Date: 2022-03-04T17:39:34+00:00

Actual source code: rosw.c

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

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

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

      9:   where F represents the stiff part of the physics and G represents the non-stiff part.
     10:   This method is designed to be linearly implicit on F and can use an approximate and lagged Jacobian.

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

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

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

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

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

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

     69:      Only an approximate Jacobian is needed.

     71:      Level: intermediate

     73: .seealso: TSROSW
     74: M*/

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

     79:      Only an approximate Jacobian is needed.

     81:      Level: intermediate

     83: .seealso: TSROSW
     84: M*/

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

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

     91:      Level: intermediate

     93: .seealso: TSROSW
     94: M*/

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

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

    101:      Level: intermediate

    103: .seealso: TSROSW
    104: M*/

    106: /*MC
    107:      TSROSWRA3PW - Three stage third order Rosenbrock-W scheme for PDAE of index 1.

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

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

    113:      References:
    114: .  1. -   Rang and Angermann, New Rosenbrock W methods of order 3 for partial differential algebraic equations of index 1, 2005.

    116:      Level: intermediate

    118: .seealso: TSROSW
    119: M*/

    121: /*MC
    122:      TSROSWRA34PW2 - Four stage third order L-stable Rosenbrock-W scheme for PDAE of index 1.

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

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

    128:      References:
    129: .  1. -   Rang and Angermann, New Rosenbrock W methods of order 3 for partial differential algebraic equations of index 1, 2005.

    131:      Level: intermediate

    133: .seealso: TSROSW
    134: M*/

    136: /*MC
    137:      TSROSWRODAS3 - Four stage third order L-stable Rosenbrock scheme

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

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

    143:      References:
    144: .  1. -   Sandu et al, Benchmarking stiff ODE solvers for atmospheric chemistry problems II, Rosenbrock solvers, 1997.

    146:      Level: intermediate

    148: .seealso: TSROSW, TSROSWSANDU3
    149: M*/

    151: /*MC
    152:      TSROSWSANDU3 - Three stage third order L-stable Rosenbrock scheme

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

    156:      The third order method is L-stable, but not stiffly accurate.
    157:      The second order embedded method is strongly A-stable with R(infty) = 0.5.
    158:      The internal stages are L-stable.
    159:      This method is called ROS3 in the paper.

    161:      References:
    162: .  1. -   Sandu et al, Benchmarking stiff ODE solvers for atmospheric chemistry problems II, Rosenbrock solvers, 1997.

    164:      Level: intermediate

    166: .seealso: TSROSW, TSROSWRODAS3
    167: M*/

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

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

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

    176:      References:
    177: .     Emil Constantinescu

    179:      Level: intermediate

    181: .seealso: TSROSW, TSROSWLASSP3P4S2C, TSROSWLLSSP3P4S2C, SSP
    182: M*/

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

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

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

    191:      References:
    192: .     Emil Constantinescu

    194:      Level: intermediate

    196: .seealso: TSROSW, TSROSWASSP3P3S1C, TSROSWLLSSP3P4S2C, TSSSP
    197: M*/

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

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

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

    206:      References:
    207: .     Emil Constantinescu

    209:      Level: intermediate

    211: .seealso: TSROSW, TSROSWASSP3P3S1C, TSROSWLASSP3P4S2C, TSSSP
    212: M*/

    214: /*MC
    215:      TSROSWGRK4T - four stage, fourth order Rosenbrock (not W) method from Kaps and Rentrop

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

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

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

    223:      References:
    224: +   1. -  Kaps and Rentrop, Generalized Runge Kutta methods of order four with stepsize control for stiff ordinary differential equations, 1979.
    225: -   2. -  Hairer and Wanner, Solving Ordinary Differential Equations II, Section 4 Table 7.2.

    227:      Hairer's code ros4.f

    229:      Level: intermediate

    231: .seealso: TSROSW, TSROSWSHAMP4, TSROSWVELDD4, TSROSW4L
    232: M*/

    234: /*MC
    235:      TSROSWSHAMP4 - four stage, fourth order Rosenbrock (not W) method from Shampine

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

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

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

    243:      References:
    244: +   1. -  Shampine, Implementation of Rosenbrock methods, 1982.
    245: -   2. -  Hairer and Wanner, Solving Ordinary Differential Equations II, Section 4 Table 7.2.

    247:      Hairer's code ros4.f

    249:      Level: intermediate

    251: .seealso: TSROSW, TSROSWGRK4T, TSROSWVELDD4, TSROSW4L
    252: M*/

    254: /*MC
    255:      TSROSWVELDD4 - four stage, fourth order Rosenbrock (not W) method from van Veldhuizen

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

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

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

    263:      References:
    264: +   1. -  van Veldhuizen, D stability and Kaps Rentrop methods, 1984.
    265: -   2. -  Hairer and Wanner, Solving Ordinary Differential Equations II, Section 4 Table 7.2.

    267:      Hairer's code ros4.f

    269:      Level: intermediate

    271: .seealso: TSROSW, TSROSWGRK4T, TSROSWSHAMP4, TSROSW4L
    272: M*/

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

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

    279:      A-stable and L-stable

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

    283:      References:
    284: .  1. -   Hairer and Wanner, Solving Ordinary Differential Equations II, Section 4 Table 7.2.

    286:      Hairer's code ros4.f

    288:      Level: intermediate

    290: .seealso: TSROSW, TSROSWGRK4T, TSROSWSHAMP4, TSROSW4L
    291: M*/

    293: /*@C
    294:   TSRosWRegisterAll - Registers all of the Rosenbrock-W methods in TSRosW

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

    298:   Level: advanced

    300: .seealso:  TSRosWRegisterDestroy()
    301: @*/
    302: PetscErrorCode TSRosWRegisterAll(void)
    303: {

    307:   if (TSRosWRegisterAllCalled) return(0);
    308:   TSRosWRegisterAllCalled = PETSC_TRUE;

    310:   {
    311:     const PetscReal A = 0;
    312:     const PetscReal Gamma = 1;
    313:     const PetscReal b = 1;
    314:     const PetscReal binterpt=1;

    316:     TSRosWRegister(TSROSWTHETA1,1,1,&A,&Gamma,&b,NULL,1,&binterpt);
    317:   }

    319:   {
    320:     const PetscReal A = 0;
    321:     const PetscReal Gamma = 0.5;
    322:     const PetscReal b = 1;
    323:     const PetscReal binterpt=1;

    325:     TSRosWRegister(TSROSWTHETA2,2,1,&A,&Gamma,&b,NULL,1,&binterpt);
    326:   }

    328:   {
    329:     /*const PetscReal g = 1. + 1./PetscSqrtReal(2.0);   Direct evaluation: 1.707106781186547524401. Used for setting up arrays of values known at compile time below. */
    330:     const PetscReal
    331:       A[2][2]     = {{0,0}, {1.,0}},
    332:       Gamma[2][2] = {{1.707106781186547524401,0}, {-2.*1.707106781186547524401,1.707106781186547524401}},
    333:       b[2]        = {0.5,0.5},
    334:       b1[2]       = {1.0,0.0};
    335:     PetscReal binterpt[2][2];
    336:     binterpt[0][0] = 1.707106781186547524401 - 1.0;
    337:     binterpt[1][0] = 2.0 - 1.707106781186547524401;
    338:     binterpt[0][1] = 1.707106781186547524401 - 1.5;
    339:     binterpt[1][1] = 1.5 - 1.707106781186547524401;

    341:     TSRosWRegister(TSROSW2P,2,2,&A[0][0],&Gamma[0][0],b,b1,2,&binterpt[0][0]);
    342:   }
    343:   {
    344:     /*const PetscReal g = 1. - 1./PetscSqrtReal(2.0);   Direct evaluation: 0.2928932188134524755992. Used for setting up arrays of values known at compile time below. */
    345:     const PetscReal
    346:       A[2][2]     = {{0,0}, {1.,0}},
    347:       Gamma[2][2] = {{0.2928932188134524755992,0}, {-2.*0.2928932188134524755992,0.2928932188134524755992}},
    348:       b[2]        = {0.5,0.5},
    349:       b1[2]       = {1.0,0.0};
    350:     PetscReal binterpt[2][2];
    351:     binterpt[0][0] = 0.2928932188134524755992 - 1.0;
    352:     binterpt[1][0] = 2.0 - 0.2928932188134524755992;
    353:     binterpt[0][1] = 0.2928932188134524755992 - 1.5;
    354:     binterpt[1][1] = 1.5 - 0.2928932188134524755992;

    356:     TSRosWRegister(TSROSW2M,2,2,&A[0][0],&Gamma[0][0],b,b1,2,&binterpt[0][0]);
    357:   }
    358:   {
    359:     /*const PetscReal g = 7.8867513459481287e-01; Directly written in-place below */
    360:     PetscReal binterpt[3][2];
    361:     const PetscReal
    362:       A[3][3] = {{0,0,0},
    363:                  {1.5773502691896257e+00,0,0},
    364:                  {0.5,0,0}},
    365:       Gamma[3][3] = {{7.8867513459481287e-01,0,0},
    366:                      {-1.5773502691896257e+00,7.8867513459481287e-01,0},
    367:                      {-6.7075317547305480e-01,-1.7075317547305482e-01,7.8867513459481287e-01}},
    368:       b[3]  = {1.0566243270259355e-01,4.9038105676657971e-02,8.4529946162074843e-01},
    369:       b2[3] = {-1.7863279495408180e-01,1./3.,8.4529946162074843e-01};

