Date: 2022-03-04T17:39:33+00:00

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:
     28: .     -ts_rk_type 1fe

     30:      Level: advanced

     32: .seealso: TSRK, TSRKType, TSRKSetType()
     33: M*/
     34: /*MC
     35:      TSRK2A - Second order RK scheme.

     37:      This method has two stages.

     39:      Options database:
     40: .     -ts_rk_type 2a

     42:      Level: advanced

     44: .seealso: TSRK, TSRKType, TSRKSetType()
     45: M*/
     46: /*MC
     47:      TSRK3 - Third order RK scheme.

     49:      This method has three stages.

     51:      Options database:
     52: .     -ts_rk_type 3

     54:      Level: advanced

     56: .seealso: TSRK, TSRKType, TSRKSetType()
     57: M*/
     58: /*MC
     59:      TSRK3BS - Third order RK scheme of Bogacki-Shampine with 2nd order embedded method.

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

     63:      Options database:
     64: .     -ts_rk_type 3bs

     66:      Level: advanced

     68:      References: https://doi.org/10.1016/0893-9659(89)90079-7

     70: .seealso: TSRK, TSRKType, TSRKSetType()
     71: M*/
     72: /*MC
     73:      TSRK4 - Fourth order RK scheme.

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

     77:      Options database:
     78: .     -ts_rk_type 4

     80:      Level: advanced

     82: .seealso: TSRK, TSRKType, TSRKSetType()
     83: M*/
     84: /*MC
     85:      TSRK5F - Fifth order Fehlberg RK scheme with a 4th order embedded method.

     87:      This method has six stages.

     89:      Options database:
     90: .     -ts_rk_type 5f

     92:      Level: advanced

     94: .seealso: TSRK, TSRKType, TSRKSetType()
     95: M*/
     96: /*MC
     97:      TSRK5DP - Fifth order Dormand-Prince RK scheme with the 4th order embedded method.

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

    101:      Options database:
    102: .     -ts_rk_type 5dp

    104:      Level: advanced

    106:      References: https://doi.org/10.1016/0771-050X(80)90013-3

    108: .seealso: TSRK, TSRKType, TSRKSetType()
    109: M*/
    110: /*MC
    111:      TSRK5BS - Fifth order Bogacki-Shampine RK scheme with 4th order embedded method.

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

    115:      Options database:
    116: .     -ts_rk_type 5bs

    118:      Level: advanced

    120:      References: https://doi.org/10.1016/0898-1221(96)00141-1

    122: .seealso: TSRK, TSRKType, TSRKSetType()
    123: M*/
    124: /*MC
    125:      TSRK6VR - Sixth order robust Verner RK scheme with fifth order embedded method.

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

    129:      Options database:
    130: .     -ts_rk_type 6vr

    132:      Level: advanced

    134:      References: http://people.math.sfu.ca/~jverner/RKV65.IIIXb.Robust.00010102836.081204.CoeffsOnlyRAT

    136: .seealso: TSRK, TSRKType, TSRKSetType()
    137: M*/
    138: /*MC
    139:      TSRK7VR - Seventh order robust Verner RK scheme with sixth order embedded method.

    141:      This method has ten stages.

    143:      Options database:
    144: .     -ts_rk_type 7vr

    146:      Level: advanced

    148:      References: http://people.math.sfu.ca/~jverner/RKV76.IIa.Robust.000027015646.081206.CoeffsOnlyRAT

    150: .seealso: TSRK, TSRKType, TSRKSetType()
    151: M*/
    152: /*MC
    153:      TSRK8VR - Eigth order robust Verner RK scheme with seventh order embedded method.

    155:      This method has thirteen stages.

    157:      Options database:
    158: .     -ts_rk_type 8vr

    160:      Level: advanced

    162:      References: http://people.math.sfu.ca/~jverner/RKV87.IIa.Robust.00000754677.081208.CoeffsOnlyRATandFLOAT

    164: .seealso: TSRK, TSRKType, TSRKSetType()
    165: M*/

    167: /*@C
    168:   TSRKRegisterAll - Registers all of the Runge-Kutta explicit methods in TSRK

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

    172:   Level: advanced

    174: .seealso:  TSRKRegisterDestroy()
    175: @*/
    176: PetscErrorCode TSRKRegisterAll(void)
    177: {

    181:   if (TSRKRegisterAllCalled) return(0);
    182:   TSRKRegisterAllCalled = PETSC_TRUE;

