Actual source code: rk.c

      1: /*
      2:   Code for time stepping with the Runge-Kutta method

      4:   Notes:
      5:   The general system is written as

      7:   Udot = F(t,U)

      9: */

     11: #include <petsc/private/tsimpl.h>
     12: #include <petscdm.h>
     13: #include <../src/ts/impls/explicit/rk/rk.h>
     14: #include <../src/ts/impls/explicit/rk/mrk.h>

     16: static TSRKType  TSRKDefault = TSRK3BS;
     17: static PetscBool TSRKRegisterAllCalled;
     18: static PetscBool TSRKPackageInitialized;

     20: static RKTableauLink RKTableauList;

     22: /*MC
     23:      TSRK1FE - First order forward Euler scheme.

     25:      This method has one stage.

     27:      Options Database Key:
     28: .     -ts_rk_type 1fe - use type 1fe

     30:      Level: advanced

     32: .seealso: [](ch_ts), `TSRK`, `TSRKType`, `TSRKSetType()`
     33: M*/
     34: /*MC
     35:      TSRK2A - Second order RK scheme (Heun's method).

     37:      This method has two stages.

     39:      Options Database Key:
     40: .     -ts_rk_type 2a - use type 2a

     42:      Level: advanced

     44: .seealso: [](ch_ts), `TSRK`, `TSRKType`, `TSRKSetType()`
     45: M*/
     46: /*MC
     47:      TSRK2B - Second order RK scheme (the midpoint method).

     49:      This method has two stages.

     51:      Options Database Key:
     52: .     -ts_rk_type 2b - use type 2b

     54:      Level: advanced

     56: .seealso: [](ch_ts), `TSRK`, `TSRKType`, `TSRKSetType()`
     57: M*/
     58: /*MC
     59:      TSRK3 - Third order RK scheme.

     61:      This method has three stages.

     63:      Options Database Key:
     64: .     -ts_rk_type 3 - use type 3

     66:      Level: advanced

     68: .seealso: [](ch_ts), `TSRK`, `TSRKType`, `TSRKSetType()`
     69: M*/
     70: /*MC
     71:      TSRK3BS - Third order RK scheme of Bogacki-Shampine with 2nd order embedded method <https://doi.org/10.1016/0893-9659(89)90079-7>

     73:      This method has four stages with the First Same As Last (FSAL) property.

     75:      Options Database Key:
     76: .     -ts_rk_type 3bs - use type 3bs

     78:      Level: advanced

     80: .seealso: [](ch_ts), `TSRK`, `TSRKType`, `TSRKSetType()`
     81: M*/
     82: /*MC
     83:      TSRK4 - Fourth order RK scheme.

     85:      This is the classical Runge-Kutta method with four stages.

     87:      Options Database Key:
     88: .     -ts_rk_type 4 - use type 4

     90:      Level: advanced

     92: .seealso: [](ch_ts), `TSRK`, `TSRKType`, `TSRKSetType()`
     93: M*/
     94: /*MC
     95:      TSRK5F - Fifth order Fehlberg RK scheme with a 4th order embedded method.

     97:      This method has six stages.

     99:      Options Database Key:
    100: .     -ts_rk_type 5f - use type 5f

    102:      Level: advanced

    104: .seealso: [](ch_ts), `TSRK`, `TSRKType`, `TSRKSetType()`
    105: M*/
    106: /*MC
    107:      TSRK5DP - Fifth order Dormand-Prince RK scheme with the 4th order embedded method <https://doi.org/10.1016/0771-050X(80)90013-3>

    109:      This method has seven stages with the First Same As Last (FSAL) property.

    111:      Options Database Key:
    112: .     -ts_rk_type 5dp - use type 5dp

    114:      Level: advanced

    116: .seealso: [](ch_ts), `TSRK`, `TSRKType`, `TSRKSetType()`
    117: M*/
    118: /*MC
    119:      TSRK5BS - Fifth order Bogacki-Shampine RK scheme with 4th order embedded method <https://doi.org/10.1016/0898-1221(96)00141-1>

    121:      This method has eight stages with the First Same As Last (FSAL) property.

    123:      Options Database Key:
    124: .     -ts_rk_type 5bs - use type 5bs

    126:      Level: advanced

    128: .seealso: [](ch_ts), `TSRK`, `TSRKType`, `TSRKSetType()`
    129: M*/
    130: /*MC
    131:      TSRK6VR - Sixth order robust Verner RK scheme with fifth order embedded method.
    132:      <http://people.math.sfu.ca/~jverner/RKV65.IIIXb.Robust.00010102836.081204.CoeffsOnlyRAT>

    134:      This method has nine stages with the First Same As Last (FSAL) property.

    136:      Options Database Key:
    137: .     -ts_rk_type 6vr - use type 6vr

    139:      Level: advanced

    141: .seealso: [](ch_ts), `TSRK`, `TSRKType`, `TSRKSetType()`
    142: M*/
    143: /*MC
    144:      TSRK7VR - Seventh order robust Verner RK scheme with sixth order embedded method.
    145:      <http://people.math.sfu.ca/~jverner/RKV65.IIIXb.Robust.00010102836.081204.CoeffsOnlyRAT>

    147:      This method has ten stages.

    149:      Options Database Key:
    150: .     -ts_rk_type 7vr - use type 7vr

    152:      Level: advanced

    154: .seealso: [](ch_ts), `TSRK`, `TSRKType`, `TSRKSetType()`
    155: M*/
    156: /*MC
    157:      TSRK8VR - Eighth order robust Verner RK scheme with seventh order embedded method.
    158:      <http://people.math.sfu.ca/~jverner/RKV87.IIa.Robust.00000754677.081208.CoeffsOnlyRATandFLOAT>

    160:      This method has thirteen stages.

    162:      Options Database Key:
    163: .     -ts_rk_type 8vr - use type 8vr

    165:      Level: advanced

    167: .seealso: [](ch_ts), `TSRK`, `TSRKType`, `TSRKSetType()`
    168: M*/

    170: /*@C
    171:   TSRKRegisterAll - Registers all of the Runge-Kutta explicit methods in `TSRK`

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

    175:   Level: advanced

    177: .seealso: [](ch_ts), `TSRKRegisterDestroy()`, `TSRKRegister()`
    178: @*/
    179: PetscErrorCode TSRKRegisterAll(void)
    180: {
    181:   PetscFunctionBegin;
    182:   if (TSRKRegisterAllCalled) PetscFunctionReturn(PETSC_SUCCESS);
    183:   TSRKRegisterAllCalled = PETSC_TRUE;

