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: .  * - 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: .  * - 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: .  * - 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: .  * - 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: +   * -  Kaps and Rentrop, Generalized Runge Kutta methods of order four with stepsize control for stiff ordinary differential equations, 1979.
    225: -   * -  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: +   * -  Shampine, Implementation of Rosenbrock methods, 1982.
    245: -   * -  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: +   * -  van Veldhuizen, D stability and Kaps Rentrop methods, 1984.
    265: -   * -  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: .  * -   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: {
    304:   if (TSRosWRegisterAllCalled) return 0;
    305:   TSRosWRegisterAllCalled = PETSC_TRUE;

    307:   {
    308:     const PetscReal A = 0;
    309:     const PetscReal Gamma = 1;
    310:     const PetscReal b = 1;
    311:     const PetscReal binterpt=1;

    313:     TSRosWRegister(TSROSWTHETA1,1,1,&A,&Gamma,&b,NULL,1,&binterpt);
    314:   }

    316:   {
    317:     const PetscReal A = 0;
    318:     const PetscReal Gamma = 0.5;
    319:     const PetscReal b = 1;
    320:     const PetscReal binterpt=1;

    322:     TSRosWRegister(TSROSWTHETA2,2,1,&A,&Gamma,&b,NULL,1,&binterpt);
    323:   }

    325:   {
    326:     /*const PetscReal g = 1. + 1./PetscSqrtReal(2.0);   Direct evaluation: 1.707106781186547524401. Used for setting up arrays of values known at compile time below. */
    327:     const PetscReal
    328:       A[2][2]     = {{0,0}, {1.,0}},
    329:       Gamma[2][2] = {{1.707106781186547524401,0}, {-2.*1.707106781186547524401,1.707106781186547524401}},
    330:       b[2]        = {0.5,0.5},
    331:       b1[2]       = {1.0,0.0};
    332:     PetscReal binterpt[2][2];
    333:     binterpt[0][0] = 1.707106781186547524401 - 1.0;
    334:     binterpt[1][0] = 2.0 - 1.707106781186547524401;
    335:     binterpt[0][1] = 1.707106781186547524401 - 1.5;
    336:     binterpt[1][1] = 1.5 - 1.707106781186547524401;

    338:     TSRosWRegister(TSROSW2P,2,2,&A[0][0],&Gamma[0][0],b,b1,2,&binterpt[0][0]);
    339:   }
    340:   {
    341:     /*const PetscReal g = 1. - 1./PetscSqrtReal(2.0);   Direct evaluation: 0.2928932188134524755992. Used for setting up arrays of values known at compile time below. */
    342:     const PetscReal
    343:       A[2][2]     = {{0,0}, {1.,0}},
    344:       Gamma[2][2] = {{0.2928932188134524755992,0}, {-2.*0.2928932188134524755992,0.2928932188134524755992}},
    345:       b[2]        = {0.5,0.5},
    346:       b1[2]       = {1.0,0.0};
    347:     PetscReal binterpt[2][2];
    348:     binterpt[0][0] = 0.2928932188134524755992 - 1.0;
    349:     binterpt[1][0] = 2.0 - 0.2928932188134524755992;
    350:     binterpt[0][1] = 0.2928932188134524755992 - 1.5;
    351:     binterpt[1][1] = 1.5 - 0.2928932188134524755992;

    353:     TSRosWRegister(TSROSW2M,2,2,&A[0][0],&Gamma[0][0],b,b1,2,&binterpt[0][0]);
    354:   }
    355:   {
    356:     /*const PetscReal g = 7.8867513459481287e-01; Directly written in-place below */
    357:     PetscReal binterpt[3][2];
    358:     const PetscReal
    359:       A[3][3] = {{0,0,0},
    360:                  {1.5773502691896257e+00,0,0},
    361:                  {0.5,0,0}},
    362:       Gamma[3][3] = {{7.8867513459481287e-01,0,0},
    363:                      {-1.5773502691896257e+00,7.8867513459481287e-01,0},
    364:                      {-6.7075317547305480e-01,-1.7075317547305482e-01,7.8867513459481287e-01}},
    365:       b[3]  = {1.0566243270259355e-01,4.9038105676657971e-02,8.4529946162074843e-01},
    366:       b2[3] = {-1.7863279495408180e-01,1./3.,8.4529946162074843e-01};

    368:       binterpt[0][0] = -0.8094010767585034;
    369:       binterpt[1][0] = -0.5;
    370:       binterpt[2][0] = 2.3094010767585034;
    371:       binterpt[0][1] = 0.9641016151377548;
    372:       binterpt[1][1] = 0.5;
    373:       binterpt[2][1] = -1.4641016151377548;

    375:       TSRosWRegister(TSROSWRA3PW,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]);
    376:   }
    377:   {
    378:     PetscReal  binterpt[4][3];
    379:     /*const PetscReal g = 4.3586652150845900e-01; Directly written in-place below */
    380:     const PetscReal
    381:       A[4][4] = {{0,0,0,0},
    382:                  {8.7173304301691801e-01,0,0,0},
    383:                  {8.4457060015369423e-01,-1.1299064236484185e-01,0,0},
    384:                  {0,0,1.,0}},
    385:       Gamma[4][4] = {{4.3586652150845900e-01,0,0,0},
    386:                      {-8.7173304301691801e-01,4.3586652150845900e-01,0,0},
    387:                      {-9.0338057013044082e-01,5.4180672388095326e-02,4.3586652150845900e-01,0},
    388:                      {2.4212380706095346e-01,-1.2232505839045147e+00,5.4526025533510214e-01,4.3586652150845900e-01}},
    389:       b[4]  = {2.4212380706095346e-01,-1.2232505839045147e+00,1.5452602553351020e+00,4.3586652150845900e-01},
    390:       b2[4] = {3.7810903145819369e-01,-9.6042292212423178e-02,5.0000000000000000e-01,2.1793326075422950e-01};

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

    405:     TSRosWRegister(TSROSWRA34PW2,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
    406:   }
    407:   {
    408:     /* const PetscReal g = 0.5;       Directly written in-place below */
    409:     const PetscReal
    410:       A[4][4] = {{0,0,0,0},
    411:                  {0,0,0,0},
    412:                  {1.,0,0,0},
    413:                  {0.75,-0.25,0.5,0}},
    414:       Gamma[4][4] = {{0.5,0,0,0},
    415:                      {1.,0.5,0,0},
    416:                      {-0.25,-0.25,0.5,0},
    417:                      {1./12,1./12,-2./3,0.5}},
    418:       b[4]  = {5./6,-1./6,-1./6,0.5},
    419:       b2[4] = {0.75,-0.25,0.5,0};

    421:     TSRosWRegister(TSROSWRODAS3,3,4,&A[0][0],&Gamma[0][0],b,b2,0,NULL);
    422:   }
    423:   {
    424:     /*const PetscReal g = 0.43586652150845899941601945119356;       Directly written in-place below */
    425:     const PetscReal
    426:       A[3][3] = {{0,0,0},
    427:                  {0.43586652150845899941601945119356,0,0},
    428:                  {0.43586652150845899941601945119356,0,0}},
    429:       Gamma[3][3] = {{0.43586652150845899941601945119356,0,0},
    430:                      {-0.19294655696029095575009695436041,0.43586652150845899941601945119356,0},
    431:                      {0,1.74927148125794685173529749738960,0.43586652150845899941601945119356}},
    432:       b[3]  = {-0.75457412385404315829818998646589,1.94100407061964420292840123379419,-0.18642994676560104463021124732829},
    433:       b2[3] = {-1.53358745784149585370766523913002,2.81745131148625772213931745457622,-0.28386385364476186843165221544619};

    435:     PetscReal binterpt[3][2];
    436:     binterpt[0][0] = 3.793692883777660870425141387941;
    437:     binterpt[1][0] = -2.918692883777660870425141387941;
    438:     binterpt[2][0] = 0.125;
    439:     binterpt[0][1] = -0.725741064379812106687651020584;
    440:     binterpt[1][1] = 0.559074397713145440020984353917;
    441:     binterpt[2][1] = 0.16666666666666666666666666666667;

