xref: /petsc/src/ts/impls/rosw/rosw.c (revision daa037dfd3c3bec8dc8659548d2b20b07c1dc6de)

/*
  Code for timestepping with Rosenbrock W methods

  Notes:
  The general system is written as

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

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

*/
#include <petsc/private/tsimpl.h>                /*I   "petscts.h"   I*/
#include <petscdm.h>

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

static TSRosWType TSRosWDefault = TSROSWRA34PW2;
static PetscBool  TSRosWRegisterAllCalled;
static PetscBool  TSRosWPackageInitialized;

typedef struct _RosWTableau *RosWTableau;
struct _RosWTableau {
  char      *name;
  PetscInt  order;              /* Classical approximation order of the method */
  PetscInt  s;                  /* Number of stages */
  PetscInt  pinterp;            /* Interpolation order */
  PetscReal *A;                 /* Propagation table, strictly lower triangular */
  PetscReal *Gamma;             /* Stage table, lower triangular with nonzero diagonal */
  PetscBool *GammaZeroDiag;     /* Diagonal entries that are zero in stage table Gamma, vector indicating explicit statages */
  PetscReal *GammaExplicitCorr; /* Coefficients for correction terms needed for explicit stages in transformed variables*/
  PetscReal *b;                 /* Step completion table */
  PetscReal *bembed;            /* Step completion table for embedded method of order one less */
  PetscReal *ASum;              /* Row sum of A */
  PetscReal *GammaSum;          /* Row sum of Gamma, only needed for non-autonomous systems */
  PetscReal *At;                /* Propagation table in transformed variables */
  PetscReal *bt;                /* Step completion table in transformed variables */
  PetscReal *bembedt;           /* Step completion table of order one less in transformed variables */
  PetscReal *GammaInv;          /* Inverse of Gamma, used for transformed variables */
  PetscReal ccfl;               /* Placeholder for CFL coefficient relative to forward Euler */
  PetscReal *binterpt;          /* Dense output formula */
};
typedef struct _RosWTableauLink *RosWTableauLink;
struct _RosWTableauLink {
  struct _RosWTableau tab;
  RosWTableauLink     next;
};
static RosWTableauLink RosWTableauList;

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

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

     Only an approximate Jacobian is needed.

     Level: intermediate

.seealso: TSROSW
M*/

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

     Only an approximate Jacobian is needed.

     Level: intermediate

.seealso: TSROSW
M*/

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

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

     Level: intermediate

.seealso: TSROSW
M*/

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

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

     Level: intermediate

.seealso: TSROSW
M*/

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

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

     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.

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

     Level: intermediate

.seealso: TSROSW
M*/

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

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

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

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

     Level: intermediate

.seealso: TSROSW
M*/

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

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

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

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

     Level: intermediate

.seealso: TSROSW, TSROSWSANDU3
M*/

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

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

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

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

     Level: intermediate

.seealso: TSROSW, TSROSWRODAS3
M*/

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

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

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

     References:
. * - Emil Constantinescu

     Level: intermediate

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

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

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

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

     References:
. * - Emil Constantinescu

     Level: intermediate

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

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

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

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

     References:
. * - Emil Constantinescu

     Level: intermediate

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

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

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

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

     This method does not provide a dense output formula.

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

     Hairer's code ros4.f

     Level: intermediate

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

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

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

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

     This method does not provide a dense output formula.

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

     Hairer's code ros4.f

     Level: intermediate

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

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

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

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

     This method does not provide a dense output formula.

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

     Hairer's code ros4.f

     Level: intermediate

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

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

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

     A-stable and L-stable

     This method does not provide a dense output formula.

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

     Hairer's code ros4.f

     Level: intermediate

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

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

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

  Level: advanced

.seealso:  TSRosWRegisterDestroy()
@*/
PetscErrorCode TSRosWRegisterAll(void)
{
  PetscFunctionBegin;
  if (TSRosWRegisterAllCalled) PetscFunctionReturn(0);
  TSRosWRegisterAllCalled = PETSC_TRUE;

  {
    const PetscReal A = 0;
    const PetscReal Gamma = 1;
    const PetscReal b = 1;
    const PetscReal binterpt=1;

    PetscCall(TSRosWRegister(TSROSWTHETA1,1,1,&A,&Gamma,&b,NULL,1,&binterpt));
  }

  {
    const PetscReal A = 0;
    const PetscReal Gamma = 0.5;
    const PetscReal b = 1;
    const PetscReal binterpt=1;

    PetscCall(TSRosWRegister(TSROSWTHETA2,2,1,&A,&Gamma,&b,NULL,1,&binterpt));
  }

  {
    /*const PetscReal g = 1. + 1./PetscSqrtReal(2.0);   Direct evaluation: 1.707106781186547524401. Used for setting up arrays of values known at compile time below. */
    const PetscReal
      A[2][2]     = {{0,0}, {1.,0}},
      Gamma[2][2] = {{1.707106781186547524401,0}, {-2.*1.707106781186547524401,1.707106781186547524401}},
      b[2]        = {0.5,0.5},
      b1[2]       = {1.0,0.0};
    PetscReal binterpt[2][2];
    binterpt[0][0] = 1.707106781186547524401 - 1.0;
    binterpt[1][0] = 2.0 - 1.707106781186547524401;
    binterpt[0][1] = 1.707106781186547524401 - 1.5;
    binterpt[1][1] = 1.5 - 1.707106781186547524401;

    PetscCall(TSRosWRegister(TSROSW2P,2,2,&A[0][0],&Gamma[0][0],b,b1,2,&binterpt[0][0]));
  }
  {
    /*const PetscReal g = 1. - 1./PetscSqrtReal(2.0);   Direct evaluation: 0.2928932188134524755992. Used for setting up arrays of values known at compile time below. */
    const PetscReal
      A[2][2]     = {{0,0}, {1.,0}},
      Gamma[2][2] = {{0.2928932188134524755992,0}, {-2.*0.2928932188134524755992,0.2928932188134524755992}},
      b[2]        = {0.5,0.5},
      b1[2]       = {1.0,0.0};
    PetscReal binterpt[2][2];
    binterpt[0][0] = 0.2928932188134524755992 - 1.0;
    binterpt[1][0] = 2.0 - 0.2928932188134524755992;
    binterpt[0][1] = 0.2928932188134524755992 - 1.5;
    binterpt[1][1] = 1.5 - 0.2928932188134524755992;

    PetscCall(TSRosWRegister(TSROSW2M,2,2,&A[0][0],&Gamma[0][0],b,b1,2,&binterpt[0][0]));
  }
  {
    /*const PetscReal g = 7.8867513459481287e-01; Directly written in-place below */
    PetscReal binterpt[3][2];
    const PetscReal
      A[3][3] = {{0,0,0},
                 {1.5773502691896257e+00,0,0},
                 {0.5,0,0}},
      Gamma[3][3] = {{7.8867513459481287e-01,0,0},
                     {-1.5773502691896257e+00,7.8867513459481287e-01,0},
                     {-6.7075317547305480e-01,-1.7075317547305482e-01,7.8867513459481287e-01}},
      b[3]  = {1.0566243270259355e-01,4.9038105676657971e-02,8.4529946162074843e-01},
      b2[3] = {-1.7863279495408180e-01,1./3.,8.4529946162074843e-01};

      binterpt[0][0] = -0.8094010767585034;
      binterpt[1][0] = -0.5;
      binterpt[2][0] = 2.3094010767585034;
      binterpt[0][1] = 0.9641016151377548;
      binterpt[1][1] = 0.5;
      binterpt[2][1] = -1.4641016151377548;

