xref: /petsc/src/ts/impls/rosw/rosw.c (revision 0feba352864cb2b6ab156bb0bcb562b8a3f04b68)

/*
  Code for timestepping with Rosenbrock W methods

  Notes:
  The general system is written as

  G(t,X,Xdot) = F(t,X)

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

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

#include <../src/mat/blockinvert.h>

static const TSRosWType TSRosWDefault = TSROSW2P;
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 */
  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 */
};
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 */
  PetscScalar *work;            /* Scalar work space of length number of stages, used to prepare VecMAXPY() */
  PetscReal   shift;
  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
     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, three 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, three 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*/

#undef __FUNCT__
#define __FUNCT__ "TSRosWRegisterAll"
/*@C
  TSRosWRegisterAll - Registers all of the additive Runge-Kutta implicit-explicit methods in TSRosW

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

  Level: advanced

.keywords: TS, TSRosW, register, all

.seealso:  TSRosWRegisterDestroy()
@*/
PetscErrorCode TSRosWRegisterAll(void)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (TSRosWRegisterAllCalled) PetscFunctionReturn(0);
  TSRosWRegisterAllCalled = PETSC_TRUE;

  {
    const PetscReal g = 1. + 1./PetscSqrtReal(2.0);
    const PetscReal
      A[2][2] = {{0,0}, {1.,0}},
      Gamma[2][2] = {{g,0}, {-2.*g,g}},
      b[2] = {0.5,0.5},
      b1[2] = {1.0,0.0};
    ierr = TSRosWRegister(TSROSW2P,2,2,&A[0][0],&Gamma[0][0],b,b1);CHKERRQ(ierr);
  }
  {
    const PetscReal g = 1. - 1./PetscSqrtReal(2.0);
    const PetscReal
      A[2][2] = {{0,0}, {1.,0}},
      Gamma[2][2] = {{g,0}, {-2.*g,g}},
      b[2] = {0.5,0.5},
      b1[2] = {1.0,0.0};
    ierr = TSRosWRegister(TSROSW2M,2,2,&A[0][0],&Gamma[0][0],b,b1);CHKERRQ(ierr);
  }
  {
    const PetscReal g = 7.8867513459481287e-01;
    const PetscReal
      A[3][3] = {{0,0,0},
                 {1.5773502691896257e+00,0,0},
                 {0.5,0,0}},
      Gamma[3][3] = {{g,0,0},
                     {-1.5773502691896257e+00,g,0},
                     {-6.7075317547305480e-01,1.7075317547305482e-01,g}},
      b[3] = {1.0566243270259355e-01,4.9038105676657971e-02,8.4529946162074843e-01},
      b2[3] = {-1.7863279495408180e-01,1./3.,8.4529946162074843e-01};
    ierr = TSRosWRegister(TSROSWRA3PW,3,3,&A[0][0],&Gamma[0][0],b,b2);CHKERRQ(ierr);
  }
  {
    const PetscReal g = 4.3586652150845900e-01;
    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] = {{g,0,0,0},
                     {-8.7173304301691801e-01,g,0,0},
                     {-9.0338057013044082e-01,5.4180672388095326e-02,g,0},
                     {2.4212380706095346e-01,-1.2232505839045147e+00,5.4526025533510214e-01,g}},
      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};
    ierr = TSRosWRegister(TSROSWRA34PW2,3,4,&A[0][0],&Gamma[0][0],b,b2);CHKERRQ(ierr);
  }
  {
    const PetscReal g = 0.5;
    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] = {{g,0,0,0},
                     {1.,g,0,0},
                     {-0.25,-0.25,g,0},
                     {1./12,1./12,-2./3,g}},
      b[4] = {5./6,-1./6,-1./6,0.5},
      b2[4] = {0.75,-0.25,0.5,0};
    ierr = TSRosWRegister(TSROSWRODAS3,3,4,&A[0][0],&Gamma[0][0],b,b2);CHKERRQ(ierr);
  }
  {
    const PetscReal g = 0.43586652150845899941601945119356;
    const PetscReal
      A[3][3] = {{0,0,0},
                 {g,0,0},
                 {g,0,0}},
      Gamma[3][3] = {{g,0,0},
                     {-0.19294655696029095575009695436041,g,0},
                     {0,1.74927148125794685173529749738960,g}},
      b[3] = {-0.75457412385404315829818998646589,1.94100407061964420292840123379419,-0.18642994676560104463021124732829},
      b2[3] = {-1.53358745784149585370766523913002,2.81745131148625772213931745457622,-0.28386385364476186843165221544619};
    ierr = TSRosWRegister(TSROSWSANDU3,3,3,&A[0][0],&Gamma[0][0],b,b2);CHKERRQ(ierr);
  }
  {
    const PetscReal s3 = PetscSqrtReal(3.),g = (3.0+s3)/6.0;
    const PetscReal
      A[3][3] = {{0,0,0},
                 {1,0,0},
                 {0.25,0.25,0}},
      Gamma[3][3] = {{0,0,0},
                     {(-3.0-s3)/6.0,g,0},
                     {(-3.0-s3)/24.0,(-3.0-s3)/8.0,g}},
        b[3] = {1./6.,1./6.,2./3.},
          b2[3] = {1./4.,1./4.,1./2.};
    ierr = TSRosWRegister(TSROSWASSP3P3S1C,3,3,&A[0][0],&Gamma[0][0],b,b2);CHKERRQ(ierr);
  }