    184: #define RC PetscRealConstant
    185:   {
    186:     const PetscReal
    187:       A[1][1] = {{0}},
    188:       b[1]    = {RC(1.0)};
    189:     TSRKRegister(TSRK1FE,1,1,&A[0][0],b,NULL,NULL,0,NULL);
    190:   }
    191:   {
    192:     const PetscReal
    193:       A[2][2]   = {{0,0},
    194:                    {RC(1.0),0}},
    195:       b[2]      =  {RC(0.5),RC(0.5)},
    196:       bembed[2] =  {RC(1.0),0};
    197:     TSRKRegister(TSRK2A,2,2,&A[0][0],b,NULL,bembed,0,NULL);
    198:   }
    199:   {
    200:     const PetscReal
    201:       A[3][3] = {{0,0,0},
    202:                  {RC(2.0)/RC(3.0),0,0},
    203:                  {RC(-1.0)/RC(3.0),RC(1.0),0}},
    204:       b[3]    =  {RC(0.25),RC(0.5),RC(0.25)};
    205:     TSRKRegister(TSRK3,3,3,&A[0][0],b,NULL,NULL,0,NULL);
    206:   }
    207:   {
    208:     const PetscReal
    209:       A[4][4]   = {{0,0,0,0},
    210:                    {RC(1.0)/RC(2.0),0,0,0},
    211:                    {0,RC(3.0)/RC(4.0),0,0},
    212:                    {RC(2.0)/RC(9.0),RC(1.0)/RC(3.0),RC(4.0)/RC(9.0),0}},
    213:       b[4]      =  {RC(2.0)/RC(9.0),RC(1.0)/RC(3.0),RC(4.0)/RC(9.0),0},
    214:       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)};
    215:     TSRKRegister(TSRK3BS,3,4,&A[0][0],b,NULL,bembed,0,NULL);
    216:   }
    217:   {
    218:     const PetscReal
    219:       A[4][4] = {{0,0,0,0},
    220:                  {RC(0.5),0,0,0},
    221:                  {0,RC(0.5),0,0},
    222:                  {0,0,RC(1.0),0}},
    223:       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)};
    224:     TSRKRegister(TSRK4,4,4,&A[0][0],b,NULL,NULL,0,NULL);
    225:   }
    226:   {
    227:     const PetscReal
    228:       A[6][6]   = {{0,0,0,0,0,0},
    229:                    {RC(0.25),0,0,0,0,0},
    230:                    {RC(3.0)/RC(32.0),RC(9.0)/RC(32.0),0,0,0,0},
    231:                    {RC(1932.0)/RC(2197.0),RC(-7200.0)/RC(2197.0),RC(7296.0)/RC(2197.0),0,0,0},
    232:                    {RC(439.0)/RC(216.0),RC(-8.0),RC(3680.0)/RC(513.0),RC(-845.0)/RC(4104.0),0,0},
    233:                    {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}},
    234:       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)},
    235:       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};
    236:     TSRKRegister(TSRK5F,5,6,&A[0][0],b,NULL,bembed,0,NULL);
    237:   }
    238:   {
    239:     const PetscReal
    240:       A[7][7]       = {{0,0,0,0,0,0,0},
    241:                        {RC(1.0)/RC(5.0),0,0,0,0,0,0},
    242:                        {RC(3.0)/RC(40.0),RC(9.0)/RC(40.0),0,0,0,0,0},
    243:                        {RC(44.0)/RC(45.0),RC(-56.0)/RC(15.0),RC(32.0)/RC(9.0),0,0,0,0},
    244:                        {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},
    245:                        {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},
    246:                        {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}},
    247:       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},
    248:       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)},
    249:       binterp[7][5] = {{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)},
    250:                        {0,0,0,0,0},
    251:                        {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)},
    252:                        {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)},
    253:                        {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)},
    254:                        {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)},
    255:                        {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)}};
    256:       TSRKRegister(TSRK5DP,5,7,&A[0][0],b,NULL,bembed,5,binterp[0]);
    257:   }
    258:   {
    259:     const PetscReal
    260:       A[8][8]   = {{0,0,0,0,0,0,0,0},
    261:                    {RC(1.0)/RC(6.0),0,0,0,0,0,0,0},
    262:                    {RC(2.0)/RC(27.0),RC(4.0)/RC(27.0),0,0,0,0,0,0},
    263:                    {RC(183.0)/RC(1372.0),RC(-162.0)/RC(343.0),RC(1053.0)/RC(1372.0),0,0,0,0,0},
    264:                    {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},
    265:                    {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},
    266:                    {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},
    267:                    {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}},
    268:       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},
    269:       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)};
    270:     TSRKRegister(TSRK5BS,5,8,&A[0][0],b,NULL,bembed,0,NULL);
    271:   }
    272:   {
    273:     const PetscReal
    274:       A[9][9]   = {{0,0,0,0,0,0,0,0,0},
    275:                    {RC(1.8000000000000000000000000000000000000000e-01),0,0,0,0,0,0,0,0},
    276:                    {RC(8.9506172839506172839506172839506172839506e-02),RC(7.7160493827160493827160493827160493827160e-02),0,0,0,0,0,0,0},
    277:                    {RC(6.2500000000000000000000000000000000000000e-02),0,RC(1.8750000000000000000000000000000000000000e-01),0,0,0,0,0,0},
    278:                    {RC(3.1651600000000000000000000000000000000000e-01),0,RC(-1.0449480000000000000000000000000000000000e+00),RC(1.2584320000000000000000000000000000000000e+00),0,0,0,0,0},
    279:                    {RC(2.7232612736485626257225065566674305502508e-01),0,RC(-8.2513360323886639676113360323886639676113e-01),RC(1.0480917678812415654520917678812415654521e+00),RC(1.0471570799276856873679117969088177628396e-01),0,0,0,0},
    280:                    {RC(-1.6699418599716514314329607278961797333198e-01),0,RC(6.3170850202429149797570850202429149797571e-01),RC(1.7461044552773876082146758838488161796432e-01),RC(-1.0665356459086066122525194734018680677781e+00),RC(1.2272108843537414965986394557823129251701e+00),0,0,0},
    281:                    {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},
    282:                    {RC(7.6388888888888888888888888888888888888889e-02),0,0,RC(3.6940836940836940836940836940836940836941e-01),0,RC(2.4801587301587301587301587301587301587302e-01),RC(2.3674242424242424242424242424242424242424e-01),RC(6.9444444444444444444444444444444444444444e-02),0}},
    283:       b[9]      =  {RC(7.6388888888888888888888888888888888888889e-02),0,0,RC(3.6940836940836940836940836940836940836941e-01),0,RC(2.4801587301587301587301587301587301587302e-01),RC(2.3674242424242424242424242424242424242424e-01),RC(6.9444444444444444444444444444444444444444e-02),0},
    284:       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)};
    285:     TSRKRegister(TSRK6VR,6,9,&A[0][0],b,NULL,bembed,0,NULL);
    286:   }
    287:   {
    288:     const PetscReal
    289:       A[10][10]  = {{0,0,0,0,0,0,0,0,0,0},
    290:                     {RC(5.0000000000000000000000000000000000000000e-03),0,0,0,0,0,0,0,0,0},
    291:                     {RC(-1.0767901234567901234567901234567901234568e+00),RC(1.1856790123456790123456790123456790123457e+00),0,0,0,0,0,0,0,0},
    292:                     {RC(4.0833333333333333333333333333333333333333e-02),0,RC(1.2250000000000000000000000000000000000000e-01),0,0,0,0,0,0,0},
    293:                     {RC(6.3607142857142857142857142857142857142857e-01),0,RC(-2.4444642857142857142857142857142857142857e+00),RC(2.2633928571428571428571428571428571428571e+00),0,0,0,0,0,0},
    294:                     {RC(-2.5351211079349245229256383554660215487207e+00),0,RC(1.0299374654449267920438514460756024913612e+01),RC(-7.9513032885990579949493217458266876536482e+00),RC(7.9301148923100592201226014271115261823800e-01),0,0,0,0,0},
    295:                     {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},
    296:                     {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},
    297:                     {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},
    298:                     {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}},
    299:       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},
    300:       bembed[10] =  {RC(4.7485247699299631037531273805727961552268e-02),0,0,RC(2.5599412588690633297154918245905393870497e-01),RC(2.7058478081067688722530891099268135732387e-01),RC(1.2505618684425992913638822323746917920448e-01),RC(2.5204468723743860507184043820197442562182e-01),0,0,RC(4.8834971521418614557381971303093137592592e-02)};
    301:     TSRKRegister(TSRK7VR,7,10,&A[0][0],b,NULL,bembed,0,NULL);
    302:   }
    303:   {
    304:     const PetscReal
    305:       A[13][13]  = {{0,0,0,0,0,0,0,0,0,0,0,0,0},
    306:                     {RC(2.5000000000000000000000000000000000000000e-01),0,0,0,0,0,0,0,0,0,0,0,0},
    307:                     {RC(8.7400846504915232052686327594877411977046e-02),RC(2.5487604938654321753087950620345685135815e-02),0,0,0,0,0,0,0,0,0,0,0},
    308:                     {RC(4.2333169291338582677165354330708661417323e-02),0,RC(1.2699950787401574803149606299212598425197e-01),0,0,0,0,0,0,0,0,0,0},
    309:                     {RC(4.2609505888742261494881445237572274090942e-01),0,RC(-1.5987952846591523265427733230657181117089e+00),RC(1.5967002257717297115939588706899953707994e+00),0,0,0,0,0,0,0,0,0},
    310:                     {RC(5.0719337296713929515090618138513639239329e-02),0,0,RC(2.5433377264600407582754714408877778031369e-01),RC(2.0394689005728199465736223777270858044698e-01),0,0,0,0,0,0,0,0},
    311:                     {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},
    312:                     {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},
    313:                     {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},
    314:                     {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},
    315:                     {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},
    316:                     {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},
    317:                     {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}},
    318:       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},
    319:       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)};
    320:     TSRKRegister(TSRK8VR,8,13,&A[0][0],b,NULL,bembed,0,NULL);
    321:   }
    322: #undef RC
    323:   return(0);
    324: }

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

    329:    Not Collective

    331:    Level: advanced

    333: .seealso: TSRKRegister(), TSRKRegisterAll()
    334: @*/
    335: PetscErrorCode TSRKRegisterDestroy(void)
    336: {
    338:   RKTableauLink  link;

    341:   while ((link = RKTableauList)) {
    342:     RKTableau t = &link->tab;
    343:     RKTableauList = link->next;
    344:     PetscFree3(t->A,t->b,t->c);
    345:     PetscFree(t->bembed);
    346:     PetscFree(t->binterp);
    347:     PetscFree(t->name);
    348:     PetscFree(link);
    349:   }
    350:   TSRKRegisterAllCalled = PETSC_FALSE;
    351:   return(0);
    352: }

    354: /*@C
    355:   TSRKInitializePackage - This function initializes everything in the TSRK package. It is called
    356:   from TSInitializePackage().

    358:   Level: developer

    360: .seealso: PetscInitialize()
    361: @*/
    362: PetscErrorCode TSRKInitializePackage(void)
    363: {

    367:   if (TSRKPackageInitialized) return(0);
    368:   TSRKPackageInitialized = PETSC_TRUE;
    369:   TSRKRegisterAll();
    370:   PetscRegisterFinalize(TSRKFinalizePackage);
    371:   return(0);
    372: }

    374: /*@C
    375:   TSRKFinalizePackage - This function destroys everything in the TSRK package. It is
    376:   called from PetscFinalize().

    378:   Level: developer

    380: .seealso: PetscFinalize()
    381: @*/
    382: PetscErrorCode TSRKFinalizePackage(void)
    383: {

    387:   TSRKPackageInitialized = PETSC_FALSE;
    388:   TSRKRegisterDestroy();
    389:   return(0);
    390: }

    392: /*@C
    393:    TSRKRegister - register an RK scheme by providing the entries in the Butcher tableau and optionally embedded approximations and interpolation

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

    397:    Input Parameters:
    398: +  name - identifier for method
    399: .  order - approximation order of method
    400: .  s - number of stages, this is the dimension of the matrices below
    401: .  A - stage coefficients (dimension s*s, row-major)
    402: .  b - step completion table (dimension s; NULL to use last row of A)
    403: .  c - abscissa (dimension s; NULL to use row sums of A)
    404: .  bembed - completion table for embedded method (dimension s; NULL if not available)
    405: .  p - Order of the interpolation scheme, equal to the number of columns of binterp
    406: -  binterp - Coefficients of the interpolation formula (dimension s*p; NULL to reuse b with p=1)

    408:    Notes:
    409:    Several RK methods are provided, this function is only needed to create new methods.