      PetscCall(TSRosWRegister(TSROSWRA3PW,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]));
  }
  {
    PetscReal  binterpt[4][3];
    /*const PetscReal g = 4.3586652150845900e-01; Directly written in-place below */
    const PetscReal
      A[4][4] = {{0,0,0,0},
                 {8.7173304301691801e-01,0,0,0},
                 {8.4457060015369423e-01,-1.1299064236484185e-01,0,0},
                 {0,0,1.,0}},
      Gamma[4][4] = {{4.3586652150845900e-01,0,0,0},
                     {-8.7173304301691801e-01,4.3586652150845900e-01,0,0},
                     {-9.0338057013044082e-01,5.4180672388095326e-02,4.3586652150845900e-01,0},
                     {2.4212380706095346e-01,-1.2232505839045147e+00,5.4526025533510214e-01,4.3586652150845900e-01}},
      b[4]  = {2.4212380706095346e-01,-1.2232505839045147e+00,1.5452602553351020e+00,4.3586652150845900e-01},
      b2[4] = {3.7810903145819369e-01,-9.6042292212423178e-02,5.0000000000000000e-01,2.1793326075422950e-01};

    binterpt[0][0]=1.0564298455794094;
    binterpt[1][0]=2.296429974281067;
    binterpt[2][0]=-1.307599564525376;
    binterpt[3][0]=-1.045260255335102;
    binterpt[0][1]=-1.3864882699759573;
    binterpt[1][1]=-8.262611700275677;
    binterpt[2][1]=7.250979895056055;
    binterpt[3][1]=2.398120075195581;
    binterpt[0][2]=0.5721822314575016;
    binterpt[1][2]=4.742931142090097;
    binterpt[2][2]=-4.398120075195578;
    binterpt[3][2]=-0.9169932983520199;

    PetscCall(TSRosWRegister(TSROSWRA34PW2,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]));
  }
  {
    /* const PetscReal g = 0.5;       Directly written in-place below */
    const PetscReal
      A[4][4] = {{0,0,0,0},
                 {0,0,0,0},
                 {1.,0,0,0},
                 {0.75,-0.25,0.5,0}},
      Gamma[4][4] = {{0.5,0,0,0},
                     {1.,0.5,0,0},
                     {-0.25,-0.25,0.5,0},
                     {1./12,1./12,-2./3,0.5}},
      b[4]  = {5./6,-1./6,-1./6,0.5},
      b2[4] = {0.75,-0.25,0.5,0};

    PetscCall(TSRosWRegister(TSROSWRODAS3,3,4,&A[0][0],&Gamma[0][0],b,b2,0,NULL));
  }
  {
    /*const PetscReal g = 0.43586652150845899941601945119356;       Directly written in-place below */
    const PetscReal
      A[3][3] = {{0,0,0},
                 {0.43586652150845899941601945119356,0,0},
                 {0.43586652150845899941601945119356,0,0}},
      Gamma[3][3] = {{0.43586652150845899941601945119356,0,0},
                     {-0.19294655696029095575009695436041,0.43586652150845899941601945119356,0},
                     {0,1.74927148125794685173529749738960,0.43586652150845899941601945119356}},
      b[3]  = {-0.75457412385404315829818998646589,1.94100407061964420292840123379419,-0.18642994676560104463021124732829},
      b2[3] = {-1.53358745784149585370766523913002,2.81745131148625772213931745457622,-0.28386385364476186843165221544619};

    PetscReal binterpt[3][2];
    binterpt[0][0] = 3.793692883777660870425141387941;
    binterpt[1][0] = -2.918692883777660870425141387941;
    binterpt[2][0] = 0.125;
    binterpt[0][1] = -0.725741064379812106687651020584;
    binterpt[1][1] = 0.559074397713145440020984353917;
    binterpt[2][1] = 0.16666666666666666666666666666667;

    PetscCall(TSRosWRegister(TSROSWSANDU3,3,3,&A[0][0],&Gamma[0][0],b,b2,2,&binterpt[0][0]));
  }
  {
    /*const PetscReal s3 = PetscSqrtReal(3.),g = (3.0+s3)/6.0;
     * Direct evaluation: s3 = 1.732050807568877293527;
     *                     g = 0.7886751345948128822546;
     * Values are directly inserted below to ensure availability at compile time (compiler warnings otherwise...) */
    const PetscReal
      A[3][3] = {{0,0,0},
                 {1,0,0},
                 {0.25,0.25,0}},
      Gamma[3][3] = {{0,0,0},
                     {(-3.0-1.732050807568877293527)/6.0,0.7886751345948128822546,0},
                     {(-3.0-1.732050807568877293527)/24.0,(-3.0-1.732050807568877293527)/8.0,0.7886751345948128822546}},
      b[3]  = {1./6.,1./6.,2./3.},
      b2[3] = {1./4.,1./4.,1./2.};
    PetscReal binterpt[3][2];

    binterpt[0][0]=0.089316397477040902157517886164709;
    binterpt[1][0]=-0.91068360252295909784248211383529;
    binterpt[2][0]=1.8213672050459181956849642276706;
    binterpt[0][1]=0.077350269189625764509148780501957;
    binterpt[1][1]=1.077350269189625764509148780502;
    binterpt[2][1]=-1.1547005383792515290182975610039;

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

  {
    const PetscReal
      A[4][4] = {{0,0,0,0},
                 {1./2.,0,0,0},
                 {1./2.,1./2.,0,0},
                 {1./6.,1./6.,1./6.,0}},
      Gamma[4][4] = {{1./2.,0,0,0},
                     {0.0,1./4.,0,0},
                     {-2.,-2./3.,2./3.,0},
                     {1./2.,5./36.,-2./9,0}},
      b[4]  = {1./6.,1./6.,1./6.,1./2.},
      b2[4] = {1./8.,3./4.,1./8.,0};
    PetscReal binterpt[4][3];

    binterpt[0][0]=6.25;
    binterpt[1][0]=-30.25;
    binterpt[2][0]=1.75;
    binterpt[3][0]=23.25;
    binterpt[0][1]=-9.75;
    binterpt[1][1]=58.75;
    binterpt[2][1]=-3.25;
    binterpt[3][1]=-45.75;
    binterpt[0][2]=3.6666666666666666666666666666667;
    binterpt[1][2]=-28.333333333333333333333333333333;
    binterpt[2][2]=1.6666666666666666666666666666667;
    binterpt[3][2]=23.;