  {
    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};
     ierr = TSRosWRegister(TSROSWLASSP3P4S2C,3,4,&A[0][0],&Gamma[0][0],b,b2);CHKERRQ(ierr);
  }

  {
    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};
     ierr = TSRosWRegister(TSROSWLLSSP3P4S2C,3,4,&A[0][0],&Gamma[0][0],b,b2);CHKERRQ(ierr);
  }

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

   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;

   ierr = TSRosWRegister(TSROSWARK3,3,4,&A[0][0],&Gamma[0][0],b,b2);CHKERRQ(ierr);
  }

  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSRosWRegisterDestroy"
/*@C
   TSRosWRegisterDestroy - Frees the list of schemes that were registered by TSRosWRegister().

   Not Collective

   Level: advanced

.keywords: TSRosW, register, destroy
.seealso: TSRosWRegister(), TSRosWRegisterAll(), TSRosWRegisterDynamic()
@*/
PetscErrorCode TSRosWRegisterDestroy(void)
{
  PetscErrorCode ierr;
  RosWTableauLink link;

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

#undef __FUNCT__
#define __FUNCT__ "TSRosWInitializePackage"
/*@C
  TSRosWInitializePackage - This function initializes everything in the TSRosW package. It is called
  from PetscDLLibraryRegister() when using dynamic libraries, and on the first call to TSCreate_RosW()
  when using static libraries.

  Input Parameter:
  path - The dynamic library path, or PETSC_NULL

  Level: developer

.keywords: TS, TSRosW, initialize, package
.seealso: PetscInitialize()
@*/
PetscErrorCode TSRosWInitializePackage(const char path[])
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (TSRosWPackageInitialized) PetscFunctionReturn(0);
  TSRosWPackageInitialized = PETSC_TRUE;
  ierr = TSRosWRegisterAll();CHKERRQ(ierr);
  ierr = PetscRegisterFinalize(TSRosWFinalizePackage);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

  Level: developer

.keywords: Petsc, destroy, package
.seealso: PetscFinalize()
@*/
PetscErrorCode TSRosWFinalizePackage(void)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  TSRosWPackageInitialized = PETSC_FALSE;
  ierr = TSRosWRegisterDestroy();CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSRosWRegister"
/*@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, PETSC_NULL if no embedded scheme is available)