    443:     TSRosWRegister(TSROSWSANDU3,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]);
    444:   }
    445:   {
    446:     /*const PetscReal s3 = PetscSqrtReal(3.),g = (3.0+s3)/6.0;
    447:      * Direct evaluation: s3 = 1.732050807568877293527;
    448:      *                     g = 0.7886751345948128822546;
    449:      * Values are directly inserted below to ensure availability at compile time (compiler warnings otherwise...) */
    450:     const PetscReal
    451:       A[3][3] = {{0,0,0},
    452:                  {1,0,0},
    453:                  {0.25,0.25,0}},
    454:       Gamma[3][3] = {{0,0,0},
    455:                      {(-3.0-1.732050807568877293527)/6.0,0.7886751345948128822546,0},
    456:                      {(-3.0-1.732050807568877293527)/24.0,(-3.0-1.732050807568877293527)/8.0,0.7886751345948128822546}},
    457:       b[3]  = {1./6.,1./6.,2./3.},
    458:       b2[3] = {1./4.,1./4.,1./2.};
    459:     PetscReal binterpt[3][2];

    461:     binterpt[0][0]=0.089316397477040902157517886164709;
    462:     binterpt[1][0]=-0.91068360252295909784248211383529;
    463:     binterpt[2][0]=1.8213672050459181956849642276706;
    464:     binterpt[0][1]=0.077350269189625764509148780501957;
    465:     binterpt[1][1]=1.077350269189625764509148780502;
    466:     binterpt[2][1]=-1.1547005383792515290182975610039;

    468:     TSRosWRegister(TSROSWASSP3P3S1C,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]);
    469:   }

    471:   {
    472:     const PetscReal
    473:       A[4][4] = {{0,0,0,0},
    474:                  {1./2.,0,0,0},
    475:                  {1./2.,1./2.,0,0},
    476:                  {1./6.,1./6.,1./6.,0}},
    477:       Gamma[4][4] = {{1./2.,0,0,0},
    478:                      {0.0,1./4.,0,0},
    479:                      {-2.,-2./3.,2./3.,0},
    480:                      {1./2.,5./36.,-2./9,0}},
    481:       b[4]  = {1./6.,1./6.,1./6.,1./2.},
    482:       b2[4] = {1./8.,3./4.,1./8.,0};
    483:     PetscReal binterpt[4][3];

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

    498:     TSRosWRegister(TSROSWLASSP3P4S2C,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
    499:   }

    501:   {
    502:     const PetscReal
    503:       A[4][4] = {{0,0,0,0},
    504:                  {1./2.,0,0,0},
    505:                  {1./2.,1./2.,0,0},
    506:                  {1./6.,1./6.,1./6.,0}},
    507:       Gamma[4][4] = {{1./2.,0,0,0},
    508:                      {0.0,3./4.,0,0},
    509:                      {-2./3.,-23./9.,2./9.,0},
    510:                      {1./18.,65./108.,-2./27,0}},
    511:       b[4]  = {1./6.,1./6.,1./6.,1./2.},
    512:       b2[4] = {3./16.,10./16.,3./16.,0};
    513:     PetscReal binterpt[4][3];

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

    528:     TSRosWRegister(TSROSWLLSSP3P4S2C,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
    529:   }

    531:   {
    532:     PetscReal A[4][4],Gamma[4][4],b[4],b2[4];
    533:     PetscReal binterpt[4][3];

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

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

    560:     b[0]=0.1876410243467238251612921333138006734899663569186926;
    561:     b[1]=-0.5952974735769549480478230473706443582188442040780541;
    562:     b[2]=0.9717899277217721234705114616271378792182450260943198;
    563:     b[3]=0.4358665215084589994160194475295062513822671686978816;

    565:     b2[0]=0.2147402862233891404862383521089097657790734483804460;
    566:     b2[1]=-0.4851622638849390928209050538171743017757490232519684;
    567:     b2[2]=0.8687250025203875511662123688667549217531982787600080;
    568:     b2[3]=0.4016969751411624011684543450940068201770721128357014;

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

    583:     TSRosWRegister(TSROSWARK3,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]);
    584:   }
    585:   TSRosWRegisterRos4(TSROSWGRK4T,0.231,PETSC_DEFAULT,PETSC_DEFAULT,0,-0.1282612945269037e+01);
    586:   TSRosWRegisterRos4(TSROSWSHAMP4,0.5,PETSC_DEFAULT,PETSC_DEFAULT,0,125./108.);
    587:   TSRosWRegisterRos4(TSROSWVELDD4,0.22570811482256823492,PETSC_DEFAULT,PETSC_DEFAULT,0,-1.355958941201148);
    588:   TSRosWRegisterRos4(TSROSW4L,0.57282,PETSC_DEFAULT,PETSC_DEFAULT,0,-1.093502252409163);
    589:   return 0;
    590: }

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

    595:    Not Collective

    597:    Level: advanced

    599: .seealso: TSRosWRegister(), TSRosWRegisterAll()
    600: @*/
    601: PetscErrorCode TSRosWRegisterDestroy(void)
    602: {
    603:   RosWTableauLink link;

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

    619: /*@C
    620:   TSRosWInitializePackage - This function initializes everything in the TSRosW package. It is called
    621:   from TSInitializePackage().

    623:   Level: developer

    625: .seealso: PetscInitialize()
    626: @*/
    627: PetscErrorCode TSRosWInitializePackage(void)
    628: {
    629:   if (TSRosWPackageInitialized) return 0;
    630:   TSRosWPackageInitialized = PETSC_TRUE;
    631:   TSRosWRegisterAll();
    632:   PetscRegisterFinalize(TSRosWFinalizePackage);
    633:   return 0;
    634: }

    636: /*@C
    637:   TSRosWFinalizePackage - This function destroys everything in the TSRosW package. It is
    638:   called from PetscFinalize().

    640:   Level: developer

    642: .seealso: PetscFinalize()
    643: @*/
    644: PetscErrorCode TSRosWFinalizePackage(void)
    645: {
    646:   TSRosWPackageInitialized = PETSC_FALSE;
    647:   TSRosWRegisterDestroy();
    648:   return 0;
    649: }

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

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

    656:    Input Parameters:
    657: +  name - identifier for method
    658: .  order - approximation order of method
    659: .  s - number of stages, this is the dimension of the matrices below
    660: .  A - Table of propagated stage coefficients (dimension s*s, row-major), strictly lower triangular
    661: .  Gamma - Table of coefficients in implicit stage equations (dimension s*s, row-major), lower triangular with nonzero diagonal
    662: .  b - Step completion table (dimension s)
    663: .  bembed - Step completion table for a scheme of order one less (dimension s, NULL if no embedded scheme is available)
    664: .  pinterp - Order of the interpolation scheme, equal to the number of columns of binterpt
    665: -  binterpt - Coefficients of the interpolation formula (dimension s*pinterp)

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

    670:    Level: advanced

    672: .seealso: TSRosW
    673: @*/
    674: PetscErrorCode TSRosWRegister(TSRosWType name,PetscInt order,PetscInt s,const PetscReal A[],const PetscReal Gamma[],const PetscReal b[],const PetscReal bembed[],
    675:                               PetscInt pinterp,const PetscReal binterpt[])
    676: {
    677:   RosWTableauLink link;
    678:   RosWTableau     t;
    679:   PetscInt        i,j,k;
    680:   PetscScalar     *GammaInv;