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

  {
    const PetscReal
      A[4][4] = {{0,0,0,0},
                 {1./2.,0,0,0},
                 {1./2.,1./2.,0,0},
                 {1./6.,1./6.,1./6.,0}},
      Gamma[4][4] = {{1./2.,0,0,0},
                     {0.0,3./4.,0,0},
                     {-2./3.,-23./9.,2./9.,0},
                     {1./18.,65./108.,-2./27,0}},
      b[4]  = {1./6.,1./6.,1./6.,1./2.},
      b2[4] = {3./16.,10./16.,3./16.,0};
    PetscReal binterpt[4][3];

    binterpt[0][0]=1.6911764705882352941176470588235;
    binterpt[1][0]=3.6813725490196078431372549019608;
    binterpt[2][0]=0.23039215686274509803921568627451;
    binterpt[3][0]=-4.6029411764705882352941176470588;
    binterpt[0][1]=-0.95588235294117647058823529411765;
    binterpt[1][1]=-6.2401960784313725490196078431373;
    binterpt[2][1]=-0.31862745098039215686274509803922;
    binterpt[3][1]=7.5147058823529411764705882352941;
    binterpt[0][2]=-0.56862745098039215686274509803922;
    binterpt[1][2]=2.7254901960784313725490196078431;
    binterpt[2][2]=0.25490196078431372549019607843137;
    binterpt[3][2]=-2.4117647058823529411764705882353;

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

  {
    PetscReal A[4][4],Gamma[4][4],b[4],b2[4];
    PetscReal binterpt[4][3];

    Gamma[0][0]=0.4358665215084589994160194475295062513822671686978816;
    Gamma[0][1]=0; Gamma[0][2]=0; Gamma[0][3]=0;
    Gamma[1][0]=-1.997527830934941248426324674704153457289527280554476;
    Gamma[1][1]=0.4358665215084589994160194475295062513822671686978816;
    Gamma[1][2]=0; Gamma[1][3]=0;
    Gamma[2][0]=-1.007948511795029620852002345345404191008352770119903;
    Gamma[2][1]=-0.004648958462629345562774289390054679806993396798458131;
    Gamma[2][2]=0.4358665215084589994160194475295062513822671686978816;
    Gamma[2][3]=0;
    Gamma[3][0]=-0.6685429734233467180451604600279552604364311322650783;
    Gamma[3][1]=0.6056625986449338476089525334450053439525178740492984;
    Gamma[3][2]=-0.9717899277217721234705114616271378792182450260943198;
    Gamma[3][3]=0;

    A[0][0]=0; A[0][1]=0; A[0][2]=0; A[0][3]=0;
    A[1][0]=0.8717330430169179988320388950590125027645343373957631;
    A[1][1]=0; A[1][2]=0; A[1][3]=0;
    A[2][0]=0.5275890119763004115618079766722914408876108660811028;
    A[2][1]=0.07241098802369958843819203208518599088698057726988732;
    A[2][2]=0; A[2][3]=0;
    A[3][0]=0.3990960076760701320627260685975778145384666450351314;
    A[3][1]=-0.4375576546135194437228463747348862825846903771419953;
    A[3][2]=1.038461646937449311660120300601880176655352737312713;
    A[3][3]=0;

    b[0]=0.1876410243467238251612921333138006734899663569186926;
    b[1]=-0.5952974735769549480478230473706443582188442040780541;
    b[2]=0.9717899277217721234705114616271378792182450260943198;
    b[3]=0.4358665215084589994160194475295062513822671686978816;

    b2[0]=0.2147402862233891404862383521089097657790734483804460;
    b2[1]=-0.4851622638849390928209050538171743017757490232519684;
    b2[2]=0.8687250025203875511662123688667549217531982787600080;
    b2[3]=0.4016969751411624011684543450940068201770721128357014;

    binterpt[0][0]=2.2565812720167954547104627844105;
    binterpt[1][0]=1.349166413351089573796243820819;
    binterpt[2][0]=-2.4695174540533503758652847586647;
    binterpt[3][0]=-0.13623023131453465264142184656474;
    binterpt[0][1]=-3.0826699111559187902922463354557;
    binterpt[1][1]=-2.4689115685996042534544925650515;
    binterpt[2][1]=5.7428279814696677152129332773553;
    binterpt[3][1]=-0.19124650171414467146619437684812;
    binterpt[0][2]=1.0137296634858471607430756831148;
    binterpt[1][2]=0.52444768167155973161042570784064;
    binterpt[2][2]=-2.3015205996945452158771370439586;
    binterpt[3][2]=0.76334325453713832352363565300308;

    PetscCall(TSRosWRegister(TSROSWARK3,3,4,&A[0][0],&Gamma[0][0],b,b2,3,&binterpt[0][0]));
  }
  PetscCall(TSRosWRegisterRos4(TSROSWGRK4T,0.231,PETSC_DEFAULT,PETSC_DEFAULT,0,-0.1282612945269037e+01));
  PetscCall(TSRosWRegisterRos4(TSROSWSHAMP4,0.5,PETSC_DEFAULT,PETSC_DEFAULT,0,125./108.));
  PetscCall(TSRosWRegisterRos4(TSROSWVELDD4,0.22570811482256823492,PETSC_DEFAULT,PETSC_DEFAULT,0,-1.355958941201148));
  PetscCall(TSRosWRegisterRos4(TSROSW4L,0.57282,PETSC_DEFAULT,PETSC_DEFAULT,0,-1.093502252409163));
  PetscFunctionReturn(0);
}

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

   Not Collective

   Level: advanced

.seealso: TSRosWRegister(), TSRosWRegisterAll()
@*/
PetscErrorCode TSRosWRegisterDestroy(void)
{
  RosWTableauLink link;

  PetscFunctionBegin;
  while ((link = RosWTableauList)) {
    RosWTableau t = &link->tab;
    RosWTableauList = link->next;
    PetscCall(PetscFree5(t->A,t->Gamma,t->b,t->ASum,t->GammaSum));
    PetscCall(PetscFree5(t->At,t->bt,t->GammaInv,t->GammaZeroDiag,t->GammaExplicitCorr));
    PetscCall(PetscFree2(t->bembed,t->bembedt));
    PetscCall(PetscFree(t->binterpt));
    PetscCall(PetscFree(t->name));
    PetscCall(PetscFree(link));
  }
  TSRosWRegisterAllCalled = PETSC_FALSE;
  PetscFunctionReturn(0);
}

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

  Level: developer

.seealso: PetscInitialize()
@*/
PetscErrorCode TSRosWInitializePackage(void)
{
  PetscFunctionBegin;
  if (TSRosWPackageInitialized) PetscFunctionReturn(0);
  TSRosWPackageInitialized = PETSC_TRUE;
  PetscCall(TSRosWRegisterAll());
  PetscCall(PetscRegisterFinalize(TSRosWFinalizePackage));
  PetscFunctionReturn(0);
}

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

  Level: developer

.seealso: PetscFinalize()
@*/
PetscErrorCode TSRosWFinalizePackage(void)
{
  PetscFunctionBegin;
  TSRosWPackageInitialized = PETSC_FALSE;
  PetscCall(TSRosWRegisterDestroy());
  PetscFunctionReturn(0);
}