    185: #define RC PetscRealConstant
    186:   {
    187:     const PetscReal A[1][1] = {{0}};
    188:     const PetscReal b[1]    = {RC(1.0)};
    189:     PetscCall(TSRKRegister(TSRK1FE, 1, 1, &A[0][0], b, NULL, NULL, 0, NULL));
    190:   }
    191:   {
    192:     const PetscReal A[2][2] = {
    193:       {0,       0},
    194:       {RC(1.0), 0}
    195:     };
    196:     const PetscReal b[2]      = {RC(0.5), RC(0.5)};
    197:     const PetscReal bembed[2] = {RC(1.0), 0};
    198:     PetscCall(TSRKRegister(TSRK2A, 2, 2, &A[0][0], b, NULL, bembed, 0, NULL));
    199:   }
    200:   {
    201:     const PetscReal A[2][2] = {
    202:       {0,       0},
    203:       {RC(0.5), 0}
    204:     };
    205:     const PetscReal b[2] = {0, RC(1.0)};
    206:     PetscCall(TSRKRegister(TSRK2B, 2, 2, &A[0][0], b, NULL, NULL, 0, NULL));
    207:   }
    208:   {
    209:     const PetscReal A[3][3] = {
    210:       {0,                  0,       0},
    211:       {RC(2.0) / RC(3.0),  0,       0},
    212:       {RC(-1.0) / RC(3.0), RC(1.0), 0}
    213:     };
    214:     const PetscReal b[3] = {RC(0.25), RC(0.5), RC(0.25)};
    215:     PetscCall(TSRKRegister(TSRK3, 3, 3, &A[0][0], b, NULL, NULL, 0, NULL));
    216:   }
    217:   {
    218:     const PetscReal A[4][4] = {
    219:       {0,                 0,                 0,                 0},
    220:       {RC(1.0) / RC(2.0), 0,                 0,                 0},
    221:       {0,                 RC(3.0) / RC(4.0), 0,                 0},
    222:       {RC(2.0) / RC(9.0), RC(1.0) / RC(3.0), RC(4.0) / RC(9.0), 0}
    223:     };
    224:     const PetscReal b[4]      = {RC(2.0) / RC(9.0), RC(1.0) / RC(3.0), RC(4.0) / RC(9.0), 0};
    225:     const PetscReal bembed[4] = {RC(7.0) / RC(24.0), RC(1.0) / RC(4.0), RC(1.0) / RC(3.0), RC(1.0) / RC(8.0)};
    226:     PetscCall(TSRKRegister(TSRK3BS, 3, 4, &A[0][0], b, NULL, bembed, 0, NULL));
    227:   }
    228:   {
    229:     const PetscReal A[4][4] = {
    230:       {0,       0,       0,       0},
    231:       {RC(0.5), 0,       0,       0},
    232:       {0,       RC(0.5), 0,       0},
    233:       {0,       0,       RC(1.0), 0}
    234:     };
    235:     const PetscReal b[4] = {RC(1.0) / RC(6.0), RC(1.0) / RC(3.0), RC(1.0) / RC(3.0), RC(1.0) / RC(6.0)};
    236:     PetscCall(TSRKRegister(TSRK4, 4, 4, &A[0][0], b, NULL, NULL, 0, NULL));
    237:   }
    238:   {
    239:     const PetscReal A[6][6] = {
    240:       {0,                       0,                        0,                        0,                       0,                    0},
    241:       {RC(0.25),                0,                        0,                        0,                       0,                    0},
    242:       {RC(3.0) / RC(32.0),      RC(9.0) / RC(32.0),       0,                        0,                       0,                    0},
    243:       {RC(1932.0) / RC(2197.0), RC(-7200.0) / RC(2197.0), RC(7296.0) / RC(2197.0),  0,                       0,                    0},
    244:       {RC(439.0) / RC(216.0),   RC(-8.0),                 RC(3680.0) / RC(513.0),   RC(-845.0) / RC(4104.0), 0,                    0},
    245:       {RC(-8.0) / RC(27.0),     RC(2.0),                  RC(-3544.0) / RC(2565.0), RC(1859.0) / RC(4104.0), RC(-11.0) / RC(40.0), 0}
    246:     };
    247:     const PetscReal b[6]      = {RC(16.0) / RC(135.0), 0, RC(6656.0) / RC(12825.0), RC(28561.0) / RC(56430.0), RC(-9.0) / RC(50.0), RC(2.0) / RC(55.0)};
    248:     const PetscReal bembed[6] = {RC(25.0) / RC(216.0), 0, RC(1408.0) / RC(2565.0), RC(2197.0) / RC(4104.0), RC(-1.0) / RC(5.0), 0};
    249:     PetscCall(TSRKRegister(TSRK5F, 5, 6, &A[0][0], b, NULL, bembed, 0, NULL));
    250:   }
    251:   {
    252:     const PetscReal A[7][7] = {
    253:       {0,                        0,                         0,                        0,                      0,                         0,                   0},
    254:       {RC(1.0) / RC(5.0),        0,                         0,                        0,                      0,                         0,                   0},
    255:       {RC(3.0) / RC(40.0),       RC(9.0) / RC(40.0),        0,                        0,                      0,                         0,                   0},
    256:       {RC(44.0) / RC(45.0),      RC(-56.0) / RC(15.0),      RC(32.0) / RC(9.0),       0,                      0,                         0,                   0},
    257:       {RC(19372.0) / RC(6561.0), RC(-25360.0) / RC(2187.0), RC(64448.0) / RC(6561.0), RC(-212.0) / RC(729.0), 0,                         0,                   0},
    258:       {RC(9017.0) / RC(3168.0),  RC(-355.0) / RC(33.0),     RC(46732.0) / RC(5247.0), RC(49.0) / RC(176.0),   RC(-5103.0) / RC(18656.0), 0,                   0},
    259:       {RC(35.0) / RC(384.0),     0,                         RC(500.0) / RC(1113.0),   RC(125.0) / RC(192.0),  RC(-2187.0) / RC(6784.0),  RC(11.0) / RC(84.0), 0}
    260:     };
    261:     const PetscReal b[7]          = {RC(35.0) / RC(384.0), 0, RC(500.0) / RC(1113.0), RC(125.0) / RC(192.0), RC(-2187.0) / RC(6784.0), RC(11.0) / RC(84.0), 0};
    262:     const PetscReal bembed[7]     = {RC(5179.0) / RC(57600.0), 0, RC(7571.0) / RC(16695.0), RC(393.0) / RC(640.0), RC(-92097.0) / RC(339200.0), RC(187.0) / RC(2100.0), RC(1.0) / RC(40.0)};
    263:     const PetscReal binterp[7][5] = {
    264:       {RC(1.0), RC(-4034104133.0) / RC(1410260304.0),   RC(105330401.0) / RC(33982176.0),    RC(-13107642775.0) / RC(11282082432.0),  RC(6542295.0) / RC(470086768.0)       },
    265:       {0,       0,                                      0,                                   0,                                       0                                     },
    266:       {0,       RC(132343189600.0) / RC(32700410799.0), RC(-833316000.0) / RC(131326951.0),  RC(91412856700.0) / RC(32700410799.0),   RC(-523383600.0) / RC(10900136933.0)  },
    267:       {0,       RC(-115792950.0) / RC(29380423.0),      RC(185270875.0) / RC(16991088.0),    RC(-12653452475.0) / RC(1880347072.0),   RC(98134425.0) / RC(235043384.0)      },
    268:       {0,       RC(70805911779.0) / RC(24914598704.0),  RC(-4531260609.0) / RC(600351776.0), RC(988140236175.0) / RC(199316789632.0), RC(-14307999165.0) / RC(24914598704.0)},
    269:       {0,       RC(-331320693.0) / RC(205662961.0),     RC(31361737.0) / RC(7433601.0),      RC(-2426908385.0) / RC(822651844.0),     RC(97305120.0) / RC(205662961.0)      },
    270:       {0,       RC(44764047.0) / RC(29380423.0),        RC(-1532549.0) / RC(353981.0),       RC(90730570.0) / RC(29380423.0),         RC(-8293050.0) / RC(29380423.0)       }
    271:     };
    272:     PetscCall(TSRKRegister(TSRK5DP, 5, 7, &A[0][0], b, NULL, bembed, 5, binterp[0]));
    273:   }
    274:   {
    275:     const PetscReal A[8][8] = {
    276:       {0,                           0,                          0,                              0,                            0,                          0,                           0,                        0},
    277:       {RC(1.0) / RC(6.0),           0,                          0,                              0,                            0,                          0,                           0,                        0},
    278:       {RC(2.0) / RC(27.0),          RC(4.0) / RC(27.0),         0,                              0,                            0,                          0,                           0,                        0},
    279:       {RC(183.0) / RC(1372.0),      RC(-162.0) / RC(343.0),     RC(1053.0) / RC(1372.0),        0,                            0,                          0,                           0,                        0},
    280:       {RC(68.0) / RC(297.0),        RC(-4.0) / RC(11.0),        RC(42.0) / RC(143.0),           RC(1960.0) / RC(3861.0),      0,                          0,                           0,                        0},
    281:       {RC(597.0) / RC(22528.0),     RC(81.0) / RC(352.0),       RC(63099.0) / RC(585728.0),     RC(58653.0) / RC(366080.0),   RC(4617.0) / RC(20480.0),   0,                           0,                        0},
    282:       {RC(174197.0) / RC(959244.0), RC(-30942.0) / RC(79937.0), RC(8152137.0) / RC(19744439.0), RC(666106.0) / RC(1039181.0), RC(-29421.0) / RC(29068.0), RC(482048.0) / RC(414219.0), 0,                        0},
    283:       {RC(587.0) / RC(8064.0),      0,                          RC(4440339.0) / RC(15491840.0), RC(24353.0) / RC(124800.0),   RC(387.0) / RC(44800.0),    RC(2152.0) / RC(5985.0),     RC(7267.0) / RC(94080.0), 0}
    284:     };
    285:     const PetscReal b[8]      = {RC(587.0) / RC(8064.0), 0, RC(4440339.0) / RC(15491840.0), RC(24353.0) / RC(124800.0), RC(387.0) / RC(44800.0), RC(2152.0) / RC(5985.0), RC(7267.0) / RC(94080.0), 0};
    286:     const PetscReal bembed[8] = {RC(2479.0) / RC(34992.0), 0, RC(123.0) / RC(416.0), RC(612941.0) / RC(3411720.0), RC(43.0) / RC(1440.0), RC(2272.0) / RC(6561.0), RC(79937.0) / RC(1113912.0), RC(3293.0) / RC(556956.0)};
    287:     PetscCall(TSRKRegister(TSRK5BS, 5, 8, &A[0][0], b, NULL, bembed, 0, NULL));
    288:   }
    289:   {
    290:     const PetscReal A[9][9] = {
    291:       {0,                                                   0,                                                  0,                                                   0,                                                   0,                                                   0,                                                   0,                                                  0,                                                  0},
    292:       {RC(1.8000000000000000000000000000000000000000e-01),  0,                                                  0,                                                   0,                                                   0,                                                   0,                                                   0,                                                  0,                                                  0},
    293:       {RC(8.9506172839506172839506172839506172839506e-02),  RC(7.7160493827160493827160493827160493827160e-02), 0,                                                   0,                                                   0,                                                   0,                                                   0,                                                  0,                                                  0},
    294:       {RC(6.2500000000000000000000000000000000000000e-02),  0,                                                  RC(1.8750000000000000000000000000000000000000e-01),  0,                                                   0,                                                   0,                                                   0,                                                  0,                                                  0},
    295:       {RC(3.1651600000000000000000000000000000000000e-01),  0,                                                  RC(-1.0449480000000000000000000000000000000000e+00), RC(1.2584320000000000000000000000000000000000e+00),  0,                                                   0,                                                   0,                                                  0,                                                  0},
    296:       {RC(2.7232612736485626257225065566674305502508e-01),  0,                                                  RC(-8.2513360323886639676113360323886639676113e-01), RC(1.0480917678812415654520917678812415654521e+00),  RC(1.0471570799276856873679117969088177628396e-01),  0,                                                   0,                                                  0,                                                  0},
    297:       {RC(-1.6699418599716514314329607278961797333198e-01), 0,                                                  RC(6.3170850202429149797570850202429149797571e-01),  RC(1.7461044552773876082146758838488161796432e-01),  RC(-1.0665356459086066122525194734018680677781e+00), RC(1.2272108843537414965986394557823129251701e+00),  0,                                                  0,                                                  0},
    298:       {RC(3.6423751686909581646423751686909581646424e-01),  0,                                                  RC(-2.0404858299595141700404858299595141700405e-01), RC(-3.4883737816068643136312309244640071707741e-01), RC(3.2619323032856867443333608747142581729048e+00),  RC(-2.7551020408163265306122448979591836734694e+00), RC(6.8181818181818181818181818181818181818182e-01), 0,                                                  0},
    299:       {RC(7.6388888888888888888888888888888888888889e-02),  0,                                                  0,                                                   RC(3.6940836940836940836940836940836940836941e-01),  0,                                                   RC(2.4801587301587301587301587301587301587302e-01),  RC(2.3674242424242424242424242424242424242424e-01), RC(6.9444444444444444444444444444444444444444e-02), 0}
    300:     };
    301:     const PetscReal b[9] = {RC(7.6388888888888888888888888888888888888889e-02), 0, 0, RC(3.6940836940836940836940836940836940836941e-01), 0, RC(2.4801587301587301587301587301587301587302e-01), RC(2.3674242424242424242424242424242424242424e-01),
    302:                             RC(6.9444444444444444444444444444444444444444e-02), 0};
    303:     const PetscReal bembed[9] = {RC(5.8700209643605870020964360587002096436059e-02), 0, 0, RC(4.8072562358276643990929705215419501133787e-01), RC(-8.5341242076919085578832094861228313083563e-01), RC(1.2046485260770975056689342403628117913832e+00), 0, RC(-5.9242373072160306202859394348756050883710e-02), RC(1.6858043453788134639198468985703028256220e-01)};
    304:     PetscCall(TSRKRegister(TSRK6VR, 6, 9, &A[0][0], b, NULL, bembed, 0, NULL));
    305:   }
    306:   {
    307:     const PetscReal A[10][10] = {
    308:       {0,                                                   0,                                                  0,                                                   0,                                                   0,                                                   0,                                                   0,                                                  0,                                                  0, 0},
    309:       {RC(5.0000000000000000000000000000000000000000e-03),  0,                                                  0,                                                   0,                                                   0,                                                   0,                                                   0,                                                  0,                                                  0, 0},
    310:       {RC(-1.0767901234567901234567901234567901234568e+00), RC(1.1856790123456790123456790123456790123457e+00), 0,                                                   0,                                                   0,                                                   0,                                                   0,                                                  0,                                                  0, 0},
    311:       {RC(4.0833333333333333333333333333333333333333e-02),  0,                                                  RC(1.2250000000000000000000000000000000000000e-01),  0,                                                   0,                                                   0,                                                   0,                                                  0,                                                  0, 0},
    312:       {RC(6.3607142857142857142857142857142857142857e-01),  0,                                                  RC(-2.4444642857142857142857142857142857142857e+00), RC(2.2633928571428571428571428571428571428571e+00),  0,                                                   0,                                                   0,                                                  0,                                                  0, 0},
    313:       {RC(-2.5351211079349245229256383554660215487207e+00), 0,                                                  RC(1.0299374654449267920438514460756024913612e+01),  RC(-7.9513032885990579949493217458266876536482e+00), RC(7.9301148923100592201226014271115261823800e-01),  0,                                                   0,                                                  0,                                                  0, 0},
    314:       {RC(1.0018765812524632961969196583094999808207e+00),  0,                                                  RC(-4.1665712824423798331313938005470971453189e+00), RC(3.8343432929128642412552665218251378665197e+00),  RC(-5.0233333560710847547464330228611765612403e-01), RC(6.6768474388416077115385092269857695410259e-01),  0,                                                  0,                                                  0, 0},
    315:       {RC(2.7255018354630767130333963819175005717348e+01),  0,                                                  RC(-4.2004617278410638355318645443909295369611e+01), RC(-1.0535713126619489917921081600546526103722e+01), RC(8.0495536711411937147983652158926826634202e+01),  RC(-6.7343882271790513468549075963212975640927e+01), RC(1.3048657610777937463471187029566964762710e+01), 0,                                                  0, 0},
    316:       {RC(-3.0397378057114965146943658658755763226883e+00), 0,                                                  RC(1.0138161410329801111857946190709700150441e+01),  RC(-6.4293056748647215721462825629555298064437e+00), RC(-1.5864371483408276587115312853798610579467e+00), RC(1.8921781841968424410864308909131353365021e+00),  RC(1.9699335407608869061292360163336442838006e-02), RC(5.4416989827933235465102724247952572977903e-03), 0, 0},
    317:       {RC(-1.4449518916777735137351003179355712360517e+00), 0,                                                  RC(8.0318913859955919224117033223019560435041e+00),  RC(-7.5831741663401346820798883023671588604984e+00), RC(3.5816169353190074211247685442452878696855e+00),  RC(-2.4369722632199529411183809065693752383733e+00), RC(8.5158999992326179339689766032486142173390e-01), 0,                                                  0, 0}
    318:     };
    319:     const PetscReal b[10] = {RC(4.7425837833706756083569172717574534698932e-02), 0, 0, RC(2.5622361659370562659961727458274623448160e-01), RC(2.6951376833074206619473817258075952886764e-01), RC(1.2686622409092782845989138364739173247882e-01), RC(2.4887225942060071622046449427647492767292e-01), RC(3.0744837408200631335304388479099184768645e-03), RC(4.8023809989496943308189063347143123323209e-02), 0};
    320:     const PetscReal bembed[10] = {RC(4.7485247699299631037531273805727961552268e-02), 0, 0, RC(2.5599412588690633297154918245905393870497e-01), RC(2.7058478081067688722530891099268135732387e-01), RC(1.2505618684425992913638822323746917920448e-01),
    321:                                   RC(2.5204468723743860507184043820197442562182e-01), 0, 0, RC(4.8834971521418614557381971303093137592592e-02)};
    322:     PetscCall(TSRKRegister(TSRK7VR, 7, 10, &A[0][0], b, NULL, bembed, 0, NULL));
    323:   }
    324:   {
    325:     const PetscReal A[13][13] = {
    326:       {0,                                                   0,                                                  0,                                                   0,                                                   0,                                                   0,                                                   0,                                                   0,                                                  0,                                                   0,                                                  0,                                                  0, 0},
    327:       {RC(2.5000000000000000000000000000000000000000e-01),  0,                                                  0,                                                   0,                                                   0,                                                   0,                                                   0,                                                   0,                                                  0,                                                   0,                                                  0,                                                  0, 0},
    328:       {RC(8.7400846504915232052686327594877411977046e-02),  RC(2.5487604938654321753087950620345685135815e-02), 0,                                                   0,                                                   0,                                                   0,                                                   0,                                                   0,                                                  0,                                                   0,                                                  0,                                                  0, 0},
    329:       {RC(4.2333169291338582677165354330708661417323e-02),  0,                                                  RC(1.2699950787401574803149606299212598425197e-01),  0,                                                   0,                                                   0,                                                   0,                                                   0,                                                  0,                                                   0,                                                  0,                                                  0, 0},
    330:       {RC(4.2609505888742261494881445237572274090942e-01),  0,                                                  RC(-1.5987952846591523265427733230657181117089e+00), RC(1.5967002257717297115939588706899953707994e+00),  0,                                                   0,                                                   0,                                                   0,                                                  0,                                                   0,                                                  0,                                                  0, 0},
    331:       {RC(5.0719337296713929515090618138513639239329e-02),  0,                                                  0,                                                   RC(2.5433377264600407582754714408877778031369e-01),  RC(2.0394689005728199465736223777270858044698e-01),  0,                                                   0,                                                   0,                                                  0,                                                   0,                                                  0,                                                  0, 0},
    332:       {RC(-2.9000374717523110970388379285425896124091e-01), 0,                                                  0,                                                   RC(1.3441873910260789889438681109414337003184e+00),  RC(-2.8647779433614427309611103827036562829470e+00), RC(2.6775942995105948517211260646164815438695e+00),  0,                                                   0,                                                  0,                                                   0,                                                  0,                                                  0, 0},
    333:       {RC(9.8535011337993546469740402980727014284756e-02),  0,                                                  0,                                                   0,                                                   RC(2.2192680630751384842024036498197387903583e-01),  RC(-1.8140622911806994312690338288073952457474e-01), RC(1.0944411472562548236922614918038631254153e-02),  0,                                                  0,                                                   0,                                                  0,                                                  0, 0},
    334:       {RC(3.8711052545731144679444618165166373405645e-01),  0,                                                  0,                                                   RC(-1.4424454974855277571256745553077927767173e+00), RC(2.9053981890699509317691346449233848441744e+00),  RC(-1.8537710696301059290843332675811978025183e+00), RC(1.4003648098728154269497325109771241479223e-01),  RC(5.7273940811495816575746774624447706488753e-01), 0,                                                   0,                                                  0,                                                  0, 0},
    335:       {RC(-1.6124403444439308100630016197913480595436e-01), 0,                                                  0,                                                   RC(-1.7339602957358984083578404473962567894901e-01), RC(-1.3012892814065147406016812745172492529744e+00), RC(1.1379503751738617308558792131431003472124e+00),  RC(-3.1747649663966880106923521138043024698980e-02), RC(9.3351293824933666439811064486056884856590e-01), RC(-8.3786318334733852703300855629616433201504e-02), 0,                                                  0,                                                  0, 0},
    336:       {RC(-1.9199444881589533281510804651483576073142e-02), 0,                                                  0,                                                   RC(2.7330857265264284907942326254016124275617e-01),  RC(-6.7534973206944372919691611210942380856240e-01), RC(3.4151849813846016071738489974728382711981e-01),  RC(-6.7950064803375772478920516198524629391910e-02), RC(9.6591752247623878884265586491216376509746e-02), RC(1.3253082511182101180721038466545389951226e-01),  RC(3.6854959360386113446906329951531666812946e-01), 0,                                                  0, 0},
    337:       {RC(6.0918774036452898676888412111588817784584e-01),  0,                                                  0,                                                   RC(-2.2725690858980016768999800931413088399719e+00), RC(4.7578983426940290068155255881914785497547e+00),  RC(-5.5161067066927584824294689667844248244842e+00), RC(2.9005963696801192709095818565946174378180e-01),  RC(5.6914239633590368229109858454801849145630e-01), RC(7.9267957603321670271339916205893327579951e-01),  RC(1.5473720453288822894126190771849898232047e-01), RC(1.6149708956621816247083215106334544434974e+00), 0, 0},
    338:       {RC(8.8735762208534719663211694051981022704884e-01),  0,                                                  0,                                                   RC(-2.9754597821085367558513632804709301581977e+00), RC(5.6007170094881630597990392548350098923829e+00),  RC(-5.9156074505366744680014930189941657351840e+00), RC(2.2029689156134927016879142540807638331238e-01),  RC(1.0155097824462216666143271340902996997549e-01), RC(1.1514345647386055909780397752125850553556e+00),  RC(1.9297101665271239396134361900805843653065e+00), 0,                                                  0, 0}
    339:     };
    340:     const PetscReal b[13] = {RC(4.4729564666695714203015840429049382466467e-02), 0, 0, 0, 0, RC(1.5691033527708199813368698010726645409175e-01), RC(1.8460973408151637740702451873526277892035e-01), RC(2.2516380602086991042479419400350721970920e-01), RC(1.4794615651970234687005179885449141753736e-01), RC(7.6055542444955825269798361910336491012732e-02), RC(1.2277290235018619610824346315921437388535e-01), RC(4.1811958638991631583384842800871882376786e-02), 0};
    341:     const PetscReal bembed[13] = {RC(4.5847111400495925878664730122010282095875e-02), 0, 0, 0, 0, RC(2.6231891404152387437443356584845803392392e-01), RC(1.9169372337852611904485738635688429008025e-01), RC(2.1709172327902618330978407422906448568196e-01), RC(1.2738189624833706796803169450656737867900e-01), RC(1.1510530385365326258240515750043192148894e-01), 0, 0, RC(4.0561327798437566841823391436583608050053e-02)};
    342:     PetscCall(TSRKRegister(TSRK8VR, 8, 13, &A[0][0], b, NULL, bembed, 0, NULL));
    343:   }
    344: #undef RC
    345:   PetscFunctionReturn(PETSC_SUCCESS);
    346: }

