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

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:   if (done) *done = PETSC_FALSE;
591:   else
592:     SETERRQ(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, tab->order, order);
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_DEFAULT, &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:   PetscCall(VecDuplicateVecs(ts->vecs_sensi[0], s * ts->numcost, &rk->VecsDeltaLam));
856:   PetscCall(VecDuplicateVecs(ts->vecs_sensi[0], ts->numcost, &rk->VecsSensiTemp));
857:   if (ts->vecs_sensip) PetscCall(VecDuplicate(ts->vecs_sensip[0], &rk->VecDeltaMu));
858:   if (ts->vecs_sensi2) {
859:     PetscCall(VecDuplicateVecs(ts->vecs_sensi[0], s * ts->numcost, &rk->VecsDeltaLam2));
860:     PetscCall(VecDuplicateVecs(ts->vecs_sensi2[0], ts->numcost, &rk->VecsSensi2Temp));
861:   }
862:   if (ts->vecs_sensi2p) PetscCall(VecDuplicate(ts->vecs_sensi2p[0], &rk->VecDeltaMu2));
863:   PetscFunctionReturn(PETSC_SUCCESS);
864: }

866: /*
867:   Assumptions:
868:     - TSStep_RK() always evaluates the step with b, not bembed.
869: */
870: static PetscErrorCode TSAdjointStep_RK(TS ts)
871: {
872:   TS_RK           *rk     = (TS_RK *)ts->data;
873:   TS               quadts = ts->quadraturets;
874:   RKTableau        tab    = rk->tableau;
875:   Mat              J, Jpre, Jquad;
876:   const PetscInt   s = tab->s;
877:   const PetscReal *A = tab->A, *b = tab->b, *c = tab->c;
878:   PetscScalar     *w = rk->work, *xarr;
879:   Vec             *Y = rk->Y, *VecsDeltaLam = rk->VecsDeltaLam, VecDeltaMu = rk->VecDeltaMu, *VecsSensiTemp = rk->VecsSensiTemp;
880:   Vec             *VecsDeltaLam2 = rk->VecsDeltaLam2, VecDeltaMu2 = rk->VecDeltaMu2, *VecsSensi2Temp = rk->VecsSensi2Temp;
881:   Vec              VecDRDUTransCol = ts->vec_drdu_col, VecDRDPTransCol = ts->vec_drdp_col;
882:   PetscInt         i, j, nadj;
883:   PetscReal        t = ts->ptime;
884:   PetscReal        h = ts->time_step;

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

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

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

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

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

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

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

1021:   for (j = 0; j < s; j++) w[j] = 1.0;
1022:   for (nadj = 0; nadj < ts->numcost; nadj++) { /* no need to do this for mu's */
1023:     PetscCall(VecMAXPY(ts->vecs_sensi[nadj], s, w, &VecsDeltaLam[nadj * s]));
1024:     if (ts->vecs_sensi2) PetscCall(VecMAXPY(ts->vecs_sensi2[nadj], s, w, &VecsDeltaLam2[nadj * s]));
1025:   }
1026:   rk->status = TS_STEP_COMPLETE;
1027:   PetscFunctionReturn(PETSC_SUCCESS);
1028: }

1030: static PetscErrorCode TSAdjointReset_RK(TS ts)
1031: {
1032:   TS_RK    *rk  = (TS_RK *)ts->data;
1033:   RKTableau tab = rk->tableau;

1035:   PetscFunctionBegin;
1036:   PetscCall(VecDestroyVecs(tab->s * ts->numcost, &rk->VecsDeltaLam));
1037:   PetscCall(VecDestroyVecs(ts->numcost, &rk->VecsSensiTemp));
1038:   PetscCall(VecDestroy(&rk->VecDeltaMu));
1039:   PetscCall(VecDestroyVecs(tab->s * ts->numcost, &rk->VecsDeltaLam2));
1040:   PetscCall(VecDestroy(&rk->VecDeltaMu2));
1041:   PetscCall(VecDestroyVecs(ts->numcost, &rk->VecsSensi2Temp));
1042:   PetscFunctionReturn(PETSC_SUCCESS);
1043: }

1045: static PetscErrorCode TSInterpolate_RK(TS ts, PetscReal itime, Vec X)
1046: {
1047:   TS_RK           *rk = (TS_RK *)ts->data;
1048:   PetscInt         s = rk->tableau->s, p = rk->tableau->p, i, j;
1049:   PetscReal        h;
1050:   PetscReal        tt, t;
1051:   PetscScalar     *b;
1052:   const PetscReal *B = rk->tableau->binterp;