    688:   TSRosWInitializePackage();
    689:   PetscNew(&link);
    690:   t        = &link->tab;
    691:   PetscStrallocpy(name,&t->name);
    692:   t->order = order;
    693:   t->s     = s;
    694:   PetscMalloc5(s*s,&t->A,s*s,&t->Gamma,s,&t->b,s,&t->ASum,s,&t->GammaSum);
    695:   PetscMalloc5(s*s,&t->At,s,&t->bt,s*s,&t->GammaInv,s,&t->GammaZeroDiag,s*s,&t->GammaExplicitCorr);
    696:   PetscArraycpy(t->A,A,s*s);
    697:   PetscArraycpy(t->Gamma,Gamma,s*s);
    698:   PetscArraycpy(t->GammaExplicitCorr,Gamma,s*s);
    699:   PetscArraycpy(t->b,b,s);
    700:   if (bembed) {
    701:     PetscMalloc2(s,&t->bembed,s,&t->bembedt);
    702:     PetscArraycpy(t->bembed,bembed,s);
    703:   }
    704:   for (i=0; i<s; i++) {
    705:     t->ASum[i]     = 0;
    706:     t->GammaSum[i] = 0;
    707:     for (j=0; j<s; j++) {
    708:       t->ASum[i]     += A[i*s+j];
    709:       t->GammaSum[i] += Gamma[i*s+j];
    710:     }
    711:   }
    712:   PetscMalloc1(s*s,&GammaInv); /* Need to use Scalar for inverse, then convert back to Real */
    713:   for (i=0; i<s*s; i++) GammaInv[i] = Gamma[i];
    714:   for (i=0; i<s; i++) {
    715:     if (Gamma[i*s+i] == 0.0) {
    716:       GammaInv[i*s+i] = 1.0;
    717:       t->GammaZeroDiag[i] = PETSC_TRUE;
    718:     } else {
    719:       t->GammaZeroDiag[i] = PETSC_FALSE;
    720:     }
    721:   }

    723:   switch (s) {
    724:   case 1: GammaInv[0] = 1./GammaInv[0]; break;
    725:   case 2: PetscKernel_A_gets_inverse_A_2(GammaInv,0,PETSC_FALSE,NULL); break;
    726:   case 3: PetscKernel_A_gets_inverse_A_3(GammaInv,0,PETSC_FALSE,NULL); break;
    727:   case 4: PetscKernel_A_gets_inverse_A_4(GammaInv,0,PETSC_FALSE,NULL); break;
    728:   case 5: {
    729:     PetscInt  ipvt5[5];
    730:     MatScalar work5[5*5];
    731:     PetscKernel_A_gets_inverse_A_5(GammaInv,ipvt5,work5,0,PETSC_FALSE,NULL); break;
    732:   }
    733:   case 6: PetscKernel_A_gets_inverse_A_6(GammaInv,0,PETSC_FALSE,NULL); break;
    734:   case 7: PetscKernel_A_gets_inverse_A_7(GammaInv,0,PETSC_FALSE,NULL); break;
    735:   default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Not implemented for %D stages",s);
    736:   }
    737:   for (i=0; i<s*s; i++) t->GammaInv[i] = PetscRealPart(GammaInv[i]);
    738:   PetscFree(GammaInv);

    740:   for (i=0; i<s; i++) {
    741:     for (k=0; k<i+1; k++) {
    742:       t->GammaExplicitCorr[i*s+k]=(t->GammaExplicitCorr[i*s+k])*(t->GammaInv[k*s+k]);
    743:       for (j=k+1; j<i+1; j++) {
    744:         t->GammaExplicitCorr[i*s+k]+=(t->GammaExplicitCorr[i*s+j])*(t->GammaInv[j*s+k]);
    745:       }
    746:     }
    747:   }

    749:   for (i=0; i<s; i++) {
    750:     for (j=0; j<s; j++) {
    751:       t->At[i*s+j] = 0;
    752:       for (k=0; k<s; k++) {
    753:         t->At[i*s+j] += t->A[i*s+k] * t->GammaInv[k*s+j];
    754:       }
    755:     }
    756:     t->bt[i] = 0;
    757:     for (j=0; j<s; j++) {
    758:       t->bt[i] += t->b[j] * t->GammaInv[j*s+i];
    759:     }
    760:     if (bembed) {
    761:       t->bembedt[i] = 0;
    762:       for (j=0; j<s; j++) {
    763:         t->bembedt[i] += t->bembed[j] * t->GammaInv[j*s+i];
    764:       }
    765:     }
    766:   }
    767:   t->ccfl = 1.0;                /* Fix this */

    769:   t->pinterp = pinterp;
    770:   PetscMalloc1(s*pinterp,&t->binterpt);
    771:   PetscArraycpy(t->binterpt,binterpt,s*pinterp);
    772:   link->next = RosWTableauList;
    773:   RosWTableauList = link;
    774:   return 0;
    775: }

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

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

    782:    Input Parameters:
    783: +  name - identifier for method
    784: .  gamma - leading coefficient (diagonal entry)
    785: .  a2 - design parameter, see Table 7.2 of Hairer&Wanner
    786: .  a3 - design parameter or PETSC_DEFAULT to satisfy one of the order five conditions (Eq 7.22)
    787: .  b3 - design parameter, see Table 7.2 of Hairer&Wanner
    788: .  beta43 - design parameter or PETSC_DEFAULT to use Equation 7.21 of Hairer&Wanner
    789: -  e4 - design parameter for embedded method, see coefficient E4 in ros4.f code from Hairer

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

    796:    Level: developer

    798: .seealso: TSRosW, TSRosWRegister()
    799: @*/
    800: PetscErrorCode TSRosWRegisterRos4(TSRosWType name,PetscReal gamma,PetscReal a2,PetscReal a3,PetscReal b3,PetscReal e4)
    801: {
    802:   /* Declare numeric constants so they can be quad precision without being truncated at double */
    803:   const PetscReal one = 1,two = 2,three = 3,four = 4,five = 5,six = 6,eight = 8,twelve = 12,twenty = 20,twentyfour = 24,
    804:     p32 = one/six - gamma + gamma*gamma,
    805:     p42 = one/eight - gamma/three,
    806:     p43 = one/twelve - gamma/three,
    807:     p44 = one/twentyfour - gamma/two + three/two*gamma*gamma - gamma*gamma*gamma,
    808:     p56 = one/twenty - gamma/four;
    809:   PetscReal   a4,a32,a42,a43,b1,b2,b4,beta2p,beta3p,beta4p,beta32,beta42,beta43,beta32beta2p,beta4jbetajp;
    810:   PetscReal   A[4][4],Gamma[4][4],b[4],bm[4];
    811:   PetscScalar M[3][3],rhs[3];

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

    818:   /* Solve order conditions 7.15a, 7.15c, 7.15e */
    819:   M[0][0] = one; M[0][1] = one;      M[0][2] = one;      /* 7.15a */
    820:   M[1][0] = 0.0; M[1][1] = a2*a2;    M[1][2] = a4*a4;    /* 7.15c */
    821:   M[2][0] = 0.0; M[2][1] = a2*a2*a2; M[2][2] = a4*a4*a4; /* 7.15e */
    822:   rhs[0]  = one - b3;
    823:   rhs[1]  = one/three - a3*a3*b3;
    824:   rhs[2]  = one/four - a3*a3*a3*b3;
    825:   PetscKernel_A_gets_inverse_A_3(&M[0][0],0,PETSC_FALSE,NULL);
    826:   b1      = PetscRealPart(M[0][0]*rhs[0] + M[0][1]*rhs[1] + M[0][2]*rhs[2]);
    827:   b2      = PetscRealPart(M[1][0]*rhs[0] + M[1][1]*rhs[1] + M[1][2]*rhs[2]);
    828:   b4      = PetscRealPart(M[2][0]*rhs[0] + M[2][1]*rhs[1] + M[2][2]*rhs[2]);