    348: /*@C
    349:   TSRKRegisterDestroy - Frees the list of schemes that were registered by `TSRKRegister()`.

    351:   Not Collective

    353:   Level: advanced

    355: .seealso: [](ch_ts), `TSRK`, `TSRKRegister()`, `TSRKRegisterAll()`
    356: @*/
    357: PetscErrorCode TSRKRegisterDestroy(void)
    358: {
    359:   RKTableauLink link;

    361:   PetscFunctionBegin;
    362:   while ((link = RKTableauList)) {
    363:     RKTableau t   = &link->tab;
    364:     RKTableauList = link->next;
    365:     PetscCall(PetscFree3(t->A, t->b, t->c));
    366:     PetscCall(PetscFree(t->bembed));
    367:     PetscCall(PetscFree(t->binterp));
    368:     PetscCall(PetscFree(t->name));
    369:     PetscCall(PetscFree(link));
    370:   }
    371:   TSRKRegisterAllCalled = PETSC_FALSE;
    372:   PetscFunctionReturn(PETSC_SUCCESS);
    373: }

    375: /*@C
    376:   TSRKInitializePackage - This function initializes everything in the `TSRK` package. It is called
    377:   from `TSInitializePackage()`.

    379:   Level: developer

    381: .seealso: [](ch_ts), `TSInitializePackage()`, `PetscInitialize()`, `TSRKFinalizePackage()`
    382: @*/
    383: PetscErrorCode TSRKInitializePackage(void)
    384: {
    385:   PetscFunctionBegin;
    386:   if (TSRKPackageInitialized) PetscFunctionReturn(PETSC_SUCCESS);
    387:   TSRKPackageInitialized = PETSC_TRUE;
    388:   PetscCall(TSRKRegisterAll());
    389:   PetscCall(PetscRegisterFinalize(TSRKFinalizePackage));
    390:   PetscFunctionReturn(PETSC_SUCCESS);
    391: }

    393: /*@C
    394:   TSRKFinalizePackage - This function destroys everything in the `TSRK` package. It is
    395:   called from `PetscFinalize()`.

    397:   Level: developer

    399: .seealso: [](ch_ts), `PetscFinalize()`, `TSRKInitializePackage()`
    400: @*/
    401: PetscErrorCode TSRKFinalizePackage(void)
    402: {
    403:   PetscFunctionBegin;
    404:   TSRKPackageInitialized = PETSC_FALSE;
    405:   PetscCall(TSRKRegisterDestroy());
    406:   PetscFunctionReturn(PETSC_SUCCESS);
    407: }

    409: /*@C
    410:   TSRKRegister - register an `TSRK` scheme by providing the entries in the Butcher tableau and optionally embedded approximations and interpolation

    412:   Not Collective, but the same schemes should be registered on all processes on which they will be used, No Fortran Support

    414:   Input Parameters:
    415: + name    - identifier for method
    416: . order   - approximation order of method
    417: . s       - number of stages, this is the dimension of the matrices below
    418: . A       - stage coefficients (dimension s*s, row-major)
    419: . b       - step completion table (dimension s; NULL to use last row of A)
    420: . c       - abscissa (dimension s; NULL to use row sums of A)
    421: . bembed  - completion table for embedded method (dimension s; NULL if not available)
    422: . p       - Order of the interpolation scheme, equal to the number of columns of binterp
    423: - binterp - Coefficients of the interpolation formula (dimension s*p; NULL to reuse b with p=1)

    425:   Level: advanced

    427:   Note:
    428:   Several `TSRK` methods are provided, this function is only needed to create new methods.