1054:   PetscFunctionBegin;
1055:   PetscCheck(B, PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "TSRK %s does not have an interpolation formula", rk->tableau->name);

1057:   switch (rk->status) {
1058:   case TS_STEP_INCOMPLETE:
1059:   case TS_STEP_PENDING:
1060:     h = ts->time_step;
1061:     t = (itime - ts->ptime) / h;
1062:     break;
1063:   case TS_STEP_COMPLETE:
1064:     h = ts->ptime - ts->ptime_prev;
1065:     t = (itime - ts->ptime) / h + 1; /* In the interval [0,1] */
1066:     break;
1067:   default:
1068:     SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_PLIB, "Invalid TSStepStatus");
1069:   }
1070:   PetscCall(PetscMalloc1(s, &b));
1071:   for (i = 0; i < s; i++) b[i] = 0;
1072:   for (j = 0, tt = t; j < p; j++, tt *= t) {
1073:     for (i = 0; i < s; i++) b[i] += h * B[i * p + j] * tt;
1074:   }
1075:   PetscCall(VecCopy(rk->Y[0], X));
1076:   PetscCall(VecMAXPY(X, s, b, rk->YdotRHS));
1077:   PetscCall(PetscFree(b));
1078:   PetscFunctionReturn(PETSC_SUCCESS);
1079: }

1081: /*------------------------------------------------------------*/

1083: static PetscErrorCode TSRKTableauReset(TS ts)
1084: {
1085:   TS_RK    *rk  = (TS_RK *)ts->data;
1086:   RKTableau tab = rk->tableau;

1088:   PetscFunctionBegin;
1089:   if (!tab) PetscFunctionReturn(PETSC_SUCCESS);
1090:   PetscCall(PetscFree(rk->work));
1091:   PetscCall(VecDestroyVecs(tab->s, &rk->Y));
1092:   PetscCall(VecDestroyVecs(tab->s, &rk->YdotRHS));
1093:   PetscFunctionReturn(PETSC_SUCCESS);
1094: }

1096: static PetscErrorCode TSReset_RK(TS ts)
1097: {
1098:   PetscFunctionBegin;
1099:   PetscCall(TSRKTableauReset(ts));
1100:   if (ts->use_splitrhsfunction) {
1101:     PetscTryMethod(ts, "TSReset_RK_MultirateSplit_C", (TS), (ts));
1102:   } else {
1103:     PetscTryMethod(ts, "TSReset_RK_MultirateNonsplit_C", (TS), (ts));
1104:   }
1105:   PetscFunctionReturn(PETSC_SUCCESS);
1106: }

1108: static PetscErrorCode DMCoarsenHook_TSRK(DM fine, DM coarse, void *ctx)
1109: {
1110:   PetscFunctionBegin;
1111:   PetscFunctionReturn(PETSC_SUCCESS);
1112: }

1114: static PetscErrorCode DMRestrictHook_TSRK(DM fine, Mat restrct, Vec rscale, Mat inject, DM coarse, void *ctx)
1115: {
1116:   PetscFunctionBegin;
1117:   PetscFunctionReturn(PETSC_SUCCESS);
1118: }

1120: static PetscErrorCode DMSubDomainHook_TSRK(DM dm, DM subdm, void *ctx)
1121: {
1122:   PetscFunctionBegin;
1123:   PetscFunctionReturn(PETSC_SUCCESS);
1124: }

1126: static PetscErrorCode DMSubDomainRestrictHook_TSRK(DM dm, VecScatter gscat, VecScatter lscat, DM subdm, void *ctx)
1127: {
1128:   PetscFunctionBegin;
1129:   PetscFunctionReturn(PETSC_SUCCESS);
1130: }

1132: static PetscErrorCode TSRKTableauSetUp(TS ts)
1133: {
1134:   TS_RK    *rk  = (TS_RK *)ts->data;
1135:   RKTableau tab = rk->tableau;

1137:   PetscFunctionBegin;
1138:   PetscCall(PetscMalloc1(tab->s, &rk->work));
1139:   PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &rk->Y));
1140:   PetscCall(VecDuplicateVecs(ts->vec_sol, tab->s, &rk->YdotRHS));
1141:   rk->newtableau = PETSC_TRUE;
1142:   PetscFunctionReturn(PETSC_SUCCESS);
1143: }

1145: static PetscErrorCode TSSetUp_RK(TS ts)
1146: {
1147:   TS quadts = ts->quadraturets;
1148:   DM dm;

