xref: /petsc/src/ts/impls/explicit/rk/rk.c (revision a685a7631af17466f49634cd8e44a2aecae18a36)

/*
  Code for time stepping with the Runge-Kutta method(single-rate RK and multi-rate RK)

  Notes:
  1) The general system is written as
     Udot = F(t,U) for single-rate RK;
  2) The general system is written as
     Udot = F(t,U) for combined RHS multi-rate RK,
     user should give the indexes for both slow and fast components;
  3) The general system is written as
     Usdot = Fs(t,Us,Uf)
     Ufdot = Ff(t,Us,Uf) for partitioned RHS multi-rate RK,
     user should partioned RHS by themselves and also provide the indexes for both slow and fast components.
*/

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

static TSRKType  TSRKDefault = TSRK3BS;
static TSRKType  TSMRKDefault = TSRK2A;
static PetscBool TSRKRegisterAllCalled;
static PetscBool TSRKPackageInitialized;

typedef struct _RKTableau *RKTableau;
struct _RKTableau {
  char       *name;
  PetscInt   order;               /* Classical approximation order of the method i              */
  PetscInt   s;                   /* Number of stages                                           */
  PetscInt   p;                   /* Interpolation order                                        */
  PetscBool  FSAL;                /* flag to indicate if tableau is FSAL                        */
  PetscReal  *A,*b,*c;            /* Tableau                                                    */
  PetscReal  *bembed;             /* Embedded formula of order one less (order-1)               */
  PetscReal  *binterp;            /* Dense output formula                                       */
  PetscReal  ccfl;                /* Placeholder for CFL coefficient relative to forward Euler  */
};
typedef struct _RKTableauLink *RKTableauLink;
struct _RKTableauLink {
  struct _RKTableau tab;
  RKTableauLink     next;
};
static RKTableauLink RKTableauList;

typedef struct {
  RKTableau    tableau;
  Vec          *Y;               /* States computed during the step                                              */
  Vec          *YdotRHS;         /* Function evaluations for the non-stiff part and contains all components      */
  Vec          *YdotRHS_fast;    /* Function evaluations for the non-stiff part and contains fast components     */
  Vec          *YdotRHS_slow;    /* Function evaluations for the non-stiff part and contains slow components     */
  Vec          *VecDeltaLam;     /* Increment of the adjoint sensitivity w.r.t IC at stage                       */
  Vec          *VecDeltaMu;      /* Increment of the adjoint sensitivity w.r.t P at stage                        */
  Vec          VecCostIntegral0; /* backup for roll-backs due to events                                          */
  PetscScalar  *work;            /* Scalar work                                                                  */
  PetscInt     slow;             /* flag indicates call slow components solver (0) or fast components solver (1) */
  PetscReal    stage_time;
  TSStepStatus status;
  PetscReal    ptime;
  PetscReal    time_step;
  PetscInt     dtratio;          /* ratio between slow time step size and fast step size                         */
} TS_RK;


/*MC
     TSRK1FE - First order forward Euler scheme.

     This method has one stage.

     Options database:
.     -ts_rk_type 1fe

     Level: advanced

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

     This method has two stages.

     Options database:
.     -ts_rk_type 2a

     Level: advanced

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

     This method has three stages.

     Options database:
.     -ts_rk_type 3

     Level: advanced

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

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

     Options database:
.     -ts_rk_type 3bs

     Level: advanced

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

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

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

     Options database:
.     -ts_rk_type 4

     Level: advanced

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

     This method has six stages.