    430: .seealso: [](ch_ts), `TSRK`
    431: @*/
    432: PetscErrorCode TSRKRegister(TSRKType name, PetscInt order, PetscInt s, const PetscReal A[], const PetscReal b[], const PetscReal c[], const PetscReal bembed[], PetscInt p, const PetscReal binterp[])
    433: {
    434:   RKTableauLink link;
    435:   RKTableau     t;
    436:   PetscInt      i, j;

    438:   PetscFunctionBegin;
    439:   PetscAssertPointer(name, 1);
    440:   PetscAssertPointer(A, 4);
    441:   if (b) PetscAssertPointer(b, 5);
    442:   if (c) PetscAssertPointer(c, 6);
    443:   if (bembed) PetscAssertPointer(bembed, 7);
    444:   if (binterp || p > 1) PetscAssertPointer(binterp, 9);
    445:   PetscCheck(s >= 0, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Expected number of stages s %" PetscInt_FMT " >= 0", s);

    447:   PetscCall(TSRKInitializePackage());
    448:   PetscCall(PetscNew(&link));
    449:   t = &link->tab;

    451:   PetscCall(PetscStrallocpy(name, &t->name));
    452:   t->order = order;
    453:   t->s     = s;
    454:   PetscCall(PetscMalloc3(s * s, &t->A, s, &t->b, s, &t->c));
    455:   PetscCall(PetscArraycpy(t->A, A, s * s));
    456:   if (b) PetscCall(PetscArraycpy(t->b, b, s));
    457:   else
    458:     for (i = 0; i < s; i++) t->b[i] = A[(s - 1) * s + i];
    459:   if (c) PetscCall(PetscArraycpy(t->c, c, s));
    460:   else
    461:     for (i = 0; i < s; i++)
    462:       for (j = 0, t->c[i] = 0; j < s; j++) t->c[i] += A[i * s + j];
    463:   t->FSAL = PETSC_TRUE;
    464:   for (i = 0; i < s; i++)
    465:     if (t->A[(s - 1) * s + i] != t->b[i]) t->FSAL = PETSC_FALSE;

    467:   if (bembed) {
    468:     PetscCall(PetscMalloc1(s, &t->bembed));
    469:     PetscCall(PetscArraycpy(t->bembed, bembed, s));
    470:   }

    472:   if (!binterp) {
    473:     p       = 1;
    474:     binterp = t->b;
    475:   }
    476:   t->p = p;
    477:   PetscCall(PetscMalloc1(s * p, &t->binterp));
    478:   PetscCall(PetscArraycpy(t->binterp, binterp, s * p));

    480:   link->next    = RKTableauList;
    481:   RKTableauList = link;
    482:   PetscFunctionReturn(PETSC_SUCCESS);
    483: }

    485: static PetscErrorCode TSRKGetTableau_RK(TS ts, PetscInt *s, const PetscReal **A, const PetscReal **b, const PetscReal **c, const PetscReal **bembed, PetscInt *p, const PetscReal **binterp, PetscBool *FSAL)
    486: {
    487:   TS_RK    *rk  = (TS_RK *)ts->data;
    488:   RKTableau tab = rk->tableau;

    490:   PetscFunctionBegin;
    491:   if (s) *s = tab->s;
    492:   if (A) *A = tab->A;
    493:   if (b) *b = tab->b;
    494:   if (c) *c = tab->c;
    495:   if (bembed) *bembed = tab->bembed;
    496:   if (p) *p = tab->p;
    497:   if (binterp) *binterp = tab->binterp;
    498:   if (FSAL) *FSAL = tab->FSAL;
    499:   PetscFunctionReturn(PETSC_SUCCESS);
    500: }

    502: /*@C
    503:   TSRKGetTableau - Get info on the `TSRK` tableau

    505:   Not Collective

    507:   Input Parameter:
    508: . ts - timestepping context

    510:   Output Parameters:
    511: + s       - number of stages, this is the dimension of the matrices below
    512: . A       - stage coefficients (dimension s*s, row-major)
    513: . b       - step completion table (dimension s)
    514: . c       - abscissa (dimension s)
    515: . bembed  - completion table for embedded method (dimension s; NULL if not available)
    516: . p       - Order of the interpolation scheme, equal to the number of columns of binterp
    517: . binterp - Coefficients of the interpolation formula (dimension s*p)
    518: - FSAL    - whether or not the scheme has the First Same As Last property

    520:   Level: developer

    522: .seealso: [](ch_ts), `TSRK`, `TSRKRegister()`, `TSRKSetType()`
    523: @*/
    524: PetscErrorCode TSRKGetTableau(TS ts, PetscInt *s, const PetscReal **A, const PetscReal **b, const PetscReal **c, const PetscReal **bembed, PetscInt *p, const PetscReal **binterp, PetscBool *FSAL)
    525: {
    526:   PetscFunctionBegin;
    528:   PetscUseMethod(ts, "TSRKGetTableau_C", (TS, PetscInt *, const PetscReal **, const PetscReal **, const PetscReal **, const PetscReal **, PetscInt *, const PetscReal **, PetscBool *), (ts, s, A, b, c, bembed, p, binterp, FSAL));
    529:   PetscFunctionReturn(PETSC_SUCCESS);
    530: }

    532: /*
    533:  This is for single-step RK method
    534:  The step completion formula is

    536:  x1 = x0 + h b^T YdotRHS

    538:  This function can be called before or after ts->vec_sol has been updated.
    539:  Suppose we have a completion formula (b) and an embedded formula (be) of different order.
    540:  We can write

    542:  x1e = x0 + h be^T YdotRHS
    543:      = x1 - h b^T YdotRHS + h be^T YdotRHS
    544:      = x1 + h (be - b)^T YdotRHS

    546:  so we can evaluate the method with different order even after the step has been optimistically completed.
    547: */
    548: static PetscErrorCode TSEvaluateStep_RK(TS ts, PetscInt order, Vec X, PetscBool *done)
    549: {
    550:   TS_RK       *rk  = (TS_RK *)ts->data;
    551:   RKTableau    tab = rk->tableau;
    552:   PetscScalar *w   = rk->work;
    553:   PetscReal    h;
    554:   PetscInt     s = tab->s, j;

    556:   PetscFunctionBegin;
    557:   switch (rk->status) {
    558:   case TS_STEP_INCOMPLETE:
    559:   case TS_STEP_PENDING:
    560:     h = ts->time_step;
    561:     break;
    562:   case TS_STEP_COMPLETE:
    563:     h = ts->ptime - ts->ptime_prev;
    564:     break;
    565:   default:
    566:     SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_PLIB, "Invalid TSStepStatus");
    567:   }
    568:   if (order == tab->order) {
    569:     if (rk->status == TS_STEP_INCOMPLETE) {
    570:       PetscCall(VecCopy(ts->vec_sol, X));
    571:       for (j = 0; j < s; j++) w[j] = h * tab->b[j] / rk->dtratio;
    572:       PetscCall(VecMAXPY(X, s, w, rk->YdotRHS));
    573:     } else PetscCall(VecCopy(ts->vec_sol, X));
    574:     PetscFunctionReturn(PETSC_SUCCESS);
    575:   } else if (order == tab->order - 1) {
    576:     if (!tab->bembed) goto unavailable;
    577:     if (rk->status == TS_STEP_INCOMPLETE) { /*Complete with the embedded method (be)*/
    578:       PetscCall(VecCopy(ts->vec_sol, X));
    579:       for (j = 0; j < s; j++) w[j] = h * tab->bembed[j];
    580:       PetscCall(VecMAXPY(X, s, w, rk->YdotRHS));
    581:     } else { /*Rollback and re-complete using (be-b) */
    582:       PetscCall(VecCopy(ts->vec_sol, X));
    583:       for (j = 0; j < s; j++) w[j] = h * (tab->bembed[j] - tab->b[j]);
    584:       PetscCall(VecMAXPY(X, s, w, rk->YdotRHS));
    585:     }
    586:     if (done) *done = PETSC_TRUE;
    587:     PetscFunctionReturn(PETSC_SUCCESS);
    588:   }
    589: unavailable:
    590:   PetscCheck(done, PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "RK '%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,
    591:              tab->order, order);
    592:   *done = PETSC_FALSE;
    593:   PetscFunctionReturn(PETSC_SUCCESS);
    594: }

    596: static PetscErrorCode TSForwardCostIntegral_RK(TS ts)
    597: {
    598:   TS_RK           *rk     = (TS_RK *)ts->data;
    599:   TS               quadts = ts->quadraturets;
    600:   RKTableau        tab    = rk->tableau;
    601:   const PetscInt   s      = tab->s;
    602:   const PetscReal *b = tab->b, *c = tab->c;
    603:   Vec             *Y = rk->Y;
    604:   PetscInt         i;

    606:   PetscFunctionBegin;
    607:   /* No need to backup quadts->vec_sol since it can be reverted in TSRollBack_RK */
    608:   for (i = s - 1; i >= 0; i--) {
    609:     /* Evolve quadrature TS solution to compute integrals */
    610:     PetscCall(TSComputeRHSFunction(quadts, rk->ptime + rk->time_step * c[i], Y[i], ts->vec_costintegrand));
    611:     PetscCall(VecAXPY(quadts->vec_sol, rk->time_step * b[i], ts->vec_costintegrand));
    612:   }
    613:   PetscFunctionReturn(PETSC_SUCCESS);
    614: }

    616: static PetscErrorCode TSAdjointCostIntegral_RK(TS ts)
    617: {
    618:   TS_RK           *rk     = (TS_RK *)ts->data;
    619:   RKTableau        tab    = rk->tableau;
    620:   TS               quadts = ts->quadraturets;
    621:   const PetscInt   s      = tab->s;
    622:   const PetscReal *b = tab->b, *c = tab->c;
    623:   Vec             *Y = rk->Y;
    624:   PetscInt         i;

    626:   PetscFunctionBegin;
    627:   for (i = s - 1; i >= 0; i--) {
    628:     /* Evolve quadrature TS solution to compute integrals */
    629:     PetscCall(TSComputeRHSFunction(quadts, ts->ptime + ts->time_step * (1.0 - c[i]), Y[i], ts->vec_costintegrand));
    630:     PetscCall(VecAXPY(quadts->vec_sol, -ts->time_step * b[i], ts->vec_costintegrand));
    631:   }
    632:   PetscFunctionReturn(PETSC_SUCCESS);
    633: }

    635: static PetscErrorCode TSRollBack_RK(TS ts)
    636: {
    637:   TS_RK           *rk     = (TS_RK *)ts->data;
    638:   TS               quadts = ts->quadraturets;
    639:   RKTableau        tab    = rk->tableau;
    640:   const PetscInt   s      = tab->s;
    641:   const PetscReal *b = tab->b, *c = tab->c;
    642:   PetscScalar     *w = rk->work;
    643:   Vec             *Y = rk->Y, *YdotRHS = rk->YdotRHS;
    644:   PetscInt         j;
    645:   PetscReal        h;

    647:   PetscFunctionBegin;
    648:   switch (rk->status) {
    649:   case TS_STEP_INCOMPLETE:
    650:   case TS_STEP_PENDING:
    651:     h = ts->time_step;
    652:     break;
    653:   case TS_STEP_COMPLETE:
    654:     h = ts->ptime - ts->ptime_prev;
    655:     break;
    656:   default:
    657:     SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_PLIB, "Invalid TSStepStatus");
    658:   }
    659:   for (j = 0; j < s; j++) w[j] = -h * b[j];
    660:   PetscCall(VecMAXPY(ts->vec_sol, s, w, YdotRHS));
    661:   if (quadts && ts->costintegralfwd) {
    662:     for (j = 0; j < s; j++) {
    663:       /* Revert the quadrature TS solution */
    664:       PetscCall(TSComputeRHSFunction(quadts, rk->ptime + h * c[j], Y[j], ts->vec_costintegrand));
    665:       PetscCall(VecAXPY(quadts->vec_sol, -h * b[j], ts->vec_costintegrand));
    666:     }
    667:   }
    668:   PetscFunctionReturn(PETSC_SUCCESS);
    669: }

    671: static PetscErrorCode TSForwardStep_RK(TS ts)
    672: {
    673:   TS_RK           *rk  = (TS_RK *)ts->data;
    674:   RKTableau        tab = rk->tableau;
    675:   Mat              J, *MatsFwdSensipTemp = rk->MatsFwdSensipTemp;
    676:   const PetscInt   s = tab->s;
    677:   const PetscReal *A = tab->A, *c = tab->c, *b = tab->b;
    678:   Vec             *Y = rk->Y;
    679:   PetscInt         i, j;
    680:   PetscReal        stage_time, h = ts->time_step;
    681:   PetscBool        zero;

    683:   PetscFunctionBegin;
    684:   PetscCall(MatCopy(ts->mat_sensip, rk->MatFwdSensip0, SAME_NONZERO_PATTERN));
    685:   PetscCall(TSGetRHSJacobian(ts, &J, NULL, NULL, NULL));