1150:   PetscFunctionBegin;
1151:   PetscCall(TSCheckImplicitTerm(ts));
1152:   PetscCall(TSRKTableauSetUp(ts));
1153:   if (quadts && ts->costintegralfwd) {
1154:     Mat Jquad;
1155:     PetscCall(TSGetRHSJacobian(quadts, &Jquad, NULL, NULL, NULL));
1156:   }
1157:   PetscCall(TSGetDM(ts, &dm));
1158:   PetscCall(DMCoarsenHookAdd(dm, DMCoarsenHook_TSRK, DMRestrictHook_TSRK, ts));
1159:   PetscCall(DMSubDomainHookAdd(dm, DMSubDomainHook_TSRK, DMSubDomainRestrictHook_TSRK, ts));
1160:   if (ts->use_splitrhsfunction) {
1161:     PetscTryMethod(ts, "TSSetUp_RK_MultirateSplit_C", (TS), (ts));
1162:   } else {
1163:     PetscTryMethod(ts, "TSSetUp_RK_MultirateNonsplit_C", (TS), (ts));
1164:   }
1165:   PetscFunctionReturn(PETSC_SUCCESS);
1166: }

1168: static PetscErrorCode TSSetFromOptions_RK(TS ts, PetscOptionItems *PetscOptionsObject)
1169: {
1170:   TS_RK *rk = (TS_RK *)ts->data;

1172:   PetscFunctionBegin;
1173:   PetscOptionsHeadBegin(PetscOptionsObject, "RK ODE solver options");
1174:   {
1175:     RKTableauLink link;
1176:     PetscInt      count, choice;
1177:     PetscBool     flg, use_multirate = PETSC_FALSE;
1178:     const char  **namelist;

1180:     for (link = RKTableauList, count = 0; link; link = link->next, count++);
1181:     PetscCall(PetscMalloc1(count, (char ***)&namelist));
1182:     for (link = RKTableauList, count = 0; link; link = link->next, count++) namelist[count] = link->tab.name;
1183:     PetscCall(PetscOptionsBool("-ts_rk_multirate", "Use interpolation-based multirate RK method", "TSRKSetMultirate", rk->use_multirate, &use_multirate, &flg));
1184:     if (flg) PetscCall(TSRKSetMultirate(ts, use_multirate));
1185:     PetscCall(PetscOptionsEList("-ts_rk_type", "Family of RK method", "TSRKSetType", (const char *const *)namelist, count, rk->tableau->name, &choice, &flg));
1186:     if (flg) PetscCall(TSRKSetType(ts, namelist[choice]));
1187:     PetscCall(PetscFree(namelist));
1188:   }
1189:   PetscOptionsHeadEnd();
1190:   PetscOptionsBegin(PetscObjectComm((PetscObject)ts), NULL, "Multirate methods options", "");
1191:   PetscCall(PetscOptionsInt("-ts_rk_dtratio", "time step ratio between slow and fast", "", rk->dtratio, &rk->dtratio, NULL));
1192:   PetscOptionsEnd();
1193:   PetscFunctionReturn(PETSC_SUCCESS);
1194: }

1196: static PetscErrorCode TSView_RK(TS ts, PetscViewer viewer)
1197: {
1198:   TS_RK    *rk = (TS_RK *)ts->data;
1199:   PetscBool iascii;

1201:   PetscFunctionBegin;
1202:   PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &iascii));
1203:   if (iascii) {
1204:     RKTableau        tab = rk->tableau;
1205:     TSRKType         rktype;
1206:     const PetscReal *c;
1207:     PetscInt         s;
1208:     char             buf[512];
1209:     PetscBool        FSAL;

1211:     PetscCall(TSRKGetType(ts, &rktype));
1212:     PetscCall(TSRKGetTableau(ts, &s, NULL, NULL, &c, NULL, NULL, NULL, &FSAL));
1213:     PetscCall(PetscViewerASCIIPrintf(viewer, "  RK type %s\n", rktype));
1214:     PetscCall(PetscViewerASCIIPrintf(viewer, "  Order: %" PetscInt_FMT "\n", tab->order));
1215:     PetscCall(PetscViewerASCIIPrintf(viewer, "  FSAL property: %s\n", FSAL ? "yes" : "no"));
1216:     PetscCall(PetscFormatRealArray(buf, sizeof(buf), "% 8.6f", s, c));
1217:     PetscCall(PetscViewerASCIIPrintf(viewer, "  Abscissa c = %s\n", buf));
1218:   }
1219:   PetscFunctionReturn(PETSC_SUCCESS);
1220: }