    371:       binterpt[0][0] = -0.8094010767585034;
    372:       binterpt[1][0] = -0.5;
    373:       binterpt[2][0] = 2.3094010767585034;
    374:       binterpt[0][1] = 0.9641016151377548;
    375:       binterpt[1][1] = 0.5;
    376:       binterpt[2][1] = -1.4641016151377548;

    378:       TSRosWRegister(TSROSWRA3PW,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]);
    379:   }
    380:   {
    381:     PetscReal  binterpt[4][3];
    382:     /*const PetscReal g = 4.3586652150845900e-01; Directly written in-place below */
    383:     const PetscReal
    384:       A[4][4] = {{0,0,0,0},
    385:                  {8.7173304301691801e-01,0,0,0},
    386:                  {8.4457060015369423e-01,-1.1299064236484185e-01,0,0},
    387:                  {0,0,1.,0}},
    388:       Gamma[4][4] = {{4.3586652150845900e-01,0,0,0},
    389:                      {-8.7173304301691801e-01,4.3586652150845900e-01,0,0},
    390:                      {-9.0338057013044082e-01,5.4180672388095326e-02,4.3586652150845900e-01,0},
    391:                      {2.4212380706095346e-01,-1.2232505839045147e+00,5.4526025533510214e-01,4.3586652150845900e-01}},
    392:       b[4]  = {2.4212380706095346e-01,-1.2232505839045147e+00,1.5452602553351020e+00,4.3586652150845900e-01},
    393:       b2[4] = {3.7810903145819369e-01,-9.6042292212423178e-02,5.0000000000000000e-01,2.1793326075422950e-01};

    395:     binterpt[0][0]=1.0564298455794094;
    396:     binterpt[1][0]=2.296429974281067;
    397:     binterpt[2][0]=-1.307599564525376;
    398:     binterpt[3][0]=-1.045260255335102;
    399:     binterpt[0][1]=-1.3864882699759573;
    400:     binterpt[1][1]=-8.262611700275677;
    401:     binterpt[2][1]=7.250979895056055;
    402:     binterpt[3][1]=2.398120075195581;
    403:     binterpt[0][2]=0.5721822314575016;
    404:     binterpt[1][2]=4.742931142090097;
    405:     binterpt[2][2]=-4.398120075195578;
    406:     binterpt[3][2]=-0.9169932983520199;

    408:     TSRosWRegister(TSROSWRA34PW2,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
    409:   }
    410:   {
    411:     /* const PetscReal g = 0.5;       Directly written in-place below */
    412:     const PetscReal
    413:       A[4][4] = {{0,0,0,0},
    414:                  {0,0,0,0},
    415:                  {1.,0,0,0},
    416:                  {0.75,-0.25,0.5,0}},
    417:       Gamma[4][4] = {{0.5,0,0,0},
    418:                      {1.,0.5,0,0},
    419:                      {-0.25,-0.25,0.5,0},
    420:                      {1./12,1./12,-2./3,0.5}},
    421:       b[4]  = {5./6,-1./6,-1./6,0.5},
    422:       b2[4] = {0.75,-0.25,0.5,0};

    424:     TSRosWRegister(TSROSWRODAS3,3,4,&A[0][0],&Gamma[0][0],b,b2,0,NULL);
    425:   }
    426:   {
    427:     /*const PetscReal g = 0.43586652150845899941601945119356;       Directly written in-place below */
    428:     const PetscReal
    429:       A[3][3] = {{0,0,0},
    430:                  {0.43586652150845899941601945119356,0,0},
    431:                  {0.43586652150845899941601945119356,0,0}},
    432:       Gamma[3][3] = {{0.43586652150845899941601945119356,0,0},
    433:                      {-0.19294655696029095575009695436041,0.43586652150845899941601945119356,0},
    434:                      {0,1.74927148125794685173529749738960,0.43586652150845899941601945119356}},
    435:       b[3]  = {-0.75457412385404315829818998646589,1.94100407061964420292840123379419,-0.18642994676560104463021124732829},
    436:       b2[3] = {-1.53358745784149585370766523913002,2.81745131148625772213931745457622,-0.28386385364476186843165221544619};

    438:     PetscReal binterpt[3][2];
    439:     binterpt[0][0] = 3.793692883777660870425141387941;
    440:     binterpt[1][0] = -2.918692883777660870425141387941;
    441:     binterpt[2][0] = 0.125;
    442:     binterpt[0][1] = -0.725741064379812106687651020584;
    443:     binterpt[1][1] = 0.559074397713145440020984353917;
    444:     binterpt[2][1] = 0.16666666666666666666666666666667;

    446:     TSRosWRegister(TSROSWSANDU3,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]);
    447:   }
    448:   {
    449:     /*const PetscReal s3 = PetscSqrtReal(3.),g = (3.0+s3)/6.0;
    450:      * Direct evaluation: s3 = 1.732050807568877293527;
    451:      *                     g = 0.7886751345948128822546;
    452:      * Values are directly inserted below to ensure availability at compile time (compiler warnings otherwise...) */
    453:     const PetscReal
    454:       A[3][3] = {{0,0,0},
    455:                  {1,0,0},
    456:                  {0.25,0.25,0}},
    457:       Gamma[3][3] = {{0,0,0},
    458:                      {(-3.0-1.732050807568877293527)/6.0,0.7886751345948128822546,0},
    459:                      {(-3.0-1.732050807568877293527)/24.0,(-3.0-1.732050807568877293527)/8.0,0.7886751345948128822546}},
    460:       b[3]  = {1./6.,1./6.,2./3.},
    461:       b2[3] = {1./4.,1./4.,1./2.};
    462:     PetscReal binterpt[3][2];

    464:     binterpt[0][0]=0.089316397477040902157517886164709;
    465:     binterpt[1][0]=-0.91068360252295909784248211383529;
    466:     binterpt[2][0]=1.8213672050459181956849642276706;
    467:     binterpt[0][1]=0.077350269189625764509148780501957;
    468:     binterpt[1][1]=1.077350269189625764509148780502;
    469:     binterpt[2][1]=-1.1547005383792515290182975610039;

    471:     TSRosWRegister(TSROSWASSP3P3S1C,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]);
    472:   }

    474:   {
    475:     const PetscReal
    476:       A[4][4] = {{0,0,0,0},
    477:                  {1./2.,0,0,0},
    478:                  {1./2.,1./2.,0,0},
    479:                  {1./6.,1./6.,1./6.,0}},
    480:       Gamma[4][4] = {{1./2.,0,0,0},
    481:                      {0.0,1./4.,0,0},
    482:                      {-2.,-2./3.,2./3.,0},
    483:                      {1./2.,5./36.,-2./9,0}},
    484:       b[4]  = {1./6.,1./6.,1./6.,1./2.},
    485:       b2[4] = {1./8.,3./4.,1./8.,0};
    486:     PetscReal binterpt[4][3];

    488:     binterpt[0][0]=6.25;
    489:     binterpt[1][0]=-30.25;
    490:     binterpt[2][0]=1.75;
    491:     binterpt[3][0]=23.25;
    492:     binterpt[0][1]=-9.75;
    493:     binterpt[1][1]=58.75;
    494:     binterpt[2][1]=-3.25;
    495:     binterpt[3][1]=-45.75;
    496:     binterpt[0][2]=3.6666666666666666666666666666667;
    497:     binterpt[1][2]=-28.333333333333333333333333333333;
    498:     binterpt[2][2]=1.6666666666666666666666666666667;
    499:     binterpt[3][2]=23.;

    501:     TSRosWRegister(TSROSWLASSP3P4S2C,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
    502:   }

    504:   {
    505:     const PetscReal
    506:       A[4][4] = {{0,0,0,0},
    507:                  {1./2.,0,0,0},
    508:                  {1./2.,1./2.,0,0},
    509:                  {1./6.,1./6.,1./6.,0}},
    510:       Gamma[4][4] = {{1./2.,0,0,0},
    511:                      {0.0,3./4.,0,0},
    512:                      {-2./3.,-23./9.,2./9.,0},
    513:                      {1./18.,65./108.,-2./27,0}},
    514:       b[4]  = {1./6.,1./6.,1./6.,1./2.},
    515:       b2[4] = {3./16.,10./16.,3./16.,0};
    516:     PetscReal binterpt[4][3];

    518:     binterpt[0][0]=1.6911764705882352941176470588235;
    519:     binterpt[1][0]=3.6813725490196078431372549019608;
    520:     binterpt[2][0]=0.23039215686274509803921568627451;
    521:     binterpt[3][0]=-4.6029411764705882352941176470588;
    522:     binterpt[0][1]=-0.95588235294117647058823529411765;
    523:     binterpt[1][1]=-6.2401960784313725490196078431373;
    524:     binterpt[2][1]=-0.31862745098039215686274509803922;
    525:     binterpt[3][1]=7.5147058823529411764705882352941;
    526:     binterpt[0][2]=-0.56862745098039215686274509803922;
    527:     binterpt[1][2]=2.7254901960784313725490196078431;
    528:     binterpt[2][2]=0.25490196078431372549019607843137;
    529:     binterpt[3][2]=-2.4117647058823529411764705882353;

    531:     TSRosWRegister(TSROSWLLSSP3P4S2C,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
    532:   }

    534:   {
    535:     PetscReal A[4][4],Gamma[4][4],b[4],b2[4];
    536:     PetscReal binterpt[4][3];