     Options database:
.     -ts_rk_type 5f

     Level: advanced

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

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

     Options database:
.     -ts_rk_type 5dp

     Level: advanced

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

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

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

     Options database:
.     -ts_rk_type 5bs

     Level: advanced

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

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

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

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

  Level: advanced

.keywords: TS, TSRK, register, all

.seealso:  TSRKRegisterDestroy()
@*/
PetscErrorCode TSRKRegisterAll(void)
{
  PetscErrorCode ierr;

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

#define RC PetscRealConstant
  {
    const PetscReal
      A[1][1] = {{0}},
      b[1]    = {RC(1.0)};
    ierr = TSRKRegister(TSRK1FE,1,1,&A[0][0],b,NULL,NULL,0,NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[2][2]   = {{0,0},
                   {RC(1.0),0}},
      b[2]      =  {RC(0.5),RC(0.5)},
      bembed[2] =  {RC(1.0),0};
    ierr = TSRKRegister(TSRK2A,2,2,&A[0][0],b,NULL,bembed,0,NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[3][3] = {{0,0,0},
                 {RC(2.0)/RC(3.0),0,0},
                 {RC(-1.0)/RC(3.0),RC(1.0),0}},
      b[3]    =  {RC(0.25),RC(0.5),RC(0.25)};
    ierr = TSRKRegister(TSRK3,3,3,&A[0][0],b,NULL,NULL,0,NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[4][4]   = {{0,0,0,0},
                   {RC(1.0)/RC(2.0),0,0,0},
                   {0,RC(3.0)/RC(4.0),0,0},
                   {RC(2.0)/RC(9.0),RC(1.0)/RC(3.0),RC(4.0)/RC(9.0),0}},
      b[4]      =  {RC(2.0)/RC(9.0),RC(1.0)/RC(3.0),RC(4.0)/RC(9.0),0},
      bembed[4] =  {RC(7.0)/RC(24.0),RC(1.0)/RC(4.0),RC(1.0)/RC(3.0),RC(1.0)/RC(8.0)};
    ierr = TSRKRegister(TSRK3BS,3,4,&A[0][0],b,NULL,bembed,0,NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[4][4] = {{0,0,0,0},
                 {RC(0.5),0,0,0},
                 {0,RC(0.5),0,0},
                 {0,0,RC(1.0),0}},
      b[4]    =  {RC(1.0)/RC(6.0),RC(1.0)/RC(3.0),RC(1.0)/RC(3.0),RC(1.0)/RC(6.0)};
    ierr = TSRKRegister(TSRK4,4,4,&A[0][0],b,NULL,NULL,0,NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[6][6]   = {{0,0,0,0,0,0},
                   {RC(0.25),0,0,0,0,0},
                   {RC(3.0)/RC(32.0),RC(9.0)/RC(32.0),0,0,0,0},
                   {RC(1932.0)/RC(2197.0),RC(-7200.0)/RC(2197.0),RC(7296.0)/RC(2197.0),0,0,0},
                   {RC(439.0)/RC(216.0),RC(-8.0),RC(3680.0)/RC(513.0),RC(-845.0)/RC(4104.0),0,0},
                   {RC(-8.0)/RC(27.0),RC(2.0),RC(-3544.0)/RC(2565.0),RC(1859.0)/RC(4104.0),RC(-11.0)/RC(40.0),0}},
      b[6]      =  {RC(16.0)/RC(135.0),0,RC(6656.0)/RC(12825.0),RC(28561.0)/RC(56430.0),RC(-9.0)/RC(50.0),RC(2.0)/RC(55.0)},
      bembed[6] =  {RC(25.0)/RC(216.0),0,RC(1408.0)/RC(2565.0),RC(2197.0)/RC(4104.0),RC(-1.0)/RC(5.0),0};
    ierr = TSRKRegister(TSRK5F,5,6,&A[0][0],b,NULL,bembed,0,NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[7][7]   = {{0,0,0,0,0,0,0},
                   {RC(1.0)/RC(5.0),0,0,0,0,0,0},
                   {RC(3.0)/RC(40.0),RC(9.0)/RC(40.0),0,0,0,0,0},
                   {RC(44.0)/RC(45.0),RC(-56.0)/RC(15.0),RC(32.0)/RC(9.0),0,0,0,0},
                   {RC(19372.0)/RC(6561.0),RC(-25360.0)/RC(2187.0),RC(64448.0)/RC(6561.0),RC(-212.0)/RC(729.0),0,0,0},
                   {RC(9017.0)/RC(3168.0),RC(-355.0)/RC(33.0),RC(46732.0)/RC(5247.0),RC(49.0)/RC(176.0),RC(-5103.0)/RC(18656.0),0,0},
                   {RC(35.0)/RC(384.0),0,RC(500.0)/RC(1113.0),RC(125.0)/RC(192.0),RC(-2187.0)/RC(6784.0),RC(11.0)/RC(84.0),0}},
      b[7]      =  {RC(35.0)/RC(384.0),0,RC(500.0)/RC(1113.0),RC(125.0)/RC(192.0),RC(-2187.0)/RC(6784.0),RC(11.0)/RC(84.0),0},
        bembed[7] =  {RC(5179.0)/RC(57600.0),0,RC(7571.0)/RC(16695.0),RC(393.0)/RC(640.0),RC(-92097.0)/RC(339200.0),RC(187.0)/RC(2100.0),RC(1.0)/RC(40.0)},
        binterp[7][5] =  {{RC(1.0),RC(-4034104133.0)/RC(1410260304.0),RC(105330401.0)/RC(33982176.0),RC(-13107642775.0)/RC(11282082432.0),RC(6542295.0)/RC(470086768.0)},
                    {0,0,0,0,0},
                    {0,RC(132343189600.0)/RC(32700410799.0),RC(-833316000.0)/RC(131326951.0),RC(91412856700.0)/RC(32700410799.0),RC(-523383600.0)/RC(10900136933.0)},
                    {0,RC(-115792950.0)/RC(29380423.0),RC(185270875.0)/RC(16991088.0),RC(-12653452475.0)/RC(1880347072.0),RC(98134425.0)/RC(235043384.0)},
                    {0,RC(70805911779.0)/RC(24914598704.0),RC(-4531260609.0)/RC(600351776.0),RC(988140236175.0)/RC(199316789632.0),RC(-14307999165.0)/RC(24914598704.0)},
                    {0,RC(-331320693.0)/RC(205662961.0),RC(31361737.0)/RC(7433601.0),RC(-2426908385.0)/RC(822651844.0),RC(97305120.0)/RC(205662961.0)},
                    {0,RC(44764047.0)/RC(29380423.0),RC(-1532549.0)/RC(353981.0),RC(90730570.0)/RC(29380423.0),RC(-8293050.0)/RC(29380423.0)}};