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

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

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

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

   Level: advanced

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

  PetscFunctionBegin;
  PetscValidCharPointer(name,1);
  PetscValidRealPointer(A,4);
  PetscValidRealPointer(Gamma,5);
  PetscValidRealPointer(b,6);
  if (bembed) PetscValidRealPointer(bembed,7);

  PetscCall(TSRosWInitializePackage());
  PetscCall(PetscNew(&link));
  t        = &link->tab;
  PetscCall(PetscStrallocpy(name,&t->name));
  t->order = order;
  t->s     = s;
  PetscCall(PetscMalloc5(s*s,&t->A,s*s,&t->Gamma,s,&t->b,s,&t->ASum,s,&t->GammaSum));
  PetscCall(PetscMalloc5(s*s,&t->At,s,&t->bt,s*s,&t->GammaInv,s,&t->GammaZeroDiag,s*s,&t->GammaExplicitCorr));
  PetscCall(PetscArraycpy(t->A,A,s*s));
  PetscCall(PetscArraycpy(t->Gamma,Gamma,s*s));
  PetscCall(PetscArraycpy(t->GammaExplicitCorr,Gamma,s*s));
  PetscCall(PetscArraycpy(t->b,b,s));
  if (bembed) {
    PetscCall(PetscMalloc2(s,&t->bembed,s,&t->bembedt));
    PetscCall(PetscArraycpy(t->bembed,bembed,s));
  }
  for (i=0; i<s; i++) {
    t->ASum[i]     = 0;
    t->GammaSum[i] = 0;
    for (j=0; j<s; j++) {
      t->ASum[i]     += A[i*s+j];
      t->GammaSum[i] += Gamma[i*s+j];
    }
  }
  PetscCall(PetscMalloc1(s*s,&GammaInv)); /* Need to use Scalar for inverse, then convert back to Real */
  for (i=0; i<s*s; i++) GammaInv[i] = Gamma[i];
  for (i=0; i<s; i++) {
    if (Gamma[i*s+i] == 0.0) {
      GammaInv[i*s+i] = 1.0;
      t->GammaZeroDiag[i] = PETSC_TRUE;
    } else {
      t->GammaZeroDiag[i] = PETSC_FALSE;
    }
  }

  switch (s) {
  case 1: GammaInv[0] = 1./GammaInv[0]; break;
  case 2: PetscCall(PetscKernel_A_gets_inverse_A_2(GammaInv,0,PETSC_FALSE,NULL)); break;
  case 3: PetscCall(PetscKernel_A_gets_inverse_A_3(GammaInv,0,PETSC_FALSE,NULL)); break;
  case 4: PetscCall(PetscKernel_A_gets_inverse_A_4(GammaInv,0,PETSC_FALSE,NULL)); break;
  case 5: {
    PetscInt  ipvt5[5];
    MatScalar work5[5*5];
    PetscCall(PetscKernel_A_gets_inverse_A_5(GammaInv,ipvt5,work5,0,PETSC_FALSE,NULL)); break;
  }
  case 6: PetscCall(PetscKernel_A_gets_inverse_A_6(GammaInv,0,PETSC_FALSE,NULL)); break;
  case 7: PetscCall(PetscKernel_A_gets_inverse_A_7(GammaInv,0,PETSC_FALSE,NULL)); break;
  default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Not implemented for %D stages",s);
  }
  for (i=0; i<s*s; i++) t->GammaInv[i] = PetscRealPart(GammaInv[i]);
  PetscCall(PetscFree(GammaInv));

  for (i=0; i<s; i++) {
    for (k=0; k<i+1; k++) {
      t->GammaExplicitCorr[i*s+k]=(t->GammaExplicitCorr[i*s+k])*(t->GammaInv[k*s+k]);
      for (j=k+1; j<i+1; j++) {
        t->GammaExplicitCorr[i*s+k]+=(t->GammaExplicitCorr[i*s+j])*(t->GammaInv[j*s+k]);
      }
    }
  }

  for (i=0; i<s; i++) {
    for (j=0; j<s; j++) {
      t->At[i*s+j] = 0;
      for (k=0; k<s; k++) {
        t->At[i*s+j] += t->A[i*s+k] * t->GammaInv[k*s+j];
      }
    }
    t->bt[i] = 0;
    for (j=0; j<s; j++) {
      t->bt[i] += t->b[j] * t->GammaInv[j*s+i];
    }
    if (bembed) {
      t->bembedt[i] = 0;
      for (j=0; j<s; j++) {
        t->bembedt[i] += t->bembed[j] * t->GammaInv[j*s+i];
      }
    }
  }
  t->ccfl = 1.0;                /* Fix this */

  t->pinterp = pinterp;
  PetscCall(PetscMalloc1(s*pinterp,&t->binterpt));
  PetscCall(PetscArraycpy(t->binterpt,binterpt,s*pinterp));
  link->next = RosWTableauList;
  RosWTableauList = link;
  PetscFunctionReturn(0);
}

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

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

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

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

   Level: developer

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

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

  /* Solve order conditions 7.15a, 7.15c, 7.15e */
  M[0][0] = one; M[0][1] = one;      M[0][2] = one;      /* 7.15a */
  M[1][0] = 0.0; M[1][1] = a2*a2;    M[1][2] = a4*a4;    /* 7.15c */
  M[2][0] = 0.0; M[2][1] = a2*a2*a2; M[2][2] = a4*a4*a4; /* 7.15e */
  rhs[0]  = one - b3;
  rhs[1]  = one/three - a3*a3*b3;
  rhs[2]  = one/four - a3*a3*a3*b3;
  PetscCall(PetscKernel_A_gets_inverse_A_3(&M[0][0],0,PETSC_FALSE,NULL));
  b1      = PetscRealPart(M[0][0]*rhs[0] + M[0][1]*rhs[1] + M[0][2]*rhs[2]);
  b2      = PetscRealPart(M[1][0]*rhs[0] + M[1][1]*rhs[1] + M[1][2]*rhs[2]);
  b4      = PetscRealPart(M[2][0]*rhs[0] + M[2][1]*rhs[1] + M[2][2]*rhs[2]);

  /* Step 3 */
  beta43       = (p56 - a2*p43) / (b4*a3*a3*(a3 - a2)); /* 7.21 */
  beta32beta2p =  p44 / (b4*beta43);                    /* 7.15h */
  beta4jbetajp = (p32 - b3*beta32beta2p) / b4;
  M[0][0]      = b2;                                    M[0][1] = b3;                 M[0][2] = b4;
  M[1][0]      = a4*a4*beta32beta2p-a3*a3*beta4jbetajp; M[1][1] = a2*a2*beta4jbetajp; M[1][2] = -a2*a2*beta32beta2p;
  M[2][0]      = b4*beta43*a3*a3-p43;                   M[2][1] = -b4*beta43*a2*a2;   M[2][2] = 0;
  rhs[0]       = one/two - gamma; rhs[1] = 0; rhs[2] = -a2*a2*p32;
  PetscCall(PetscKernel_A_gets_inverse_A_3(&M[0][0],0,PETSC_FALSE,NULL));
  beta2p       = PetscRealPart(M[0][0]*rhs[0] + M[0][1]*rhs[1] + M[0][2]*rhs[2]);
  beta3p       = PetscRealPart(M[1][0]*rhs[0] + M[1][1]*rhs[1] + M[1][2]*rhs[2]);
  beta4p       = PetscRealPart(M[2][0]*rhs[0] + M[2][1]*rhs[1] + M[2][2]*rhs[2]);