    411:    Level: advanced

    413: .seealso: TSRK
    414: @*/
    415: PetscErrorCode TSRKRegister(TSRKType name,PetscInt order,PetscInt s,
    416:                             const PetscReal A[],const PetscReal b[],const PetscReal c[],
    417:                             const PetscReal bembed[],PetscInt p,const PetscReal binterp[])
    418: {
    419:   PetscErrorCode  ierr;
    420:   RKTableauLink   link;
    421:   RKTableau       t;
    422:   PetscInt        i,j;

    432:   TSRKInitializePackage();
    433:   PetscNew(&link);
    434:   t = &link->tab;

    436:   PetscStrallocpy(name,&t->name);
    437:   t->order = order;
    438:   t->s = s;
    439:   PetscMalloc3(s*s,&t->A,s,&t->b,s,&t->c);
    440:   PetscArraycpy(t->A,A,s*s);
    441:   if (b)  { PetscArraycpy(t->b,b,s); }
    442:   else for (i=0; i<s; i++) t->b[i] = A[(s-1)*s+i];
    443:   if (c)  { PetscArraycpy(t->c,c,s); }
    444:   else for (i=0; i<s; i++) for (j=0,t->c[i]=0; j<s; j++) t->c[i] += A[i*s+j];
    445:   t->FSAL = PETSC_TRUE;
    446:   for (i=0; i<s; i++) if (t->A[(s-1)*s+i] != t->b[i]) t->FSAL = PETSC_FALSE;

    448:   if (bembed) {
    449:     PetscMalloc1(s,&t->bembed);
    450:     PetscArraycpy(t->bembed,bembed,s);
    451:   }

    453:   if (!binterp) { p = 1; binterp = t->b; }
    454:   t->p = p;
    455:   PetscMalloc1(s*p,&t->binterp);
    456:   PetscArraycpy(t->binterp,binterp,s*p);

    458:   link->next = RKTableauList;
    459:   RKTableauList = link;
    460:   return(0);
    461: }

    463: PetscErrorCode TSRKGetTableau_RK(TS ts, PetscInt *s, const PetscReal **A, const PetscReal **b, const PetscReal **c, const PetscReal **bembed,
    464:                                         PetscInt *p, const PetscReal **binterp, PetscBool *FSAL)
    465: {
    466:   TS_RK     *rk   = (TS_RK*)ts->data;
    467:   RKTableau tab  = rk->tableau;

    470:   if (s) *s = tab->s;
    471:   if (A) *A = tab->A;
    472:   if (b) *b = tab->b;
    473:   if (c) *c = tab->c;
    474:   if (bembed) *bembed = tab->bembed;
    475:   if (p) *p = tab->p;
    476:   if (binterp) *binterp = tab->binterp;
    477:   if (FSAL) *FSAL = tab->FSAL;
    478:   return(0);
    479: }

    481: /*@C
    482:    TSRKGetTableau - Get info on the RK tableau

    484:    Not Collective

    486:    Input Parameter:
    487: .  ts - timestepping context

    489:    Output Parameters:
    490: +  s - number of stages, this is the dimension of the matrices below
    491: .  A - stage coefficients (dimension s*s, row-major)
    492: .  b - step completion table (dimension s)
    493: .  c - abscissa (dimension s)
    494: .  bembed - completion table for embedded method (dimension s; NULL if not available)
    495: .  p - Order of the interpolation scheme, equal to the number of columns of binterp
    496: .  binterp - Coefficients of the interpolation formula (dimension s*p)
    497: -  FSAL - wheather or not the scheme has the First Same As Last property

    499:    Level: developer

    501: .seealso: TSRK
    502: @*/
    503: PetscErrorCode TSRKGetTableau(TS ts, PetscInt *s, const PetscReal **A, const PetscReal **b, const PetscReal **c, const PetscReal **bembed,
    504:                                      PetscInt *p, const PetscReal **binterp, PetscBool *FSAL)
    505: {

    510:   PetscUseMethod(ts,"TSRKGetTableau_C",(TS,PetscInt*,const PetscReal**,const PetscReal**,const PetscReal**,const PetscReal**,
    511:                                                   PetscInt*,const PetscReal**,PetscBool*),(ts,s,A,b,c,bembed,p,binterp,FSAL));
    512:   return(0);
    513: }

    515: /*
    516:  This is for single-step RK method
    517:  The step completion formula is

    519:  x1 = x0 + h b^T YdotRHS

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

    525:  x1e = x0 + h be^T YdotRHS
    526:      = x1 - h b^T YdotRHS + h be^T YdotRHS
    527:      = x1 + h (be - b)^T YdotRHS

    529:  so we can evaluate the method with different order even after the step has been optimistically completed.
    530: */
    531: static PetscErrorCode TSEvaluateStep_RK(TS ts,PetscInt order,Vec X,PetscBool *done)
    532: {
    533:   TS_RK          *rk   = (TS_RK*)ts->data;
    534:   RKTableau      tab  = rk->tableau;
    535:   PetscScalar    *w    = rk->work;
    536:   PetscReal      h;
    537:   PetscInt       s    = tab->s,j;

    541:   switch (rk->status) {
    542:   case TS_STEP_INCOMPLETE:
    543:   case TS_STEP_PENDING:
    544:     h = ts->time_step; break;
    545:   case TS_STEP_COMPLETE:
    546:     h = ts->ptime - ts->ptime_prev; break;
    547:   default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
    548:   }
    549:   if (order == tab->order) {
    550:     if (rk->status == TS_STEP_INCOMPLETE) {
    551:       VecCopy(ts->vec_sol,X);
    552:       for (j=0; j<s; j++) w[j] = h*tab->b[j]/rk->dtratio;
    553:       VecMAXPY(X,s,w,rk->YdotRHS);
    554:     } else {VecCopy(ts->vec_sol,X);}
    555:     return(0);
    556:   } else if (order == tab->order-1) {
    557:     if (!tab->bembed) goto unavailable;
    558:     if (rk->status == TS_STEP_INCOMPLETE) { /*Complete with the embedded method (be)*/
    559:       VecCopy(ts->vec_sol,X);
    560:       for (j=0; j<s; j++) w[j] = h*tab->bembed[j];
    561:       VecMAXPY(X,s,w,rk->YdotRHS);
    562:     } else {  /*Rollback and re-complete using (be-b) */
    563:       VecCopy(ts->vec_sol,X);
    564:       for (j=0; j<s; j++) w[j] = h*(tab->bembed[j] - tab->b[j]);
    565:       VecMAXPY(X,s,w,rk->YdotRHS);
    566:     }
    567:     if (done) *done = PETSC_TRUE;
    568:     return(0);
    569:   }
    570: unavailable:
    571:   if (done) *done = PETSC_FALSE;
    572:   else SETERRQ3(PetscObjectComm((PetscObject)ts),PETSC_ERR_SUP,"RK '%s' of order %D cannot evaluate step at order %D. Consider using -ts_adapt_type none or a different method that has an embedded estimate.",tab->name,tab->order,order);
    573:   return(0);
    574: }

    576: static PetscErrorCode TSForwardCostIntegral_RK(TS ts)
    577: {
    578:   TS_RK           *rk = (TS_RK*)ts->data;
    579:   TS              quadts = ts->quadraturets;
    580:   RKTableau       tab = rk->tableau;
    581:   const PetscInt  s = tab->s;
    582:   const PetscReal *b = tab->b,*c = tab->c;
    583:   Vec             *Y = rk->Y;
    584:   PetscInt        i;
    585:   PetscErrorCode  ierr;

    588:   /* No need to backup quadts->vec_sol since it can be reverted in TSRollBack_RK */
    589:   for (i=s-1; i>=0; i--) {
    590:     /* Evolve quadrature TS solution to compute integrals */
    591:     TSComputeRHSFunction(quadts,rk->ptime+rk->time_step*c[i],Y[i],ts->vec_costintegrand);
    592:     VecAXPY(quadts->vec_sol,rk->time_step*b[i],ts->vec_costintegrand);
    593:   }
    594:   return(0);
    595: }

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

    609:   for (i=s-1; i>=0; i--) {
    610:     /* Evolve quadrature TS solution to compute integrals */
    611:     TSComputeRHSFunction(quadts,ts->ptime+ts->time_step*(1.0-c[i]),Y[i],ts->vec_costintegrand);
    612:     VecAXPY(quadts->vec_sol,-ts->time_step*b[i],ts->vec_costintegrand);
    613:   }
    614:   return(0);
    615: }

    617: static PetscErrorCode TSRollBack_RK(TS ts)
    618: {
    619:   TS_RK           *rk = (TS_RK*)ts->data;
    620:   TS              quadts = ts->quadraturets;
    621:   RKTableau       tab = rk->tableau;
    622:   const PetscInt  s  = tab->s;
    623:   const PetscReal *b = tab->b,*c = tab->c;
    624:   PetscScalar     *w = rk->work;
    625:   Vec             *Y = rk->Y,*YdotRHS = rk->YdotRHS;
    626:   PetscInt        j;
    627:   PetscReal       h;
    628:   PetscErrorCode  ierr;