    830:   /* Step 3 */
    831:   beta43       = (p56 - a2*p43) / (b4*a3*a3*(a3 - a2)); /* 7.21 */
    832:   beta32beta2p =  p44 / (b4*beta43);                    /* 7.15h */
    833:   beta4jbetajp = (p32 - b3*beta32beta2p) / b4;
    834:   M[0][0]      = b2;                                    M[0][1] = b3;                 M[0][2] = b4;
    835:   M[1][0]      = a4*a4*beta32beta2p-a3*a3*beta4jbetajp; M[1][1] = a2*a2*beta4jbetajp; M[1][2] = -a2*a2*beta32beta2p;
    836:   M[2][0]      = b4*beta43*a3*a3-p43;                   M[2][1] = -b4*beta43*a2*a2;   M[2][2] = 0;
    837:   rhs[0]       = one/two - gamma; rhs[1] = 0; rhs[2] = -a2*a2*p32;
    838:   PetscKernel_A_gets_inverse_A_3(&M[0][0],0,PETSC_FALSE,NULL);
    839:   beta2p       = PetscRealPart(M[0][0]*rhs[0] + M[0][1]*rhs[1] + M[0][2]*rhs[2]);
    840:   beta3p       = PetscRealPart(M[1][0]*rhs[0] + M[1][1]*rhs[1] + M[1][2]*rhs[2]);
    841:   beta4p       = PetscRealPart(M[2][0]*rhs[0] + M[2][1]*rhs[1] + M[2][2]*rhs[2]);

    843:   /* Step 4: back-substitute */
    844:   beta32 = beta32beta2p / beta2p;
    845:   beta42 = (beta4jbetajp - beta43*beta3p) / beta2p;

    847:   /* Step 5: 7.15f and 7.20, then 7.16 */
    848:   a43 = 0;
    849:   a32 = p42 / (b3*a3*beta2p + b4*a4*beta2p);
    850:   a42 = a32;

    852:   A[0][0]     = 0;          A[0][1] = 0;   A[0][2] = 0;   A[0][3] = 0;
    853:   A[1][0]     = a2;         A[1][1] = 0;   A[1][2] = 0;   A[1][3] = 0;
    854:   A[2][0]     = a3-a32;     A[2][1] = a32; A[2][2] = 0;   A[2][3] = 0;
    855:   A[3][0]     = a4-a43-a42; A[3][1] = a42; A[3][2] = a43; A[3][3] = 0;
    856:   Gamma[0][0] = gamma;                        Gamma[0][1] = 0;              Gamma[0][2] = 0;              Gamma[0][3] = 0;
    857:   Gamma[1][0] = beta2p-A[1][0];               Gamma[1][1] = gamma;          Gamma[1][2] = 0;              Gamma[1][3] = 0;
    858:   Gamma[2][0] = beta3p-beta32-A[2][0];        Gamma[2][1] = beta32-A[2][1]; Gamma[2][2] = gamma;          Gamma[2][3] = 0;
    859:   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;
    860:   b[0] = b1; b[1] = b2; b[2] = b3; b[3] = b4;

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

    868:   {
    869:     const PetscReal misfit = a2*a2*bm[1] + a3*a3*bm[2] + a4*a4*bm[3] - one/three;
    871:   }
    872:   TSRosWRegister(name,4,4,&A[0][0],&Gamma[0][0],b,bm,0,NULL);
    873:   return 0;
    874: }

    876: /*
    877:  The step completion formula is

    879:  x1 = x0 + b^T Y

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

    884:  x1e = x0 + be^T Y
    885:      = x1 - b^T Y + be^T Y
    886:      = x1 + (be - b)^T Y

    888:  so we can evaluate the method of different order even after the step has been optimistically completed.
    889: */
    890: static PetscErrorCode TSEvaluateStep_RosW(TS ts,PetscInt order,Vec U,PetscBool *done)
    891: {
    892:   TS_RosW        *ros = (TS_RosW*)ts->data;
    893:   RosWTableau    tab  = ros->tableau;
    894:   PetscScalar    *w   = ros->work;
    895:   PetscInt       i;

    897:   if (order == tab->order) {
    898:     if (ros->status == TS_STEP_INCOMPLETE) { /* Use standard completion formula */
    899:       VecCopy(ts->vec_sol,U);
    900:       for (i=0; i<tab->s; i++) w[i] = tab->bt[i];
    901:       VecMAXPY(U,tab->s,w,ros->Y);
    902:     } else VecCopy(ts->vec_sol,U);
    903:     if (done) *done = PETSC_TRUE;
    904:     return 0;
    905:   } else if (order == tab->order-1) {
    906:     if (!tab->bembedt) goto unavailable;
    907:     if (ros->status == TS_STEP_INCOMPLETE) { /* Use embedded completion formula */
    908:       VecCopy(ts->vec_sol,U);
    909:       for (i=0; i<tab->s; i++) w[i] = tab->bembedt[i];
    910:       VecMAXPY(U,tab->s,w,ros->Y);
    911:     } else {                    /* Use rollback-and-recomplete formula (bembedt - bt) */
    912:       for (i=0; i<tab->s; i++) w[i] = tab->bembedt[i] - tab->bt[i];
    913:       VecCopy(ts->vec_sol,U);
    914:       VecMAXPY(U,tab->s,w,ros->Y);
    915:     }
    916:     if (done) *done = PETSC_TRUE;
    917:     return 0;
    918:   }
    919:   unavailable:
    920:   if (done) *done = PETSC_FALSE;
    921:   else SETERRQ(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);
    922:   return 0;
    923: }

    925: static PetscErrorCode TSRollBack_RosW(TS ts)
    926: {
    927:   TS_RosW        *ros = (TS_RosW*)ts->data;

    929:   VecCopy(ros->vec_sol_prev,ts->vec_sol);
    930:   return 0;
    931: }

    933: static PetscErrorCode TSStep_RosW(TS ts)
    934: {
    935:   TS_RosW         *ros = (TS_RosW*)ts->data;
    936:   RosWTableau     tab  = ros->tableau;
    937:   const PetscInt  s    = tab->s;
    938:   const PetscReal *At  = tab->At,*Gamma = tab->Gamma,*ASum = tab->ASum,*GammaInv = tab->GammaInv;
    939:   const PetscReal *GammaExplicitCorr = tab->GammaExplicitCorr;
    940:   const PetscBool *GammaZeroDiag = tab->GammaZeroDiag;
    941:   PetscScalar     *w   = ros->work;
    942:   Vec             *Y   = ros->Y,Ydot = ros->Ydot,Zdot = ros->Zdot,Zstage = ros->Zstage;
    943:   SNES            snes;
    944:   TSAdapt         adapt;
    945:   PetscInt        i,j,its,lits;
    946:   PetscInt        rejections = 0;
    947:   PetscBool       stageok,accept = PETSC_TRUE;
    948:   PetscReal       next_time_step = ts->time_step;
    949:   PetscInt        lag;

    951:   if (!ts->steprollback) {
    952:     VecCopy(ts->vec_sol,ros->vec_sol_prev);
    953:   }

    955:   ros->status = TS_STEP_INCOMPLETE;
    956:   while (!ts->reason && ros->status != TS_STEP_COMPLETE) {
    957:     const PetscReal h = ts->time_step;
    958:     for (i=0; i<s; i++) {
    959:       ros->stage_time = ts->ptime + h*ASum[i];
    960:       TSPreStage(ts,ros->stage_time);
    961:       if (GammaZeroDiag[i]) {
    962:         ros->stage_explicit = PETSC_TRUE;
    963:         ros->scoeff         = 1.;
    964:       } else {
    965:         ros->stage_explicit = PETSC_FALSE;
    966:         ros->scoeff         = 1./Gamma[i*s+i];
    967:       }