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

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

  A[0][0]     = 0;          A[0][1] = 0;   A[0][2] = 0;   A[0][3] = 0;
  A[1][0]     = a2;         A[1][1] = 0;   A[1][2] = 0;   A[1][3] = 0;
  A[2][0]     = a3-a32;     A[2][1] = a32; A[2][2] = 0;   A[2][3] = 0;
  A[3][0]     = a4-a43-a42; A[3][1] = a42; A[3][2] = a43; A[3][3] = 0;
  Gamma[0][0] = gamma;                        Gamma[0][1] = 0;              Gamma[0][2] = 0;              Gamma[0][3] = 0;
  Gamma[1][0] = beta2p-A[1][0];               Gamma[1][1] = gamma;          Gamma[1][2] = 0;              Gamma[1][3] = 0;
  Gamma[2][0] = beta3p-beta32-A[2][0];        Gamma[2][1] = beta32-A[2][1]; Gamma[2][2] = gamma;          Gamma[2][3] = 0;
  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;
  b[0] = b1; b[1] = b2; b[2] = b3; b[3] = b4;

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

  {
    const PetscReal misfit = a2*a2*bm[1] + a3*a3*bm[2] + a4*a4*bm[3] - one/three;
    PetscCheck(PetscAbs(misfit) <= PETSC_SMALL,PETSC_COMM_SELF,PETSC_ERR_SUP,"Assumptions violated, could not construct a third order embedded method");
  }
  PetscCall(TSRosWRegister(name,4,4,&A[0][0],&Gamma[0][0],b,bm,0,NULL));
  PetscFunctionReturn(0);
}

/*
 The step completion formula is

 x1 = x0 + b^T Y

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

 x1e = x0 + be^T Y
     = x1 - b^T Y + be^T Y
     = x1 + (be - b)^T Y

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

  PetscFunctionBegin;
  if (order == tab->order) {
    if (ros->status == TS_STEP_INCOMPLETE) { /* Use standard completion formula */
      PetscCall(VecCopy(ts->vec_sol,U));
      for (i=0; i<tab->s; i++) w[i] = tab->bt[i];
      PetscCall(VecMAXPY(U,tab->s,w,ros->Y));
    } else PetscCall(VecCopy(ts->vec_sol,U));
    if (done) *done = PETSC_TRUE;
    PetscFunctionReturn(0);
  } else if (order == tab->order-1) {
    if (!tab->bembedt) goto unavailable;
    if (ros->status == TS_STEP_INCOMPLETE) { /* Use embedded completion formula */
      PetscCall(VecCopy(ts->vec_sol,U));
      for (i=0; i<tab->s; i++) w[i] = tab->bembedt[i];
      PetscCall(VecMAXPY(U,tab->s,w,ros->Y));
    } else {                    /* Use rollback-and-recomplete formula (bembedt - bt) */
      for (i=0; i<tab->s; i++) w[i] = tab->bembedt[i] - tab->bt[i];
      PetscCall(VecCopy(ts->vec_sol,U));
      PetscCall(VecMAXPY(U,tab->s,w,ros->Y));
    }
    if (done) *done = PETSC_TRUE;
    PetscFunctionReturn(0);
  }
  unavailable:
  if (done) *done = PETSC_FALSE;
  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);
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  PetscCall(VecCopy(ros->vec_sol_prev,ts->vec_sol));
  PetscFunctionReturn(0);
}

static PetscErrorCode TSStep_RosW(TS ts)
{
  TS_RosW         *ros = (TS_RosW*)ts->data;
  RosWTableau     tab  = ros->tableau;
  const PetscInt  s    = tab->s;
  const PetscReal *At  = tab->At,*Gamma = tab->Gamma,*ASum = tab->ASum,*GammaInv = tab->GammaInv;
  const PetscReal *GammaExplicitCorr = tab->GammaExplicitCorr;
  const PetscBool *GammaZeroDiag = tab->GammaZeroDiag;
  PetscScalar     *w   = ros->work;
  Vec             *Y   = ros->Y,Ydot = ros->Ydot,Zdot = ros->Zdot,Zstage = ros->Zstage;
  SNES            snes;
  TSAdapt         adapt;
  PetscInt        i,j,its,lits;
  PetscInt        rejections = 0;
  PetscBool       stageok,accept = PETSC_TRUE;
  PetscReal       next_time_step = ts->time_step;
  PetscInt        lag;

  PetscFunctionBegin;
  if (!ts->steprollback) {
    PetscCall(VecCopy(ts->vec_sol,ros->vec_sol_prev));
  }

  ros->status = TS_STEP_INCOMPLETE;
  while (!ts->reason && ros->status != TS_STEP_COMPLETE) {
    const PetscReal h = ts->time_step;
    for (i=0; i<s; i++) {
      ros->stage_time = ts->ptime + h*ASum[i];
      PetscCall(TSPreStage(ts,ros->stage_time));
      if (GammaZeroDiag[i]) {
        ros->stage_explicit = PETSC_TRUE;
        ros->scoeff         = 1.;
      } else {
        ros->stage_explicit = PETSC_FALSE;
        ros->scoeff         = 1./Gamma[i*s+i];
      }

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

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

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

      if (!ros->stage_explicit) {
        PetscCall(TSGetSNES(ts,&snes));
        if (!ros->recompute_jacobian && !i) {
          PetscCall(SNESGetLagJacobian(snes,&lag));
          if (lag == 1) {  /* use did not set a nontrival lag, so lag over all stages */
            PetscCall(SNESSetLagJacobian(snes,-2)); /* Recompute the Jacobian on this solve, but not again for the rest of the stages */
          }
        }
        PetscCall(SNESSolve(snes,NULL,Y[i]));
        if (!ros->recompute_jacobian && i == s-1 && lag == 1) {
          PetscCall(SNESSetLagJacobian(snes,lag)); /* Set lag back to 1 so we know user did not set it */
        }
        PetscCall(SNESGetIterationNumber(snes,&its));
        PetscCall(SNESGetLinearSolveIterations(snes,&lits));
        ts->snes_its += its; ts->ksp_its += lits;
      } else {
        Mat J,Jp;
        PetscCall(VecZeroEntries(Ydot)); /* Evaluate Y[i]=G(t,Ydot=0,Zstage) */
        PetscCall(TSComputeIFunction(ts,ros->stage_time,Zstage,Ydot,Y[i],PETSC_FALSE));
        PetscCall(VecScale(Y[i],-1.0));
        PetscCall(VecAXPY(Y[i],-1.0,Zdot)); /*Y[i] = F(Zstage)-Zdot[=GammaInv*Y]*/