1222: static PetscErrorCode TSLoad_RK(TS ts, PetscViewer viewer)
1223: {
1224:   TSAdapt adapt;

1226:   PetscFunctionBegin;
1227:   PetscCall(TSGetAdapt(ts, &adapt));
1228:   PetscCall(TSAdaptLoad(adapt, viewer));
1229:   PetscFunctionReturn(PETSC_SUCCESS);
1230: }

1232: /*@
1233:   TSRKGetOrder - Get the order of the `TSRK` scheme

1235:   Not Collective

1237:   Input Parameter:
1238: . ts - timestepping context

1240:   Output Parameter:
1241: . order - order of `TSRK` scheme

1243:   Level: intermediate

1245: .seealso: [](ch_ts), `TSRK`, `TSRKGetType()`
1246: @*/
1247: PetscErrorCode TSRKGetOrder(TS ts, PetscInt *order)
1248: {
1249:   PetscFunctionBegin;
1251:   PetscAssertPointer(order, 2);
1252:   PetscUseMethod(ts, "TSRKGetOrder_C", (TS, PetscInt *), (ts, order));
1253:   PetscFunctionReturn(PETSC_SUCCESS);
1254: }

1256: /*@C
1257:   TSRKSetType - Set the type of the `TSRK` scheme

1259:   Logically Collective

1261:   Input Parameters:
1262: + ts     - timestepping context
1263: - rktype - type of `TSRK` scheme

1265:   Options Database Key:
1266: . -ts_rk_type - <1fe,2a,3,3bs,4,5f,5dp,5bs>

1268:   Level: intermediate

1270: .seealso: [](ch_ts), `TSRKGetType()`, `TSRK`, `TSRKType`, `TSRK1FE`, `TSRK2A`, `TSRK2B`, `TSRK3`, `TSRK3BS`, `TSRK4`, `TSRK5F`, `TSRK5DP`, `TSRK5BS`, `TSRK6VR`, `TSRK7VR`, `TSRK8VR`
1271: @*/
1272: PetscErrorCode TSRKSetType(TS ts, TSRKType rktype)
1273: {
1274:   PetscFunctionBegin;
1276:   PetscAssertPointer(rktype, 2);
1277:   PetscTryMethod(ts, "TSRKSetType_C", (TS, TSRKType), (ts, rktype));
1278:   PetscFunctionReturn(PETSC_SUCCESS);
1279: }

1281: /*@C
1282:   TSRKGetType - Get the type of `TSRK` scheme

1284:   Not Collective

1286:   Input Parameter:
1287: . ts - timestepping context

1289:   Output Parameter:
1290: . rktype - type of `TSRK`-scheme

1292:   Level: intermediate

1294: .seealso: [](ch_ts), `TSRKSetType()`
1295: @*/
1296: PetscErrorCode TSRKGetType(TS ts, TSRKType *rktype)
1297: {
1298:   PetscFunctionBegin;
1300:   PetscUseMethod(ts, "TSRKGetType_C", (TS, TSRKType *), (ts, rktype));
1301:   PetscFunctionReturn(PETSC_SUCCESS);
1302: }

1304: static PetscErrorCode TSRKGetOrder_RK(TS ts, PetscInt *order)
1305: {
1306:   TS_RK *rk = (TS_RK *)ts->data;

1308:   PetscFunctionBegin;
1309:   *order = rk->tableau->order;
1310:   PetscFunctionReturn(PETSC_SUCCESS);
1311: }

1313: static PetscErrorCode TSRKGetType_RK(TS ts, TSRKType *rktype)
1314: {
1315:   TS_RK *rk = (TS_RK *)ts->data;

1317:   PetscFunctionBegin;
1318:   *rktype = rk->tableau->name;
1319:   PetscFunctionReturn(PETSC_SUCCESS);
1320: }

1322: static PetscErrorCode TSRKSetType_RK(TS ts, TSRKType rktype)
1323: {
1324:   TS_RK        *rk = (TS_RK *)ts->data;
1325:   PetscBool     match;
1326:   RKTableauLink link;