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

   Level: advanced

.keywords: TS, register

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

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

  ierr = PetscMalloc(sizeof(*link),&link);CHKERRQ(ierr);
  ierr = PetscMemzero(link,sizeof(*link));CHKERRQ(ierr);
  t = &link->tab;
  ierr = PetscStrallocpy(name,&t->name);CHKERRQ(ierr);
  t->order = order;
  t->s = s;
  ierr = PetscMalloc5(s*s,PetscReal,&t->A,s*s,PetscReal,&t->Gamma,s,PetscReal,&t->b,s,PetscReal,&t->ASum,s,PetscReal,&t->GammaSum);CHKERRQ(ierr);
  ierr = PetscMalloc5(s*s,PetscReal,&t->At,s,PetscReal,&t->bt,s*s,PetscReal,&t->GammaInv,s,PetscBool,&t->GammaZeroDiag,s*s,PetscReal,&t->GammaExplicitCorr);CHKERRQ(ierr);
  ierr = PetscMemcpy(t->A,A,s*s*sizeof(A[0]));CHKERRQ(ierr);
  ierr = PetscMemcpy(t->Gamma,Gamma,s*s*sizeof(Gamma[0]));CHKERRQ(ierr);
  ierr = PetscMemcpy(t->GammaExplicitCorr,Gamma,s*s*sizeof(Gamma[0]));CHKERRQ(ierr);
  ierr = PetscMemcpy(t->b,b,s*sizeof(b[0]));CHKERRQ(ierr);
  if (bembed) {
    ierr = PetscMalloc2(s,PetscReal,&t->bembed,s,PetscReal,&t->bembedt);CHKERRQ(ierr);
    ierr = PetscMemcpy(t->bembed,bembed,s*sizeof(bembed[0]));CHKERRQ(ierr);
  }
  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];
    }
  }
  ierr = PetscMalloc(s*s*sizeof(PetscScalar),&GammaInv);CHKERRQ(ierr); /* 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: ierr = Kernel_A_gets_inverse_A_2(GammaInv,0);CHKERRQ(ierr); break;
  case 3: ierr = Kernel_A_gets_inverse_A_3(GammaInv,0);CHKERRQ(ierr); break;
  case 4: ierr = Kernel_A_gets_inverse_A_4(GammaInv,0);CHKERRQ(ierr); break;
  case 5: {
    PetscInt ipvt5[5];
    MatScalar work5[5*5];
    ierr = Kernel_A_gets_inverse_A_5(GammaInv,ipvt5,work5,0);CHKERRQ(ierr); break;
  }
  case 6: ierr = Kernel_A_gets_inverse_A_6(GammaInv,0);CHKERRQ(ierr); break;
  case 7: ierr = Kernel_A_gets_inverse_A_7(GammaInv,0);CHKERRQ(ierr); break;
  default: SETERRQ1(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]);
  ierr = PetscFree(GammaInv);CHKERRQ(ierr);

  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 */

  link->next = RosWTableauList;
  RosWTableauList = link;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSEvaluateStep_RosW"
/*
 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 X,PetscBool *done)
{
  TS_RosW        *ros = (TS_RosW*)ts->data;
  RosWTableau    tab  = ros->tableau;
  PetscScalar    *w = ros->work;
  PetscInt       i;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (order == tab->order) {
    if (ros->status == TS_STEP_INCOMPLETE) { /* Use standard completion formula */
      ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
      for (i=0; i<tab->s; i++) w[i] = tab->bt[i];
      ierr = VecMAXPY(X,tab->s,w,ros->Y);CHKERRQ(ierr);
    } else {ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);}
    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 */
      ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
      for (i=0; i<tab->s; i++) w[i] = tab->bembedt[i];
      ierr = VecMAXPY(X,tab->s,w,ros->Y);CHKERRQ(ierr);
    } else {                    /* Use rollback-and-recomplete formula (bembedt - bt) */
      for (i=0; i<tab->s; i++) w[i] = tab->bembedt[i] - tab->bt[i];
      ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
      ierr = VecMAXPY(X,tab->s,w,ros->Y);CHKERRQ(ierr);
    }
    if (done) *done = PETSC_TRUE;
    PetscFunctionReturn(0);
  }
  unavailable:
  if (done) *done = PETSC_FALSE;
  else SETERRQ3(((PetscObject)ts)->comm,PETSC_ERR_SUP,"Rosenbrock-W '%s' of order %D cannot evaluate step at order %D",tab->name,tab->order,order);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSStep_RosW"
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,reject,next_scheme;
  PetscReal       next_time_step;
  PetscBool       accept;
  PetscErrorCode  ierr;
  MatStructure    str;