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

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

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

    980:       if (!ros->stage_explicit) {
    981:         TSGetSNES(ts,&snes);
    982:         if (!ros->recompute_jacobian && !i) {
    983:           SNESGetLagJacobian(snes,&lag);
    984:           if (lag == 1) {  /* use did not set a nontrivial lag, so lag over all stages */
    985:             SNESSetLagJacobian(snes,-2); /* Recompute the Jacobian on this solve, but not again for the rest of the stages */
    986:           }
    987:         }
    988:         SNESSolve(snes,NULL,Y[i]);
    989:         if (!ros->recompute_jacobian && i == s-1 && lag == 1) {
    990:           SNESSetLagJacobian(snes,lag); /* Set lag back to 1 so we know user did not set it */
    991:         }
    992:         SNESGetIterationNumber(snes,&its);
    993:         SNESGetLinearSolveIterations(snes,&lits);
    994:         ts->snes_its += its; ts->ksp_its += lits;
    995:       } else {
    996:         Mat J,Jp;
    997:         VecZeroEntries(Ydot); /* Evaluate Y[i]=G(t,Ydot=0,Zstage) */
    998:         TSComputeIFunction(ts,ros->stage_time,Zstage,Ydot,Y[i],PETSC_FALSE);
    999:         VecScale(Y[i],-1.0);
    1000:         VecAXPY(Y[i],-1.0,Zdot); /*Y[i] = F(Zstage)-Zdot[=GammaInv*Y]*/

    1002:         VecZeroEntries(Zstage); /* Zstage = GammaExplicitCorr[i,j] * Y[j] */
    1003:         for (j=0; j<i; j++) w[j] = GammaExplicitCorr[i*s+j];
    1004:         VecMAXPY(Zstage,i,w,Y);

    1006:         /* Y[i] = Y[i] + Jac*Zstage[=Jac*GammaExplicitCorr[i,j] * Y[j]] */
    1007:         TSGetIJacobian(ts,&J,&Jp,NULL,NULL);
    1008:         TSComputeIJacobian(ts,ros->stage_time,ts->vec_sol,Ydot,0,J,Jp,PETSC_FALSE);
    1009:         MatMult(J,Zstage,Zdot);
    1010:         VecAXPY(Y[i],-1.0,Zdot);
    1011:         ts->ksp_its += 1;

    1013:         VecScale(Y[i],h);
    1014:       }
    1015:       TSPostStage(ts,ros->stage_time,i,Y);
    1016:       TSGetAdapt(ts,&adapt);
    1017:       TSAdaptCheckStage(adapt,ts,ros->stage_time,Y[i],&stageok);
    1018:       if (!stageok) goto reject_step;
    1019:     }

    1021:     ros->status = TS_STEP_INCOMPLETE;
    1022:     TSEvaluateStep_RosW(ts,tab->order,ts->vec_sol,NULL);
    1023:     ros->status = TS_STEP_PENDING;
    1024:     TSGetAdapt(ts,&adapt);
    1025:     TSAdaptCandidatesClear(adapt);
    1026:     TSAdaptCandidateAdd(adapt,tab->name,tab->order,1,tab->ccfl,(PetscReal)tab->s,PETSC_TRUE);
    1027:     TSAdaptChoose(adapt,ts,ts->time_step,NULL,&next_time_step,&accept);
    1028:     ros->status = accept ? TS_STEP_COMPLETE : TS_STEP_INCOMPLETE;
    1029:     if (!accept) { /* Roll back the current step */
    1030:       TSRollBack_RosW(ts);
    1031:       ts->time_step = next_time_step;
    1032:       goto reject_step;
    1033:     }

    1035:     ts->ptime += ts->time_step;
    1036:     ts->time_step = next_time_step;
    1037:     break;

    1039:   reject_step:
    1040:     ts->reject++; accept = PETSC_FALSE;
    1041:     if (!ts->reason && ++rejections > ts->max_reject && ts->max_reject >= 0) {
    1042:       ts->reason = TS_DIVERGED_STEP_REJECTED;
    1043:       PetscInfo(ts,"Step=%D, step rejections %D greater than current TS allowed, stopping solve\n",ts->steps,rejections);
    1044:     }
    1045:   }
    1046:   return 0;
    1047: }

    1049: static PetscErrorCode TSInterpolate_RosW(TS ts,PetscReal itime,Vec U)
    1050: {
    1051:   TS_RosW         *ros = (TS_RosW*)ts->data;
    1052:   PetscInt        s    = ros->tableau->s,pinterp = ros->tableau->pinterp,i,j;
    1053:   PetscReal       h;
    1054:   PetscReal       tt,t;
    1055:   PetscScalar     *bt;
    1056:   const PetscReal *Bt = ros->tableau->binterpt;
    1057:   const PetscReal *GammaInv = ros->tableau->GammaInv;
    1058:   PetscScalar     *w        = ros->work;
    1059:   Vec             *Y        = ros->Y;

    1063:   switch (ros->status) {
    1064:   case TS_STEP_INCOMPLETE:
    1065:   case TS_STEP_PENDING:
    1066:     h = ts->time_step;
    1067:     t = (itime - ts->ptime)/h;
    1068:     break;
    1069:   case TS_STEP_COMPLETE:
    1070:     h = ts->ptime - ts->ptime_prev;
    1071:     t = (itime - ts->ptime)/h + 1; /* In the interval [0,1] */
    1072:     break;
    1073:   default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
    1074:   }
    1075:   PetscMalloc1(s,&bt);
    1076:   for (i=0; i<s; i++) bt[i] = 0;
    1077:   for (j=0,tt=t; j<pinterp; j++,tt*=t) {
    1078:     for (i=0; i<s; i++) {
    1079:       bt[i] += Bt[i*pinterp+j] * tt;
    1080:     }
    1081:   }

    1083:   /* y(t+tt*h) = y(t) + Sum bt(tt) * GammaInv * Ydot */
    1084:   /* U <- 0*/
    1085:   VecZeroEntries(U);
    1086:   /* U <- Sum bt_i * GammaInv(i,1:i) * Y(1:i) */
    1087:   for (j=0; j<s; j++) w[j] = 0;
    1088:   for (j=0; j<s; j++) {
    1089:     for (i=j; i<s; i++) {
    1090:       w[j] +=  bt[i]*GammaInv[i*s+j];
    1091:     }
    1092:   }
    1093:   VecMAXPY(U,i,w,Y);
    1094:   /* U <- y(t) + U */
    1095:   VecAXPY(U,1,ros->vec_sol_prev);

    1097:   PetscFree(bt);
    1098:   return 0;
    1099: }

    1101: /*------------------------------------------------------------*/

    1103: static PetscErrorCode TSRosWTableauReset(TS ts)
    1104: {
    1105:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1106:   RosWTableau    tab  = ros->tableau;

    1108:   if (!tab) return 0;
    1109:   VecDestroyVecs(tab->s,&ros->Y);
    1110:   PetscFree(ros->work);
    1111:   return 0;
    1112: }

    1114: static PetscErrorCode TSReset_RosW(TS ts)
    1115: {
    1116:   TS_RosW        *ros = (TS_RosW*)ts->data;

    1118:   TSRosWTableauReset(ts);
    1119:   VecDestroy(&ros->Ydot);
    1120:   VecDestroy(&ros->Ystage);
    1121:   VecDestroy(&ros->Zdot);
    1122:   VecDestroy(&ros->Zstage);
    1123:   VecDestroy(&ros->vec_sol_prev);
    1124:   return 0;
    1125: }

    1127: static PetscErrorCode TSRosWGetVecs(TS ts,DM dm,Vec *Ydot,Vec *Zdot,Vec *Ystage,Vec *Zstage)
    1128: {
    1129:   TS_RosW        *rw = (TS_RosW*)ts->data;

    1131:   if (Ydot) {
    1132:     if (dm && dm != ts->dm) {
    1133:       DMGetNamedGlobalVector(dm,"TSRosW_Ydot",Ydot);
    1134:     } else *Ydot = rw->Ydot;
    1135:   }
    1136:   if (Zdot) {
    1137:     if (dm && dm != ts->dm) {
    1138:       DMGetNamedGlobalVector(dm,"TSRosW_Zdot",Zdot);
    1139:     } else *Zdot = rw->Zdot;
    1140:   }
    1141:   if (Ystage) {
    1142:     if (dm && dm != ts->dm) {
    1143:       DMGetNamedGlobalVector(dm,"TSRosW_Ystage",Ystage);
    1144:     } else *Ystage = rw->Ystage;
    1145:   }
    1146:   if (Zstage) {
    1147:     if (dm && dm != ts->dm) {
    1148:       DMGetNamedGlobalVector(dm,"TSRosW_Zstage",Zstage);
    1149:     } else *Zstage = rw->Zstage;
    1150:   }
    1151:   return 0;
    1152: }