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

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

        PetscCall(VecScale(Y[i],h));
      }
      PetscCall(TSPostStage(ts,ros->stage_time,i,Y));
      PetscCall(TSGetAdapt(ts,&adapt));
      PetscCall(TSAdaptCheckStage(adapt,ts,ros->stage_time,Y[i],&stageok));
      if (!stageok) goto reject_step;
    }

    ros->status = TS_STEP_INCOMPLETE;
    PetscCall(TSEvaluateStep_RosW(ts,tab->order,ts->vec_sol,NULL));
    ros->status = TS_STEP_PENDING;
    PetscCall(TSGetAdapt(ts,&adapt));
    PetscCall(TSAdaptCandidatesClear(adapt));
    PetscCall(TSAdaptCandidateAdd(adapt,tab->name,tab->order,1,tab->ccfl,(PetscReal)tab->s,PETSC_TRUE));
    PetscCall(TSAdaptChoose(adapt,ts,ts->time_step,NULL,&next_time_step,&accept));
    ros->status = accept ? TS_STEP_COMPLETE : TS_STEP_INCOMPLETE;
    if (!accept) { /* Roll back the current step */
      PetscCall(TSRollBack_RosW(ts));
      ts->time_step = next_time_step;
      goto reject_step;
    }

    ts->ptime += ts->time_step;
    ts->time_step = next_time_step;
    break;

  reject_step:
    ts->reject++; accept = PETSC_FALSE;
    if (!ts->reason && ++rejections > ts->max_reject && ts->max_reject >= 0) {
      ts->reason = TS_DIVERGED_STEP_REJECTED;
      PetscCall(PetscInfo(ts,"Step=%D, step rejections %D greater than current TS allowed, stopping solve\n",ts->steps,rejections));
    }
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TSInterpolate_RosW(TS ts,PetscReal itime,Vec U)
{
  TS_RosW         *ros = (TS_RosW*)ts->data;
  PetscInt        s    = ros->tableau->s,pinterp = ros->tableau->pinterp,i,j;
  PetscReal       h;
  PetscReal       tt,t;
  PetscScalar     *bt;
  const PetscReal *Bt = ros->tableau->binterpt;
  const PetscReal *GammaInv = ros->tableau->GammaInv;
  PetscScalar     *w        = ros->work;
  Vec             *Y        = ros->Y;

  PetscFunctionBegin;
  PetscCheck(Bt,PetscObjectComm((PetscObject)ts),PETSC_ERR_SUP,"TSRosW %s does not have an interpolation formula",ros->tableau->name);

  switch (ros->status) {
  case TS_STEP_INCOMPLETE:
  case TS_STEP_PENDING:
    h = ts->time_step;
    t = (itime - ts->ptime)/h;
    break;
  case TS_STEP_COMPLETE:
    h = ts->ptime - ts->ptime_prev;
    t = (itime - ts->ptime)/h + 1; /* In the interval [0,1] */
    break;
  default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
  }
  PetscCall(PetscMalloc1(s,&bt));
  for (i=0; i<s; i++) bt[i] = 0;
  for (j=0,tt=t; j<pinterp; j++,tt*=t) {
    for (i=0; i<s; i++) {
      bt[i] += Bt[i*pinterp+j] * tt;
    }
  }

  /* y(t+tt*h) = y(t) + Sum bt(tt) * GammaInv * Ydot */
  /* U <- 0*/
  PetscCall(VecZeroEntries(U));
  /* U <- Sum bt_i * GammaInv(i,1:i) * Y(1:i) */
  for (j=0; j<s; j++) w[j] = 0;
  for (j=0; j<s; j++) {
    for (i=j; i<s; i++) {
      w[j] +=  bt[i]*GammaInv[i*s+j];
    }
  }
  PetscCall(VecMAXPY(U,i,w,Y));
  /* U <- y(t) + U */
  PetscCall(VecAXPY(U,1,ros->vec_sol_prev));

  PetscCall(PetscFree(bt));
  PetscFunctionReturn(0);
}

/*------------------------------------------------------------*/

static PetscErrorCode TSRosWTableauReset(TS ts)
{
  TS_RosW        *ros = (TS_RosW*)ts->data;
  RosWTableau    tab  = ros->tableau;

  PetscFunctionBegin;
  if (!tab) PetscFunctionReturn(0);
  PetscCall(VecDestroyVecs(tab->s,&ros->Y));
  PetscCall(PetscFree(ros->work));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  PetscCall(TSRosWTableauReset(ts));
  PetscCall(VecDestroy(&ros->Ydot));
  PetscCall(VecDestroy(&ros->Ystage));
  PetscCall(VecDestroy(&ros->Zdot));
  PetscCall(VecDestroy(&ros->Zstage));
  PetscCall(VecDestroy(&ros->vec_sol_prev));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  if (Ydot) {
    if (dm && dm != ts->dm) {
      PetscCall(DMGetNamedGlobalVector(dm,"TSRosW_Ydot",Ydot));
    } else *Ydot = rw->Ydot;
  }
  if (Zdot) {
    if (dm && dm != ts->dm) {
      PetscCall(DMGetNamedGlobalVector(dm,"TSRosW_Zdot",Zdot));
    } else *Zdot = rw->Zdot;
  }
  if (Ystage) {
    if (dm && dm != ts->dm) {
      PetscCall(DMGetNamedGlobalVector(dm,"TSRosW_Ystage",Ystage));
    } else *Ystage = rw->Ystage;
  }
  if (Zstage) {
    if (dm && dm != ts->dm) {
      PetscCall(DMGetNamedGlobalVector(dm,"TSRosW_Zstage",Zstage));
    } else *Zstage = rw->Zstage;
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TSRosWRestoreVecs(TS ts,DM dm,Vec *Ydot,Vec *Zdot, Vec *Ystage, Vec *Zstage)
{
  PetscFunctionBegin;
  if (Ydot) {
    if (dm && dm != ts->dm) {
      PetscCall(DMRestoreNamedGlobalVector(dm,"TSRosW_Ydot",Ydot));
    }
  }
  if (Zdot) {
    if (dm && dm != ts->dm) {
      PetscCall(DMRestoreNamedGlobalVector(dm,"TSRosW_Zdot",Zdot));
    }
  }
  if (Ystage) {
    if (dm && dm != ts->dm) {
      PetscCall(DMRestoreNamedGlobalVector(dm,"TSRosW_Ystage",Ystage));
    }
  }
  if (Zstage) {
    if (dm && dm != ts->dm) {
      PetscCall(DMRestoreNamedGlobalVector(dm,"TSRosW_Zstage",Zstage));
    }
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode DMCoarsenHook_TSRosW(DM fine,DM coarse,void *ctx)
{
  PetscFunctionBegin;
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  PetscCall(TSRosWGetVecs(ts,fine,&Ydot,&Ystage,&Zdot,&Zstage));
  PetscCall(TSRosWGetVecs(ts,coarse,&Ydotc,&Ystagec,&Zdotc,&Zstagec));
  PetscCall(MatRestrict(restrct,Ydot,Ydotc));
  PetscCall(VecPointwiseMult(Ydotc,rscale,Ydotc));
  PetscCall(MatRestrict(restrct,Ystage,Ystagec));
  PetscCall(VecPointwiseMult(Ystagec,rscale,Ystagec));
  PetscCall(MatRestrict(restrct,Zdot,Zdotc));
  PetscCall(VecPointwiseMult(Zdotc,rscale,Zdotc));
  PetscCall(MatRestrict(restrct,Zstage,Zstagec));
  PetscCall(VecPointwiseMult(Zstagec,rscale,Zstagec));
  PetscCall(TSRosWRestoreVecs(ts,fine,&Ydot,&Ystage,&Zdot,&Zstage));
  PetscCall(TSRosWRestoreVecs(ts,coarse,&Ydotc,&Ystagec,&Zdotc,&Zstagec));
  PetscFunctionReturn(0);
}

static PetscErrorCode DMSubDomainHook_TSRosW(DM fine,DM coarse,void *ctx)
{
  PetscFunctionBegin;
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  PetscCall(TSRosWGetVecs(ts,dm,&Ydot,&Ystage,&Zdot,&Zstage));
  PetscCall(TSRosWGetVecs(ts,subdm,&Ydots,&Ystages,&Zdots,&Zstages));