    538:     Gamma[0][0]=0.4358665215084589994160194475295062513822671686978816;
    539:     Gamma[0][1]=0; Gamma[0][2]=0; Gamma[0][3]=0;
    540:     Gamma[1][0]=-1.997527830934941248426324674704153457289527280554476;
    541:     Gamma[1][1]=0.4358665215084589994160194475295062513822671686978816;
    542:     Gamma[1][2]=0; Gamma[1][3]=0;
    543:     Gamma[2][0]=-1.007948511795029620852002345345404191008352770119903;
    544:     Gamma[2][1]=-0.004648958462629345562774289390054679806993396798458131;
    545:     Gamma[2][2]=0.4358665215084589994160194475295062513822671686978816;
    546:     Gamma[2][3]=0;
    547:     Gamma[3][0]=-0.6685429734233467180451604600279552604364311322650783;
    548:     Gamma[3][1]=0.6056625986449338476089525334450053439525178740492984;
    549:     Gamma[3][2]=-0.9717899277217721234705114616271378792182450260943198;
    550:     Gamma[3][3]=0;

    552:     A[0][0]=0; A[0][1]=0; A[0][2]=0; A[0][3]=0;
    553:     A[1][0]=0.8717330430169179988320388950590125027645343373957631;
    554:     A[1][1]=0; A[1][2]=0; A[1][3]=0;
    555:     A[2][0]=0.5275890119763004115618079766722914408876108660811028;
    556:     A[2][1]=0.07241098802369958843819203208518599088698057726988732;
    557:     A[2][2]=0; A[2][3]=0;
    558:     A[3][0]=0.3990960076760701320627260685975778145384666450351314;
    559:     A[3][1]=-0.4375576546135194437228463747348862825846903771419953;
    560:     A[3][2]=1.038461646937449311660120300601880176655352737312713;
    561:     A[3][3]=0;

    563:     b[0]=0.1876410243467238251612921333138006734899663569186926;
    564:     b[1]=-0.5952974735769549480478230473706443582188442040780541;
    565:     b[2]=0.9717899277217721234705114616271378792182450260943198;
    566:     b[3]=0.4358665215084589994160194475295062513822671686978816;

    568:     b2[0]=0.2147402862233891404862383521089097657790734483804460;
    569:     b2[1]=-0.4851622638849390928209050538171743017757490232519684;
    570:     b2[2]=0.8687250025203875511662123688667549217531982787600080;
    571:     b2[3]=0.4016969751411624011684543450940068201770721128357014;

    573:     binterpt[0][0]=2.2565812720167954547104627844105;
    574:     binterpt[1][0]=1.349166413351089573796243820819;
    575:     binterpt[2][0]=-2.4695174540533503758652847586647;
    576:     binterpt[3][0]=-0.13623023131453465264142184656474;
    577:     binterpt[0][1]=-3.0826699111559187902922463354557;
    578:     binterpt[1][1]=-2.4689115685996042534544925650515;
    579:     binterpt[2][1]=5.7428279814696677152129332773553;
    580:     binterpt[3][1]=-0.19124650171414467146619437684812;
    581:     binterpt[0][2]=1.0137296634858471607430756831148;
    582:     binterpt[1][2]=0.52444768167155973161042570784064;
    583:     binterpt[2][2]=-2.3015205996945452158771370439586;
    584:     binterpt[3][2]=0.76334325453713832352363565300308;

    586:     TSRosWRegister(TSROSWARK3,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
    587:   }
    588:   TSRosWRegisterRos4(TSROSWGRK4T,0.231,PETSC_DEFAULT,PETSC_DEFAULT,0,-0.1282612945269037e+01);
    589:   TSRosWRegisterRos4(TSROSWSHAMP4,0.5,PETSC_DEFAULT,PETSC_DEFAULT,0,125./108.);
    590:   TSRosWRegisterRos4(TSROSWVELDD4,0.22570811482256823492,PETSC_DEFAULT,PETSC_DEFAULT,0,-1.355958941201148);
    591:   TSRosWRegisterRos4(TSROSW4L,0.57282,PETSC_DEFAULT,PETSC_DEFAULT,0,-1.093502252409163);
    592:   return(0);
    593: }

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

    598:    Not Collective

    600:    Level: advanced

    602: .seealso: TSRosWRegister(), TSRosWRegisterAll()
    603: @*/
    604: PetscErrorCode TSRosWRegisterDestroy(void)
    605: {
    606:   PetscErrorCode  ierr;
    607:   RosWTableauLink link;

    610:   while ((link = RosWTableauList)) {
    611:     RosWTableau t = &link->tab;
    612:     RosWTableauList = link->next;
    613:     PetscFree5(t->A,t->Gamma,t->b,t->ASum,t->GammaSum);
    614:     PetscFree5(t->At,t->bt,t->GammaInv,t->GammaZeroDiag,t->GammaExplicitCorr);
    615:     PetscFree2(t->bembed,t->bembedt);
    616:     PetscFree(t->binterpt);
    617:     PetscFree(t->name);
    618:     PetscFree(link);
    619:   }
    620:   TSRosWRegisterAllCalled = PETSC_FALSE;
    621:   return(0);
    622: }

    624: /*@C
    625:   TSRosWInitializePackage - This function initializes everything in the TSRosW package. It is called
    626:   from TSInitializePackage().

    628:   Level: developer

    630: .seealso: PetscInitialize()
    631: @*/
    632: PetscErrorCode TSRosWInitializePackage(void)
    633: {

    637:   if (TSRosWPackageInitialized) return(0);
    638:   TSRosWPackageInitialized = PETSC_TRUE;
    639:   TSRosWRegisterAll();
    640:   PetscRegisterFinalize(TSRosWFinalizePackage);
    641:   return(0);
    642: }

    644: /*@C
    645:   TSRosWFinalizePackage - This function destroys everything in the TSRosW package. It is
    646:   called from PetscFinalize().

    648:   Level: developer

    650: .seealso: PetscFinalize()
    651: @*/
    652: PetscErrorCode TSRosWFinalizePackage(void)
    653: {

    657:   TSRosWPackageInitialized = PETSC_FALSE;
    658:   TSRosWRegisterDestroy();
    659:   return(0);
    660: }

    662: /*@C
    663:    TSRosWRegister - register a Rosenbrock W scheme by providing the entries in the Butcher tableau and optionally embedded approximations and interpolation

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

    667:    Input Parameters:
    668: +  name - identifier for method
    669: .  order - approximation order of method
    670: .  s - number of stages, this is the dimension of the matrices below
    671: .  A - Table of propagated stage coefficients (dimension s*s, row-major), strictly lower triangular
    672: .  Gamma - Table of coefficients in implicit stage equations (dimension s*s, row-major), lower triangular with nonzero diagonal
    673: .  b - Step completion table (dimension s)
    674: .  bembed - Step completion table for a scheme of order one less (dimension s, NULL if no embedded scheme is available)
    675: .  pinterp - Order of the interpolation scheme, equal to the number of columns of binterpt
    676: -  binterpt - Coefficients of the interpolation formula (dimension s*pinterp)

    678:    Notes:
    679:    Several Rosenbrock W methods are provided, this function is only needed to create new methods.

    681:    Level: advanced

    683: .seealso: TSRosW
    684: @*/
    685: PetscErrorCode TSRosWRegister(TSRosWType name,PetscInt order,PetscInt s,const PetscReal A[],const PetscReal Gamma[],const PetscReal b[],const PetscReal bembed[],
    686:                               PetscInt pinterp,const PetscReal binterpt[])
    687: {
    688:   PetscErrorCode  ierr;
    689:   RosWTableauLink link;
    690:   RosWTableau     t;
    691:   PetscInt        i,j,k;
    692:   PetscScalar     *GammaInv;

    701:   TSRosWInitializePackage();
    702:   PetscNew(&link);
    703:   t        = &link->tab;
    704:   PetscStrallocpy(name,&t->name);
    705:   t->order = order;
    706:   t->s     = s;
    707:   PetscMalloc5(s*s,&t->A,s*s,&t->Gamma,s,&t->b,s,&t->ASum,s,&t->GammaSum);
    708:   PetscMalloc5(s*s,&t->At,s,&t->bt,s*s,&t->GammaInv,s,&t->GammaZeroDiag,s*s,&t->GammaExplicitCorr);
    709:   PetscArraycpy(t->A,A,s*s);
    710:   PetscArraycpy(t->Gamma,Gamma,s*s);
    711:   PetscArraycpy(t->GammaExplicitCorr,Gamma,s*s);
    712:   PetscArraycpy(t->b,b,s);
    713:   if (bembed) {
    714:     PetscMalloc2(s,&t->bembed,s,&t->bembedt);
    715:     PetscArraycpy(t->bembed,bembed,s);
    716:   }
    717:   for (i=0; i<s; i++) {
    718:     t->ASum[i]     = 0;
    719:     t->GammaSum[i] = 0;
    720:     for (j=0; j<s; j++) {
    721:       t->ASum[i]     += A[i*s+j];
    722:       t->GammaSum[i] += Gamma[i*s+j];
    723:     }
    724:   }
    725:   PetscMalloc1(s*s,&GammaInv); /* Need to use Scalar for inverse, then convert back to Real */
    726:   for (i=0; i<s*s; i++) GammaInv[i] = Gamma[i];
    727:   for (i=0; i<s; i++) {
    728:     if (Gamma[i*s+i] == 0.0) {
    729:       GammaInv[i*s+i] = 1.0;
    730:       t->GammaZeroDiag[i] = PETSC_TRUE;
    731:     } else {
    732:       t->GammaZeroDiag[i] = PETSC_FALSE;
    733:     }
    734:   }