    1154: static PetscErrorCode TSRosWRestoreVecs(TS ts,DM dm,Vec *Ydot,Vec *Zdot, Vec *Ystage, Vec *Zstage)
    1155: {
    1156:   if (Ydot) {
    1157:     if (dm && dm != ts->dm) {
    1158:       DMRestoreNamedGlobalVector(dm,"TSRosW_Ydot",Ydot);
    1159:     }
    1160:   }
    1161:   if (Zdot) {
    1162:     if (dm && dm != ts->dm) {
    1163:       DMRestoreNamedGlobalVector(dm,"TSRosW_Zdot",Zdot);
    1164:     }
    1165:   }
    1166:   if (Ystage) {
    1167:     if (dm && dm != ts->dm) {
    1168:       DMRestoreNamedGlobalVector(dm,"TSRosW_Ystage",Ystage);
    1169:     }
    1170:   }
    1171:   if (Zstage) {
    1172:     if (dm && dm != ts->dm) {
    1173:       DMRestoreNamedGlobalVector(dm,"TSRosW_Zstage",Zstage);
    1174:     }
    1175:   }
    1176:   return 0;
    1177: }

    1179: static PetscErrorCode DMCoarsenHook_TSRosW(DM fine,DM coarse,void *ctx)
    1180: {
    1181:   return 0;
    1182: }

    1184: static PetscErrorCode DMRestrictHook_TSRosW(DM fine,Mat restrct,Vec rscale,Mat inject,DM coarse,void *ctx)
    1185: {
    1186:   TS             ts = (TS)ctx;
    1187:   Vec            Ydot,Zdot,Ystage,Zstage;
    1188:   Vec            Ydotc,Zdotc,Ystagec,Zstagec;

    1190:   TSRosWGetVecs(ts,fine,&Ydot,&Ystage,&Zdot,&Zstage);
    1191:   TSRosWGetVecs(ts,coarse,&Ydotc,&Ystagec,&Zdotc,&Zstagec);
    1192:   MatRestrict(restrct,Ydot,Ydotc);
    1193:   VecPointwiseMult(Ydotc,rscale,Ydotc);
    1194:   MatRestrict(restrct,Ystage,Ystagec);
    1195:   VecPointwiseMult(Ystagec,rscale,Ystagec);
    1196:   MatRestrict(restrct,Zdot,Zdotc);
    1197:   VecPointwiseMult(Zdotc,rscale,Zdotc);
    1198:   MatRestrict(restrct,Zstage,Zstagec);
    1199:   VecPointwiseMult(Zstagec,rscale,Zstagec);
    1200:   TSRosWRestoreVecs(ts,fine,&Ydot,&Ystage,&Zdot,&Zstage);
    1201:   TSRosWRestoreVecs(ts,coarse,&Ydotc,&Ystagec,&Zdotc,&Zstagec);
    1202:   return 0;
    1203: }

    1205: static PetscErrorCode DMSubDomainHook_TSRosW(DM fine,DM coarse,void *ctx)
    1206: {
    1207:   return 0;
    1208: }

    1210: static PetscErrorCode DMSubDomainRestrictHook_TSRosW(DM dm,VecScatter gscat,VecScatter lscat,DM subdm,void *ctx)
    1211: {
    1212:   TS             ts = (TS)ctx;
    1213:   Vec            Ydot,Zdot,Ystage,Zstage;
    1214:   Vec            Ydots,Zdots,Ystages,Zstages;

    1216:   TSRosWGetVecs(ts,dm,&Ydot,&Ystage,&Zdot,&Zstage);
    1217:   TSRosWGetVecs(ts,subdm,&Ydots,&Ystages,&Zdots,&Zstages);

    1219:   VecScatterBegin(gscat,Ydot,Ydots,INSERT_VALUES,SCATTER_FORWARD);
    1220:   VecScatterEnd(gscat,Ydot,Ydots,INSERT_VALUES,SCATTER_FORWARD);

    1222:   VecScatterBegin(gscat,Ystage,Ystages,INSERT_VALUES,SCATTER_FORWARD);
    1223:   VecScatterEnd(gscat,Ystage,Ystages,INSERT_VALUES,SCATTER_FORWARD);

    1225:   VecScatterBegin(gscat,Zdot,Zdots,INSERT_VALUES,SCATTER_FORWARD);
    1226:   VecScatterEnd(gscat,Zdot,Zdots,INSERT_VALUES,SCATTER_FORWARD);

    1228:   VecScatterBegin(gscat,Zstage,Zstages,INSERT_VALUES,SCATTER_FORWARD);
    1229:   VecScatterEnd(gscat,Zstage,Zstages,INSERT_VALUES,SCATTER_FORWARD);

    1231:   TSRosWRestoreVecs(ts,dm,&Ydot,&Ystage,&Zdot,&Zstage);
    1232:   TSRosWRestoreVecs(ts,subdm,&Ydots,&Ystages,&Zdots,&Zstages);
    1233:   return 0;
    1234: }

    1236: /*
    1237:   This defines the nonlinear equation that is to be solved with SNES
    1238:   G(U) = F[t0+Theta*dt, U, (U-U0)*shift] = 0
    1239: */
    1240: static PetscErrorCode SNESTSFormFunction_RosW(SNES snes,Vec U,Vec F,TS ts)
    1241: {
    1242:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1243:   Vec            Ydot,Zdot,Ystage,Zstage;
    1244:   PetscReal      shift = ros->scoeff / ts->time_step;
    1245:   DM             dm,dmsave;

    1247:   SNESGetDM(snes,&dm);
    1248:   TSRosWGetVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
    1249:   VecWAXPY(Ydot,shift,U,Zdot);    /* Ydot = shift*U + Zdot */
    1250:   VecWAXPY(Ystage,1.0,U,Zstage);  /* Ystage = U + Zstage */
    1251:   dmsave = ts->dm;
    1252:   ts->dm = dm;
    1253:   TSComputeIFunction(ts,ros->stage_time,Ystage,Ydot,F,PETSC_FALSE);
    1254:   ts->dm = dmsave;
    1255:   TSRosWRestoreVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
    1256:   return 0;
    1257: }

    1259: static PetscErrorCode SNESTSFormJacobian_RosW(SNES snes,Vec U,Mat A,Mat B,TS ts)
    1260: {
    1261:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1262:   Vec            Ydot,Zdot,Ystage,Zstage;
    1263:   PetscReal      shift = ros->scoeff / ts->time_step;
    1264:   DM             dm,dmsave;

    1266:   /* ros->Ydot and ros->Ystage have already been computed in SNESTSFormFunction_RosW (SNES guarantees this) */
    1267:   SNESGetDM(snes,&dm);
    1268:   TSRosWGetVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
    1269:   dmsave = ts->dm;
    1270:   ts->dm = dm;
    1271:   TSComputeIJacobian(ts,ros->stage_time,Ystage,Ydot,shift,A,B,PETSC_TRUE);
    1272:   ts->dm = dmsave;
    1273:   TSRosWRestoreVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage);
    1274:   return 0;
    1275: }

    1277: static PetscErrorCode TSRosWTableauSetUp(TS ts)
    1278: {
    1279:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1280:   RosWTableau    tab  = ros->tableau;

    1282:   VecDuplicateVecs(ts->vec_sol,tab->s,&ros->Y);
    1283:   PetscMalloc1(tab->s,&ros->work);
    1284:   return 0;
    1285: }

    1287: static PetscErrorCode TSSetUp_RosW(TS ts)
    1288: {
    1289:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1290:   DM             dm;
    1291:   SNES           snes;
    1292:   TSRHSJacobian  rhsjacobian;