  PetscFunctionBegin;
  ierr = TSGetSNES(ts,&snes);CHKERRQ(ierr);
  next_time_step = ts->time_step;
  accept = PETSC_TRUE;
  ros->status = TS_STEP_INCOMPLETE;

  for (reject=0; reject<ts->max_reject; reject++,ts->reject++) {
    const PetscReal h = ts->time_step;
    for (i=0; i<s; i++) {
      ros->stage_time = ts->ptime + h*ASum[i];
      if (GammaZeroDiag[i]) {
        ros->stage_explicit = PETSC_TRUE;
        ros->shift = 1./h;
      } else {
        ros->stage_explicit = PETSC_FALSE;
        ros->shift = 1./(h*Gamma[i*s+i]);
      }

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

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

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

      if (!ros->stage_explicit) {
        if (!ros->recompute_jacobian && !i) {
          ierr = SNESSetLagJacobian(snes,-2);CHKERRQ(ierr); /* Recompute the Jacobian on this solve, but not again */
        }
        ierr = SNESSolve(snes,PETSC_NULL,Y[i]);CHKERRQ(ierr);
        ierr = SNESGetIterationNumber(snes,&its);CHKERRQ(ierr);
        ierr = SNESGetLinearSolveIterations(snes,&lits);CHKERRQ(ierr);
        ts->nonlinear_its += its; ts->linear_its += lits;
      } else {
        ierr = VecZeroEntries(Ydot);CHKERRQ(ierr); /* Evaluate Y[i]=G(t,Ydot=0,Zstage) */
        ierr = TSComputeIFunction(ts,ros->stage_time,Zstage,Ydot,Y[i],PETSC_FALSE);CHKERRQ(ierr);
        ierr = VecScale(Y[i],-1.0);
        ierr = VecAXPY(Y[i],-1.0,Zdot);CHKERRQ(ierr); /*Y[i]=F(Zstage)-Zdot[=GammaInv*Y]*/

        ierr = VecZeroEntries(Zstage);CHKERRQ(ierr); /* Zstage = GammaExplicitCorr[i,j] * Y[j] */
        for (j=0; j<i; j++) w[j] = GammaExplicitCorr[i*s+j];
        ierr = VecMAXPY(Zstage,i,w,Y);CHKERRQ(ierr);
        /*Y[i] += Y[i] + Jac*Zstage[=Jac*GammaExplicitCorr[i,j] * Y[j]] */
        str = SAME_NONZERO_PATTERN;
        Mat J,Jp;
        ierr = TSGetIJacobian(ts,&J,&Jp,PETSC_NULL,PETSC_NULL);
        ierr = TSComputeIJacobian(ts,ros->stage_time,ts->vec_sol,Ydot,0,&J,&Jp,&str,PETSC_FALSE);CHKERRQ(ierr);
        ierr = MatMult(J,Zstage,Zdot);

        ierr = VecAXPY(Y[i],-1.0,Zdot);CHKERRQ(ierr);
        ierr = VecScale(Y[i],h);
        ts->linear_its += 1;
      }
    }
    ierr = TSEvaluateStep(ts,tab->order,ts->vec_sol,PETSC_NULL);CHKERRQ(ierr);
    ros->status = TS_STEP_PENDING;

    /* Register only the current method as a candidate because we're not supporting multiple candidates yet. */
    ierr = TSGetAdapt(ts,&adapt);CHKERRQ(ierr);
    ierr = TSAdaptCandidatesClear(adapt);CHKERRQ(ierr);
    ierr = TSAdaptCandidateAdd(adapt,tab->name,tab->order,1,tab->ccfl,1.*tab->s,PETSC_TRUE);CHKERRQ(ierr);
    ierr = TSAdaptChoose(adapt,ts,ts->time_step,&next_scheme,&next_time_step,&accept);CHKERRQ(ierr);
    if (accept) {
      /* ignore next_scheme for now */
      ts->ptime += ts->time_step;
      ts->time_step = next_time_step;
      ts->steps++;
      ros->status = TS_STEP_COMPLETE;
      break;
    } else {                    /* Roll back the current step */
      for (i=0; i<s; i++) w[i] = -tab->bt[i];
      ierr = VecMAXPY(ts->vec_sol,s,w,Y);CHKERRQ(ierr);
      ts->time_step = next_time_step;
      ros->status = TS_STEP_INCOMPLETE;
    }
  }
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSInterpolate_RosW"
static PetscErrorCode TSInterpolate_RosW(TS ts,PetscReal itime,Vec X)
{
  TS_RosW        *ros = (TS_RosW*)ts->data;