    736:   switch (s) {
    737:   case 1: GammaInv[0] = 1./GammaInv[0]; break;
    738:   case 2: PetscKernel_A_gets_inverse_A_2(GammaInv,0,PETSC_FALSE,NULL); break;
    739:   case 3: PetscKernel_A_gets_inverse_A_3(GammaInv,0,PETSC_FALSE,NULL); break;
    740:   case 4: PetscKernel_A_gets_inverse_A_4(GammaInv,0,PETSC_FALSE,NULL); break;
    741:   case 5: {
    742:     PetscInt  ipvt5[5];
    743:     MatScalar work5[5*5];
    744:     PetscKernel_A_gets_inverse_A_5(GammaInv,ipvt5,work5,0,PETSC_FALSE,NULL); break;
    745:   }
    746:   case 6: PetscKernel_A_gets_inverse_A_6(GammaInv,0,PETSC_FALSE,NULL); break;
    747:   case 7: PetscKernel_A_gets_inverse_A_7(GammaInv,0,PETSC_FALSE,NULL); break;
    748:   default: SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_SUP,"Not implemented for %D stages",s);
    749:   }
    750:   for (i=0; i<s*s; i++) t->GammaInv[i] = PetscRealPart(GammaInv[i]);
    751:   PetscFree(GammaInv);

    753:   for (i=0; i<s; i++) {
    754:     for (k=0; k<i+1; k++) {
    755:       t->GammaExplicitCorr[i*s+k]=(t->GammaExplicitCorr[i*s+k])*(t->GammaInv[k*s+k]);
    756:       for (j=k+1; j<i+1; j++) {
    757:         t->GammaExplicitCorr[i*s+k]+=(t->GammaExplicitCorr[i*s+j])*(t->GammaInv[j*s+k]);
    758:       }
    759:     }
    760:   }

    762:   for (i=0; i<s; i++) {
    763:     for (j=0; j<s; j++) {
    764:       t->At[i*s+j] = 0;
    765:       for (k=0; k<s; k++) {
    766:         t->At[i*s+j] += t->A[i*s+k] * t->GammaInv[k*s+j];
    767:       }
    768:     }
    769:     t->bt[i] = 0;
    770:     for (j=0; j<s; j++) {
    771:       t->bt[i] += t->b[j] * t->GammaInv[j*s+i];
    772:     }
    773:     if (bembed) {
    774:       t->bembedt[i] = 0;
    775:       for (j=0; j<s; j++) {
    776:         t->bembedt[i] += t->bembed[j] * t->GammaInv[j*s+i];
    777:       }
    778:     }
    779:   }
    780:   t->ccfl = 1.0;                /* Fix this */

    782:   t->pinterp = pinterp;
    783:   PetscMalloc1(s*pinterp,&t->binterpt);
    784:   PetscArraycpy(t->binterpt,binterpt,s*pinterp);
    785:   link->next = RosWTableauList;
    786:   RosWTableauList = link;
    787:   return(0);
    788: }

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

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

    795:    Input Parameters:
    796: +  name - identifier for method
    797: .  gamma - leading coefficient (diagonal entry)
    798: .  a2 - design parameter, see Table 7.2 of Hairer&Wanner
    799: .  a3 - design parameter or PETSC_DEFAULT to satisfy one of the order five conditions (Eq 7.22)
    800: .  b3 - design parameter, see Table 7.2 of Hairer&Wanner
    801: .  beta43 - design parameter or PETSC_DEFAULT to use Equation 7.21 of Hairer&Wanner
    802: -  e4 - design parameter for embedded method, see coefficient E4 in ros4.f code from Hairer

    804:    Notes:
    805:    This routine encodes the design of fourth order Rosenbrock methods as described in Hairer and Wanner volume 2.
    806:    It is used here to implement several methods from the book and can be used to experiment with new methods.
    807:    It was written this way instead of by copying coefficients in order to provide better than double precision satisfaction of the order conditions.

    809:    Level: developer

    811: .seealso: TSRosW, TSRosWRegister()
    812: @*/
    813: PetscErrorCode TSRosWRegisterRos4(TSRosWType name,PetscReal gamma,PetscReal a2,PetscReal a3,PetscReal b3,PetscReal e4)
    814: {
    816:   /* Declare numeric constants so they can be quad precision without being truncated at double */
    817:   const PetscReal one = 1,two = 2,three = 3,four = 4,five = 5,six = 6,eight = 8,twelve = 12,twenty = 20,twentyfour = 24,
    818:     p32 = one/six - gamma + gamma*gamma,
    819:     p42 = one/eight - gamma/three,
    820:     p43 = one/twelve - gamma/three,
    821:     p44 = one/twentyfour - gamma/two + three/two*gamma*gamma - gamma*gamma*gamma,
    822:     p56 = one/twenty - gamma/four;
    823:   PetscReal   a4,a32,a42,a43,b1,b2,b4,beta2p,beta3p,beta4p,beta32,beta42,beta43,beta32beta2p,beta4jbetajp;
    824:   PetscReal   A[4][4],Gamma[4][4],b[4],bm[4];
    825:   PetscScalar M[3][3],rhs[3];

    828:   /* Step 1: choose Gamma (input) */
    829:   /* Step 2: choose a2,a3,a4; b1,b2,b3,b4 to satisfy order conditions */
    830:   if (a3 == PETSC_DEFAULT) a3 = (one/five - a2/four)/(one/four - a2/three); /* Eq 7.22 */
    831:   a4 = a3;                                                  /* consequence of 7.20 */

    833:   /* Solve order conditions 7.15a, 7.15c, 7.15e */
    834:   M[0][0] = one; M[0][1] = one;      M[0][2] = one;      /* 7.15a */
    835:   M[1][0] = 0.0; M[1][1] = a2*a2;    M[1][2] = a4*a4;    /* 7.15c */
    836:   M[2][0] = 0.0; M[2][1] = a2*a2*a2; M[2][2] = a4*a4*a4; /* 7.15e */
    837:   rhs[0]  = one - b3;
    838:   rhs[1]  = one/three - a3*a3*b3;
    839:   rhs[2]  = one/four - a3*a3*a3*b3;
    840:   PetscKernel_A_gets_inverse_A_3(&M[0][0],0,PETSC_FALSE,NULL);
    841:   b1      = PetscRealPart(M[0][0]*rhs[0] + M[0][1]*rhs[1] + M[0][2]*rhs[2]);
    842:   b2      = PetscRealPart(M[1][0]*rhs[0] + M[1][1]*rhs[1] + M[1][2]*rhs[2]);
    843:   b4      = PetscRealPart(M[2][0]*rhs[0] + M[2][1]*rhs[1] + M[2][2]*rhs[2]);

    845:   /* Step 3 */
    846:   beta43       = (p56 - a2*p43) / (b4*a3*a3*(a3 - a2)); /* 7.21 */
    847:   beta32beta2p =  p44 / (b4*beta43);                    /* 7.15h */
    848:   beta4jbetajp = (p32 - b3*beta32beta2p) / b4;
    849:   M[0][0]      = b2;                                    M[0][1] = b3;                 M[0][2] = b4;
    850:   M[1][0]      = a4*a4*beta32beta2p-a3*a3*beta4jbetajp; M[1][1] = a2*a2*beta4jbetajp; M[1][2] = -a2*a2*beta32beta2p;
    851:   M[2][0]      = b4*beta43*a3*a3-p43;                   M[2][1] = -b4*beta43*a2*a2;   M[2][2] = 0;
    852:   rhs[0]       = one/two - gamma; rhs[1] = 0; rhs[2] = -a2*a2*p32;
    853:   PetscKernel_A_gets_inverse_A_3(&M[0][0],0,PETSC_FALSE,NULL);
    854:   beta2p       = PetscRealPart(M[0][0]*rhs[0] + M[0][1]*rhs[1] + M[0][2]*rhs[2]);
    855:   beta3p       = PetscRealPart(M[1][0]*rhs[0] + M[1][1]*rhs[1] + M[1][2]*rhs[2]);
    856:   beta4p       = PetscRealPart(M[2][0]*rhs[0] + M[2][1]*rhs[1] + M[2][2]*rhs[2]);

    858:   /* Step 4: back-substitute */
    859:   beta32 = beta32beta2p / beta2p;
    860:   beta42 = (beta4jbetajp - beta43*beta3p) / beta2p;

    862:   /* Step 5: 7.15f and 7.20, then 7.16 */
    863:   a43 = 0;
    864:   a32 = p42 / (b3*a3*beta2p + b4*a4*beta2p);
    865:   a42 = a32;

    867:   A[0][0]     = 0;          A[0][1] = 0;   A[0][2] = 0;   A[0][3] = 0;
    868:   A[1][0]     = a2;         A[1][1] = 0;   A[1][2] = 0;   A[1][3] = 0;
    869:   A[2][0]     = a3-a32;     A[2][1] = a32; A[2][2] = 0;   A[2][3] = 0;
    870:   A[3][0]     = a4-a43-a42; A[3][1] = a42; A[3][2] = a43; A[3][3] = 0;
    871:   Gamma[0][0] = gamma;                        Gamma[0][1] = 0;              Gamma[0][2] = 0;              Gamma[0][3] = 0;
    872:   Gamma[1][0] = beta2p-A[1][0];               Gamma[1][1] = gamma;          Gamma[1][2] = 0;              Gamma[1][3] = 0;
    873:   Gamma[2][0] = beta3p-beta32-A[2][0];        Gamma[2][1] = beta32-A[2][1]; Gamma[2][2] = gamma;          Gamma[2][3] = 0;
    874:   Gamma[3][0] = beta4p-beta42-beta43-A[3][0]; Gamma[3][1] = beta42-A[3][1]; Gamma[3][2] = beta43-A[3][2]; Gamma[3][3] = gamma;
    875:   b[0] = b1; b[1] = b2; b[2] = b3; b[3] = b4;