    687:   for (i = 0; i < s; i++) {
    688:     stage_time = ts->ptime + h * c[i];
    689:     zero       = PETSC_FALSE;
    690:     if (b[i] == 0 && i == s - 1) zero = PETSC_TRUE;
    691:     /* TLM Stage values */
    692:     if (!i) {
    693:       PetscCall(MatCopy(ts->mat_sensip, rk->MatsFwdStageSensip[i], SAME_NONZERO_PATTERN));
    694:     } else if (!zero) {
    695:       PetscCall(MatZeroEntries(rk->MatsFwdStageSensip[i]));
    696:       for (j = 0; j < i; j++) PetscCall(MatAXPY(rk->MatsFwdStageSensip[i], h * A[i * s + j], MatsFwdSensipTemp[j], SAME_NONZERO_PATTERN));
    697:       PetscCall(MatAXPY(rk->MatsFwdStageSensip[i], 1., ts->mat_sensip, SAME_NONZERO_PATTERN));
    698:     } else {
    699:       PetscCall(MatZeroEntries(rk->MatsFwdStageSensip[i]));
    700:     }

    702:     PetscCall(TSComputeRHSJacobian(ts, stage_time, Y[i], J, J));
    703:     PetscCall(MatMatMult(J, rk->MatsFwdStageSensip[i], MAT_REUSE_MATRIX, PETSC_DETERMINE, &MatsFwdSensipTemp[i]));
    704:     if (ts->Jacprhs) {
    705:       PetscCall(TSComputeRHSJacobianP(ts, stage_time, Y[i], ts->Jacprhs)); /* get f_p */
    706:       if (ts->vecs_sensi2p) {                                              /* TLM used for 2nd-order adjoint */
    707:         PetscScalar *xarr;
    708:         PetscCall(MatDenseGetColumn(MatsFwdSensipTemp[i], 0, &xarr));
    709:         PetscCall(VecPlaceArray(rk->VecDeltaFwdSensipCol, xarr));
    710:         PetscCall(MatMultAdd(ts->Jacprhs, ts->vec_dir, rk->VecDeltaFwdSensipCol, rk->VecDeltaFwdSensipCol));
    711:         PetscCall(VecResetArray(rk->VecDeltaFwdSensipCol));
    712:         PetscCall(MatDenseRestoreColumn(MatsFwdSensipTemp[i], &xarr));
    713:       } else {
    714:         PetscCall(MatAXPY(MatsFwdSensipTemp[i], 1., ts->Jacprhs, SUBSET_NONZERO_PATTERN));
    715:       }
    716:     }
    717:   }

    719:   for (i = 0; i < s; i++) PetscCall(MatAXPY(ts->mat_sensip, h * b[i], rk->MatsFwdSensipTemp[i], SAME_NONZERO_PATTERN));
    720:   rk->status = TS_STEP_COMPLETE;
    721:   PetscFunctionReturn(PETSC_SUCCESS);
    722: }

    724: static PetscErrorCode TSForwardGetStages_RK(TS ts, PetscInt *ns, Mat **stagesensip)
    725: {
    726:   TS_RK    *rk  = (TS_RK *)ts->data;
    727:   RKTableau tab = rk->tableau;

    729:   PetscFunctionBegin;
    730:   if (ns) *ns = tab->s;
    731:   if (stagesensip) *stagesensip = rk->MatsFwdStageSensip;
    732:   PetscFunctionReturn(PETSC_SUCCESS);
    733: }

    735: static PetscErrorCode TSForwardSetUp_RK(TS ts)
    736: {
    737:   TS_RK    *rk  = (TS_RK *)ts->data;
    738:   RKTableau tab = rk->tableau;
    739:   PetscInt  i;

    741:   PetscFunctionBegin;
    742:   /* backup sensitivity results for roll-backs */
    743:   PetscCall(MatDuplicate(ts->mat_sensip, MAT_DO_NOT_COPY_VALUES, &rk->MatFwdSensip0));

    745:   PetscCall(PetscMalloc1(tab->s, &rk->MatsFwdStageSensip));
    746:   PetscCall(PetscMalloc1(tab->s, &rk->MatsFwdSensipTemp));
    747:   for (i = 0; i < tab->s; i++) {
    748:     PetscCall(MatDuplicate(ts->mat_sensip, MAT_DO_NOT_COPY_VALUES, &rk->MatsFwdStageSensip[i]));
    749:     PetscCall(MatDuplicate(ts->mat_sensip, MAT_DO_NOT_COPY_VALUES, &rk->MatsFwdSensipTemp[i]));
    750:   }
    751:   PetscCall(VecDuplicate(ts->vec_sol, &rk->VecDeltaFwdSensipCol));
    752:   PetscFunctionReturn(PETSC_SUCCESS);
    753: }

    755: static PetscErrorCode TSForwardReset_RK(TS ts)
    756: {
    757:   TS_RK    *rk  = (TS_RK *)ts->data;
    758:   RKTableau tab = rk->tableau;
    759:   PetscInt  i;

    761:   PetscFunctionBegin;
    762:   PetscCall(MatDestroy(&rk->MatFwdSensip0));
    763:   if (rk->MatsFwdStageSensip) {
    764:     for (i = 0; i < tab->s; i++) PetscCall(MatDestroy(&rk->MatsFwdStageSensip[i]));
    765:     PetscCall(PetscFree(rk->MatsFwdStageSensip));
    766:   }
    767:   if (rk->MatsFwdSensipTemp) {
    768:     for (i = 0; i < tab->s; i++) PetscCall(MatDestroy(&rk->MatsFwdSensipTemp[i]));
    769:     PetscCall(PetscFree(rk->MatsFwdSensipTemp));
    770:   }
    771:   PetscCall(VecDestroy(&rk->VecDeltaFwdSensipCol));
    772:   PetscFunctionReturn(PETSC_SUCCESS);
    773: }

    775: static PetscErrorCode TSStep_RK(TS ts)
    776: {
    777:   TS_RK           *rk  = (TS_RK *)ts->data;
    778:   RKTableau        tab = rk->tableau;
    779:   const PetscInt   s   = tab->s;
    780:   const PetscReal *A = tab->A, *c = tab->c;
    781:   PetscScalar     *w = rk->work;
    782:   Vec             *Y = rk->Y, *YdotRHS = rk->YdotRHS;
    783:   PetscBool        FSAL = (PetscBool)(tab->FSAL && !rk->newtableau);
    784:   TSAdapt          adapt;
    785:   PetscInt         i, j;
    786:   PetscInt         rejections = 0;
    787:   PetscBool        stageok, accept = PETSC_TRUE;
    788:   PetscReal        next_time_step = ts->time_step;

    790:   PetscFunctionBegin;
    791:   if (ts->steprollback || ts->steprestart) FSAL = PETSC_FALSE;
    792:   if (FSAL) PetscCall(VecCopy(YdotRHS[s - 1], YdotRHS[0]));
    793:   rk->newtableau = PETSC_FALSE;

    795:   rk->status = TS_STEP_INCOMPLETE;
    796:   while (!ts->reason && rk->status != TS_STEP_COMPLETE) {
    797:     PetscReal t = ts->ptime;
    798:     PetscReal h = ts->time_step;
    799:     for (i = 0; i < s; i++) {
    800:       rk->stage_time = t + h * c[i];
    801:       PetscCall(TSPreStage(ts, rk->stage_time));
    802:       PetscCall(VecCopy(ts->vec_sol, Y[i]));
    803:       for (j = 0; j < i; j++) w[j] = h * A[i * s + j];
    804:       PetscCall(VecMAXPY(Y[i], i, w, YdotRHS));
    805:       PetscCall(TSPostStage(ts, rk->stage_time, i, Y));
    806:       PetscCall(TSGetAdapt(ts, &adapt));
    807:       PetscCall(TSAdaptCheckStage(adapt, ts, rk->stage_time, Y[i], &stageok));
    808:       if (!stageok) goto reject_step;
    809:       if (FSAL && !i) continue;
    810:       PetscCall(TSComputeRHSFunction(ts, t + h * c[i], Y[i], YdotRHS[i]));
    811:     }

    813:     rk->status = TS_STEP_INCOMPLETE;
    814:     PetscCall(TSEvaluateStep(ts, tab->order, ts->vec_sol, NULL));
    815:     rk->status = TS_STEP_PENDING;
    816:     PetscCall(TSGetAdapt(ts, &adapt));
    817:     PetscCall(TSAdaptCandidatesClear(adapt));
    818:     PetscCall(TSAdaptCandidateAdd(adapt, tab->name, tab->order, 1, tab->ccfl, (PetscReal)tab->s, PETSC_TRUE));
    819:     PetscCall(TSAdaptChoose(adapt, ts, ts->time_step, NULL, &next_time_step, &accept));
    820:     rk->status = accept ? TS_STEP_COMPLETE : TS_STEP_INCOMPLETE;
    821:     if (!accept) { /* Roll back the current step */
    822:       PetscCall(TSRollBack_RK(ts));
    823:       ts->time_step = next_time_step;
    824:       goto reject_step;
    825:     }

    827:     if (ts->costintegralfwd) { /* Save the info for the later use in cost integral evaluation */
    828:       rk->ptime     = ts->ptime;
    829:       rk->time_step = ts->time_step;
    830:     }

    832:     ts->ptime += ts->time_step;
    833:     ts->time_step = next_time_step;
    834:     break;

    836:   reject_step:
    837:     ts->reject++;
    838:     accept = PETSC_FALSE;
    839:     if (!ts->reason && ++rejections > ts->max_reject && ts->max_reject >= 0) {
    840:       ts->reason = TS_DIVERGED_STEP_REJECTED;
    841:       PetscCall(PetscInfo(ts, "Step=%" PetscInt_FMT ", step rejections %" PetscInt_FMT " greater than current TS allowed, stopping solve\n", ts->steps, rejections));
    842:     }
    843:   }
    844:   PetscFunctionReturn(PETSC_SUCCESS);
    845: }

    847: static PetscErrorCode TSAdjointSetUp_RK(TS ts)
    848: {
    849:   TS_RK    *rk  = (TS_RK *)ts->data;
    850:   RKTableau tab = rk->tableau;
    851:   PetscInt  s   = tab->s;

    853:   PetscFunctionBegin;
    854:   if (ts->adjointsetupcalled) PetscFunctionReturn(PETSC_SUCCESS);
    855:   ts->adjointsetupcalled = PETSC_TRUE;
    856:   PetscCall(VecDuplicateVecs(ts->vecs_sensi[0], s * ts->numcost, &rk->VecsDeltaLam));
    857:   PetscCall(VecDuplicateVecs(ts->vecs_sensi[0], ts->numcost, &rk->VecsSensiTemp));
    858:   if (ts->vecs_sensip) PetscCall(VecDuplicate(ts->vecs_sensip[0], &rk->VecDeltaMu));
    859:   if (ts->vecs_sensi2) {
    860:     PetscCall(VecDuplicateVecs(ts->vecs_sensi[0], s * ts->numcost, &rk->VecsDeltaLam2));
    861:     PetscCall(VecDuplicateVecs(ts->vecs_sensi2[0], ts->numcost, &rk->VecsSensi2Temp));
    862:   }
    863:   if (ts->vecs_sensi2p) PetscCall(VecDuplicate(ts->vecs_sensi2p[0], &rk->VecDeltaMu2));
    864:   PetscFunctionReturn(PETSC_SUCCESS);
    865: }

    867: /*
    868:   Assumptions:
    869:     - TSStep_RK() always evaluates the step with b, not bembed.
    870: */
    871: static PetscErrorCode TSAdjointStep_RK(TS ts)
    872: {
    873:   TS_RK           *rk     = (TS_RK *)ts->data;
    874:   TS               quadts = ts->quadraturets;
    875:   RKTableau        tab    = rk->tableau;
    876:   Mat              J, Jpre, Jquad;
    877:   const PetscInt   s = tab->s;
    878:   const PetscReal *A = tab->A, *b = tab->b, *c = tab->c;
    879:   PetscScalar     *w = rk->work, *xarr;
    880:   Vec             *Y = rk->Y, *VecsDeltaLam = rk->VecsDeltaLam, VecDeltaMu = rk->VecDeltaMu, *VecsSensiTemp = rk->VecsSensiTemp;
    881:   Vec             *VecsDeltaLam2 = rk->VecsDeltaLam2, VecDeltaMu2 = rk->VecDeltaMu2, *VecsSensi2Temp = rk->VecsSensi2Temp;
    882:   Vec              VecDRDUTransCol = ts->vec_drdu_col, VecDRDPTransCol = ts->vec_drdp_col;
    883:   PetscInt         i, j, nadj;
    884:   PetscReal        t = ts->ptime;
    885:   PetscReal        h = ts->time_step;