        ierr = TSRKRegister(TSRK5DP,5,7,&A[0][0],b,NULL,bembed,5,binterp[0]);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[8][8]   = {{0,0,0,0,0,0,0,0},
                   {RC(1.0)/RC(6.0),0,0,0,0,0,0,0},
                   {RC(2.0)/RC(27.0),RC(4.0)/RC(27.0),0,0,0,0,0,0},
                   {RC(183.0)/RC(1372.0),RC(-162.0)/RC(343.0),RC(1053.0)/RC(1372.0),0,0,0,0,0},
                   {RC(68.0)/RC(297.0),RC(-4.0)/RC(11.0),RC(42.0)/RC(143.0),RC(1960.0)/RC(3861.0),0,0,0,0},
                   {RC(597.0)/RC(22528.0),RC(81.0)/RC(352.0),RC(63099.0)/RC(585728.0),RC(58653.0)/RC(366080.0),RC(4617.0)/RC(20480.0),0,0,0},
                   {RC(174197.0)/RC(959244.0),RC(-30942.0)/RC(79937.0),RC(8152137.0)/RC(19744439.0),RC(666106.0)/RC(1039181.0),RC(-29421.0)/RC(29068.0),RC(482048.0)/RC(414219.0),0,0},
                   {RC(587.0)/RC(8064.0),0,RC(4440339.0)/RC(15491840.0),RC(24353.0)/RC(124800.0),RC(387.0)/RC(44800.0),RC(2152.0)/RC(5985.0),RC(7267.0)/RC(94080.0),0}},
      b[8]      =  {RC(587.0)/RC(8064.0),0,RC(4440339.0)/RC(15491840.0),RC(24353.0)/RC(124800.0),RC(387.0)/RC(44800.0),RC(2152.0)/RC(5985.0),RC(7267.0)/RC(94080.0),0},
      bembed[8] =  {RC(2479.0)/RC(34992.0),0,RC(123.0)/RC(416.0),RC(612941.0)/RC(3411720.0),RC(43.0)/RC(1440.0),RC(2272.0)/RC(6561.0),RC(79937.0)/RC(1113912.0),RC(3293.0)/RC(556956.0)};
    ierr = TSRKRegister(TSRK5BS,5,8,&A[0][0],b,NULL,bembed,0,NULL);CHKERRQ(ierr);
  }
#undef RC
  PetscFunctionReturn(0);
}