    631:   switch (rk->status) {
    632:   case TS_STEP_INCOMPLETE:
    633:   case TS_STEP_PENDING:
    634:     h = ts->time_step; break;
    635:   case TS_STEP_COMPLETE:
    636:     h = ts->ptime - ts->ptime_prev; break;
    637:   default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
    638:   }
    639:   for (j=0; j<s; j++) w[j] = -h*b[j];
    640:   VecMAXPY(ts->vec_sol,s,w,YdotRHS);
    641:   if (quadts && ts->costintegralfwd) {
    642:     for (j=0; j<s; j++) {
    643:       /* Revert the quadrature TS solution */
    644:       TSComputeRHSFunction(quadts,rk->ptime+h*c[j],Y[j],ts->vec_costintegrand);
    645:       VecAXPY(quadts->vec_sol,-h*b[j],ts->vec_costintegrand);
    646:     }
    647:   }
    648:   return(0);
    649: }

    651: static PetscErrorCode TSForwardStep_RK(TS ts)
    652: {
    653:   TS_RK           *rk = (TS_RK*)ts->data;
    654:   RKTableau       tab = rk->tableau;
    655:   Mat             J,*MatsFwdSensipTemp = rk->MatsFwdSensipTemp;
    656:   const PetscInt  s = tab->s;
    657:   const PetscReal *A = tab->A,*c = tab->c,*b = tab->b;
    658:   Vec             *Y = rk->Y;
    659:   PetscInt        i,j;
    660:   PetscReal       stage_time,h = ts->time_step;
    661:   PetscBool       zero;
    662:   PetscErrorCode  ierr;

    665:   MatCopy(ts->mat_sensip,rk->MatFwdSensip0,SAME_NONZERO_PATTERN);
    666:   TSGetRHSJacobian(ts,&J,NULL,NULL,NULL);

    668:   for (i=0; i<s; i++) {
    669:     stage_time = ts->ptime + h*c[i];
    670:     zero = PETSC_FALSE;
    671:     if (b[i] == 0 && i == s-1) zero = PETSC_TRUE;
    672:     /* TLM Stage values */
    673:     if (!i) {
    674:       MatCopy(ts->mat_sensip,rk->MatsFwdStageSensip[i],SAME_NONZERO_PATTERN);
    675:     } else if (!zero) {
    676:       MatZeroEntries(rk->MatsFwdStageSensip[i]);
    677:       for (j=0; j<i; j++) {
    678:         MatAXPY(rk->MatsFwdStageSensip[i],h*A[i*s+j],MatsFwdSensipTemp[j],SAME_NONZERO_PATTERN);
    679:       }
    680:       MatAXPY(rk->MatsFwdStageSensip[i],1.,ts->mat_sensip,SAME_NONZERO_PATTERN);
    681:     } else {
    682:       MatZeroEntries(rk->MatsFwdStageSensip[i]);
    683:     }

    685:     TSComputeRHSJacobian(ts,stage_time,Y[i],J,J);
    686:     MatMatMult(J,rk->MatsFwdStageSensip[i],MAT_REUSE_MATRIX,PETSC_DEFAULT,&MatsFwdSensipTemp[i]);
    687:     if (ts->Jacprhs) {
    688:       TSComputeRHSJacobianP(ts,stage_time,Y[i],ts->Jacprhs); /* get f_p */
    689:       if (ts->vecs_sensi2p) { /* TLM used for 2nd-order adjoint */
    690:         PetscScalar *xarr;
    691:         MatDenseGetColumn(MatsFwdSensipTemp[i],0,&xarr);
    692:         VecPlaceArray(rk->VecDeltaFwdSensipCol,xarr);
    693:         MatMultAdd(ts->Jacprhs,ts->vec_dir,rk->VecDeltaFwdSensipCol,rk->VecDeltaFwdSensipCol);
    694:         VecResetArray(rk->VecDeltaFwdSensipCol);
    695:         MatDenseRestoreColumn(MatsFwdSensipTemp[i],&xarr);
    696:       } else {
    697:         MatAXPY(MatsFwdSensipTemp[i],1.,ts->Jacprhs,SUBSET_NONZERO_PATTERN);
    698:       }
    699:     }
    700:   }

    702:   for (i=0; i<s; i++) {
    703:     MatAXPY(ts->mat_sensip,h*b[i],rk->MatsFwdSensipTemp[i],SAME_NONZERO_PATTERN);
    704:   }
    705:   rk->status = TS_STEP_COMPLETE;
    706:   return(0);
    707: }

    709: static PetscErrorCode TSForwardGetStages_RK(TS ts,PetscInt *ns,Mat **stagesensip)
    710: {
    711:   TS_RK     *rk = (TS_RK*)ts->data;
    712:   RKTableau tab  = rk->tableau;

    715:   if (ns) *ns = tab->s;
    716:   if (stagesensip) *stagesensip = rk->MatsFwdStageSensip;
    717:   return(0);
    718: }

    720: static PetscErrorCode TSForwardSetUp_RK(TS ts)
    721: {
    722:   TS_RK          *rk = (TS_RK*)ts->data;
    723:   RKTableau      tab  = rk->tableau;
    724:   PetscInt       i;

    728:   /* backup sensitivity results for roll-backs */
    729:   MatDuplicate(ts->mat_sensip,MAT_DO_NOT_COPY_VALUES,&rk->MatFwdSensip0);

    731:   PetscMalloc1(tab->s,&rk->MatsFwdStageSensip);
    732:   PetscMalloc1(tab->s,&rk->MatsFwdSensipTemp);
    733:   for (i=0; i<tab->s; i++) {
    734:     MatDuplicate(ts->mat_sensip,MAT_DO_NOT_COPY_VALUES,&rk->MatsFwdStageSensip[i]);
    735:     MatDuplicate(ts->mat_sensip,MAT_DO_NOT_COPY_VALUES,&rk->MatsFwdSensipTemp[i]);
    736:   }
    737:   VecDuplicate(ts->vec_sol,&rk->VecDeltaFwdSensipCol);
    738:   return(0);
    739: }

    741: static PetscErrorCode TSForwardReset_RK(TS ts)
    742: {
    743:   TS_RK          *rk = (TS_RK*)ts->data;
    744:   RKTableau      tab  = rk->tableau;
    745:   PetscInt       i;

    749:   MatDestroy(&rk->MatFwdSensip0);
    750:   if (rk->MatsFwdStageSensip) {
    751:     for (i=0; i<tab->s; i++) {
    752:       MatDestroy(&rk->MatsFwdStageSensip[i]);
    753:     }
    754:     PetscFree(rk->MatsFwdStageSensip);
    755:   }
    756:   if (rk->MatsFwdSensipTemp) {
    757:     for (i=0; i<tab->s; i++) {
    758:       MatDestroy(&rk->MatsFwdSensipTemp[i]);
    759:     }
    760:     PetscFree(rk->MatsFwdSensipTemp);
    761:   }
    762:   VecDestroy(&rk->VecDeltaFwdSensipCol);
    763:   return(0);
    764: }

    766: static PetscErrorCode TSStep_RK(TS ts)
    767: {
    768:   TS_RK           *rk  = (TS_RK*)ts->data;
    769:   RKTableau       tab  = rk->tableau;
    770:   const PetscInt  s = tab->s;
    771:   const PetscReal *A = tab->A,*c = tab->c;
    772:   PetscScalar     *w = rk->work;
    773:   Vec             *Y = rk->Y,*YdotRHS = rk->YdotRHS;
    774:   PetscBool       FSAL = tab->FSAL;
    775:   TSAdapt         adapt;
    776:   PetscInt        i,j;
    777:   PetscInt        rejections = 0;
    778:   PetscBool       stageok,accept = PETSC_TRUE;
    779:   PetscReal       next_time_step = ts->time_step;
    780:   PetscErrorCode  ierr;

    783:   if (ts->steprollback || ts->steprestart) FSAL = PETSC_FALSE;
    784:   if (FSAL) { VecCopy(YdotRHS[s-1],YdotRHS[0]); }