  PetscFunctionBegin;
  SETERRQ1(((PetscObject)ts)->comm,PETSC_ERR_SUP,"TSRosW %s does not have an interpolation formula",ros->tableau->name);
  PetscFunctionReturn(0);
}

/*------------------------------------------------------------*/
#undef __FUNCT__
#define __FUNCT__ "TSReset_RosW"
static PetscErrorCode TSReset_RosW(TS ts)
{
  TS_RosW        *ros = (TS_RosW*)ts->data;
  PetscInt       s;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!ros->tableau) PetscFunctionReturn(0);
   s = ros->tableau->s;
  ierr = VecDestroyVecs(s,&ros->Y);CHKERRQ(ierr);
  ierr = VecDestroy(&ros->Ydot);CHKERRQ(ierr);
  ierr = VecDestroy(&ros->Ystage);CHKERRQ(ierr);
  ierr = VecDestroy(&ros->Zdot);CHKERRQ(ierr);
  ierr = VecDestroy(&ros->Zstage);CHKERRQ(ierr);
  ierr = PetscFree(ros->work);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSDestroy_RosW"
static PetscErrorCode TSDestroy_RosW(TS ts)
{
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  ierr = TSReset_RosW(ts);CHKERRQ(ierr);
  ierr = PetscFree(ts->data);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunctionDynamic((PetscObject)ts,"TSRosWGetType_C","",PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunctionDynamic((PetscObject)ts,"TSRosWSetType_C","",PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunctionDynamic((PetscObject)ts,"TSRosWSetRecomputeJacobian_C","",PETSC_NULL);CHKERRQ(ierr);
  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
*/
#undef __FUNCT__
#define __FUNCT__ "SNESTSFormFunction_RosW"
static PetscErrorCode SNESTSFormFunction_RosW(SNES snes,Vec X,Vec F,TS ts)
{
  TS_RosW        *ros = (TS_RosW*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = VecWAXPY(ros->Ydot,ros->shift,X,ros->Zdot);CHKERRQ(ierr); /* Ydot = shift*X + Zdot */
  ierr = VecWAXPY(ros->Ystage,1.0,X,ros->Zstage);CHKERRQ(ierr);    /* Ystage = X + Zstage */
  ierr = TSComputeIFunction(ts,ros->stage_time,ros->Ystage,ros->Ydot,F,PETSC_FALSE);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESTSFormJacobian_RosW"
static PetscErrorCode SNESTSFormJacobian_RosW(SNES snes,Vec X,Mat *A,Mat *B,MatStructure *str,TS ts)
{
  TS_RosW        *ros = (TS_RosW*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  /* ros->Ydot and ros->Ystage have already been computed in SNESTSFormFunction_RosW (SNES guarantees this) */
  ierr = TSComputeIJacobian(ts,ros->stage_time,ros->Ystage,ros->Ydot,ros->shift,A,B,str,PETSC_TRUE);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSSetUp_RosW"
static PetscErrorCode TSSetUp_RosW(TS ts)
{
  TS_RosW        *ros = (TS_RosW*)ts->data;
  RosWTableau    tab  = ros->tableau;
  PetscInt       s    = tab->s;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!ros->tableau) {
    ierr = TSRosWSetType(ts,TSRosWDefault);CHKERRQ(ierr);
  }
  ierr = VecDuplicateVecs(ts->vec_sol,s,&ros->Y);CHKERRQ(ierr);
  ierr = VecDuplicate(ts->vec_sol,&ros->Ydot);CHKERRQ(ierr);
  ierr = VecDuplicate(ts->vec_sol,&ros->Ystage);CHKERRQ(ierr);
  ierr = VecDuplicate(ts->vec_sol,&ros->Zdot);CHKERRQ(ierr);
  ierr = VecDuplicate(ts->vec_sol,&ros->Zstage);CHKERRQ(ierr);
  ierr = PetscMalloc(s*sizeof(ros->work[0]),&ros->work);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}
/*------------------------------------------------------------*/