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

   Not Collective

   Level: advanced

.keywords: TSRK, register, destroy
.seealso: TSRKRegister(), TSRKRegisterAll()
@*/
PetscErrorCode TSRKRegisterDestroy(void)
{
  PetscErrorCode ierr;
  RKTableauLink  link;

  PetscFunctionBegin;
  while ((link = RKTableauList)) {
    RKTableau t = &link->tab;
    RKTableauList = link->next;
    ierr = PetscFree3(t->A,t->b,t->c);  CHKERRQ(ierr);
    ierr = PetscFree (t->bembed);       CHKERRQ(ierr);
    ierr = PetscFree (t->binterp);      CHKERRQ(ierr);
    ierr = PetscFree (t->name);         CHKERRQ(ierr);
    ierr = PetscFree (link);            CHKERRQ(ierr);
  }
  TSRKRegisterAllCalled = PETSC_FALSE;
  PetscFunctionReturn(0);
}

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

  Level: developer

.keywords: TS, TSRK, initialize, package
.seealso: PetscInitialize()
@*/
PetscErrorCode TSRKInitializePackage(void)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (TSRKPackageInitialized) PetscFunctionReturn(0);
  TSRKPackageInitialized = PETSC_TRUE;
  ierr = TSRKRegisterAll();CHKERRQ(ierr);
  ierr = PetscRegisterFinalize(TSRKFinalizePackage);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

  Level: developer

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

  PetscFunctionBegin;
  TSRKPackageInitialized = PETSC_FALSE;
  ierr = TSRKRegisterDestroy();CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@C
   TSRKRegister - register an RK 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 - stage coefficients (dimension s*s, row-major)
.  b - step completion table (dimension s; NULL to use last row of A)
.  c - abscissa (dimension s; NULL to use row sums of A)
.  bembed - completion table for embedded method (dimension s; NULL if not available)
.  p - Order of the interpolation scheme, equal to the number of columns of binterp
-  binterp - Coefficients of the interpolation formula (dimension s*p; NULL to reuse b with p=1)

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

   Level: advanced

.keywords: TS, register

.seealso: TSRK
@*/
PetscErrorCode TSRKRegister(TSRKType name,PetscInt order,PetscInt s,
                            const PetscReal A[],const PetscReal b[],const PetscReal c[],
                            const PetscReal bembed[],PetscInt p,const PetscReal binterp[])
{
  PetscErrorCode  ierr;
  RKTableauLink   link;
  RKTableau       t;
  PetscInt        i,j;

  PetscFunctionBegin;
  PetscValidCharPointer(name,1);
  PetscValidRealPointer(A,4);
  if (b) PetscValidRealPointer(b,5);
  if (c) PetscValidRealPointer(c,6);
  if (bembed) PetscValidRealPointer(bembed,7);
  if (binterp || p > 1) PetscValidRealPointer(binterp,9);

  ierr = TSRKInitializePackage();CHKERRQ(ierr);
  ierr = PetscNew(&link);CHKERRQ(ierr);
  t = &link->tab;