    1294:   TSRosWTableauSetUp(ts);
    1295:   VecDuplicate(ts->vec_sol,&ros->Ydot);
    1296:   VecDuplicate(ts->vec_sol,&ros->Ystage);
    1297:   VecDuplicate(ts->vec_sol,&ros->Zdot);
    1298:   VecDuplicate(ts->vec_sol,&ros->Zstage);
    1299:   VecDuplicate(ts->vec_sol,&ros->vec_sol_prev);
    1300:   TSGetDM(ts,&dm);
    1301:   DMCoarsenHookAdd(dm,DMCoarsenHook_TSRosW,DMRestrictHook_TSRosW,ts);
    1302:   DMSubDomainHookAdd(dm,DMSubDomainHook_TSRosW,DMSubDomainRestrictHook_TSRosW,ts);
    1303:   /* Rosenbrock methods are linearly implicit, so set that unless the user has specifically asked for something else */
    1304:   TSGetSNES(ts,&snes);
    1305:   if (!((PetscObject)snes)->type_name) {
    1306:     SNESSetType(snes,SNESKSPONLY);
    1307:   }
    1308:   DMTSGetRHSJacobian(dm,&rhsjacobian,NULL);
    1309:   if (rhsjacobian == TSComputeRHSJacobianConstant) {
    1310:     Mat Amat,Pmat;

    1312:     /* Set the SNES matrix to be different from the RHS matrix because there is no way to reconstruct shift*M-J */
    1313:     SNESGetJacobian(snes,&Amat,&Pmat,NULL,NULL);
    1314:     if (Amat && Amat == ts->Arhs) {
    1315:       if (Amat == Pmat) {
    1316:         MatDuplicate(ts->Arhs,MAT_COPY_VALUES,&Amat);
    1317:         SNESSetJacobian(snes,Amat,Amat,NULL,NULL);
    1318:       } else {
    1319:         MatDuplicate(ts->Arhs,MAT_COPY_VALUES,&Amat);
    1320:         SNESSetJacobian(snes,Amat,NULL,NULL,NULL);
    1321:         if (Pmat && Pmat == ts->Brhs) {
    1322:           MatDuplicate(ts->Brhs,MAT_COPY_VALUES,&Pmat);
    1323:           SNESSetJacobian(snes,NULL,Pmat,NULL,NULL);
    1324:           MatDestroy(&Pmat);
    1325:         }
    1326:       }
    1327:       MatDestroy(&Amat);
    1328:     }
    1329:   }
    1330:   return 0;
    1331: }
    1332: /*------------------------------------------------------------*/

    1334: static PetscErrorCode TSSetFromOptions_RosW(PetscOptionItems *PetscOptionsObject,TS ts)
    1335: {
    1336:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1337:   SNES           snes;

    1339:   PetscOptionsHead(PetscOptionsObject,"RosW ODE solver options");
    1340:   {
    1341:     RosWTableauLink link;
    1342:     PetscInt        count,choice;
    1343:     PetscBool       flg;
    1344:     const char      **namelist;

    1346:     for (link=RosWTableauList,count=0; link; link=link->next,count++) ;
    1347:     PetscMalloc1(count,(char***)&namelist);
    1348:     for (link=RosWTableauList,count=0; link; link=link->next,count++) namelist[count] = link->tab.name;
    1349:     PetscOptionsEList("-ts_rosw_type","Family of Rosenbrock-W method","TSRosWSetType",(const char*const*)namelist,count,ros->tableau->name,&choice,&flg);
    1350:     if (flg) TSRosWSetType(ts,namelist[choice]);
    1351:     PetscFree(namelist);

    1353:     PetscOptionsBool("-ts_rosw_recompute_jacobian","Recompute the Jacobian at each stage","TSRosWSetRecomputeJacobian",ros->recompute_jacobian,&ros->recompute_jacobian,NULL);
    1354:   }
    1355:   PetscOptionsTail();
    1356:   /* Rosenbrock methods are linearly implicit, so set that unless the user has specifically asked for something else */
    1357:   TSGetSNES(ts,&snes);
    1358:   if (!((PetscObject)snes)->type_name) {
    1359:     SNESSetType(snes,SNESKSPONLY);
    1360:   }
    1361:   return 0;
    1362: }

    1364: static PetscErrorCode TSView_RosW(TS ts,PetscViewer viewer)
    1365: {
    1366:   TS_RosW        *ros = (TS_RosW*)ts->data;
    1367:   PetscBool      iascii;

    1369:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
    1370:   if (iascii) {
    1371:     RosWTableau tab  = ros->tableau;
    1372:     TSRosWType  rostype;
    1373:     char        buf[512];
    1374:     PetscInt    i;
    1375:     PetscReal   abscissa[512];
    1376:     TSRosWGetType(ts,&rostype);
    1377:     PetscViewerASCIIPrintf(viewer,"  Rosenbrock-W %s\n",rostype);
    1378:     PetscFormatRealArray(buf,sizeof(buf),"% 8.6f",tab->s,tab->ASum);
    1379:     PetscViewerASCIIPrintf(viewer,"  Abscissa of A       = %s\n",buf);
    1380:     for (i=0; i<tab->s; i++) abscissa[i] = tab->ASum[i] + tab->Gamma[i];
    1381:     PetscFormatRealArray(buf,sizeof(buf),"% 8.6f",tab->s,abscissa);
    1382:     PetscViewerASCIIPrintf(viewer,"  Abscissa of A+Gamma = %s\n",buf);
    1383:   }
    1384:   return 0;
    1385: }

    1387: static PetscErrorCode TSLoad_RosW(TS ts,PetscViewer viewer)
    1388: {
    1389:   SNES           snes;
    1390:   TSAdapt        adapt;

    1392:   TSGetAdapt(ts,&adapt);
    1393:   TSAdaptLoad(adapt,viewer);
    1394:   TSGetSNES(ts,&snes);
    1395:   SNESLoad(snes,viewer);
    1396:   /* function and Jacobian context for SNES when used with TS is always ts object */
    1397:   SNESSetFunction(snes,NULL,NULL,ts);
    1398:   SNESSetJacobian(snes,NULL,NULL,NULL,ts);
    1399:   return 0;
    1400: }

    1402: /*@C
    1403:   TSRosWSetType - Set the type of Rosenbrock-W scheme

    1405:   Logically collective

    1407:   Input Parameters:
    1408: +  ts - timestepping context
    1409: -  roswtype - type of Rosenbrock-W scheme

    1411:   Level: beginner

    1413: .seealso: TSRosWGetType(), TSROSW, TSROSW2M, TSROSW2P, TSROSWRA3PW, TSROSWRA34PW2, TSROSWRODAS3, TSROSWSANDU3, TSROSWASSP3P3S1C, TSROSWLASSP3P4S2C, TSROSWLLSSP3P4S2C, TSROSWARK3
    1414: @*/
    1415: PetscErrorCode TSRosWSetType(TS ts,TSRosWType roswtype)
    1416: {
    1419:   PetscTryMethod(ts,"TSRosWSetType_C",(TS,TSRosWType),(ts,roswtype));
    1420:   return 0;
    1421: }

    1423: /*@C
    1424:   TSRosWGetType - Get the type of Rosenbrock-W scheme

    1426:   Logically collective

    1428:   Input Parameter:
    1429: .  ts - timestepping context

    1431:   Output Parameter:
    1432: .  rostype - type of Rosenbrock-W scheme

    1434:   Level: intermediate

    1436: .seealso: TSRosWGetType()
    1437: @*/
    1438: PetscErrorCode TSRosWGetType(TS ts,TSRosWType *rostype)
    1439: {
    1441:   PetscUseMethod(ts,"TSRosWGetType_C",(TS,TSRosWType*),(ts,rostype));
    1442:   return 0;
    1443: }

    1445: /*@C
    1446:   TSRosWSetRecomputeJacobian - Set whether to recompute the Jacobian at each stage. The default is to update the Jacobian once per step.