1328:   PetscFunctionBegin;
1329:   if (rk->tableau) {
1330:     PetscCall(PetscStrcmp(rk->tableau->name, rktype, &match));
1331:     if (match) PetscFunctionReturn(PETSC_SUCCESS);
1332:   }
1333:   for (link = RKTableauList; link; link = link->next) {
1334:     PetscCall(PetscStrcmp(link->tab.name, rktype, &match));
1335:     if (match) {
1336:       if (ts->setupcalled) PetscCall(TSRKTableauReset(ts));
1337:       rk->tableau = &link->tab;
1338:       if (ts->setupcalled) PetscCall(TSRKTableauSetUp(ts));
1339:       ts->default_adapt_type = rk->tableau->bembed ? TSADAPTBASIC : TSADAPTNONE;
1340:       PetscFunctionReturn(PETSC_SUCCESS);
1341:     }
1342:   }
1343:   SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_ARG_UNKNOWN_TYPE, "Could not find '%s'", rktype);
1344: }

1346: static PetscErrorCode TSGetStages_RK(TS ts, PetscInt *ns, Vec **Y)
1347: {
1348:   TS_RK *rk = (TS_RK *)ts->data;

1350:   PetscFunctionBegin;
1351:   if (ns) *ns = rk->tableau->s;
1352:   if (Y) *Y = rk->Y;
1353:   PetscFunctionReturn(PETSC_SUCCESS);
1354: }

1356: static PetscErrorCode TSDestroy_RK(TS ts)
1357: {
1358:   PetscFunctionBegin;
1359:   PetscCall(TSReset_RK(ts));
1360:   if (ts->dm) {
1361:     PetscCall(DMCoarsenHookRemove(ts->dm, DMCoarsenHook_TSRK, DMRestrictHook_TSRK, ts));
1362:     PetscCall(DMSubDomainHookRemove(ts->dm, DMSubDomainHook_TSRK, DMSubDomainRestrictHook_TSRK, ts));
1363:   }
1364:   PetscCall(PetscFree(ts->data));
1365:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetOrder_C", NULL));
1366:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetType_C", NULL));
1367:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKSetType_C", NULL));
1368:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetTableau_C", NULL));
1369:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKSetMultirate_C", NULL));
1370:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetMultirate_C", NULL));
1371:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSSetUp_RK_MultirateSplit_C", NULL));
1372:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSReset_RK_MultirateSplit_C", NULL));
1373:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSSetUp_RK_MultirateNonsplit_C", NULL));
1374:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSReset_RK_MultirateNonsplit_C", NULL));
1375:   PetscFunctionReturn(PETSC_SUCCESS);
1376: }

1378: /*
1379:   This defines the nonlinear equation that is to be solved with SNES
1380:   We do not need to solve the equation; we just use SNES to approximate the Jacobian
1381: */
1382: static PetscErrorCode SNESTSFormFunction_RK(SNES snes, Vec x, Vec y, TS ts)
1383: {
1384:   TS_RK *rk = (TS_RK *)ts->data;
1385:   DM     dm, dmsave;

1387:   PetscFunctionBegin;
1388:   PetscCall(SNESGetDM(snes, &dm));
1389:   /* DM monkey-business allows user code to call TSGetDM() inside of functions evaluated on levels of FAS */
1390:   dmsave = ts->dm;
1391:   ts->dm = dm;
1392:   PetscCall(TSComputeRHSFunction(ts, rk->stage_time, x, y));
1393:   ts->dm = dmsave;
1394:   PetscFunctionReturn(PETSC_SUCCESS);
1395: }

1397: static PetscErrorCode SNESTSFormJacobian_RK(SNES snes, Vec x, Mat A, Mat B, TS ts)
1398: {
1399:   TS_RK *rk = (TS_RK *)ts->data;
1400:   DM     dm, dmsave;

1402:   PetscFunctionBegin;
1403:   PetscCall(SNESGetDM(snes, &dm));
1404:   dmsave = ts->dm;
1405:   ts->dm = dm;
1406:   PetscCall(TSComputeRHSJacobian(ts, rk->stage_time, x, A, B));
1407:   ts->dm = dmsave;
1408:   PetscFunctionReturn(PETSC_SUCCESS);
1409: }

1411: /*@C
1412:   TSRKSetMultirate - Use the interpolation-based multirate `TSRK` method

1414:   Logically Collective

1416:   Input Parameters:
1417: + ts            - timestepping context
1418: - 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

1420:   Options Database Key:
1421: . -ts_rk_multirate - <true,false>

1423:   Level: intermediate

1425:   Note:
1426:   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).