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

    883:   {
    884:     const PetscReal misfit = a2*a2*bm[1] + a3*a3*bm[2] + a4*a4*bm[3] - one/three;
    885:     if (PetscAbs(misfit) > PETSC_SMALL) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Assumptions violated, could not construct a third order embedded method");
    886:   }
    887:   TSRosWRegister(name,4,4,&A[0][0],&Gamma[0][0],b,bm,0,NULL);
    888:   return(0);
    889: }

    891: /*
    892:  The step completion formula is

    894:  x1 = x0 + b^T Y

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

    899:  x1e = x0 + be^T Y
    900:      = x1 - b^T Y + be^T Y
    901:      = x1 + (be - b)^T Y

    903:  so we can evaluate the method of different order even after the step has been optimistically completed.
    904: */
    905: static PetscErrorCode TSEvaluateStep_RosW(TS ts,PetscInt order,Vec U,PetscBool *done)
    906: {
    907:   TS_RosW        *ros = (TS_RosW*)ts->data;
    908:   RosWTableau    tab  = ros->tableau;
    909:   PetscScalar    *w   = ros->work;
    910:   PetscInt       i;

    914:   if (order == tab->order) {
    915:     if (ros->status == TS_STEP_INCOMPLETE) { /* Use standard completion formula */
    916:       VecCopy(ts->vec_sol,U);
    917:       for (i=0; i<tab->s; i++) w[i] = tab->bt[i];
    918:       VecMAXPY(U,tab->s,w,ros->Y);
    919:     } else {VecCopy(ts->vec_sol,U);}
    920:     if (done) *done = PETSC_TRUE;
    921:     return(0);
    922:   } else if (order == tab->order-1) {
    923:     if (!tab->bembedt) goto unavailable;
    924:     if (ros->status == TS_STEP_INCOMPLETE) { /* Use embedded completion formula */
    925:       VecCopy(ts->vec_sol,U);
    926:       for (i=0; i<tab->s; i++) w[i] = tab->bembedt[i];
    927:       VecMAXPY(U,tab->s,w,ros->Y);
    928:     } else {                    /* Use rollback-and-recomplete formula (bembedt - bt) */
    929:       for (i=0; i<tab->s; i++) w[i] = tab->bembedt[i] - tab->bt[i];
    930:       VecCopy(ts->vec_sol,U);
    931:       VecMAXPY(U,tab->s,w,ros->Y);
    932:     }
    933:     if (done) *done = PETSC_TRUE;
    934:     return(0);
    935:   }
    936:   unavailable:
    937:   if (done) *done = PETSC_FALSE;
    938:   else SETERRQ3(PetscObjectComm((PetscObject)ts),PETSC_ERR_SUP,"Rosenbrock-W '%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);
    939:   return(0);
    940: }

    942: static PetscErrorCode TSRollBack_RosW(TS ts)
    943: {
    944:   TS_RosW        *ros = (TS_RosW*)ts->data;

    948:   VecCopy(ros->vec_sol_prev,ts->vec_sol);
    949:   return(0);
    950: }

    952: static PetscErrorCode TSStep_RosW(TS ts)
    953: {
    954:   TS_RosW         *ros = (TS_RosW*)ts->data;
    955:   RosWTableau     tab  = ros->tableau;
    956:   const PetscInt  s    = tab->s;
    957:   const PetscReal *At  = tab->At,*Gamma = tab->Gamma,*ASum = tab->ASum,*GammaInv = tab->GammaInv;
    958:   const PetscReal *GammaExplicitCorr = tab->GammaExplicitCorr;
    959:   const PetscBool *GammaZeroDiag = tab->GammaZeroDiag;
    960:   PetscScalar     *w   = ros->work;
    961:   Vec             *Y   = ros->Y,Ydot = ros->Ydot,Zdot = ros->Zdot,Zstage = ros->Zstage;
    962:   SNES            snes;
    963:   TSAdapt         adapt;
    964:   PetscInt        i,j,its,lits;
    965:   PetscInt        rejections = 0;
    966:   PetscBool       stageok,accept = PETSC_TRUE;
    967:   PetscReal       next_time_step = ts->time_step;
    968:   PetscErrorCode  ierr;
    969:   PetscInt        lag;

    972:   if (!ts->steprollback) {
    973:     VecCopy(ts->vec_sol,ros->vec_sol_prev);
    974:   }

    976:   ros->status = TS_STEP_INCOMPLETE;
    977:   while (!ts->reason && ros->status != TS_STEP_COMPLETE) {
    978:     const PetscReal h = ts->time_step;
    979:     for (i=0; i<s; i++) {
    980:       ros->stage_time = ts->ptime + h*ASum[i];
    981:       TSPreStage(ts,ros->stage_time);
    982:       if (GammaZeroDiag[i]) {
    983:         ros->stage_explicit = PETSC_TRUE;
    984:         ros->scoeff         = 1.;
    985:       } else {
    986:         ros->stage_explicit = PETSC_FALSE;
    987:         ros->scoeff         = 1./Gamma[i*s+i];
    988:       }

    990:       VecCopy(ts->vec_sol,Zstage);
    991:       for (j=0; j<i; j++) w[j] = At[i*s+j];
    992:       VecMAXPY(Zstage,i,w,Y);

    994:       for (j=0; j<i; j++) w[j] = 1./h * GammaInv[i*s+j];
    995:       VecZeroEntries(Zdot);
    996:       VecMAXPY(Zdot,i,w,Y);

    998:       /* Initial guess taken from last stage */
    999:       VecZeroEntries(Y[i]);

    1001:       if (!ros->stage_explicit) {
    1002:         TSGetSNES(ts,&snes);
    1003:         if (!ros->recompute_jacobian && !i) {
    1004:           SNESGetLagJacobian(snes,&lag);
    1005:           if (lag == 1) {  /* use did not set a nontrival lag, so lag over all stages */
    1006:             SNESSetLagJacobian(snes,-2); /* Recompute the Jacobian on this solve, but not again for the rest of the stages */
    1007:           }
    1008:         }
    1009:         SNESSolve(snes,NULL,Y[i]);
    1010:         if (!ros->recompute_jacobian && i == s-1 && lag == 1) {
    1011:           SNESSetLagJacobian(snes,lag); /* Set lag back to 1 so we know user did not set it */
    1012:         }
    1013:         SNESGetIterationNumber(snes,&its);
    1014:         SNESGetLinearSolveIterations(snes,&lits);
    1015:         ts->snes_its += its; ts->ksp_its += lits;
    1016:       } else {
    1017:         Mat J,Jp;
    1018:         VecZeroEntries(Ydot); /* Evaluate Y[i]=G(t,Ydot=0,Zstage) */
    1019:         TSComputeIFunction(ts,ros->stage_time,Zstage,Ydot,Y[i],PETSC_FALSE);
    1020:         VecScale(Y[i],-1.0);
    1021:         VecAXPY(Y[i],-1.0,Zdot); /*Y[i] = F(Zstage)-Zdot[=GammaInv*Y]*/

    1023:         VecZeroEntries(Zstage); /* Zstage = GammaExplicitCorr[i,j] * Y[j] */
    1024:         for (j=0; j<i; j++) w[j] = GammaExplicitCorr[i*s+j];
    1025:         VecMAXPY(Zstage,i,w,Y);

    1027:         /* Y[i] = Y[i] + Jac*Zstage[=Jac*GammaExplicitCorr[i,j] * Y[j]] */
    1028:         TSGetIJacobian(ts,&J,&Jp,NULL,NULL);
    1029:         TSComputeIJacobian(ts,ros->stage_time,ts->vec_sol,Ydot,0,J,Jp,PETSC_FALSE);
    1030:         MatMult(J,Zstage,Zdot);
    1031:         VecAXPY(Y[i],-1.0,Zdot);
    1032:         ts->ksp_its += 1;

    1034:         VecScale(Y[i],h);
    1035:       }
    1036:       TSPostStage(ts,ros->stage_time,i,Y);
    1037:       TSGetAdapt(ts,&adapt);
    1038:       TSAdaptCheckStage(adapt,ts,ros->stage_time,Y[i],&stageok);
    1039:       if (!stageok) goto reject_step;
    1040:     }

    1042:     ros->status = TS_STEP_INCOMPLETE;
    1043:     TSEvaluateStep_RosW(ts,tab->order,ts->vec_sol,NULL);
    1044:     ros->status = TS_STEP_PENDING;
    1045:     TSGetAdapt(ts,&adapt);
    1046:     TSAdaptCandidatesClear(adapt);
    1047:     TSAdaptCandidateAdd(adapt,tab->name,tab->order,1,tab->ccfl,(PetscReal)tab->s,PETSC_TRUE);
    1048:     TSAdaptChoose(adapt,ts,ts->time_step,NULL,&next_time_step,&accept);
    1049:     ros->status = accept ? TS_STEP_COMPLETE : TS_STEP_INCOMPLETE;
    1050:     if (!accept) { /* Roll back the current step */
    1051:       TSRollBack_RosW(ts);
    1052:       ts->time_step = next_time_step;
    1053:       goto reject_step;
    1054:     }

    1056:     ts->ptime += ts->time_step;
    1057:     ts->time_step = next_time_step;
    1058:     break;