    786:   rk->status = TS_STEP_INCOMPLETE;
    787:   while (!ts->reason && rk->status != TS_STEP_COMPLETE) {
    788:     PetscReal t = ts->ptime;
    789:     PetscReal h = ts->time_step;
    790:     for (i=0; i<s; i++) {
    791:       rk->stage_time = t + h*c[i];
    792:       TSPreStage(ts,rk->stage_time);
    793:       VecCopy(ts->vec_sol,Y[i]);
    794:       for (j=0; j<i; j++) w[j] = h*A[i*s+j];
    795:       VecMAXPY(Y[i],i,w,YdotRHS);
    796:       TSPostStage(ts,rk->stage_time,i,Y);
    797:       TSGetAdapt(ts,&adapt);
    798:       TSAdaptCheckStage(adapt,ts,rk->stage_time,Y[i],&stageok);
    799:       if (!stageok) goto reject_step;
    800:       if (FSAL && !i) continue;
    801:       TSComputeRHSFunction(ts,t+h*c[i],Y[i],YdotRHS[i]);
    802:     }

    804:     rk->status = TS_STEP_INCOMPLETE;
    805:     TSEvaluateStep(ts,tab->order,ts->vec_sol,NULL);
    806:     rk->status = TS_STEP_PENDING;
    807:     TSGetAdapt(ts,&adapt);
    808:     TSAdaptCandidatesClear(adapt);
    809:     TSAdaptCandidateAdd(adapt,tab->name,tab->order,1,tab->ccfl,(PetscReal)tab->s,PETSC_TRUE);
    810:     TSAdaptChoose(adapt,ts,ts->time_step,NULL,&next_time_step,&accept);
    811:     rk->status = accept ? TS_STEP_COMPLETE : TS_STEP_INCOMPLETE;
    812:     if (!accept) { /* Roll back the current step */
    813:       TSRollBack_RK(ts);
    814:       ts->time_step = next_time_step;
    815:       goto reject_step;
    816:     }

    818:     if (ts->costintegralfwd) { /* Save the info for the later use in cost integral evaluation */
    819:       rk->ptime     = ts->ptime;
    820:       rk->time_step = ts->time_step;
    821:     }

    823:     ts->ptime += ts->time_step;
    824:     ts->time_step = next_time_step;
    825:     break;

    827:     reject_step:
    828:     ts->reject++; accept = PETSC_FALSE;
    829:     if (!ts->reason && ++rejections > ts->max_reject && ts->max_reject >= 0) {
    830:       ts->reason = TS_DIVERGED_STEP_REJECTED;
    831:       PetscInfo2(ts,"Step=%D, step rejections %D greater than current TS allowed, stopping solve\n",ts->steps,rejections);
    832:     }
    833:   }
    834:   return(0);
    835: }

    837: static PetscErrorCode TSAdjointSetUp_RK(TS ts)
    838: {
    839:   TS_RK          *rk  = (TS_RK*)ts->data;
    840:   RKTableau      tab = rk->tableau;
    841:   PetscInt       s   = tab->s;

    845:   if (ts->adjointsetupcalled++) return(0);
    846:   VecDuplicateVecs(ts->vecs_sensi[0],s*ts->numcost,&rk->VecsDeltaLam);
    847:   VecDuplicateVecs(ts->vecs_sensi[0],ts->numcost,&rk->VecsSensiTemp);
    848:   if (ts->vecs_sensip) {
    849:     VecDuplicate(ts->vecs_sensip[0],&rk->VecDeltaMu);
    850:   }
    851:   if (ts->vecs_sensi2) {
    852:     VecDuplicateVecs(ts->vecs_sensi[0],s*ts->numcost,&rk->VecsDeltaLam2);
    853:     VecDuplicateVecs(ts->vecs_sensi2[0],ts->numcost,&rk->VecsSensi2Temp);
    854:   }
    855:   if (ts->vecs_sensi2p) {
    856:     VecDuplicate(ts->vecs_sensi2p[0],&rk->VecDeltaMu2);
    857:   }
    858:   return(0);
    859: }

    861: /*
    862:   Assumptions:
    863:     - TSStep_RK() always evaluates the step with b, not bembed.
    864: */
    865: static PetscErrorCode TSAdjointStep_RK(TS ts)
    866: {
    867:   TS_RK            *rk = (TS_RK*)ts->data;
    868:   TS               quadts = ts->quadraturets;
    869:   RKTableau        tab = rk->tableau;
    870:   Mat              J,Jpre,Jquad;
    871:   const PetscInt   s = tab->s;
    872:   const PetscReal  *A = tab->A,*b = tab->b,*c = tab->c;
    873:   PetscScalar      *w = rk->work,*xarr;
    874:   Vec              *Y = rk->Y,*VecsDeltaLam = rk->VecsDeltaLam,VecDeltaMu = rk->VecDeltaMu,*VecsSensiTemp = rk->VecsSensiTemp;
    875:   Vec              *VecsDeltaLam2 = rk->VecsDeltaLam2,VecDeltaMu2 = rk->VecDeltaMu2,*VecsSensi2Temp = rk->VecsSensi2Temp;
    876:   Vec              VecDRDUTransCol = ts->vec_drdu_col,VecDRDPTransCol = ts->vec_drdp_col;
    877:   PetscInt         i,j,nadj;
    878:   PetscReal        t = ts->ptime;
    879:   PetscReal        h = ts->time_step;
    880:   PetscErrorCode   ierr;

    883:   rk->status = TS_STEP_INCOMPLETE;

    885:   TSGetRHSJacobian(ts,&J,&Jpre,NULL,NULL);
    886:   if (quadts) {
    887:     TSGetRHSJacobian(quadts,&Jquad,NULL,NULL,NULL);
    888:   }
    889:   for (i=s-1; i>=0; i--) {
    890:     if (tab->FSAL && i == s-1) {
    891:       /* VecsDeltaLam[nadj*s+s-1] are initialized with zeros and the values never change.*/
    892:       continue;
    893:     }
    894:     rk->stage_time = t + h*(1.0-c[i]);
    895:     TSComputeSNESJacobian(ts,Y[i],J,Jpre);
    896:     if (quadts) {
    897:       TSComputeRHSJacobian(quadts,rk->stage_time,Y[i],Jquad,Jquad); /* get r_u^T */
    898:     }
    899:     if (ts->vecs_sensip) {
    900:       TSComputeRHSJacobianP(ts,rk->stage_time,Y[i],ts->Jacprhs); /* get f_p */
    901:       if (quadts) {
    902:         TSComputeRHSJacobianP(quadts,rk->stage_time,Y[i],quadts->Jacprhs); /* get f_p for the quadrature */
    903:       }
    904:     }

    906:     if (b[i]) {
    907:       for (j=i+1; j<s; j++) w[j-i-1] = A[j*s+i]/b[i]; /* coefficients for computing VecsSensiTemp */
    908:     } else {
    909:       for (j=i+1; j<s; j++) w[j-i-1] = A[j*s+i]; /* coefficients for computing VecsSensiTemp */
    910:     }

    912:     for (nadj=0; nadj<ts->numcost; nadj++) {
    913:       /* Stage values of lambda */
    914:       if (b[i]) {
    915:         /* lambda_{n+1} + \sum_{j=i+1}^s a_{ji}/b[i]*lambda_{s,j} */
    916:         VecCopy(ts->vecs_sensi[nadj],VecsSensiTemp[nadj]); /* VecDeltaLam is an vec array of size s by numcost */
    917:         VecMAXPY(VecsSensiTemp[nadj],s-i-1,w,&VecsDeltaLam[nadj*s+i+1]);
    918:         MatMultTranspose(J,VecsSensiTemp[nadj],VecsDeltaLam[nadj*s+i]); /* VecsSensiTemp will be reused by 2nd-order adjoint */
    919:         VecScale(VecsDeltaLam[nadj*s+i],-h*b[i]);
    920:         if (quadts) {
    921:           MatDenseGetColumn(Jquad,nadj,&xarr);
    922:           VecPlaceArray(VecDRDUTransCol,xarr);
    923:           VecAXPY(VecsDeltaLam[nadj*s+i],-h*b[i],VecDRDUTransCol);
    924:           VecResetArray(VecDRDUTransCol);
    925:           MatDenseRestoreColumn(Jquad,&xarr);
    926:         }
    927:       } else {
    928:         /* \sum_{j=i+1}^s a_{ji}*lambda_{s,j} */
    929:         VecSet(VecsSensiTemp[nadj],0);
    930:         VecMAXPY(VecsSensiTemp[nadj],s-i-1,w,&VecsDeltaLam[nadj*s+i+1]);
    931:         MatMultTranspose(J,VecsSensiTemp[nadj],VecsDeltaLam[nadj*s+i]);
    932:         VecScale(VecsDeltaLam[nadj*s+i],-h);
    933:       }