1428: .seealso: [](ch_ts), `TSRK`, `TSRKGetMultirate()`
1429: @*/
1430: PetscErrorCode TSRKSetMultirate(TS ts, PetscBool use_multirate)
1431: {
1432:   PetscFunctionBegin;
1433:   PetscTryMethod(ts, "TSRKSetMultirate_C", (TS, PetscBool), (ts, use_multirate));
1434:   PetscFunctionReturn(PETSC_SUCCESS);
1435: }

1437: /*@C
1438:   TSRKGetMultirate - Gets whether to use the interpolation-based multirate `TSRK` method

1440:   Not Collective

1442:   Input Parameter:
1443: . ts - timestepping context

1445:   Output Parameter:
1446: . use_multirate - `PETSC_TRUE` if the multirate RK method is enabled, `PETSC_FALSE` otherwise

1448:   Level: intermediate

1450: .seealso: [](ch_ts), `TSRK`, `TSRKSetMultirate()`
1451: @*/
1452: PetscErrorCode TSRKGetMultirate(TS ts, PetscBool *use_multirate)
1453: {
1454:   PetscFunctionBegin;
1455:   PetscUseMethod(ts, "TSRKGetMultirate_C", (TS, PetscBool *), (ts, use_multirate));
1456:   PetscFunctionReturn(PETSC_SUCCESS);
1457: }

1459: /*MC
1460:       TSRK - ODE and DAE solver using Runge-Kutta schemes

1462:   The user should provide the right-hand side of the equation
1463:   using `TSSetRHSFunction()`.

1465:   Level: beginner

1467:   Notes:
1468:   The default is `TSRK3BS`, it can be changed with `TSRKSetType()` or -ts_rk_type

1470: .seealso: [](ch_ts), `TSCreate()`, `TS`, `TSRK`, `TSSetType()`, `TSRKSetType()`, `TSRKGetType()`, `TSRK2D`, `TSRK2E`, `TSRK3`,
1471:           `TSRK4`, `TSRK5`, `TSRKPRSSP2`, `TSRKBPR3`, `TSRKType`, `TSRKRegister()`, `TSRKSetMultirate()`, `TSRKGetMultirate()`, `TSType`
1472: M*/
1473: PETSC_EXTERN PetscErrorCode TSCreate_RK(TS ts)
1474: {
1475:   TS_RK *rk;

1477:   PetscFunctionBegin;
1478:   PetscCall(TSRKInitializePackage());

1480:   ts->ops->reset          = TSReset_RK;
1481:   ts->ops->destroy        = TSDestroy_RK;
1482:   ts->ops->view           = TSView_RK;
1483:   ts->ops->load           = TSLoad_RK;
1484:   ts->ops->setup          = TSSetUp_RK;
1485:   ts->ops->interpolate    = TSInterpolate_RK;
1486:   ts->ops->step           = TSStep_RK;
1487:   ts->ops->evaluatestep   = TSEvaluateStep_RK;
1488:   ts->ops->rollback       = TSRollBack_RK;
1489:   ts->ops->setfromoptions = TSSetFromOptions_RK;
1490:   ts->ops->getstages      = TSGetStages_RK;

1492:   ts->ops->snesfunction    = SNESTSFormFunction_RK;
1493:   ts->ops->snesjacobian    = SNESTSFormJacobian_RK;
1494:   ts->ops->adjointintegral = TSAdjointCostIntegral_RK;
1495:   ts->ops->adjointsetup    = TSAdjointSetUp_RK;
1496:   ts->ops->adjointstep     = TSAdjointStep_RK;
1497:   ts->ops->adjointreset    = TSAdjointReset_RK;

1499:   ts->ops->forwardintegral  = TSForwardCostIntegral_RK;
1500:   ts->ops->forwardsetup     = TSForwardSetUp_RK;
1501:   ts->ops->forwardreset     = TSForwardReset_RK;
1502:   ts->ops->forwardstep      = TSForwardStep_RK;
1503:   ts->ops->forwardgetstages = TSForwardGetStages_RK;

1505:   PetscCall(PetscNew(&rk));
1506:   ts->data = (void *)rk;

1508:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetOrder_C", TSRKGetOrder_RK));
1509:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetType_C", TSRKGetType_RK));
1510:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKSetType_C", TSRKSetType_RK));
1511:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetTableau_C", TSRKGetTableau_RK));
1512:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKSetMultirate_C", TSRKSetMultirate_RK));
1513:   PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSRKGetMultirate_C", TSRKGetMultirate_RK));

1515:   PetscCall(TSRKSetType(ts, TSRKDefault));
1516:   rk->dtratio = 1;
1517:   PetscFunctionReturn(PETSC_SUCCESS);
1518: }