  ierr = PetscStrallocpy(name,&t->name);CHKERRQ(ierr);
  t->order = order;
  t->s = s;
  ierr = PetscMalloc3(s*s,&t->A,s,&t->b,s,&t->c);CHKERRQ(ierr);
  ierr = PetscMemcpy(t->A,A,s*s*sizeof(A[0]));CHKERRQ(ierr);
  if (b)  { ierr = PetscMemcpy(t->b,b,s*sizeof(b[0]));CHKERRQ(ierr); }
  else for (i=0; i<s; i++) t->b[i] = A[(s-1)*s+i];
  if (c)  { ierr = PetscMemcpy(t->c,c,s*sizeof(c[0]));CHKERRQ(ierr); }
  else for (i=0; i<s; i++) for (j=0,t->c[i]=0; j<s; j++) t->c[i] += A[i*s+j];
  t->FSAL = PETSC_TRUE;
  for (i=0; i<s; i++) if (t->A[(s-1)*s+i] != t->b[i]) t->FSAL = PETSC_FALSE;

  if (bembed) {
    ierr = PetscMalloc1(s,&t->bembed);CHKERRQ(ierr);
    ierr = PetscMemcpy(t->bembed,bembed,s*sizeof(bembed[0]));CHKERRQ(ierr);
  }

  if (!binterp) { p = 1; binterp = t->b; }
  t->p = p;
  ierr = PetscMalloc1(s*p,&t->binterp);CHKERRQ(ierr);
  ierr = PetscMemcpy(t->binterp,binterp,s*p*sizeof(binterp[0]));CHKERRQ(ierr);

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

/*
 This is for single-step RK method
 The step completion formula is

 x1 = x0 + h b^T YdotRHS

 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 + h be^T YdotRHS
     = x1 - h b^T YdotRHS + h be^T YdotRHS
     = x1 + h (be - b)^T YdotRHS

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

  PetscFunctionBegin;
  switch (rk->status) {
  case TS_STEP_INCOMPLETE:
  case TS_STEP_PENDING:
    h = ts->time_step; break;
  case TS_STEP_COMPLETE:
    h = ts->ptime - ts->ptime_prev; break;
  default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
  }
  if (order == tab->order) {
    if (rk->status == TS_STEP_INCOMPLETE) {
      ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
      for (j=0; j<s; j++) w[j] = h*tab->b[j];
      ierr = VecMAXPY(X,s,w,rk->YdotRHS);CHKERRQ(ierr);
    } else {ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);}
    PetscFunctionReturn(0);
  } else if (order == tab->order-1) {
    if (!tab->bembed) goto unavailable;
    if (rk->status == TS_STEP_INCOMPLETE) { /*Complete with the embedded method (be)*/
      ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
      for (j=0; j<s; j++) w[j] = h*tab->bembed[j];
      ierr = VecMAXPY(X,s,w,rk->YdotRHS);CHKERRQ(ierr);
    } else {  /*Rollback and re-complete using (be-b) */
      ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
      for (j=0; j<s; j++) w[j] = h*(tab->bembed[j] - tab->b[j]);
      ierr = VecMAXPY(X,s,w,rk->YdotRHS);CHKERRQ(ierr);
    }
    if (done) *done = PETSC_TRUE;
    PetscFunctionReturn(0);
  }
unavailable:
  if (done) *done = PETSC_FALSE;
  else SETERRQ3(PetscObjectComm((PetscObject)ts),PETSC_ERR_SUP,"RK '%s' of order %D cannot evaluate step at order %D. Consider using -ts_adapt_type none or a different method that has an embedded estimate.",tab->name,tab->order,order);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSForwardCostIntegral_RK(TS ts)
{
  TS_RK           *rk = (TS_RK*)ts->data;
  RKTableau       tab = rk->tableau;
  const PetscInt  s = tab->s;
  const PetscReal *b = tab->b,*c = tab->c;
  Vec             *Y = rk->Y;
  PetscInt        i;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  /* backup cost integral */
  ierr = VecCopy(ts->vec_costintegral,rk->VecCostIntegral0);CHKERRQ(ierr);
  for (i=s-1; i>=0; i--) {
    /* Evolve ts->vec_costintegral to compute integrals */
    ierr = TSComputeCostIntegrand(ts,rk->ptime+rk->time_step*c[i],Y[i],ts->vec_costintegrand);CHKERRQ(ierr);
    ierr = VecAXPY(ts->vec_costintegral,rk->time_step*b[i],ts->vec_costintegrand);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TSAdjointCostIntegral_RK(TS ts)
{
  TS_RK           *rk = (TS_RK*)ts->data;
  RKTableau       tab = rk->tableau;
  const PetscInt  s = tab->s;
  const PetscReal *b = tab->b,*c = tab->c;
  Vec             *Y = rk->Y;
  PetscInt        i;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  for (i=s-1; i>=0; i--) {
    /* Evolve ts->vec_costintegral to compute integrals */
    ierr = TSComputeCostIntegrand(ts,ts->ptime+ts->time_step*(1.0-c[i]),Y[i],ts->vec_costintegrand);CHKERRQ(ierr);
    ierr = VecAXPY(ts->vec_costintegral,-ts->time_step*b[i],ts->vec_costintegrand);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TSRollBack_RK(TS ts)
{
  TS_RK           *rk = (TS_RK*)ts->data;
  RKTableau       tab = rk->tableau;
  const PetscInt  s  = tab->s;
  const PetscReal *b = tab->b;
  PetscScalar     *w = rk->work;
  Vec             *YdotRHS = rk->YdotRHS;
  PetscInt        j;
  PetscReal       h;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  switch (rk->status) {
  case TS_STEP_INCOMPLETE:
  case TS_STEP_PENDING:
    h = ts->time_step; break;
  case TS_STEP_COMPLETE:
    h = ts->ptime - ts->ptime_prev; break;
  default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
  }
  for (j=0; j<s; j++) w[j] = -h*b[j];
  ierr = VecMAXPY(ts->vec_sol,s,w,YdotRHS);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSStep_RK(TS ts)
{
  TS_RK            *rk  = (TS_RK*)ts->data;
  RKTableau        tab  = rk->tableau;
  const PetscInt   s = tab->s;
  const PetscReal  *A = tab->A,*c = tab->c;
  PetscScalar      *w = rk->work;
  Vec              *Y = rk->Y,*YdotRHS = rk->YdotRHS;
  PetscBool        FSAL = tab->FSAL;
  TSAdapt          adapt;
  PetscInt         i,j;
  PetscInt         rejections = 0;
  PetscBool        stageok,accept = PETSC_TRUE;
  PetscReal        next_time_step = ts->time_step;
  PetscErrorCode   ierr;