    887:   PetscFunctionBegin;
    888:   rk->status = TS_STEP_INCOMPLETE;

    890:   PetscCall(TSGetRHSJacobian(ts, &J, &Jpre, NULL, NULL));
    891:   if (quadts) PetscCall(TSGetRHSJacobian(quadts, &Jquad, NULL, NULL, NULL));
    892:   for (i = s - 1; i >= 0; i--) {
    893:     if (tab->FSAL && i == s - 1) {
    894:       /* VecsDeltaLam[nadj*s+s-1] are initialized with zeros and the values never change.*/
    895:       continue;
    896:     }
    897:     rk->stage_time = t + h * (1.0 - c[i]);
    898:     PetscCall(TSComputeSNESJacobian(ts, Y[i], J, Jpre));
    899:     if (quadts) PetscCall(TSComputeRHSJacobian(quadts, rk->stage_time, Y[i], Jquad, Jquad)); /* get r_u^T */
    900:     if (ts->vecs_sensip) {
    901:       PetscCall(TSComputeRHSJacobianP(ts, rk->stage_time, Y[i], ts->Jacprhs));                     /* get f_p */
    902:       if (quadts) PetscCall(TSComputeRHSJacobianP(quadts, rk->stage_time, Y[i], quadts->Jacprhs)); /* get f_p for the quadrature */
    903:     }

    905:     if (b[i]) {
    906:       for (j = i + 1; j < s; j++) w[j - i - 1] = A[j * s + i] / b[i]; /* coefficients for computing VecsSensiTemp */
    907:     } else {
    908:       for (j = i + 1; j < s; j++) w[j - i - 1] = A[j * s + i]; /* coefficients for computing VecsSensiTemp */
    909:     }

    911:     for (nadj = 0; nadj < ts->numcost; nadj++) {
    912:       /* Stage values of lambda */
    913:       if (b[i]) {
    914:         /* lambda_{n+1} + \sum_{j=i+1}^s a_{ji}/b[i]*lambda_{s,j} */
    915:         PetscCall(VecCopy(ts->vecs_sensi[nadj], VecsSensiTemp[nadj])); /* VecDeltaLam is an vec array of size s by numcost */
    916:         PetscCall(VecMAXPY(VecsSensiTemp[nadj], s - i - 1, w, &VecsDeltaLam[nadj * s + i + 1]));
    917:         PetscCall(MatMultTranspose(J, VecsSensiTemp[nadj], VecsDeltaLam[nadj * s + i])); /* VecsSensiTemp will be reused by 2nd-order adjoint */
    918:         PetscCall(VecScale(VecsDeltaLam[nadj * s + i], -h * b[i]));
    919:         if (quadts) {
    920:           PetscCall(MatDenseGetColumn(Jquad, nadj, &xarr));
    921:           PetscCall(VecPlaceArray(VecDRDUTransCol, xarr));
    922:           PetscCall(VecAXPY(VecsDeltaLam[nadj * s + i], -h * b[i], VecDRDUTransCol));
    923:           PetscCall(VecResetArray(VecDRDUTransCol));
    924:           PetscCall(MatDenseRestoreColumn(Jquad, &xarr));
    925:         }
    926:       } else {
    927:         /* \sum_{j=i+1}^s a_{ji}*lambda_{s,j} */
    928:         PetscCall(VecSet(VecsSensiTemp[nadj], 0));
    929:         PetscCall(VecMAXPY(VecsSensiTemp[nadj], s - i - 1, w, &VecsDeltaLam[nadj * s + i + 1]));
    930:         PetscCall(MatMultTranspose(J, VecsSensiTemp[nadj], VecsDeltaLam[nadj * s + i]));
    931:         PetscCall(VecScale(VecsDeltaLam[nadj * s + i], -h));
    932:       }

    934:       /* Stage values of mu */
    935:       if (ts->vecs_sensip) {
    936:         PetscCall(MatMultTranspose(ts->Jacprhs, VecsSensiTemp[nadj], VecDeltaMu));
    937:         if (b[i]) {
    938:           PetscCall(VecScale(VecDeltaMu, -h * b[i]));
    939:           if (quadts) {
    940:             PetscCall(MatDenseGetColumn(quadts->Jacprhs, nadj, &xarr));
    941:             PetscCall(VecPlaceArray(VecDRDPTransCol, xarr));
    942:             PetscCall(VecAXPY(VecDeltaMu, -h * b[i], VecDRDPTransCol));
    943:             PetscCall(VecResetArray(VecDRDPTransCol));
    944:             PetscCall(MatDenseRestoreColumn(quadts->Jacprhs, &xarr));
    945:           }
    946:         } else {
    947:           PetscCall(VecScale(VecDeltaMu, -h));
    948:         }
    949:         PetscCall(VecAXPY(ts->vecs_sensip[nadj], 1., VecDeltaMu)); /* update sensip for each stage */
    950:       }
    951:     }

    953:     if (ts->vecs_sensi2 && ts->forward_solve) { /* 2nd-order adjoint, TLM mode has to be turned on */
    954:       /* Get w1 at t_{n+1} from TLM matrix */
    955:       PetscCall(MatDenseGetColumn(rk->MatsFwdStageSensip[i], 0, &xarr));
    956:       PetscCall(VecPlaceArray(ts->vec_sensip_col, xarr));
    957:       /* lambda_s^T F_UU w_1 */
    958:       PetscCall(TSComputeRHSHessianProductFunctionUU(ts, rk->stage_time, Y[i], VecsSensiTemp, ts->vec_sensip_col, ts->vecs_guu));
    959:       if (quadts) {
    960:         /* R_UU w_1 */
    961:         PetscCall(TSComputeRHSHessianProductFunctionUU(quadts, rk->stage_time, Y[i], NULL, ts->vec_sensip_col, ts->vecs_guu));
    962:       }
    963:       if (ts->vecs_sensip) {
    964:         /* lambda_s^T F_UP w_2 */
    965:         PetscCall(TSComputeRHSHessianProductFunctionUP(ts, rk->stage_time, Y[i], VecsSensiTemp, ts->vec_dir, ts->vecs_gup));
    966:         if (quadts) {
    967:           /* R_UP w_2 */
    968:           PetscCall(TSComputeRHSHessianProductFunctionUP(quadts, rk->stage_time, Y[i], NULL, ts->vec_sensip_col, ts->vecs_gup));
    969:         }
    970:       }
    971:       if (ts->vecs_sensi2p) {
    972:         /* lambda_s^T F_PU w_1 */
    973:         PetscCall(TSComputeRHSHessianProductFunctionPU(ts, rk->stage_time, Y[i], VecsSensiTemp, ts->vec_sensip_col, ts->vecs_gpu));
    974:         /* lambda_s^T F_PP w_2 */
    975:         PetscCall(TSComputeRHSHessianProductFunctionPP(ts, rk->stage_time, Y[i], VecsSensiTemp, ts->vec_dir, ts->vecs_gpp));
    976:         if (b[i] && quadts) {
    977:           /* R_PU w_1 */
    978:           PetscCall(TSComputeRHSHessianProductFunctionPU(quadts, rk->stage_time, Y[i], NULL, ts->vec_sensip_col, ts->vecs_gpu));
    979:           /* R_PP w_2 */
    980:           PetscCall(TSComputeRHSHessianProductFunctionPP(quadts, rk->stage_time, Y[i], NULL, ts->vec_dir, ts->vecs_gpp));
    981:         }
    982:       }
    983:       PetscCall(VecResetArray(ts->vec_sensip_col));
    984:       PetscCall(MatDenseRestoreColumn(rk->MatsFwdStageSensip[i], &xarr));

    986:       for (nadj = 0; nadj < ts->numcost; nadj++) {
    987:         /* Stage values of lambda */
    988:         if (b[i]) {
    989:           /* J_i^T*(Lambda_{n+1}+\sum_{j=i+1}^s a_{ji}/b_i*Lambda_{s,j} */
    990:           PetscCall(VecCopy(ts->vecs_sensi2[nadj], VecsSensi2Temp[nadj]));
    991:           PetscCall(VecMAXPY(VecsSensi2Temp[nadj], s - i - 1, w, &VecsDeltaLam2[nadj * s + i + 1]));
    992:           PetscCall(MatMultTranspose(J, VecsSensi2Temp[nadj], VecsDeltaLam2[nadj * s + i]));
    993:           PetscCall(VecScale(VecsDeltaLam2[nadj * s + i], -h * b[i]));
    994:           PetscCall(VecAXPY(VecsDeltaLam2[nadj * s + i], -h * b[i], ts->vecs_guu[nadj]));
    995:           if (ts->vecs_sensip) PetscCall(VecAXPY(VecsDeltaLam2[nadj * s + i], -h * b[i], ts->vecs_gup[nadj]));
    996:         } else {
    997:           /* \sum_{j=i+1}^s a_{ji}*Lambda_{s,j} */
    998:           PetscCall(VecSet(VecsDeltaLam2[nadj * s + i], 0));
    999:           PetscCall(VecMAXPY(VecsSensi2Temp[nadj], s - i - 1, w, &VecsDeltaLam2[nadj * s + i + 1]));
    1000:           PetscCall(MatMultTranspose(J, VecsSensi2Temp[nadj], VecsDeltaLam2[nadj * s + i]));
    1001:           PetscCall(VecScale(VecsDeltaLam2[nadj * s + i], -h));
    1002:           PetscCall(VecAXPY(VecsDeltaLam2[nadj * s + i], -h, ts->vecs_guu[nadj]));
    1003:           if (ts->vecs_sensip) PetscCall(VecAXPY(VecsDeltaLam2[nadj * s + i], -h, ts->vecs_gup[nadj]));
    1004:         }
    1005:         if (ts->vecs_sensi2p) { /* 2nd-order adjoint for parameters */
    1006:           PetscCall(MatMultTranspose(ts->Jacprhs, VecsSensi2Temp[nadj], VecDeltaMu2));
    1007:           if (b[i]) {
    1008:             PetscCall(VecScale(VecDeltaMu2, -h * b[i]));
    1009:             PetscCall(VecAXPY(VecDeltaMu2, -h * b[i], ts->vecs_gpu[nadj]));
    1010:             PetscCall(VecAXPY(VecDeltaMu2, -h * b[i], ts->vecs_gpp[nadj]));
    1011:           } else {
    1012:             PetscCall(VecScale(VecDeltaMu2, -h));
    1013:             PetscCall(VecAXPY(VecDeltaMu2, -h, ts->vecs_gpu[nadj]));
    1014:             PetscCall(VecAXPY(VecDeltaMu2, -h, ts->vecs_gpp[nadj]));
    1015:           }
    1016:           PetscCall(VecAXPY(ts->vecs_sensi2p[nadj], 1, VecDeltaMu2)); /* update sensi2p for each stage */
    1017:         }
    1018:       }
    1019:     }
    1020:   }

    1022:   for (j = 0; j < s; j++) w[j] = 1.0;
    1023:   for (nadj = 0; nadj < ts->numcost; nadj++) { /* no need to do this for mu's */
    1024:     PetscCall(VecMAXPY(ts->vecs_sensi[nadj], s, w, &VecsDeltaLam[nadj * s]));
    1025:     if (ts->vecs_sensi2) PetscCall(VecMAXPY(ts->vecs_sensi2[nadj], s, w, &VecsDeltaLam2[nadj * s]));
    1026:   }
    1027:   rk->status = TS_STEP_COMPLETE;
    1028:   PetscFunctionReturn(PETSC_SUCCESS);
    1029: }

    1031: static PetscErrorCode TSAdjointReset_RK(TS ts)
    1032: {
    1033:   TS_RK    *rk  = (TS_RK *)ts->data;
    1034:   RKTableau tab = rk->tableau;