  PetscCall(VecScatterBegin(gscat,Ydot,Ydots,INSERT_VALUES,SCATTER_FORWARD));
  PetscCall(VecScatterEnd(gscat,Ydot,Ydots,INSERT_VALUES,SCATTER_FORWARD));

  PetscCall(VecScatterBegin(gscat,Ystage,Ystages,INSERT_VALUES,SCATTER_FORWARD));
  PetscCall(VecScatterEnd(gscat,Ystage,Ystages,INSERT_VALUES,SCATTER_FORWARD));

  PetscCall(VecScatterBegin(gscat,Zdot,Zdots,INSERT_VALUES,SCATTER_FORWARD));
  PetscCall(VecScatterEnd(gscat,Zdot,Zdots,INSERT_VALUES,SCATTER_FORWARD));

  PetscCall(VecScatterBegin(gscat,Zstage,Zstages,INSERT_VALUES,SCATTER_FORWARD));
  PetscCall(VecScatterEnd(gscat,Zstage,Zstages,INSERT_VALUES,SCATTER_FORWARD));

  PetscCall(TSRosWRestoreVecs(ts,dm,&Ydot,&Ystage,&Zdot,&Zstage));
  PetscCall(TSRosWRestoreVecs(ts,subdm,&Ydots,&Ystages,&Zdots,&Zstages));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  PetscCall(SNESGetDM(snes,&dm));
  PetscCall(TSRosWGetVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage));
  PetscCall(VecWAXPY(Ydot,shift,U,Zdot));    /* Ydot = shift*U + Zdot */
  PetscCall(VecWAXPY(Ystage,1.0,U,Zstage));  /* Ystage = U + Zstage */
  dmsave = ts->dm;
  ts->dm = dm;
  PetscCall(TSComputeIFunction(ts,ros->stage_time,Ystage,Ydot,F,PETSC_FALSE));
  ts->dm = dmsave;
  PetscCall(TSRosWRestoreVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  /* ros->Ydot and ros->Ystage have already been computed in SNESTSFormFunction_RosW (SNES guarantees this) */
  PetscCall(SNESGetDM(snes,&dm));
  PetscCall(TSRosWGetVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage));
  dmsave = ts->dm;
  ts->dm = dm;
  PetscCall(TSComputeIJacobian(ts,ros->stage_time,Ystage,Ydot,shift,A,B,PETSC_TRUE));
  ts->dm = dmsave;
  PetscCall(TSRosWRestoreVecs(ts,dm,&Ydot,&Zdot,&Ystage,&Zstage));
  PetscFunctionReturn(0);
}

static PetscErrorCode TSRosWTableauSetUp(TS ts)
{
  TS_RosW        *ros = (TS_RosW*)ts->data;
  RosWTableau    tab  = ros->tableau;

  PetscFunctionBegin;
  PetscCall(VecDuplicateVecs(ts->vec_sol,tab->s,&ros->Y));
  PetscCall(PetscMalloc1(tab->s,&ros->work));
  PetscFunctionReturn(0);
}

static PetscErrorCode TSSetUp_RosW(TS ts)
{
  TS_RosW        *ros = (TS_RosW*)ts->data;
  DM             dm;
  SNES           snes;
  TSRHSJacobian  rhsjacobian;

  PetscFunctionBegin;
  PetscCall(TSRosWTableauSetUp(ts));
  PetscCall(VecDuplicate(ts->vec_sol,&ros->Ydot));
  PetscCall(VecDuplicate(ts->vec_sol,&ros->Ystage));
  PetscCall(VecDuplicate(ts->vec_sol,&ros->Zdot));
  PetscCall(VecDuplicate(ts->vec_sol,&ros->Zstage));
  PetscCall(VecDuplicate(ts->vec_sol,&ros->vec_sol_prev));
  PetscCall(TSGetDM(ts,&dm));
  PetscCall(DMCoarsenHookAdd(dm,DMCoarsenHook_TSRosW,DMRestrictHook_TSRosW,ts));
  PetscCall(DMSubDomainHookAdd(dm,DMSubDomainHook_TSRosW,DMSubDomainRestrictHook_TSRosW,ts));
  /* Rosenbrock methods are linearly implicit, so set that unless the user has specifically asked for something else */
  PetscCall(TSGetSNES(ts,&snes));
  if (!((PetscObject)snes)->type_name) {
    PetscCall(SNESSetType(snes,SNESKSPONLY));
  }
  PetscCall(DMTSGetRHSJacobian(dm,&rhsjacobian,NULL));
  if (rhsjacobian == TSComputeRHSJacobianConstant) {
    Mat Amat,Pmat;

    /* Set the SNES matrix to be different from the RHS matrix because there is no way to reconstruct shift*M-J */
    PetscCall(SNESGetJacobian(snes,&Amat,&Pmat,NULL,NULL));
    if (Amat && Amat == ts->Arhs) {
      if (Amat == Pmat) {
        PetscCall(MatDuplicate(ts->Arhs,MAT_COPY_VALUES,&Amat));
        PetscCall(SNESSetJacobian(snes,Amat,Amat,NULL,NULL));
      } else {
        PetscCall(MatDuplicate(ts->Arhs,MAT_COPY_VALUES,&Amat));
        PetscCall(SNESSetJacobian(snes,Amat,NULL,NULL,NULL));
        if (Pmat && Pmat == ts->Brhs) {
          PetscCall(MatDuplicate(ts->Brhs,MAT_COPY_VALUES,&Pmat));
          PetscCall(SNESSetJacobian(snes,NULL,Pmat,NULL,NULL));
          PetscCall(MatDestroy(&Pmat));
        }
      }
      PetscCall(MatDestroy(&Amat));
    }
  }
  PetscFunctionReturn(0);
}
/*------------------------------------------------------------*/

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

  PetscFunctionBegin;
  PetscCall(PetscOptionsHead(PetscOptionsObject,"RosW ODE solver options"));
  {
    RosWTableauLink link;
    PetscInt        count,choice;
    PetscBool       flg;
    const char      **namelist;