    1060:   reject_step:
    1061:     ts->reject++; accept = PETSC_FALSE;
    1062:     if (!ts->reason && ++rejections > ts->max_reject && ts->max_reject >= 0) {
    1063:       ts->reason = TS_DIVERGED_STEP_REJECTED;
    1064:       PetscInfo2(ts,"Step=%D, step rejections %D greater than current TS allowed, stopping solve\n",ts->steps,rejections);
    1065:     }
    1066:   }
    1067:   return(0);
    1068: }

    1070: static PetscErrorCode TSInterpolate_RosW(TS ts,PetscReal itime,Vec U)
    1071: {
    1072:   TS_RosW         *ros = (TS_RosW*)ts->data;
    1073:   PetscInt        s    = ros->tableau->s,pinterp = ros->tableau->pinterp,i,j;
    1074:   PetscReal       h;
    1075:   PetscReal       tt,t;
    1076:   PetscScalar     *bt;
    1077:   const PetscReal *Bt = ros->tableau->binterpt;
    1078:   PetscErrorCode  ierr;
    1079:   const PetscReal *GammaInv = ros->tableau->GammaInv;
    1080:   PetscScalar     *w        = ros->work;
    1081:   Vec             *Y        = ros->Y;

    1084:   if (!Bt) SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_SUP,"TSRosW %s does not have an interpolation formula",ros->tableau->name);

    1086:   switch (ros->status) {
    1087:   case TS_STEP_INCOMPLETE:
    1088:   case TS_STEP_PENDING:
    1089:     h = ts->time_step;
    1090:     t = (itime - ts->ptime)/h;
    1091:     break;
    1092:   case TS_STEP_COMPLETE:
    1093:     h = ts->ptime - ts->ptime_prev;
    1094:     t = (itime - ts->ptime)/h + 1; /* In the interval [0,1] */
    1095:     break;
    1096:   default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
    1097:   }
    1098:   PetscMalloc1(s,&bt);
    1099:   for (i=0; i<s; i++) bt[i] = 0;
    1100:   for (j=0,tt=t; j<pinterp; j++,tt*=t) {
    1101:     for (i=0; i<s; i++) {
    1102:       bt[i] += Bt[i*pinterp+j] * tt;
    1103:     }
    1104:   }

    1106:   /* y(t+tt*h) = y(t) + Sum bt(tt) * GammaInv * Ydot */
    1107:   /* U <- 0*/
    1108:   VecZeroEntries(U);
    1109:   /* U <- Sum bt_i * GammaInv(i,1:i) * Y(1:i) */
    1110:   for (j=0; j<s; j++) w[j] = 0;
    1111:   for (j=0; j<s; j++) {
    1112:     for (i=j; i<s; i++) {
    1113:       w[j] +=  bt[i]*GammaInv[i*s+j];
    1114:     }
    1115:   }
    1116:   VecMAXPY(U,i,w,Y);
    1117:   /* U <- y(t) + U */
    1118:   VecAXPY(U,1,ros->vec_sol_prev);

    1120:   PetscFree(bt);
    1121:   return(0);
    1122: }

    1124: /*------------------------------------------------------------*/

    1126: static PetscErrorCode TSRosWTableauReset(TS ts)
    1127: {
    1128:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1129:   RosWTableau    tab  = ros->tableau;

    1133:   if (!tab) return(0);
    1134:   VecDestroyVecs(tab->s,&ros->Y);
    1135:   PetscFree(ros->work);
    1136:   return(0);
    1137: }

    1139: static PetscErrorCode TSReset_RosW(TS ts)
    1140: {
    1141:   TS_RosW        *ros = (TS_RosW*)ts->data;

    1145:   TSRosWTableauReset(ts);
    1146:   VecDestroy(&ros->Ydot);
    1147:   VecDestroy(&ros->Ystage);
    1148:   VecDestroy(&ros->Zdot);
    1149:   VecDestroy(&ros->Zstage);
    1150:   VecDestroy(&ros->vec_sol_prev);
    1151:   return(0);
    1152: }

    1154: static PetscErrorCode TSRosWGetVecs(TS ts,DM dm,Vec *Ydot,Vec *Zdot,Vec *Ystage,Vec *Zstage)
    1155: {
    1156:   TS_RosW        *rw = (TS_RosW*)ts->data;

    1160:   if (Ydot) {
    1161:     if (dm && dm != ts->dm) {
    1162:       DMGetNamedGlobalVector(dm,"TSRosW_Ydot",Ydot);
    1163:     } else *Ydot = rw->Ydot;
    1164:   }
    1165:   if (Zdot) {
    1166:     if (dm && dm != ts->dm) {
    1167:       DMGetNamedGlobalVector(dm,"TSRosW_Zdot",Zdot);
    1168:     } else *Zdot = rw->Zdot;
    1169:   }
    1170:   if (Ystage) {
    1171:     if (dm && dm != ts->dm) {
    1172:       DMGetNamedGlobalVector(dm,"TSRosW_Ystage",Ystage);
    1173:     } else *Ystage = rw->Ystage;
    1174:   }
    1175:   if (Zstage) {
    1176:     if (dm && dm != ts->dm) {
    1177:       DMGetNamedGlobalVector(dm,"TSRosW_Zstage",Zstage);
    1178:     } else *Zstage = rw->Zstage;
    1179:   }
    1180:   return(0);
    1181: }

    1183: static PetscErrorCode TSRosWRestoreVecs(TS ts,DM dm,Vec *Ydot,Vec *Zdot, Vec *Ystage, Vec *Zstage)
    1184: {

    1188:   if (Ydot) {
    1189:     if (dm && dm != ts->dm) {
    1190:       DMRestoreNamedGlobalVector(dm,"TSRosW_Ydot",Ydot);
    1191:     }
    1192:   }
    1193:   if (Zdot) {
    1194:     if (dm && dm != ts->dm) {
    1195:       DMRestoreNamedGlobalVector(dm,"TSRosW_Zdot",Zdot);
    1196:     }
    1197:   }
    1198:   if (Ystage) {
    1199:     if (dm && dm != ts->dm) {
    1200:       DMRestoreNamedGlobalVector(dm,"TSRosW_Ystage",Ystage);
    1201:     }
    1202:   }
    1203:   if (Zstage) {
    1204:     if (dm && dm != ts->dm) {
    1205:       DMRestoreNamedGlobalVector(dm,"TSRosW_Zstage",Zstage);
    1206:     }
    1207:   }
    1208:   return(0);
    1209: }

    1211: static PetscErrorCode DMCoarsenHook_TSRosW(DM fine,DM coarse,void *ctx)
    1212: {
    1214:   return(0);
    1215: }

    1217: static PetscErrorCode DMRestrictHook_TSRosW(DM fine,Mat restrct,Vec rscale,Mat inject,DM coarse,void *ctx)
    1218: {
    1219:   TS             ts = (TS)ctx;
    1221:   Vec            Ydot,Zdot,Ystage,Zstage;
    1222:   Vec            Ydotc,Zdotc,Ystagec,Zstagec;

    1225:   TSRosWGetVecs(ts,fine,&Ydot,&Ystage,&Zdot,&Zstage);
    1226:   TSRosWGetVecs(ts,coarse,&Ydotc,&Ystagec,&Zdotc,&Zstagec);
    1227:   MatRestrict(restrct,Ydot,Ydotc);
    1228:   VecPointwiseMult(Ydotc,rscale,Ydotc);
    1229:   MatRestrict(restrct,Ystage,Ystagec);
    1230:   VecPointwiseMult(Ystagec,rscale,Ystagec);
    1231:   MatRestrict(restrct,Zdot,Zdotc);
    1232:   VecPointwiseMult(Zdotc,rscale,Zdotc);
    1233:   MatRestrict(restrct,Zstage,Zstagec);
    1234:   VecPointwiseMult(Zstagec,rscale,Zstagec);
    1235:   TSRosWRestoreVecs(ts,fine,&Ydot,&Ystage,&Zdot,&Zstage);
    1236:   TSRosWRestoreVecs(ts,coarse,&Ydotc,&Ystagec,&Zdotc,&Zstagec);
    1237:   return(0);
    1238: }

    1240: static PetscErrorCode DMSubDomainHook_TSRosW(DM fine,DM coarse,void *ctx)
    1241: {
    1243:   return(0);
    1244: }

    1246: static PetscErrorCode DMSubDomainRestrictHook_TSRosW(DM dm,VecScatter gscat,VecScatter lscat,DM subdm,void *ctx)
    1247: {
    1248:   TS             ts = (TS)ctx;
    1250:   Vec            Ydot,Zdot,Ystage,Zstage;
    1251:   Vec            Ydots,Zdots,Ystages,Zstages;

    1254:   TSRosWGetVecs(ts,dm,&Ydot,&Ystage,&Zdot,&Zstage);
    1255:   TSRosWGetVecs(ts,subdm,&Ydots,&Ystages,&Zdots,&Zstages);

    1257:   VecScatterBegin(gscat,Ydot,Ydots,INSERT_VALUES,SCATTER_FORWARD);
    1258:   VecScatterEnd(gscat,Ydot,Ydots,INSERT_VALUES,SCATTER_FORWARD);

    1260:   VecScatterBegin(gscat,Ystage,Ystages,INSERT_VALUES,SCATTER_FORWARD);
    1261:   VecScatterEnd(gscat,Ystage,Ystages,INSERT_VALUES,SCATTER_FORWARD);

    1263:   VecScatterBegin(gscat,Zdot,Zdots,INSERT_VALUES,SCATTER_FORWARD);
    1264:   VecScatterEnd(gscat,Zdot,Zdots,INSERT_VALUES,SCATTER_FORWARD);