    1448:   Logically collective

    1450:   Input Parameters:
    1451: +  ts - timestepping context
    1452: -  flg - PETSC_TRUE to recompute the Jacobian at each stage

    1454:   Level: intermediate

    1456: .seealso: TSRosWGetType()
    1457: @*/
    1458: PetscErrorCode TSRosWSetRecomputeJacobian(TS ts,PetscBool flg)
    1459: {
    1461:   PetscTryMethod(ts,"TSRosWSetRecomputeJacobian_C",(TS,PetscBool),(ts,flg));
    1462:   return 0;
    1463: }

    1465: static PetscErrorCode  TSRosWGetType_RosW(TS ts,TSRosWType *rostype)
    1466: {
    1467:   TS_RosW        *ros = (TS_RosW*)ts->data;

    1469:   *rostype = ros->tableau->name;
    1470:   return 0;
    1471: }

    1473: static PetscErrorCode  TSRosWSetType_RosW(TS ts,TSRosWType rostype)
    1474: {
    1475:   TS_RosW         *ros = (TS_RosW*)ts->data;
    1476:   PetscBool       match;
    1477:   RosWTableauLink link;

    1479:   if (ros->tableau) {
    1480:     PetscStrcmp(ros->tableau->name,rostype,&match);
    1481:     if (match) return 0;
    1482:   }
    1483:   for (link = RosWTableauList; link; link=link->next) {
    1484:     PetscStrcmp(link->tab.name,rostype,&match);
    1485:     if (match) {
    1486:       if (ts->setupcalled) TSRosWTableauReset(ts);
    1487:       ros->tableau = &link->tab;
    1488:       if (ts->setupcalled) TSRosWTableauSetUp(ts);
    1489:       ts->default_adapt_type = ros->tableau->bembed ? TSADAPTBASIC : TSADAPTNONE;
    1490:       return 0;
    1491:     }
    1492:   }
    1493:   SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_ARG_UNKNOWN_TYPE,"Could not find '%s'",rostype);
    1494: }

    1496: static PetscErrorCode  TSRosWSetRecomputeJacobian_RosW(TS ts,PetscBool flg)
    1497: {
    1498:   TS_RosW *ros = (TS_RosW*)ts->data;

    1500:   ros->recompute_jacobian = flg;
    1501:   return 0;
    1502: }

    1504: static PetscErrorCode TSDestroy_RosW(TS ts)
    1505: {
    1506:   TSReset_RosW(ts);
    1507:   if (ts->dm) {
    1508:     DMCoarsenHookRemove(ts->dm,DMCoarsenHook_TSRosW,DMRestrictHook_TSRosW,ts);
    1509:     DMSubDomainHookRemove(ts->dm,DMSubDomainHook_TSRosW,DMSubDomainRestrictHook_TSRosW,ts);
    1510:   }
    1511:   PetscFree(ts->data);
    1512:   PetscObjectComposeFunction((PetscObject)ts,"TSRosWGetType_C",NULL);
    1513:   PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetType_C",NULL);
    1514:   PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetRecomputeJacobian_C",NULL);
    1515:   return 0;
    1516: }

    1518: /* ------------------------------------------------------------ */
    1519: /*MC
    1520:       TSROSW - ODE solver using Rosenbrock-W schemes

    1522:   These methods are intended for problems with well-separated time scales, especially when a slow scale is strongly
    1523:   nonlinear such that it is expensive to solve with a fully implicit method. The user should provide the stiff part
    1524:   of the equation using TSSetIFunction() and the non-stiff part with TSSetRHSFunction().

    1526:   Notes:
    1527:   This method currently only works with autonomous ODE and DAE.

    1529:   Consider trying TSARKIMEX if the stiff part is strongly nonlinear.

    1531:   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

    1533:   Developer Notes:
    1534:   Rosenbrock-W methods are typically specified for autonomous ODE

    1536: $  udot = f(u)

    1538:   by the stage equations

    1540: $  k_i = h f(u_0 + sum_j alpha_ij k_j) + h J sum_j gamma_ij k_j

    1542:   and step completion formula

    1544: $  u_1 = u_0 + sum_j b_j k_j

    1546:   with step size h and coefficients alpha_ij, gamma_ij, and b_i. Implementing the method in this form would require f(u)
    1547:   and the Jacobian J to be available, in addition to the shifted matrix I - h gamma_ii J. Following Hairer and Wanner,
    1548:   we define new variables for the stage equations

    1550: $  y_i = gamma_ij k_j

    1552:   The k_j can be recovered because Gamma is invertible. Let C be the lower triangular part of Gamma^{-1} and define

    1554: $  A = Alpha Gamma^{-1}, bt^T = b^T Gamma^{-1}

    1556:   to rewrite the method as

    1558: $  [M/(h gamma_ii) - J] y_i = f(u_0 + sum_j a_ij y_j) + M sum_j (c_ij/h) y_j
    1559: $  u_1 = u_0 + sum_j bt_j y_j

    1561:    where we have introduced the mass matrix M. Continue by defining

    1563: $  ydot_i = 1/(h gamma_ii) y_i - sum_j (c_ij/h) y_j

    1565:    or, more compactly in tensor notation

    1567: $  Ydot = 1/h (Gamma^{-1} \otimes I) Y .

    1569:    Note that Gamma^{-1} is lower triangular. With this definition of Ydot in terms of known quantities and the current
    1570:    stage y_i, the stage equations reduce to performing one Newton step (typically with a lagged Jacobian) on the
    1571:    equation

    1573: $  g(u_0 + sum_j a_ij y_j + y_i, ydot_i) = 0

    1575:    with initial guess y_i = 0.

    1577:   Level: beginner

    1579: .seealso:  TSCreate(), TS, TSSetType(), TSRosWSetType(), TSRosWRegister(), TSROSWTHETA1, TSROSWTHETA2, TSROSW2M, TSROSW2P, TSROSWRA3PW, TSROSWRA34PW2, TSROSWRODAS3,
    1580:            TSROSWSANDU3, TSROSWASSP3P3S1C, TSROSWLASSP3P4S2C, TSROSWLLSSP3P4S2C, TSROSWGRK4T, TSROSWSHAMP4, TSROSWVELDD4, TSROSW4L
    1581: M*/
    1582: PETSC_EXTERN PetscErrorCode TSCreate_RosW(TS ts)
    1583: {
    1584:   TS_RosW        *ros;

    1586:   TSRosWInitializePackage();

    1588:   ts->ops->reset          = TSReset_RosW;
    1589:   ts->ops->destroy        = TSDestroy_RosW;
    1590:   ts->ops->view           = TSView_RosW;
    1591:   ts->ops->load           = TSLoad_RosW;
    1592:   ts->ops->setup          = TSSetUp_RosW;
    1593:   ts->ops->step           = TSStep_RosW;
    1594:   ts->ops->interpolate    = TSInterpolate_RosW;
    1595:   ts->ops->evaluatestep   = TSEvaluateStep_RosW;
    1596:   ts->ops->rollback       = TSRollBack_RosW;
    1597:   ts->ops->setfromoptions = TSSetFromOptions_RosW;
    1598:   ts->ops->snesfunction   = SNESTSFormFunction_RosW;
    1599:   ts->ops->snesjacobian   = SNESTSFormJacobian_RosW;

    1601:   ts->usessnes = PETSC_TRUE;

    1603:   PetscNewLog(ts,&ros);
    1604:   ts->data = (void*)ros;

    1606:   PetscObjectComposeFunction((PetscObject)ts,"TSRosWGetType_C",TSRosWGetType_RosW);
    1607:   PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetType_C",TSRosWSetType_RosW);
    1608:   PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetRecomputeJacobian_C",TSRosWSetRecomputeJacobian_RosW);

    1610:   TSRosWSetType(ts,TSRosWDefault);
    1611:   return 0;
    1612: }