  PetscFunctionBegin;
  if (ts->steprollback || ts->steprestart) FSAL = PETSC_FALSE;
  if (FSAL) { ierr = VecCopy(YdotRHS[s-1],YdotRHS[0]);CHKERRQ(ierr); }

  rk->status = TS_STEP_INCOMPLETE;
  while (!ts->reason && rk->status != TS_STEP_COMPLETE) {
    PetscReal t = ts->ptime;
    PetscReal h = ts->time_step;
    for (i=0; i<s; i++) {
      rk->stage_time = t + h*c[i];
      ierr = TSPreStage(ts,rk->stage_time); CHKERRQ(ierr);
      ierr = VecCopy(ts->vec_sol,Y[i]);CHKERRQ(ierr);
      for (j=0; j<i; j++) w[j] = h*A[i*s+j];
      ierr = VecMAXPY(Y[i],i,w,YdotRHS);CHKERRQ(ierr);
      ierr = TSPostStage(ts,rk->stage_time,i,Y); CHKERRQ(ierr);
      ierr = TSGetAdapt(ts,&adapt);CHKERRQ(ierr);
      ierr = TSAdaptCheckStage(adapt,ts,rk->stage_time,Y[i],&stageok);CHKERRQ(ierr);
      if (!stageok) goto reject_step;
      if (FSAL && !i) continue;
      ierr = TSComputeRHSFunction(ts,t+h*c[i],Y[i],YdotRHS[i]);CHKERRQ(ierr);
    }

