESS);
    1810: }