    1036:   PetscFunctionBegin;
    1037:   PetscCall(VecDestroyVecs(tab->s * ts->numcost, &rk->VecsDeltaLam));
    1038:   PetscCall(VecDestroyVecs(ts->numcost, &rk->VecsSensiTemp));
    1039:   PetscCall(VecDestroy(&rk->VecDeltaMu));
    1040:   PetscCall(VecDestroyVecs(tab->s * ts->numcost, &rk->VecsDeltaLam2));
    1041:   PetscCall(VecDestroy(&rk->VecDeltaMu2));
    1042:   PetscCall(VecDestroyVecs(ts->numcost, &rk->VecsSensi2Temp));
    1043:   PetscFunctionReturn(PETSC_SUCCESS);
    1044: }

    1046: static PetscErrorCode TSInterpolate_RK(TS ts, PetscReal itime, Vec X)
    1047: {
    1048:   TS_RK           *rk = (TS_RK *)ts->data;
    1049:   PetscInt         s = rk->tableau->s, p = rk->tableau->p, i, j;
    1050:   PetscReal        h;
    1051:   PetscReal        tt, t;
    1052:   PetscScalar     *b;
    1053:   const PetscReal *B = rk->tableau->binterp;

    1055:   PetscFunctionBegin;
    1056:   PetscCheck(B, PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "TSRK %s does not have an interpolation formula", rk->tableau->name);

    1058:   switch (rk->status) {
    1059:   case TS_STEP_INCOMPLETE:
    1060:   case TS_STEP_PENDING:
    1061:     h = ts->time_step;
    1062:     t = (itime - ts->ptime) / h;
    1063:     break;
    1064:   case TS_STEP_COMPLETE:
    1065:     h = ts->ptime - ts->ptime_prev;
    1066:     t = (itime - ts->ptime) / h + 1; /* In the interval [0,1] */
    1067:     break;
    1068:   default:
    1069:     SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_PLIB, "Invalid TSStepStatus");
    1070:   }
    1071:   PetscCall(PetscMalloc1(s, &b));
    1072:   for (i = 0; i < s; i++) b[i] = 0;
    1073:   for (j = 0, tt = t; j < p; j++, tt *= t) {
    1074:     for (i = 0; i < s; i++) b[i] += h * B[i * p + j] * tt;
    1075:   }
    1076:   PetscCall(VecCopy(rk->Y[0], X));
    1077:   PetscCall(VecMAXPY(X, s, b, rk->YdotRHS));
    1078:   PetscCall(PetscFree(b));
    1079:   PetscFunctionReturn(PETSC_SUCCESS);
    1080: }

    1082: /*------------------------------------------------------------*/

    1084: static PetscErrorCode TSRKTableauReset(TS ts)
    1085: {
    1086:   TS_RK    *rk  = (TS_RK *)ts->data;
    1087:   RKTableau tab = rk->tableau;

    1089:   PetscFunctionBegin;
    1090:   if (!tab) PetscFunctionReturn(PETSC_SUCCESS);
    1091:   PetscCall(PetscFree(rk->work));
    1092:   PetscCall(VecDestroyVecs(tab->s, &rk->Y));
    1093:   PetscCall(VecDestroyVecs(tab->s, &rk->YdotRHS));
    1094:   PetscFunctionReturn(PETSC_SUCCESS);
    1095: }

    1097: static PetscErrorCode TSReset_RK(TS ts)
    1098: {
    1099:   PetscFunctionBegin;
    1100:   PetscCall(TSRKTableauReset(ts));
    1101:   if (ts->use_splitrhsfunction) {
    1102:     PetscTryMethod(ts, "TSReset_RK_MultirateSplit_C", (TS), (ts));
    1103:   } else {
    1104:     PetscTryMethod(ts, "TSReset_RK_MultirateNonsplit_C", (TS), (ts));
    1105:   }
    1106:   PetscFunctionReturn(PETSC_SUCCESS);
    1107: }

    1109: static PetscErrorCode DMCoarsenHook_TSRK(DM fine, DM coarse, void *ctx)
    1110: {
    1111:   PetscFunctionBegin;
    1112:   PetscFunctionReturn(PETSC_SUCCESS);
    1113: }

    1115: static PetscErrorCode DMRestrictHook_TSRK(DM fine, Mat restrct, Vec rscale, Mat inject, DM coarse, void *ctx)
    1116: {
    1117:   PetscFunctionBegin;
    1118:   PetscFunctionReturn(PETSC_SUCCESS);
    1119: }

    1121: static PetscErrorCode DMSubDomainHook_TSRK(DM dm, DM subdm, void *ctx)
    1122: {
    1123:   PetscFunctionBegin;
    1124:   PetscFunctionReturn(PETSC_SUCCESS);
    1125: }

    1127: static PetscErrorCode DMSubDomainRestrictHook_TSRK(DM dm, VecScatter gscat, VecScatter lscat, DM subdm, void *ctx)
    1128: {
    1129:   PetscFunctionBegin;
    1130:   PetscFunctionReturn(PETSC_SUCCESS);
    1131: }

    1133: static PetscErrorCode TSRKTableauSetUp(TS ts)
    1134: {
    1135:   TS_RK    *rk  = (TS_RK *)ts->data;
    1136:   RKTableau tab = rk->tableau;

    1138:   PetscFunctionBegin;
    1139:   PetscCall(PetscMalloc1(tab->s, &rk->work));
    1140:   PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &rk->Y));
    1141:   PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &rk->YdotRHS));
    1142:   rk->newtableau = PETSC_TRUE;
    1143:   PetscFunctionReturn(PETSC_SUCCESS);
    1144: }

    1146: static PetscErrorCode TSSetUp_RK(TS ts)
    1147: {
    1148:   TS quadts = ts->quadraturets;
    1149:   DM dm;

    1151:   PetscFunctionBegin;
    1152:   PetscCall(TSCheckImplicitTerm(ts));
    1153:   PetscCall(TSRKTableauSetUp(ts));
    1154:   if (quadts && ts->costintegralfwd) {
    1155:     Mat Jquad;
    1156:     PetscCall(TSGetRHSJacobian(quadts, &Jquad, NULL, NULL, NULL));
    1157:   }
    1158:   PetscCall(TSGetDM(ts, &dm));
    1159:   PetscCall(DMCoarsenHookAdd(dm, DMCoarsenHook_TSRK, DMRestrictHook_TSRK, ts));
    1160:   PetscCall(DMSubDomainHookAdd(dm, DMSubDomainHook_TSRK, DMSubDomainRestrictHook_TSRK, ts));
    1161:   if (ts->use_splitrhsfunction) {
    1162:     PetscTryMethod(ts, "TSSetUp_RK_MultirateSplit_C", (TS), (ts));
    1163:   } else {
    1164:     PetscTryMethod(ts, "TSSetUp_RK_MultirateNonsplit_C", (TS), (ts));
    1165:   }
    1166:   PetscFunctionReturn(PETSC_SUCCESS);
    1167: }

    1169: static PetscErrorCode TSSetFromOptions_RK(TS ts, PetscOptionItems PetscOptionsObject)
    1170: {
    1171:   TS_RK *rk = (TS_RK *)ts->data;

    1173:   PetscFunctionBegin;
    1174:   PetscOptionsHeadBegin(PetscOptionsObject, "RK ODE solver options");
    1175:   {
    1176:     RKTableauLink link;
    1177:     PetscInt      count, choice;
    1178:     PetscBool     flg, use_multirate = PETSC_FALSE;
    1179:     const char  **namelist;

    1181:     for (link = RKTableauList, count = 0; link; link = link->next, count++);
    1182:     PetscCall(PetscMalloc1(count, (char ***)&namelist));
    1183:     for (link = RKTableauList, count = 0; link; link = link->next, count++) namelist[count] = link->tab.name;
    1184:     PetscCall(PetscOptionsBool("-ts_rk_multirate", "Use interpolation-based multirate RK method", "TSRKSetMultirate", rk->use_multirate, &use_multirate, &flg));
    1185:     if (flg) PetscCall(TSRKSetMultirate(ts, use_multirate));
    1186:     PetscCall(PetscOptionsEList("-ts_rk_type", "Family of RK method", "TSRKSetType", (const char *const *)namelist, count, rk->tableau->name, &choice, &flg));
    1187:     if (flg) PetscCall(TSRKSetType(ts, namelist[choice]));
    1188:     PetscCall(PetscFree(namelist));
    1189:   }
    1190:   PetscOptionsHeadEnd();
    1191:   PetscOptionsBegin(PetscObjectComm((PetscObject)ts), NULL, "Multirate methods options", "");
    1192:   PetscCall(PetscOptionsInt("-ts_rk_dtratio", "time step ratio between slow and fast", "", rk->dtratio, &rk->dtratio, NULL));
    1193:   PetscOptionsEnd();
    1194:   PetscFunctionReturn(PETSC_SUCCESS);
    1195: }

    1197: static PetscErrorCode TSView_RK(TS ts, PetscViewer viewer)
    1198: {
    1199:   TS_RK    *rk = (TS_RK *)ts->data;
    1200:   PetscBool isascii;

    1202:   PetscFunctionBegin;
    1203:   PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &isascii));
    1204:   if (isascii) {
    1205:     RKTableau        tab = rk->tableau;
    1206:     TSRKType         rktype;
    1207:     const PetscReal *c;
    1208:     PetscInt         s;
    1209:     char             buf[512];
    1210:     PetscBool        FSAL;

    1212:     PetscCall(TSRKGetType(ts, &rktype));
    1213:     PetscCall(TSRKGetTableau(ts, &s, NULL, NULL, &c, NULL, NULL, NULL, &FSAL));
    1214:     PetscCall(PetscViewerASCIIPrintf(viewer, "  RK type %s\n", rktype));
    1215:     PetscCall(PetscViewerASCIIPrintf(viewer, "  Order: %" PetscInt_FMT "\n", tab->order));
    1216:     PetscCall(PetscViewerASCIIPrintf(viewer, "  FSAL property: %s\n", FSAL ? "yes" : "no"));
    1217:     PetscCall(PetscFormatRealArray(buf, sizeof(buf), "% 8.6f", s, c));
    1218:     PetscCall(PetscViewerASCIIPrintf(viewer, "  Abscissa c = %s\n", buf));
    1219:   }
    1220:   PetscFunctionReturn(PETSC_SUCCESS);
    1221: }

    1223: static PetscErrorCode TSLoad_RK(TS ts, PetscViewer viewer)
    1224: {
    1225:   TSAdapt adapt;

    1227:   PetscFunctionBegin;
    1228:   PetscCall(TSGetAdapt(ts, &adapt));
    1229:   PetscCall(TSAdaptLoad(adapt, viewer));
    1230:   PetscFunctionReturn(PETSC_SUCCESS);
    1231: }

    1233: /*@
    1234:   TSRKGetOrder - Get the order of the `TSRK` scheme

    1236:   Not Collective

    1238:   Input Parameter:
    1239: . ts - timestepping context

    1241:   Output Parameter:
    1242: . order - order of `TSRK` scheme

    1244:   Level: intermediate

    1246: .seealso: [](ch_ts), `TSRK`, `TSRKGetType()`
    1247: @*/
    1248: PetscErrorCode TSRKGetOrder(TS ts, PetscInt *order)
    1249: {
    1250:   PetscFunctionBegin;
    1252:   PetscAssertPointer(order, 2);
    1253:   PetscUseMethod(ts, "TSRKGetOrder_C", (TS, PetscInt *), (ts, order));
    1254:   PetscFunctionReturn(PETSC_SUCCESS);
    1255: }

    1257: /*@
    1258:   TSRKSetType - Set the type of the `TSRK` scheme

    1260:   Logically Collective

    1262:   Input Parameters:
    1263: + ts     - timestepping context
    1264: - rktype - type of `TSRK` scheme

    1266:   Options Database Key:
    1267: . -ts_rk_type - <1fe,2a,3,3bs,4,5f,5dp,5bs>