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

    if (ts->costintegralfwd) {  /*Save the info for the later use in cost integral evaluation*/
      rk->ptime     = ts->ptime;
      rk->time_step = ts->time_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;
      ierr = PetscInfo2(ts,"Step=%D, step rejections %D greater than current TS allowed, stopping solve\n",ts->steps,rejections);CHKERRQ(ierr);
    }
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TSInterpolate_RK(TS ts,PetscReal itime,Vec X)
{
  TS_RK            *rk = (TS_RK*)ts->data;
  PetscInt         s  = rk->tableau->s,p = rk->tableau->p,i,j;
  PetscReal        h;
  PetscReal        tt,t;
  PetscScalar      *b;
  const PetscReal  *B = rk->tableau->binterp;
  PetscErrorCode   ierr;

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

  switch (rk->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");
  }
  ierr = PetscMalloc1(s,&b);CHKERRQ(ierr);
  for (i=0; i<s; i++) b[i] = 0;
  for (j=0,tt=t; j<p; j++,tt*=t) {
    for (i=0; i<s; i++) {
      b[i]  += h * B[i*p+j] * tt;
    }
  }
  ierr = VecCopy(rk->Y[0],X);CHKERRQ(ierr);
  ierr = VecMAXPY(X,s,b,rk->YdotRHS);CHKERRQ(ierr);
  ierr = PetscFree(b);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*
  This is interpolate formula for partitioned RHS multirate RK method
 */

static PetscErrorCode TSInterpolate_PARTITIONEDMRK(TS ts,PetscReal itime,Vec X)
{
  TS_RK            *rk = (TS_RK*)ts->data;
  PetscInt         s  = rk->tableau->s,p = rk->tableau->p,i,j;
  Vec              Yslow;    /* vector holds the slow components in Y[0] */
  PetscReal        h;
  PetscReal        tt,t;
  PetscScalar      *b;
  const PetscReal  *B = rk->tableau->binterp;
  PetscErrorCode   ierr;

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

  switch (rk->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");
  }
  ierr = PetscMalloc1(s,&b);CHKERRQ(ierr);
  for (i=0; i<s; i++) b[i] = 0;
  for (j=0,tt=t; j<p; j++,tt*=t) {
    for (i=0; i<s; i++) {
      b[i]  += h * B[i*p+j] * tt;
    }
  }
  ierr = VecGetSubVector(rk->Y[0],ts->iss,&Yslow);CHKERRQ(ierr);
  ierr = VecCopy(Yslow,X);CHKERRQ(ierr);
  ierr = VecMAXPY(X,s,b,rk->YdotRHS_slow);CHKERRQ(ierr);
  ierr = VecRestoreSubVector(rk->Y[0],ts->iss,&Yslow);CHKERRQ(ierr);
  ierr = PetscFree(b);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*
 This is for combined RHS multirate RK method
 The step completion formula is