    935:       /* Stage values of mu */
    936:       if (ts->vecs_sensip) {
    937:         if (b[i]) {
    938:           MatMultTranspose(ts->Jacprhs,VecsSensiTemp[nadj],VecDeltaMu);
    939:           VecScale(VecDeltaMu,-h*b[i]);
    940:           if (quadts) {
    941:             MatDenseGetColumn(quadts->Jacprhs,nadj,&xarr);
    942:             VecPlaceArray(VecDRDPTransCol,xarr);
    943:             VecAXPY(VecDeltaMu,-h*b[i],VecDRDPTransCol);
    944:             VecResetArray(VecDRDPTransCol);
    945:             MatDenseRestoreColumn(quadts->Jacprhs,&xarr);
    946:           }
    947:         } else {
    948:           VecScale(VecDeltaMu,-h);
    949:         }
    950:         VecAXPY(ts->vecs_sensip[nadj],1.,VecDeltaMu); /* update sensip for each stage */
    951:       }
    952:     }

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

    987:       for (nadj=0; nadj<ts->numcost; nadj++) {
    988:         /* Stage values of lambda */
    989:         if (b[i]) {
    990:           /* J_i^T*(Lambda_{n+1}+\sum_{j=i+1}^s a_{ji}/b_i*Lambda_{s,j} */
    991:           VecCopy(ts->vecs_sensi2[nadj],VecsSensi2Temp[nadj]);
    992:           VecMAXPY(VecsSensi2Temp[nadj],s-i-1,w,&VecsDeltaLam2[nadj*s+i+1]);
    993:           MatMultTranspose(J,VecsSensi2Temp[nadj],VecsDeltaLam2[nadj*s+i]);
    994:           VecScale(VecsDeltaLam2[nadj*s+i],-h*b[i]);
    995:           VecAXPY(VecsDeltaLam2[nadj*s+i],-h*b[i],ts->vecs_guu[nadj]);
    996:           if (ts->vecs_sensip) {
    997:             VecAXPY(VecsDeltaLam2[nadj*s+i],-h*b[i],ts->vecs_gup[nadj]);
    998:           }
    999:         } else {
    1000:           /* \sum_{j=i+1}^s a_{ji}*Lambda_{s,j} */
    1001:           VecSet(VecsDeltaLam2[nadj*s+i],0);
    1002:           VecMAXPY(VecsSensi2Temp[nadj],s-i-1,w,&VecsDeltaLam2[nadj*s+i+1]);
    1003:           MatMultTranspose(J,VecsSensi2Temp[nadj],VecsDeltaLam2[nadj*s+i]);
    1004:           VecScale(VecsDeltaLam2[nadj*s+i],-h);
    1005:           VecAXPY(VecsDeltaLam2[nadj*s+i],-h,ts->vecs_guu[nadj]);
    1006:           if (ts->vecs_sensip) {
    1007:             VecAXPY(VecsDeltaLam2[nadj*s+i],-h,ts->vecs_gup[nadj]);
    1008:           }
    1009:         }
    1010:         if (ts->vecs_sensi2p) { /* 2nd-order adjoint for parameters */
    1011:           MatMultTranspose(ts->Jacprhs,VecsSensi2Temp[nadj],VecDeltaMu2);
    1012:           if (b[i]) {
    1013:             VecScale(VecDeltaMu2,-h*b[i]);
    1014:             VecAXPY(VecDeltaMu2,-h*b[i],ts->vecs_gpu[nadj]);
    1015:             VecAXPY(VecDeltaMu2,-h*b[i],ts->vecs_gpp[nadj]);
    1016:           } else {
    1017:             VecScale(VecDeltaMu2,-h);
    1018:             VecAXPY(VecDeltaMu2,-h,ts->vecs_gpu[nadj]);
    1019:             VecAXPY(VecDeltaMu2,-h,ts->vecs_gpp[nadj]);
    1020:           }
    1021:           VecAXPY(ts->vecs_sensi2p[nadj],1,VecDeltaMu2); /* update sensi2p for each stage */
    1022:         }
    1023:       }
    1024:     }
    1025:   }

    1027:   for (j=0; j<s; j++) w[j] = 1.0;
    1028:   for (nadj=0; nadj<ts->numcost; nadj++) { /* no need to do this for mu's */
    1029:     VecMAXPY(ts->vecs_sensi[nadj],s,w,&VecsDeltaLam[nadj*s]);
    1030:     if (ts->vecs_sensi2) {
    1031:       VecMAXPY(ts->vecs_sensi2[nadj],s,w,&VecsDeltaLam2[nadj*s]);
    1032:     }
    1033:   }
    1034:   rk->status = TS_STEP_COMPLETE;
    1035:   return(0);
    1036: }

    1038: static PetscErrorCode TSAdjointReset_RK(TS ts)
    1039: {
    1040:   TS_RK          *rk = (TS_RK*)ts->data;
    1041:   RKTableau      tab = rk->tableau;

    1045:   VecDestroyVecs(tab->s*ts->numcost,&rk->VecsDeltaLam);
    1046:   VecDestroyVecs(ts->numcost,&rk->VecsSensiTemp);
    1047:   VecDestroy(&rk->VecDeltaMu);
    1048:   VecDestroyVecs(tab->s*ts->numcost,&rk->VecsDeltaLam2);
    1049:   VecDestroy(&rk->VecDeltaMu2);
    1050:   VecDestroyVecs(ts->numcost,&rk->VecsSensi2Temp);
    1051:   return(0);
    1052: }

    1054: static PetscErrorCode TSInterpolate_RK(TS ts,PetscReal itime,Vec X)
    1055: {
    1056:   TS_RK            *rk = (TS_RK*)ts->data;
    1057:   PetscInt         s  = rk->tableau->s,p = rk->tableau->p,i,j;
    1058:   PetscReal        h;
    1059:   PetscReal        tt,t;
    1060:   PetscScalar      *b;
    1061:   const PetscReal  *B = rk->tableau->binterp;
    1062:   PetscErrorCode   ierr;

    1065:   if (!B) SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_SUP,"TSRK %s does not have an interpolation formula",rk->tableau->name);

    1067:   switch (rk->status) {
    1068:     case TS_STEP_INCOMPLETE:
    1069:     case TS_STEP_PENDING:
    1070:       h = ts->time_step;
    1071:       t = (itime - ts->ptime)/h;
    1072:       break;
    1073:     case TS_STEP_COMPLETE:
    1074:       h = ts->ptime - ts->ptime_prev;
    1075:       t = (itime - ts->ptime)/h + 1; /* In the interval [0,1] */
    1076:       break;
    1077:     default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
    1078:   }
    1079:   PetscMalloc1(s,&b);
    1080:   for (i=0; i<s; i++) b[i] = 0;
    1081:   for (j=0,tt=t; j<p; j++,tt*=t) {
    1082:     for (i=0; i<s; i++) {
    1083:       b[i]  += h * B[i*p+j] * tt;
    1084:     }
    1085:   }
    1086:   VecCopy(rk->Y[0],X);
    1087:   VecMAXPY(X,s,b,rk->YdotRHS);
    1088:   PetscFree(b);
    1089:   return(0);
    1090: }

    1092: /*------------------------------------------------------------*/

    1094: static PetscErrorCode TSRKTableauReset(TS ts)
    1095: {
    1096:   TS_RK          *rk = (TS_RK*)ts->data;
    1097:   RKTableau      tab = rk->tableau;

    1101:   if (!tab) return(0);
    1102:   PetscFree(rk->work);
    1103:   VecDestroyVecs(tab->s,&rk->Y);
    1104:   VecDestroyVecs(tab->s,&rk->YdotRHS);
    1105:   VecDestroyVecs(tab->s*ts->numcost,&rk->VecsDeltaLam);
    1106:   VecDestroyVecs(ts->numcost,&rk->VecsSensiTemp);
    1107:   VecDestroy(&rk->VecDeltaMu);
    1108:   return(0);
    1109: }

    1111: static PetscErrorCode TSReset_RK(TS ts)
    1112: {

    1116:   TSRKTableauReset(ts);
    1117:   if (ts->use_splitrhsfunction) {
    1118:     PetscTryMethod(ts,"TSReset_RK_MultirateSplit_C",(TS),(ts));
    1119:   } else {
    1120:     PetscTryMethod(ts,"TSReset_RK_MultirateNonsplit_C",(TS),(ts));
    1121:   }
    1122:   return(0);
    1123: }

    1125: static PetscErrorCode DMCoarsenHook_TSRK(DM fine,DM coarse,void *ctx)
    1126: {
    1128:   return(0);
    1129: }

    1131: static PetscErrorCode DMRestrictHook_TSRK(DM fine,Mat restrct,Vec rscale,Mat inject,DM coarse,void *ctx)
    1132: {
    1134:   return(0);
    1135: }

    1137: static PetscErrorCode DMSubDomainHook_TSRK(DM dm,DM subdm,void *ctx)
    1138: {
    1140:   return(0);
    1141: }