    1266:   VecScatterBegin(gscat,Zstage,Zstages,INSERT_VALUES,SCATTER_FORWARD);
    1267:   VecScatterEnd(gscat,Zstage,Zstages,INSERT_VALUES,SCATTER_FORWARD);

    1269:   TSRosWRestoreVecs(ts,dm,&Ydot,&Ystage,&Zdot,&Zstage);
    1270:   TSRosWRestoreVecs(ts,subdm,&Ydots,&Ystages,&Zdots,&Zstages);
    1271:   return(0);
    1272: }

    1274: /*
    1275:   This defines the nonlinear equation that is to be solved with SNES
    1276:   G(U) = F[t0+Theta*dt, U, (U-U0)*shift] = 0
    1277: */
    1278: static PetscErrorCode SNESTSFormFunction_RosW(SNES snes,Vec U,Vec F,TS ts)
    1279: {
    1280:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1282:   Vec            Ydot,Zdot,Ystage,Zstage;
    1283:   PetscReal      shift = ros->scoeff / ts->time_step;
    1284:   DM             dm,dmsave;

    1287:   SNESGetDM(snes,&dm);
    1288:   TSRosWGetVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
    1289:   VecWAXPY(Ydot,shift,U,Zdot);    /* Ydot = shift*U + Zdot */
    1290:   VecWAXPY(Ystage,1.0,U,Zstage);  /* Ystage = U + Zstage */
    1291:   dmsave = ts->dm;
    1292:   ts->dm = dm;
    1293:   TSComputeIFunction(ts,ros->stage_time,Ystage,Ydot,F,PETSC_FALSE);
    1294:   ts->dm = dmsave;
    1295:   TSRosWRestoreVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
    1296:   return(0);
    1297: }

    1299: static PetscErrorCode SNESTSFormJacobian_RosW(SNES snes,Vec U,Mat A,Mat B,TS ts)
    1300: {
    1301:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1302:   Vec            Ydot,Zdot,Ystage,Zstage;
    1303:   PetscReal      shift = ros->scoeff / ts->time_step;
    1305:   DM             dm,dmsave;

    1308:   /* ros->Ydot and ros->Ystage have already been computed in SNESTSFormFunction_RosW (SNES guarantees this) */
    1309:   SNESGetDM(snes,&dm);
    1310:   TSRosWGetVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
    1311:   dmsave = ts->dm;
    1312:   ts->dm = dm;
    1313:   TSComputeIJacobian(ts,ros->stage_time,Ystage,Ydot,shift,A,B,PETSC_TRUE);
    1314:   ts->dm = dmsave;
    1315:   TSRosWRestoreVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
    1316:   return(0);
    1317: }

    1319: static PetscErrorCode TSRosWTableauSetUp(TS ts)
    1320: {
    1321:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1322:   RosWTableau    tab  = ros->tableau;

    1326:   VecDuplicateVecs(ts->vec_sol,tab->s,&ros->Y);
    1327:   PetscMalloc1(tab->s,&ros->work);
    1328:   return(0);
    1329: }

    1331: static PetscErrorCode TSSetUp_RosW(TS ts)
    1332: {
    1333:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1335:   DM             dm;
    1336:   SNES           snes;
    1337:   TSRHSJacobian  rhsjacobian;

    1340:   TSRosWTableauSetUp(ts);
    1341:   VecDuplicate(ts->vec_sol,&ros->Ydot);
    1342:   VecDuplicate(ts->vec_sol,&ros->Ystage);
    1343:   VecDuplicate(ts->vec_sol,&ros->Zdot);
    1344:   VecDuplicate(ts->vec_sol,&ros->Zstage);
    1345:   VecDuplicate(ts->vec_sol,&ros->vec_sol_prev);
    1346:   TSGetDM(ts,&dm);
    1347:   DMCoarsenHookAdd(dm,DMCoarsenHook_TSRosW,DMRestrictHook_TSRosW,ts);
    1348:   DMSubDomainHookAdd(dm,DMSubDomainHook_TSRosW,DMSubDomainRestrictHook_TSRosW,ts);
    1349:   /* Rosenbrock methods are linearly implicit, so set that unless the user has specifically asked for something else */
    1350:   TSGetSNES(ts,&snes);
    1351:   if (!((PetscObject)snes)->type_name) {
    1352:     SNESSetType(snes,SNESKSPONLY);
    1353:   }
    1354:   DMTSGetRHSJacobian(dm,&rhsjacobian,NULL);
    1355:   if (rhsjacobian == TSComputeRHSJacobianConstant) {
    1356:     Mat Amat,Pmat;

    1358:     /* Set the SNES matrix to be different from the RHS matrix because there is no way to reconstruct shift*M-J */
    1359:     SNESGetJacobian(snes,&Amat,&Pmat,NULL,NULL);
    1360:     if (Amat && Amat == ts->Arhs) {
    1361:       if (Amat == Pmat) {
    1362:         MatDuplicate(ts->Arhs,MAT_COPY_VALUES,&Amat);
    1363:         SNESSetJacobian(snes,Amat,Amat,NULL,NULL);
    1364:       } else {
    1365:         MatDuplicate(ts->Arhs,MAT_COPY_VALUES,&Amat);
    1366:         SNESSetJacobian(snes,Amat,NULL,NULL,NULL);
    1367:         if (Pmat && Pmat == ts->Brhs) {
    1368:           MatDuplicate(ts->Brhs,MAT_COPY_VALUES,&Pmat);
    1369:           SNESSetJacobian(snes,NULL,Pmat,NULL,NULL);
    1370:           MatDestroy(&Pmat);
    1371:         }
    1372:       }
    1373:       MatDestroy(&Amat);
    1374:     }
    1375:   }
    1376:   return(0);
    1377: }
    1378: /*------------------------------------------------------------*/

    1380: static PetscErrorCode TSSetFromOptions_RosW(PetscOptionItems *PetscOptionsObject,TS ts)
    1381: {
    1382:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1384:   SNES           snes;

    1387:   PetscOptionsHead(PetscOptionsObject,"RosW ODE solver options");
    1388:   {
    1389:     RosWTableauLink link;
    1390:     PetscInt        count,choice;
    1391:     PetscBool       flg;
    1392:     const char      **namelist;

    1394:     for (link=RosWTableauList,count=0; link; link=link->next,count++) ;
    1395:     PetscMalloc1(count,(char***)&namelist);
    1396:     for (link=RosWTableauList,count=0; link; link=link->next,count++) namelist[count] = link->tab.name;
    1397:     PetscOptionsEList("-ts_rosw_type","Family of Rosenbrock-W method","TSRosWSetType",(const char*const*)namelist,count,ros->tableau->name,&choice,&flg);
    1398:     if (flg) {TSRosWSetType(ts,namelist[choice]);}
    1399:     PetscFree(namelist);

    1401:     PetscOptionsBool("-ts_rosw_recompute_jacobian","Recompute the Jacobian at each stage","TSRosWSetRecomputeJacobian",ros->recompute_jacobian,&ros->recompute_jacobian,NULL);
    1402:   }
    1403:   PetscOptionsTail();
    1404:   /* Rosenbrock methods are linearly implicit, so set that unless the user has specifically asked for something else */
    1405:   TSGetSNES(ts,&snes);
    1406:   if (!((PetscObject)snes)->type_name) {
    1407:     SNESSetType(snes,SNESKSPONLY);
    1408:   }
    1409:   return(0);
    1410: }

    1412: static PetscErrorCode TSView_RosW(TS ts,PetscViewer viewer)
    1413: {
    1414:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1415:   PetscBool      iascii;

    1419:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
    1420:   if (iascii) {
    1421:     RosWTableau tab  = ros->tableau;
    1422:     TSRosWType  rostype;
    1423:     char        buf[512];
    1424:     PetscInt    i;
    1425:     PetscReal   abscissa[512];
    1426:     TSRosWGetType(ts,&rostype);
    1427:     PetscViewerASCIIPrintf(viewer,"  Rosenbrock-W %s\n",rostype);
    1428:     PetscFormatRealArray(buf,sizeof(buf),"% 8.6f",tab->s,tab->ASum);
    1429:     PetscViewerASCIIPrintf(viewer,"  Abscissa of A       = %s\n",buf);
    1430:     for (i=0; i<tab->s; i++) abscissa[i] = tab->ASum[i] + tab->Gamma[i];
    1431:     PetscFormatRealArray(buf,sizeof(buf),"% 8.6f",tab->s,abscissa);
    1432:     PetscViewerASCIIPrintf(viewer,"  Abscissa of A+Gamma = %s\n",buf);
    1433:   }
    1434:   return(0);
    1435: }

    1437: static PetscErrorCode TSLoad_RosW(TS ts,PetscViewer viewer)
    1438: {
    1440:   SNES           snes;
    1441:   TSAdapt        adapt;

    1444:   TSGetAdapt(ts,&adapt);
    1445:   TSAdaptLoad(adapt,viewer);
    1446:   TSGetSNES(ts,&snes);
    1447:   SNESLoad(snes,viewer);
    1448:   /* function and Jacobian context for SNES when used with TS is always ts object */
    1449:   SNESSetFunction(snes,NULL,NULL,ts);
    1450:   SNESSetJacobian(snes,NULL,NULL,NULL,ts);
    1451:   return(0);
    1452: }

    1454: /*@C
    1455:   TSRosWSetType - Set the type of Rosenbrock-W scheme

    1457:   Logically collective

    1459:   Input Parameters:
    1460: +  ts - timestepping context
    1461: -  roswtype - type of Rosenbrock-W scheme