#undef __FUNCT__
#define __FUNCT__ "TSSetFromOptions_RosW"
static PetscErrorCode TSSetFromOptions_RosW(TS ts)
{
  TS_RosW        *ros = (TS_RosW*)ts->data;
  PetscErrorCode ierr;
  char           rostype[256];

  PetscFunctionBegin;
  ierr = PetscOptionsHead("RosW ODE solver options");CHKERRQ(ierr);
  {
    RosWTableauLink link;
    PetscInt count,choice;
    PetscBool flg;
    const char **namelist;
    SNES snes;

    ierr = PetscStrncpy(rostype,TSRosWDefault,sizeof rostype);CHKERRQ(ierr);
    for (link=RosWTableauList,count=0; link; link=link->next,count++) ;
    ierr = PetscMalloc(count*sizeof(char*),&namelist);CHKERRQ(ierr);
    for (link=RosWTableauList,count=0; link; link=link->next,count++) namelist[count] = link->tab.name;
    ierr = PetscOptionsEList("-ts_rosw_type","Family of Rosenbrock-W method","TSRosWSetType",(const char*const*)namelist,count,rostype,&choice,&flg);CHKERRQ(ierr);
    ierr = TSRosWSetType(ts,flg ? namelist[choice] : rostype);CHKERRQ(ierr);
    ierr = PetscFree(namelist);CHKERRQ(ierr);

    ierr = PetscOptionsBool("-ts_rosw_recompute_jacobian","Recompute the Jacobian at each stage","TSRosWSetRecomputeJacobian",ros->recompute_jacobian,&ros->recompute_jacobian,PETSC_NULL);CHKERRQ(ierr);

    /* Rosenbrock methods are linearly implicit, so set that unless the user has specifically asked for something else */
    ierr = TSGetSNES(ts,&snes);CHKERRQ(ierr);
    if (!((PetscObject)snes)->type_name) {
      ierr = SNESSetType(snes,SNESKSPONLY);CHKERRQ(ierr);
    }
    ierr = SNESSetFromOptions(snes);CHKERRQ(ierr);
  }
  ierr = PetscOptionsTail();CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "PetscFormatRealArray"
static PetscErrorCode PetscFormatRealArray(char buf[],size_t len,const char *fmt,PetscInt n,const PetscReal x[])
{
  PetscErrorCode ierr;
  PetscInt i;
  size_t left,count;
  char *p;

  PetscFunctionBegin;
  for (i=0,p=buf,left=len; i<n; i++) {
    ierr = PetscSNPrintfCount(p,left,fmt,&count,x[i]);CHKERRQ(ierr);
    if (count >= left) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Insufficient space in buffer");
    left -= count;
    p += count;
    *p++ = ' ';
  }
  p[i ? 0 : -1] = 0;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSView_RosW"
static PetscErrorCode TSView_RosW(TS ts,PetscViewer viewer)
{
  TS_RosW        *ros = (TS_RosW*)ts->data;
  RosWTableau    tab  = ros->tableau;
  PetscBool      iascii;
  PetscErrorCode ierr;

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

#undef __FUNCT__
#define __FUNCT__ "TSRosWSetType"
/*@C
  TSRosWSetType - Set the type of Rosenbrock-W scheme

  Logically collective

  Input Parameter:
+  ts - timestepping context
-  rostype - type of Rosenbrock-W scheme

  Level: intermediate

.seealso: TSRosWGetType()
@*/
PetscErrorCode TSRosWSetType(TS ts,const TSRosWType rostype)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  ierr = PetscTryMethod(ts,"TSRosWSetType_C",(TS,const TSRosWType),(ts,rostype));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSRosWGetType"
/*@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,const TSRosWType *rostype)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  ierr = PetscUseMethod(ts,"TSRosWGetType_C",(TS,const TSRosWType*),(ts,rostype));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