    1143: static PetscErrorCode DMSubDomainRestrictHook_TSRK(DM dm,VecScatter gscat,VecScatter lscat,DM subdm,void *ctx)
    1144: {

    1147:   return(0);
    1148: }

    1150: static PetscErrorCode TSRKTableauSetUp(TS ts)
    1151: {
    1152:   TS_RK          *rk  = (TS_RK*)ts->data;
    1153:   RKTableau      tab = rk->tableau;

    1157:   PetscMalloc1(tab->s,&rk->work);
    1158:   VecDuplicateVecs(ts->vec_sol,tab->s,&rk->Y);
    1159:   VecDuplicateVecs(ts->vec_sol,tab->s,&rk->YdotRHS);
    1160:   return(0);
    1161: }

    1163: static PetscErrorCode TSSetUp_RK(TS ts)
    1164: {
    1165:   TS             quadts = ts->quadraturets;
    1167:   DM             dm;

    1170:   TSCheckImplicitTerm(ts);
    1171:   TSRKTableauSetUp(ts);
    1172:   if (quadts && ts->costintegralfwd) {
    1173:     Mat Jquad;
    1174:     TSGetRHSJacobian(quadts,&Jquad,NULL,NULL,NULL);
    1175:   }
    1176:   TSGetDM(ts,&dm);
    1177:   DMCoarsenHookAdd(dm,DMCoarsenHook_TSRK,DMRestrictHook_TSRK,ts);
    1178:   DMSubDomainHookAdd(dm,DMSubDomainHook_TSRK,DMSubDomainRestrictHook_TSRK,ts);
    1179:   if (ts->use_splitrhsfunction) {
    1180:     PetscTryMethod(ts,"TSSetUp_RK_MultirateSplit_C",(TS),(ts));
    1181:   } else {
    1182:     PetscTryMethod(ts,"TSSetUp_RK_MultirateNonsplit_C",(TS),(ts));
    1183:   }
    1184:   return(0);
    1185: }

    1187: static PetscErrorCode TSSetFromOptions_RK(PetscOptionItems *PetscOptionsObject,TS ts)
    1188: {
    1189:   TS_RK          *rk = (TS_RK*)ts->data;

    1193:   PetscOptionsHead(PetscOptionsObject,"RK ODE solver options");
    1194:   {
    1195:     RKTableauLink link;
    1196:     PetscInt      count,choice;
    1197:     PetscBool     flg,use_multirate = PETSC_FALSE;
    1198:     const char    **namelist;

    1200:     for (link=RKTableauList,count=0; link; link=link->next,count++) ;
    1201:     PetscMalloc1(count,(char***)&namelist);
    1202:     for (link=RKTableauList,count=0; link; link=link->next,count++) namelist[count] = link->tab.name;
    1203:     PetscOptionsBool("-ts_rk_multirate","Use interpolation-based multirate RK method","TSRKSetMultirate",rk->use_multirate,&use_multirate,&flg);
    1204:     if (flg) {
    1205:       TSRKSetMultirate(ts,use_multirate);
    1206:     }
    1207:     PetscOptionsEList("-ts_rk_type","Family of RK method","TSRKSetType",(const char*const*)namelist,count,rk->tableau->name,&choice,&flg);
    1208:     if (flg) {TSRKSetType(ts,namelist[choice]);}
    1209:     PetscFree(namelist);
    1210:   }
    1211:   PetscOptionsTail();
    1212:   PetscOptionsBegin(PetscObjectComm((PetscObject)ts),NULL,"Multirate methods options","");
    1213:   PetscOptionsInt("-ts_rk_dtratio","time step ratio between slow and fast","",rk->dtratio,&rk->dtratio,NULL);
    1214:   PetscOptionsEnd();
    1215:   return(0);
    1216: }

    1218: static PetscErrorCode TSView_RK(TS ts,PetscViewer viewer)
    1219: {
    1220:   TS_RK          *rk = (TS_RK*)ts->data;
    1221:   PetscBool      iascii;

    1225:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
    1226:   if (iascii) {
    1227:     RKTableau       tab  = rk->tableau;
    1228:     TSRKType        rktype;
    1229:     const PetscReal *c;
    1230:     PetscInt        s;
    1231:     char            buf[512];
    1232:     PetscBool       FSAL;

    1234:     TSRKGetType(ts,&rktype);
    1235:     TSRKGetTableau(ts,&s,NULL,NULL,&c,NULL,NULL,NULL,&FSAL);
    1236:     PetscViewerASCIIPrintf(viewer,"  RK type %s\n",rktype);
    1237:     PetscViewerASCIIPrintf(viewer,"  Order: %D\n",tab->order);
    1238:     PetscViewerASCIIPrintf(viewer,"  FSAL property: %s\n",FSAL ? "yes" : "no");
    1239:     PetscFormatRealArray(buf,sizeof(buf),"% 8.6f",s,c);
    1240:     PetscViewerASCIIPrintf(viewer,"  Abscissa c = %s\n",buf);
    1241:   }
    1242:   return(0);
    1243: }

    1245: static PetscErrorCode TSLoad_RK(TS ts,PetscViewer viewer)
    1246: {
    1248:   TSAdapt        adapt;

    1251:   TSGetAdapt(ts,&adapt);
    1252:   TSAdaptLoad(adapt,viewer);
    1253:   return(0);
    1254: }

    1256: /*@
    1257:   TSRKGetOrder - Get the order of RK scheme

    1259:   Not collective

    1261:   Input Parameter:
    1262: .  ts - timestepping context

    1264:   Output Parameter:
    1265: .  order - order of RK-scheme

    1267:   Level: intermediate

    1269: .seealso: TSRKGetType()
    1270: @*/
    1271: PetscErrorCode TSRKGetOrder(TS ts,PetscInt *order)
    1272: {

    1278:   PetscUseMethod(ts,"TSRKGetOrder_C",(TS,PetscInt*),(ts,order));
    1279:   return(0);
    1280: }

    1282: /*@C
    1283:   TSRKSetType - Set the type of RK scheme

    1285:   Logically collective

    1287:   Input Parameters:
    1288: +  ts - timestepping context
    1289: -  rktype - type of RK-scheme

    1291:   Options Database:
    1292: .   -ts_rk_type - <1fe,2a,3,3bs,4,5f,5dp,5bs>

    1294:   Level: intermediate

    1296: .seealso: TSRKGetType(), TSRK, TSRKType, TSRK1FE, TSRK2A, TSRK3, TSRK3BS, TSRK4, TSRK5F, TSRK5DP, TSRK5BS, TSRK6VR, TSRK7VR, TSRK8VR
    1297: @*/
    1298: PetscErrorCode TSRKSetType(TS ts,TSRKType rktype)
    1299: {

    1305:   PetscTryMethod(ts,"TSRKSetType_C",(TS,TSRKType),(ts,rktype));
    1306:   return(0);
    1307: }

    1309: /*@C
    1310:   TSRKGetType - Get the type of RK scheme

    1312:   Not collective

    1314:   Input Parameter:
    1315: .  ts - timestepping context

    1317:   Output Parameter:
    1318: .  rktype - type of RK-scheme

    1320:   Level: intermediate

    1322: .seealso: TSRKSetType()
    1323: @*/
    1324: PetscErrorCode TSRKGetType(TS ts,TSRKType *rktype)
    1325: {

    1330:   PetscUseMethod(ts,"TSRKGetType_C",(TS,TSRKType*),(ts,rktype));
    1331:   return(0);
    1332: }

    1334: static PetscErrorCode TSRKGetOrder_RK(TS ts,PetscInt *order)
    1335: {
    1336:   TS_RK *rk = (TS_RK*)ts->data;

    1339:   *order = rk->tableau->order;
    1340:   return(0);
    1341: }

    1343: static PetscErrorCode TSRKGetType_RK(TS ts,TSRKType *rktype)
    1344: {
    1345:   TS_RK *rk = (TS_RK*)ts->data;

    1348:   *rktype = rk->tableau->name;
    1349:   return(0);
    1350: }

    1352: static PetscErrorCode TSRKSetType_RK(TS ts,TSRKType rktype)
    1353: {
    1354:   TS_RK          *rk = (TS_RK*)ts->data;
    1356:   PetscBool      match;
    1357:   RKTableauLink  link;