    1269:   Level: intermediate

    1271: .seealso: [](ch_ts), `TSRKGetType()`, `TSRK`, `TSRKType`, `TSRK1FE`, `TSRK2A`, `TSRK2B`, `TSRK3`, `TSRK3BS`, `TSRK4`, `TSRK5F`, `TSRK5DP`, `TSRK5BS`, `TSRK6VR`, `TSRK7VR`, `TSRK8VR`
    1272: @*/
    1273: PetscErrorCode TSRKSetType(TS ts, TSRKType rktype)
    1274: {
    1275:   PetscFunctionBegin;
    1277:   PetscAssertPointer(rktype, 2);
    1278:   PetscTryMethod(ts, "TSRKSetType_C", (TS, TSRKType), (ts, rktype));
    1279:   PetscFunctionReturn(PETSC_SUCCESS);
    1280: }

    1282: /*@
    1283:   TSRKGetType - Get the type of `TSRK` scheme

    1285:   Not Collective

    1287:   Input Parameter:
    1288: . ts - timestepping context

    1290:   Output Parameter:
    1291: . rktype - type of `TSRK`-scheme

    1293:   Level: intermediate

    1295: .seealso: [](ch_ts), `TSRKSetType()`
    1296: @*/
    1297: PetscErrorCode TSRKGetType(TS ts, TSRKType *rktype)
    1298: {
    1299:   PetscFunctionBegin;
    1301:   PetscUseMethod(ts, "TSRKGetType_C", (TS, TSRKType *), (ts, rktype));
    1302:   PetscFunctionReturn(PETSC_SUCCESS);
    1303: }

    1305: static PetscErrorCode TSRKGetOrder_RK(TS ts, PetscInt *order)
    1306: {
    1307:   TS_RK *rk = (TS_RK *)ts->data;

    1309:   PetscFunctionBegin;
    1310:   *order = rk->tableau->order;
    1311:   PetscFunctionReturn(PETSC_SUCCESS);
    1312: }

    1314: static PetscErrorCode TSRKGetType_RK(TS ts, TSRKType *rktype)
    1315: {
    1316:   TS_RK *rk = (TS_RK *)ts->data;

    1318:   PetscFunctionBegin;
    1319:   *rktype = rk->tableau->name;
    1320:   PetscFunctionReturn(PETSC_SUCCESS);
    1321: }

    1323: static PetscErrorCode TSRKSetType_RK(TS ts, TSRKType rktype)
    1324: {
    1325:   TS_RK        *rk = (TS_RK *)ts->data;
    1326:   PetscBool     match;
    1327:   RKTableauLink link;

    1329:   PetscFunctionBegin;
    1330:   if (rk->tableau) {
    1331:     PetscCall(PetscStrcmp(rk->tableau->name, rktype, &match));
    1332:     if (match) PetscFunctionReturn(PETSC_SUCCESS);
    1333:   }
    1334:   for (link = RKTableauList; link; link = link->next) {
    1335:     PetscCall(PetscStrcmp(link->tab.name, rktype, &match));
    1336:     if (match) {
    1337:       if (ts->setupcalled) PetscCall(TSRKTableauReset(ts));
    1338:       rk->tableau = &link->tab;
    1339:       if (ts->setupcalled) PetscCall(TSRKTableauSetUp(ts));
    1340:       ts->default_adapt_type = rk->tableau->bembed ? TSADAPTBASIC : TSADAPTNONE;
    1341:       PetscFunctionReturn(PETSC_SUCCESS);
    1342:     }
    1343:   }
    1344:   SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_ARG_UNKNOWN_TYPE, "Could not find '%s'", rktype);
    1345: }

    1347: static PetscErrorCode TSGetStages_RK(TS ts, PetscInt *ns, Vec **Y)
    1348: {
    1349:   TS_RK *rk = (TS_RK *)ts->data;

    1351:   PetscFunctionBegin;
    1352:   if (ns) *ns = rk->tableau->s;
    1353:   if (Y) *Y = rk->Y;
    1354:   PetscFunctionReturn(PETSC_SUCCESS);
    1355: }

    1357: static PetscErrorCode TSDestroy_RK(TS ts)
    1358: {
    1359:   PetscFunctionBegin;
    1360:   PetscCall(TSReset_RK(ts));
    1361:   if (ts->dm) {
    1362:     PetscCall(DMCoarsenHookRemove(ts->dm, DMCoarsenHook_TSRK, DMRestrictHook_TSRK, ts));
    1363:     PetscCall(DMSubDomainHookRemove(ts->dm, DMSubDomainHook_TSRK, DMSubDomainRestrictHook_TSRK, ts));
    1364:   }
    1365:   PetscCall(PetscFree(ts->data));
    1366:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetOrder_C", NULL));
    1367:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetType_C", NULL));
    1368:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKSetType_C", NULL));
    1369:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetTableau_C", NULL));
    1370:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKSetMultirate_C", NULL));
    1371:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetMultirate_C", NULL));
    1372:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSSetUp_RK_MultirateSplit_C", NULL));
    1373:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSReset_RK_MultirateSplit_C", NULL));
    1374:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSSetUp_RK_MultirateNonsplit_C", NULL));
    1375:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSReset_RK_MultirateNonsplit_C", NULL));
    1376:   PetscFunctionReturn(PETSC_SUCCESS);
    1377: }

    1379: /*
    1380:   This defines the nonlinear equation that is to be solved with SNES
    1381:   We do not need to solve the equation; we just use SNES to approximate the Jacobian
    1382: */
    1383: static PetscErrorCode SNESTSFormFunction_RK(SNES snes, Vec x, Vec y, TS ts)
    1384: {
    1385:   TS_RK *rk = (TS_RK *)ts->data;
    1386:   DM     dm, dmsave;

    1388:   PetscFunctionBegin;
    1389:   PetscCall(SNESGetDM(snes, &dm));
    1390:   /* DM monkey-business allows user code to call TSGetDM() inside of functions evaluated on levels of FAS */
    1391:   dmsave = ts->dm;
    1392:   ts->dm = dm;
    1393:   PetscCall(TSComputeRHSFunction(ts, rk->stage_time, x, y));
    1394:   ts->dm = dmsave;
    1395:   PetscFunctionReturn(PETSC_SUCCESS);
    1396: }

    1398: static PetscErrorCode SNESTSFormJacobian_RK(SNES snes, Vec x, Mat A, Mat B, TS ts)
    1399: {
    1400:   TS_RK *rk = (TS_RK *)ts->data;
    1401:   DM     dm, dmsave;

    1403:   PetscFunctionBegin;
    1404:   PetscCall(SNESGetDM(snes, &dm));
    1405:   dmsave = ts->dm;
    1406:   ts->dm = dm;
    1407:   PetscCall(TSComputeRHSJacobian(ts, rk->stage_time, x, A, B));
    1408:   ts->dm = dmsave;
    1409:   PetscFunctionReturn(PETSC_SUCCESS);
    1410: }

    1412: /*@
    1413:   TSRKSetMultirate - Use the interpolation-based multirate `TSRK` method

    1415:   Logically Collective

    1417:   Input Parameters:
    1418: + ts            - timestepping context
    1419: - use_multirate - `PETSC_TRUE` enables the multirate `TSRK` method, sets the basic method to be RK2A and sets the ratio between slow stepsize and fast stepsize to be 2

    1421:   Options Database Key:
    1422: . -ts_rk_multirate - <true,false>

    1424:   Level: intermediate

    1426:   Note:
    1427:   The multirate method requires interpolation. The default interpolation works for 1st- and 2nd- order RK, but not for high-order RKs except `TSRK5DP` which comes with the interpolation coefficients (binterp).

    1429: .seealso: [](ch_ts), `TSRK`, `TSRKGetMultirate()`
    1430: @*/
    1431: PetscErrorCode TSRKSetMultirate(TS ts, PetscBool use_multirate)
    1432: {
    1433:   PetscFunctionBegin;
    1434:   PetscTryMethod(ts, "TSRKSetMultirate_C", (TS, PetscBool), (ts, use_multirate));
    1435:   PetscFunctionReturn(PETSC_SUCCESS);
    1436: }

    1438: /*@
    1439:   TSRKGetMultirate - Gets whether to use the interpolation-based multirate `TSRK` method

    1441:   Not Collective

    1443:   Input Parameter:
    1444: . ts - timestepping context

    1446:   Output Parameter:
    1447: . use_multirate - `PETSC_TRUE` if the multirate RK method is enabled, `PETSC_FALSE` otherwise

    1449:   Level: intermediate

    1451: .seealso: [](ch_ts), `TSRK`, `TSRKSetMultirate()`
    1452: @*/
    1453: PetscErrorCode TSRKGetMultirate(TS ts, PetscBool *use_multirate)
    1454: {
    1455:   PetscFunctionBegin;
    1456:   PetscUseMethod(ts, "TSRKGetMultirate_C", (TS, PetscBool *), (ts, use_multirate));
    1457:   PetscFunctionReturn(PETSC_SUCCESS);
    1458: }

    1460: /*MC
    1461:       TSRK - ODE and DAE solver using Runge-Kutta schemes

    1463:   The user should provide the right-hand side of the equation
    1464:   using `TSSetRHSFunction()`.

    1466:   Level: beginner

    1468:   Notes:
    1469:   The default is `TSRK3BS`, it can be changed with `TSRKSetType()` or -ts_rk_type

    1471: .seealso: [](ch_ts), `TSCreate()`, `TS`, `TSRK`, `TSSetType()`, `TSRKSetType()`, `TSRKGetType()`, `TSRK2D`, `TSRK2E`, `TSRK3`,
    1472:           `TSRK4`, `TSRK5`, `TSRKPRSSP2`, `TSRKBPR3`, `TSRKType`, `TSRKRegister()`, `TSRKSetMultirate()`, `TSRKGetMultirate()`, `TSType`
    1473: M*/
    1474: PETSC_EXTERN PetscErrorCode TSCreate_RK(TS ts)
    1475: {
    1476:   TS_RK *rk;

    1478:   PetscFunctionBegin;
    1479:   PetscCall(TSRKInitializePackage());

    1481:   ts->ops->reset          = TSReset_RK;
    1482:   ts->ops->destroy        = TSDestroy_RK;
    1483:   ts->ops->view           = TSView_RK;
    1484:   ts->ops->load           = TSLoad_RK;
    1485:   ts->ops->setup          = TSSetUp_RK;
    1486:   ts->ops->interpolate    = TSInterpolate_RK;
    1487:   ts->ops->step           = TSStep_RK;
    1488:   ts->ops->evaluatestep   = TSEvaluateStep_RK;
    1489:   ts->ops->rollback       = TSRollBack_RK;
    1490:   ts->ops->setfromoptions = TSSetFromOptions_RK;
    1491:   ts->ops->getstages      = TSGetStages_RK;

    1493:   ts->ops->snesfunction    = SNESTSFormFunction_RK;
    1494:   ts->ops->snesjacobian    = SNESTSFormJacobian_RK;
    1495:   ts->ops->adjointintegral = TSAdjointCostIntegral_RK;
    1496:   ts->ops->adjointsetup    = TSAdjointSetUp_RK;
    1497:   ts->ops->adjointstep     = TSAdjointStep_RK;
    1498:   ts->ops->adjointreset    = TSAdjointReset_RK;

    1500:   ts->ops->forwardintegral  = TSForwardCostIntegral_RK;
    1501:   ts->ops->forwardsetup     = TSForwardSetUp_RK;
    1502:   ts->ops->forwardreset     = TSForwardReset_RK;
    1503:   ts->ops->forwardstep      = TSForwardStep_RK;
    1504:   ts->ops->forwardgetstages = TSForwardGetStages_RK;

    1506:   PetscCall(PetscNew(&rk));
    1507:   ts->data = (void *)rk;

    1509:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetOrder_C", TSRKGetOrder_RK));
    1510:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetType_C", TSRKGetType_RK));
    1511:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKSetType_C", TSRKSetType_RK));
    1512:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetTableau_C", TSRKGetTableau_RK));
    1513:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKSetMultirate_C", TSRKSetMultirate_RK));
    1514:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetMultirate_C", TSRKGetMultirate_RK));

    1516:   PetscCall(TSRKSetType(ts, TSRKDefault));
    1517:   rk->dtratio = 1;
    1518:   PetscFunctionReturn(PETSC_SUCCESS);
    1519: }