  Logically collective

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

  Level: intermediate

.seealso: TSRosWGetType()
@*/
PetscErrorCode TSRosWSetRecomputeJacobian(TS ts,PetscBool flg)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  ierr = PetscTryMethod(ts,"TSRosWSetRecomputeJacobian_C",(TS,PetscBool),(ts,flg));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

EXTERN_C_BEGIN
#undef __FUNCT__
#define __FUNCT__ "TSRosWGetType_RosW"
PetscErrorCode  TSRosWGetType_RosW(TS ts,const TSRosWType *rostype)
{
  TS_RosW        *ros = (TS_RosW*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!ros->tableau) {ierr = TSRosWSetType(ts,TSRosWDefault);CHKERRQ(ierr);}
  *rostype = ros->tableau->name;
  PetscFunctionReturn(0);
}
#undef __FUNCT__
#define __FUNCT__ "TSRosWSetType_RosW"
PetscErrorCode  TSRosWSetType_RosW(TS ts,const TSRosWType rostype)
{
  TS_RosW         *ros = (TS_RosW*)ts->data;
  PetscErrorCode  ierr;
  PetscBool       match;
  RosWTableauLink link;

  PetscFunctionBegin;
  if (ros->tableau) {
    ierr = PetscStrcmp(ros->tableau->name,rostype,&match);CHKERRQ(ierr);
    if (match) PetscFunctionReturn(0);
  }
  for (link = RosWTableauList; link; link=link->next) {
    ierr = PetscStrcmp(link->tab.name,rostype,&match);CHKERRQ(ierr);
    if (match) {
      ierr = TSReset_RosW(ts);CHKERRQ(ierr);
      ros->tableau = &link->tab;
      PetscFunctionReturn(0);
    }
  }
  SETERRQ1(((PetscObject)ts)->comm,PETSC_ERR_ARG_UNKNOWN_TYPE,"Could not find '%s'",rostype);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSRosWSetRecomputeJacobian_RosW"
PetscErrorCode  TSRosWSetRecomputeJacobian_RosW(TS ts,PetscBool flg)
{
  TS_RosW *ros = (TS_RosW*)ts->data;

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

/* ------------------------------------------------------------ */
/*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.

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

$  xdot = f(x)

  by the stage equations

$  k_i = h f(x_0 + sum_j alpha_ij k_j) + h J sum_j gamma_ij k_j

  and step completion formula

$  x_1 = x_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(x)
  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^{-i}

  to rewrite the method as

$  [M/(h gamma_ii) - J] y_i = f(x_0 + sum_j a_ij y_j) + M sum_j (c_ij/h) y_j
$  x_1 = x_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(x_0 + sum_j a_ij y_j + y_i, ydot_i) = 0

   with initial guess y_i = 0.

  Level: beginner

.seealso:  TSCreate(), TS, TSSetType(), TSRosWRegister()

M*/
EXTERN_C_BEGIN
#undef __FUNCT__
#define __FUNCT__ "TSCreate_RosW"
PetscErrorCode  TSCreate_RosW(TS ts)
{
  TS_RosW        *ros;
  PetscErrorCode ierr;

  PetscFunctionBegin;
#if !defined(PETSC_USE_DYNAMIC_LIBRARIES)
  ierr = TSRosWInitializePackage(PETSC_NULL);CHKERRQ(ierr);
#endif

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

  ierr = PetscNewLog(ts,TS_RosW,&ros);CHKERRQ(ierr);
  ts->data = (void*)ros;

  ierr = PetscObjectComposeFunctionDynamic((PetscObject)ts,"TSRosWGetType_C","TSRosWGetType_RosW",TSRosWGetType_RosW);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunctionDynamic((PetscObject)ts,"TSRosWSetType_C","TSRosWSetType_RosW",TSRosWSetType_RosW);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunctionDynamic((PetscObject)ts,"TSRosWSetRecomputeJacobian_C","TSRosWSetRecomputeJacobian_RosW",TSRosWSetRecomputeJacobian_RosW);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}
EXTERN_C_END