    1360:   if (rk->tableau) {
    1361:     PetscStrcmp(rk->tableau->name,rktype,&match);
    1362:     if (match) return(0);
    1363:   }
    1364:   for (link = RKTableauList; link; link=link->next) {
    1365:     PetscStrcmp(link->tab.name,rktype,&match);
    1366:     if (match) {
    1367:       if (ts->setupcalled) {TSRKTableauReset(ts);}
    1368:       rk->tableau = &link->tab;
    1369:       if (ts->setupcalled) {TSRKTableauSetUp(ts);}
    1370:       ts->default_adapt_type = rk->tableau->bembed ? TSADAPTBASIC : TSADAPTNONE;
    1371:       return(0);
    1372:     }
    1373:   }
    1374:   SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_ARG_UNKNOWN_TYPE,"Could not find '%s'",rktype);
    1375: }

    1377: static PetscErrorCode  TSGetStages_RK(TS ts,PetscInt *ns,Vec **Y)
    1378: {
    1379:   TS_RK *rk = (TS_RK*)ts->data;

    1382:   if (ns) *ns = rk->tableau->s;
    1383:   if (Y)   *Y = rk->Y;
    1384:   return(0);
    1385: }

    1387: static PetscErrorCode TSDestroy_RK(TS ts)
    1388: {

    1392:   TSReset_RK(ts);
    1393:   if (ts->dm) {
    1394:     DMCoarsenHookRemove(ts->dm,DMCoarsenHook_TSRK,DMRestrictHook_TSRK,ts);
    1395:     DMSubDomainHookRemove(ts->dm,DMSubDomainHook_TSRK,DMSubDomainRestrictHook_TSRK,ts);
    1396:   }
    1397:   PetscFree(ts->data);
    1398:   PetscObjectComposeFunction((PetscObject)ts,"TSRKGetOrder_C",NULL);
    1399:   PetscObjectComposeFunction((PetscObject)ts,"TSRKGetType_C",NULL);
    1400:   PetscObjectComposeFunction((PetscObject)ts,"TSRKSetType_C",NULL);
    1401:   PetscObjectComposeFunction((PetscObject)ts,"TSRKGetTableau_C",NULL);
    1402:   PetscObjectComposeFunction((PetscObject)ts,"TSRKSetMultirate_C",NULL);
    1403:   PetscObjectComposeFunction((PetscObject)ts,"TSRKGetMultirate_C",NULL);
    1404:   return(0);
    1405: }

    1407: /*
    1408:   This defines the nonlinear equation that is to be solved with SNES
    1409:   We do not need to solve the equation; we just use SNES to approximate the Jacobian
    1410: */
    1411: static PetscErrorCode SNESTSFormFunction_RK(SNES snes,Vec x,Vec y,TS ts)
    1412: {
    1413:   TS_RK          *rk = (TS_RK*)ts->data;
    1414:   DM             dm,dmsave;

    1418:   SNESGetDM(snes,&dm);
    1419:   /* DM monkey-business allows user code to call TSGetDM() inside of functions evaluated on levels of FAS */
    1420:   dmsave = ts->dm;
    1421:   ts->dm = dm;
    1422:   TSComputeRHSFunction(ts,rk->stage_time,x,y);
    1423:   ts->dm = dmsave;
    1424:   return(0);
    1425: }

    1427: static PetscErrorCode SNESTSFormJacobian_RK(SNES snes,Vec x,Mat A,Mat B,TS ts)
    1428: {
    1429:   TS_RK          *rk = (TS_RK*)ts->data;
    1430:   DM             dm,dmsave;

    1434:   SNESGetDM(snes,&dm);
    1435:   dmsave = ts->dm;
    1436:   ts->dm = dm;
    1437:   TSComputeRHSJacobian(ts,rk->stage_time,x,A,B);
    1438:   ts->dm = dmsave;
    1439:   return(0);
    1440: }

    1442: /*@C
    1443:   TSRKSetMultirate - Use the interpolation-based multirate RK method

    1445:   Logically collective

    1447:   Input Parameters:
    1448: +  ts - timestepping context
    1449: -  use_multirate - PETSC_TRUE enables the multirate RK method, sets the basic method to be RK2A and sets the ratio between slow stepsize and fast stepsize to be 2

    1451:   Options Database:
    1452: .   -ts_rk_multirate - <true,false>

    1454:   Notes:
    1455:   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 coeffcients (binterp).

    1457:   Level: intermediate

    1459: .seealso: TSRKGetMultirate()
    1460: @*/
    1461: PetscErrorCode TSRKSetMultirate(TS ts,PetscBool use_multirate)
    1462: {

    1466:   PetscTryMethod(ts,"TSRKSetMultirate_C",(TS,PetscBool),(ts,use_multirate));
    1467:   return(0);
    1468: }

    1470: /*@C
    1471:   TSRKGetMultirate - Gets whether to Use the interpolation-based multirate RK method

    1473:   Not collective

    1475:   Input Parameter:
    1476: .  ts - timestepping context

    1478:   Output Parameter:
    1479: .  use_multirate - PETSC_TRUE if the multirate RK method is enabled, PETSC_FALSE otherwise

    1481:   Level: intermediate

    1483: .seealso: TSRKSetMultirate()
    1484: @*/
    1485: PetscErrorCode TSRKGetMultirate(TS ts,PetscBool *use_multirate)
    1486: {

    1490:   PetscUseMethod(ts,"TSRKGetMultirate_C",(TS,PetscBool*),(ts,use_multirate));
    1491:   return(0);
    1492: }

    1494: /*MC
    1495:       TSRK - ODE and DAE solver using Runge-Kutta schemes

    1497:   The user should provide the right hand side of the equation
    1498:   using TSSetRHSFunction().

    1500:   Notes:
    1501:   The default is TSRK3BS, it can be changed with TSRKSetType() or -ts_rk_type

    1503:   Level: beginner

    1505: .seealso:  TSCreate(), TS, TSSetType(), TSRKSetType(), TSRKGetType(), TSRK2D, TTSRK2E, TSRK3,
    1506:            TSRK4, TSRK5, TSRKPRSSP2, TSRKBPR3, TSRKType, TSRKRegister(), TSRKSetMultirate(), TSRKGetMultirate()

    1508: M*/
    1509: PETSC_EXTERN PetscErrorCode TSCreate_RK(TS ts)
    1510: {
    1511:   TS_RK          *rk;

    1515:   TSRKInitializePackage();

    1517:   ts->ops->reset          = TSReset_RK;
    1518:   ts->ops->destroy        = TSDestroy_RK;
    1519:   ts->ops->view           = TSView_RK;
    1520:   ts->ops->load           = TSLoad_RK;
    1521:   ts->ops->setup          = TSSetUp_RK;
    1522:   ts->ops->interpolate    = TSInterpolate_RK;
    1523:   ts->ops->step           = TSStep_RK;
    1524:   ts->ops->evaluatestep   = TSEvaluateStep_RK;
    1525:   ts->ops->rollback       = TSRollBack_RK;
    1526:   ts->ops->setfromoptions = TSSetFromOptions_RK;
    1527:   ts->ops->getstages      = TSGetStages_RK;

    1529:   ts->ops->snesfunction    = SNESTSFormFunction_RK;
    1530:   ts->ops->snesjacobian    = SNESTSFormJacobian_RK;
    1531:   ts->ops->adjointintegral = TSAdjointCostIntegral_RK;
    1532:   ts->ops->adjointsetup    = TSAdjointSetUp_RK;
    1533:   ts->ops->adjointstep     = TSAdjointStep_RK;
    1534:   ts->ops->adjointreset    = TSAdjointReset_RK;

    1536:   ts->ops->forwardintegral = TSForwardCostIntegral_RK;
    1537:   ts->ops->forwardsetup    = TSForwardSetUp_RK;
    1538:   ts->ops->forwardreset    = TSForwardReset_RK;
    1539:   ts->ops->forwardstep     = TSForwardStep_RK;
    1540:   ts->ops->forwardgetstages= TSForwardGetStages_RK;

    1542:   PetscNewLog(ts,&rk);
    1543:   ts->data = (void*)rk;

    1545:   PetscObjectComposeFunction((PetscObject)ts,"TSRKGetOrder_C",TSRKGetOrder_RK);
    1546:   PetscObjectComposeFunction((PetscObject)ts,"TSRKGetType_C",TSRKGetType_RK);
    1547:   PetscObjectComposeFunction((PetscObject)ts,"TSRKSetType_C",TSRKSetType_RK);
    1548:   PetscObjectComposeFunction((PetscObject)ts,"TSRKGetTableau_C",TSRKGetTableau_RK);
    1549:   PetscObjectComposeFunction((PetscObject)ts,"TSRKSetMultirate_C",TSRKSetMultirate_RK);
    1550:   PetscObjectComposeFunction((PetscObject)ts,"TSRKGetMultirate_C",TSRKGetMultirate_RK);

    1552:   TSRKSetType(ts,TSRKDefault);
    1553:   rk->dtratio = 1;
    1554:   return(0);
    1555: }