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

    PetscCall(PetscOptionsBool("-ts_rosw_recompute_jacobian","Recompute the Jacobian at each stage","TSRosWSetRecomputeJacobian",ros->recompute_jacobian,&ros->recompute_jacobian,NULL));
  }
  PetscCall(PetscOptionsTail());
  /* Rosenbrock methods are linearly implicit, so set that unless the user has specifically asked for something else */
  PetscCall(TSGetSNES(ts,&snes));
  if (!((PetscObject)snes)->type_name) {
    PetscCall(SNESSetType(snes,SNESKSPONLY));
  }
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii));
  if (iascii) {
    RosWTableau tab  = ros->tableau;
    TSRosWType  rostype;
    char        buf[512];
    PetscInt    i;
    PetscReal   abscissa[512];
    PetscCall(TSRosWGetType(ts,&rostype));
    PetscCall(PetscViewerASCIIPrintf(viewer,"  Rosenbrock-W %s\n",rostype));
    PetscCall(PetscFormatRealArray(buf,sizeof(buf),"% 8.6f",tab->s,tab->ASum));
    PetscCall(PetscViewerASCIIPrintf(viewer,"  Abscissa of A       = %s\n",buf));
    for (i=0; i<tab->s; i++) abscissa[i] = tab->ASum[i] + tab->Gamma[i];
    PetscCall(PetscFormatRealArray(buf,sizeof(buf),"% 8.6f",tab->s,abscissa));
    PetscCall(PetscViewerASCIIPrintf(viewer,"  Abscissa of A+Gamma = %s\n",buf));
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TSLoad_RosW(TS ts,PetscViewer viewer)
{
  SNES           snes;
  TSAdapt        adapt;

  PetscFunctionBegin;
  PetscCall(TSGetAdapt(ts,&adapt));
  PetscCall(TSAdaptLoad(adapt,viewer));
  PetscCall(TSGetSNES(ts,&snes));
  PetscCall(SNESLoad(snes,viewer));
  /* function and Jacobian context for SNES when used with TS is always ts object */
  PetscCall(SNESSetFunction(snes,NULL,NULL,ts));
  PetscCall(SNESSetJacobian(snes,NULL,NULL,NULL,ts));
  PetscFunctionReturn(0);
}

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

  Logically collective

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

  Level: beginner

.seealso: TSRosWGetType(), TSROSW, TSROSW2M, TSROSW2P, TSROSWRA3PW, TSROSWRA34PW2, TSROSWRODAS3, TSROSWSANDU3, TSROSWASSP3P3S1C, TSROSWLASSP3P4S2C, TSROSWLLSSP3P4S2C, TSROSWARK3
@*/
PetscErrorCode TSRosWSetType(TS ts,TSRosWType roswtype)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  PetscValidCharPointer(roswtype,2);
  PetscTryMethod(ts,"TSRosWSetType_C",(TS,TSRosWType),(ts,roswtype));
  PetscFunctionReturn(0);
}

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

  Logically collective

  Input Parameter:
.  ts - timestepping context

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

  Level: intermediate

.seealso: TSRosWGetType()
@*/
PetscErrorCode TSRosWGetType(TS ts,TSRosWType *rostype)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  PetscUseMethod(ts,"TSRosWGetType_C",(TS,TSRosWType*),(ts,rostype));
  PetscFunctionReturn(0);
}

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

  Logically collective

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

  Level: intermediate

.seealso: TSRosWGetType()
@*/
PetscErrorCode TSRosWSetRecomputeJacobian(TS ts,PetscBool flg)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  PetscTryMethod(ts,"TSRosWSetRecomputeJacobian_C",(TS,PetscBool),(ts,flg));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  *rostype = ros->tableau->name;
  PetscFunctionReturn(0);
}

static PetscErrorCode  TSRosWSetType_RosW(TS ts,TSRosWType rostype)
{
  TS_RosW         *ros = (TS_RosW*)ts->data;
  PetscBool       match;
  RosWTableauLink link;

  PetscFunctionBegin;
  if (ros->tableau) {
    PetscCall(PetscStrcmp(ros->tableau->name,rostype,&match));
    if (match) PetscFunctionReturn(0);
  }
  for (link = RosWTableauList; link; link=link->next) {
    PetscCall(PetscStrcmp(link->tab.name,rostype,&match));
    if (match) {
      if (ts->setupcalled) PetscCall(TSRosWTableauReset(ts));
      ros->tableau = &link->tab;
      if (ts->setupcalled) PetscCall(TSRosWTableauSetUp(ts));
      ts->default_adapt_type = ros->tableau->bembed ? TSADAPTBASIC : TSADAPTNONE;
      PetscFunctionReturn(0);
    }
  }
  SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_ARG_UNKNOWN_TYPE,"Could not find '%s'",rostype);
}

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

  PetscFunctionBegin;
  ros->recompute_jacobian = flg;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSDestroy_RosW(TS ts)
{
  PetscFunctionBegin;
  PetscCall(TSReset_RosW(ts));
  if (ts->dm) {
    PetscCall(DMCoarsenHookRemove(ts->dm,DMCoarsenHook_TSRosW,DMRestrictHook_TSRosW,ts));
    PetscCall(DMSubDomainHookRemove(ts->dm,DMSubDomainHook_TSRosW,DMSubDomainRestrictHook_TSRosW,ts));
  }
  PetscCall(PetscFree(ts->data));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSRosWGetType_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetType_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetRecomputeJacobian_C",NULL));
  PetscFunctionReturn(0);
}

/* ------------------------------------------------------------ */
/*MC
      TSROSW - ODE solver using Rosenbrock-W schemes

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

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

  Consider trying TSARKIMEX if the stiff part is strongly nonlinear.

  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

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

$  udot = f(u)

  by the stage equations

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

  and step completion formula

$  u_1 = u_0 + sum_j b_j k_j

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

$  y_i = gamma_ij k_j

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

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

  to rewrite the method as

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

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

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

   or, more compactly in tensor notation

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

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

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

   with initial guess y_i = 0.

  Level: beginner

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

  PetscFunctionBegin;
  PetscCall(TSRosWInitializePackage());

  ts->ops->reset          = TSReset_RosW;
  ts->ops->destroy        = TSDestroy_RosW;
  ts->ops->view           = TSView_RosW;
  ts->ops->load           = TSLoad_RosW;
  ts->ops->setup          = TSSetUp_RosW;
  ts->ops->step           = TSStep_RosW;
  ts->ops->interpolate    = TSInterpolate_RosW;
  ts->ops->evaluatestep   = TSEvaluateStep_RosW;
  ts->ops->rollback       = TSRollBack_RosW;
  ts->ops->setfromoptions = TSSetFromOptions_RosW;
  ts->ops->snesfunction   = SNESTSFormFunction_RosW;
  ts->ops->snesjacobian   = SNESTSFormJacobian_RosW;

  ts->usessnes = PETSC_TRUE;

  PetscCall(PetscNewLog(ts,&ros));
  ts->data = (void*)ros;

  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSRosWGetType_C",TSRosWGetType_RosW));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetType_C",TSRosWSetType_RosW));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSRosWSetRecomputeJacobian_C",TSRosWSetRecomputeJacobian_RosW));

  PetscCall(TSRosWSetType(ts,TSRosWDefault));
  PetscFunctionReturn(0);
}