    1463:   Level: beginner

    1465: .seealso: TSRosWGetType(), TSROSW, TSROSW2M, TSROSW2P, TSROSWRA3PW, TSROSWRA34PW2, TSROSWRODAS3, TSROSWSANDU3, TSROSWASSP3P3S1C, TSROSWLASSP3P4S2C, TSROSWLLSSP3P4S2C, TSROSWARK3
    1466: @*/
    1467: PetscErrorCode TSRosWSetType(TS ts,TSRosWType roswtype)
    1468: {

    1474:   PetscTryMethod(ts,"TSRosWSetType_C",(TS,TSRosWType),(ts,roswtype));
    1475:   return(0);
    1476: }

    1478: /*@C
    1479:   TSRosWGetType - Get the type of Rosenbrock-W scheme

    1481:   Logically collective

    1483:   Input Parameter:
    1484: .  ts - timestepping context

    1486:   Output Parameter:
    1487: .  rostype - type of Rosenbrock-W scheme

    1489:   Level: intermediate

    1491: .seealso: TSRosWGetType()
    1492: @*/
    1493: PetscErrorCode TSRosWGetType(TS ts,TSRosWType *rostype)
    1494: {

    1499:   PetscUseMethod(ts,"TSRosWGetType_C",(TS,TSRosWType*),(ts,rostype));
    1500:   return(0);
    1501: }

    1503: /*@C
    1504:   TSRosWSetRecomputeJacobian - Set whether to recompute the Jacobian at each stage. The default is to update the Jacobian once per step.

    1506:   Logically collective

    1508:   Input Parameters:
    1509: +  ts - timestepping context
    1510: -  flg - PETSC_TRUE to recompute the Jacobian at each stage

    1512:   Level: intermediate

    1514: .seealso: TSRosWGetType()
    1515: @*/
    1516: PetscErrorCode TSRosWSetRecomputeJacobian(TS ts,PetscBool flg)
    1517: {

    1522:   PetscTryMethod(ts,"TSRosWSetRecomputeJacobian_C",(TS,PetscBool),(ts,flg));
    1523:   return(0);
    1524: }

    1526: static PetscErrorCode  TSRosWGetType_RosW(TS ts,TSRosWType *rostype)
    1527: {
    1528:   TS_RosW        *ros = (TS_RosW*)ts->data;

    1531:   *rostype = ros->tableau->name;
    1532:   return(0);
    1533: }

    1535: static PetscErrorCode  TSRosWSetType_RosW(TS ts,TSRosWType rostype)
    1536: {
    1537:   TS_RosW         *ros = (TS_RosW*)ts->data;
    1538:   PetscErrorCode  ierr;
    1539:   PetscBool       match;
    1540:   RosWTableauLink link;

    1543:   if (ros->tableau) {
    1544:     PetscStrcmp(ros->tableau->name,rostype,&match);
    1545:     if (match) return(0);
    1546:   }
    1547:   for (link = RosWTableauList; link; link=link->next) {
    1548:     PetscStrcmp(link->tab.name,rostype,&match);
    1549:     if (match) {
    1550:       if (ts->setupcalled) {TSRosWTableauReset(ts);}
    1551:       ros->tableau = &link->tab;
    1552:       if (ts->setupcalled) {TSRosWTableauSetUp(ts);}
    1553:       ts->default_adapt_type = ros->tableau->bembed ? TSADAPTBASIC : TSADAPTNONE;
    1554:       return(0);
    1555:     }
    1556:   }
    1557:   SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_ARG_UNKNOWN_TYPE,"Could not find '%s'",rostype);
    1558: }

    1560: static PetscErrorCode  TSRosWSetRecomputeJacobian_RosW(TS ts,PetscBool flg)
    1561: {
    1562:   TS_RosW *ros = (TS_RosW*)ts->data;

    1565:   ros->recompute_jacobian = flg;
    1566:   return(0);
    1567: }

    1569: static PetscErrorCode TSDestroy_RosW(TS ts)
    1570: {

    1574:   TSReset_RosW(ts);
    1575:   if (ts->dm) {
    1576:     DMCoarsenHookRemove(ts->dm,DMCoarsenHook_TSRosW,DMRestrictHook_TSRosW,ts);
    1577:     DMSubDomainHookRemove(ts->dm,DMSubDomainHook_TSRosW,DMSubDomainRestrictHook_TSRosW,ts);
    1578:   }
    1579:   PetscFree(ts->data);
    1580:   PetscObjectComposeFunction((PetscObject)ts,"TSRosWGetType_C",NULL);
    1581:   PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetType_C",NULL);
    1582:   PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetRecomputeJacobian_C",NULL);
    1583:   return(0);
    1584: }

    1586: /* ------------------------------------------------------------ */
    1587: /*MC
    1588:       TSROSW - ODE solver using Rosenbrock-W schemes

    1590:   These methods are intended for problems with well-separated time scales, especially when a slow scale is strongly
    1591:   nonlinear such that it is expensive to solve with a fully implicit method. The user should provide the stiff part
    1592:   of the equation using TSSetIFunction() and the non-stiff part with TSSetRHSFunction().

    1594:   Notes:
    1595:   This method currently only works with autonomous ODE and DAE.

    1597:   Consider trying TSARKIMEX if the stiff part is strongly nonlinear.

    1599:   Since this uses a single linear solve per time-step if you wish to lag the jacobian or preconditioner computation you must use also -snes_lag_jacobian_persists true or -snes_lag_jacobian_preconditioner true

    1601:   Developer Notes:
    1602:   Rosenbrock-W methods are typically specified for autonomous ODE

    1604: $  udot = f(u)

    1606:   by the stage equations

    1608: $  k_i = h f(u_0 + sum_j alpha_ij k_j) + h J sum_j gamma_ij k_j

    1610:   and step completion formula

    1612: $  u_1 = u_0 + sum_j b_j k_j

    1614:   with step size h and coefficients alpha_ij, gamma_ij, and b_i. Implementing the method in this form would require f(u)
    1615:   and the Jacobian J to be available, in addition to the shifted matrix I - h gamma_ii J. Following Hairer and Wanner,
    1616:   we define new variables for the stage equations

    1618: $  y_i = gamma_ij k_j

    1620:   The k_j can be recovered because Gamma is invertible. Let C be the lower triangular part of Gamma^{-1} and define

    1622: $  A = Alpha Gamma^{-1}, bt^T = b^T Gamma^{-1}

    1624:   to rewrite the method as

    1626: $  [M/(h gamma_ii) - J] y_i = f(u_0 + sum_j a_ij y_j) + M sum_j (c_ij/h) y_j
    1627: $  u_1 = u_0 + sum_j bt_j y_j

    1629:    where we have introduced the mass matrix M. Continue by defining

    1631: $  ydot_i = 1/(h gamma_ii) y_i - sum_j (c_ij/h) y_j

    1633:    or, more compactly in tensor notation

    1635: $  Ydot = 1/h (Gamma^{-1} \otimes I) Y .

    1637:    Note that Gamma^{-1} is lower triangular. With this definition of Ydot in terms of known quantities and the current
    1638:    stage y_i, the stage equations reduce to performing one Newton step (typically with a lagged Jacobian) on the
    1639:    equation

    1641: $  g(u_0 + sum_j a_ij y_j + y_i, ydot_i) = 0

    1643:    with initial guess y_i = 0.

    1645:   Level: beginner

    1647: .seealso:  TSCreate(), TS, TSSetType(), TSRosWSetType(), TSRosWRegister(), TSROSWTHETA1, TSROSWTHETA2, TSROSW2M, TSROSW2P, TSROSWRA3PW, TSROSWRA34PW2, TSROSWRODAS3,
    1648:            TSROSWSANDU3, TSROSWASSP3P3S1C, TSROSWLASSP3P4S2C, TSROSWLLSSP3P4S2C, TSROSWGRK4T, TSROSWSHAMP4, TSROSWVELDD4, TSROSW4L
    1649: M*/
    1650: PETSC_EXTERN PetscErrorCode TSCreate_RosW(TS ts)
    1651: {
    1652:   TS_RosW        *ros;

    1656:   TSRosWInitializePackage();

    1658:   ts->ops->reset          = TSReset_RosW;
    1659:   ts->ops->destroy        = TSDestroy_RosW;
    1660:   ts->ops->view           = TSView_RosW;
    1661:   ts->ops->load           = TSLoad_RosW;
    1662:   ts->ops->setup          = TSSetUp_RosW;
    1663:   ts->ops->step           = TSStep_RosW;
    1664:   ts->ops->interpolate    = TSInterpolate_RosW;
    1665:   ts->ops->evaluatestep   = TSEvaluateStep_RosW;
    1666:   ts->ops->rollback       = TSRollBack_RosW;
    1667:   ts->ops->setfromoptions = TSSetFromOptions_RosW;
    1668:   ts->ops->snesfunction   = SNESTSFormFunction_RosW;
    1669:   ts->ops->snesjacobian   = SNESTSFormJacobian_RosW;

    1671:   ts->usessnes = PETSC_TRUE;

    1673:   PetscNewLog(ts,&ros);
    1674:   ts->data = (void*)ros;

    1676:   PetscObjectComposeFunction((PetscObject)ts,"TSRosWGetType_C",TSRosWGetType_RosW);
    1677:   PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetType_C",TSRosWSetType_RosW);
    1678:   PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetRecomputeJacobian_C",TSRosWSetRecomputeJacobian_RosW);

    1680:   TSRosWSetType(ts,TSRosWDefault);
    1681:   return(0);
    1682: }