 x1 = x0 + h b^T YdotRHS

*/
static PetscErrorCode TSEvaluateStep_RKMULTIRATE(TS ts,PetscInt order,Vec X,PetscBool *done)
{
  TS_RK           *rk = (TS_RK*)ts->data;
  RKTableau       tab  = rk->tableau;
  Vec             Xtemp;                          /* temporary work vector for X                                   */
  PetscScalar     *w = rk->work;
  PetscScalar     *x_ptr,*xtemp_ptr;              /* location to put the pointer to X and Xtemp respectively       */
  PetscReal       h = ts->time_step;
  PetscInt        s = tab->s,j;
  PetscInt        len_slow,len_fast;              /* the number of slow components and fast components respectively */
  const PetscInt  *is_slow,*is_fast;              /* the index for slow components and fast components respectively */
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
  ierr = VecDuplicate(ts->vec_sol,&Xtemp);CHKERRQ(ierr);
  ierr = VecCopy(ts->vec_sol,Xtemp);CHKERRQ(ierr);
  if (!rk->slow){
    for (j=0; j<s; j++) w[j] = h*tab->b[j];
    /* update the value of slow components, and discard the updating value of fast components */
    ierr = VecMAXPY(Xtemp,s,w,rk->YdotRHS);CHKERRQ(ierr);
    ierr = VecGetArray(X,&x_ptr);CHKERRQ(ierr);
    ierr = VecGetArray(Xtemp,&xtemp_ptr);CHKERRQ(ierr);
    ierr = ISGetSize(ts->iss,&len_slow);CHKERRQ(ierr);
    ierr = ISGetIndices(ts->iss,&is_slow);CHKERRQ(ierr);
    ierr = ISRestoreIndices(ts->iss,&is_slow);CHKERRQ(ierr);
    for (j=0; j<len_slow; j++) x_ptr[is_slow[j]] = xtemp_ptr[is_slow[j]];
    ierr = VecRestoreArray(X,&x_ptr);CHKERRQ(ierr);
    ierr = VecRestoreArray(Xtemp,&xtemp_ptr);CHKERRQ(ierr);
  } else {
    /* update the value of fast components, and discard the updating value of slow components */
    for (j=0; j<s; j++) w[j] = h/rk->dtratio*tab->b[j];
    ierr = VecMAXPY(Xtemp,s,w,rk->YdotRHS);CHKERRQ(ierr);
    ierr = VecGetArray(X,&x_ptr);CHKERRQ(ierr);
    ierr = VecGetArray(Xtemp,&xtemp_ptr);CHKERRQ(ierr);
    ierr = ISGetSize(ts->isf,&len_fast);CHKERRQ(ierr);
    ierr = ISGetIndices(ts->isf,&is_fast);CHKERRQ(ierr);
    ierr = ISRestoreIndices(ts->isf,&is_fast);CHKERRQ(ierr);
    for (j=0; j<len_fast; j++) x_ptr[is_fast[j]] = xtemp_ptr[is_fast[j]];
    ierr = VecRestoreArray(X,&x_ptr);CHKERRQ(ierr);
    ierr = VecRestoreArray(Xtemp,&xtemp_ptr);CHKERRQ(ierr);
  }
  /* free temporary vector space */
  ierr = VecDestroy(&Xtemp);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSStep_RKMULTIRATE(TS ts)
{
  TS_RK             *rk = (TS_RK*)ts->data;
  RKTableau         tab  = rk->tableau;
  Vec               *Y = rk->Y,*YdotRHS = rk->YdotRHS;
  Vec               stage_fast,stage_slow;                              /* vectors store the stage value of fast and slow components respectively           */
  Vec               presol_fast,newslo_fast;                            /* vectors store the previous and new time solution of fast components respectively */
  Vec               YdotRHS_prev,prev_sol;                              /* vectors store the previous YdotRHS and solution respectively                     */
  const PetscInt    s = tab->s,*is_slow,*is_fast;                       /* is_slow: index of slow components; is_fast: index of fast components             */
  const PetscReal   *A = tab->A,*c = tab->c;
  PetscScalar       *w = rk->work;
  PetscScalar       *y_ptr,*faststage_ptr,*slowstage_ptr;               /* location to put the pointer to Y, stage_fast and stage_slow respectively         */
  PetscScalar       *ydotrhsprev_ptr,*ydotrhs_ptr;                      /* location to put the pointer to YdotRHS_prev and YdotRHS respectively             */
  PetscInt          i,j,k,len_slow,len_fast;                            /* len_slow: the number of slow comonents; len_fast: the number of fast components  */
  PetscReal         next_time_step = ts->time_step,t = ts->ptime,h = ts->time_step;
  PetscErrorCode    ierr;

  PetscFunctionBegin;
  rk->status = TS_STEP_INCOMPLETE;
  ierr = VecDuplicate(ts->vec_sol,&stage_fast);CHKERRQ(ierr);
  ierr = VecDuplicate(ts->vec_sol,&stage_slow);CHKERRQ(ierr);
  ierr = VecDuplicate(ts->vec_sol,&YdotRHS_prev);CHKERRQ(ierr);
  ierr = VecDuplicate(ts->vec_sol,&prev_sol);CHKERRQ(ierr);
  ierr = ISGetSize(ts->iss,&len_slow);CHKERRQ(ierr);
  ierr = ISGetSize(ts->isf,&len_fast);CHKERRQ(ierr);
  ierr = ISGetIndices(ts->iss,&is_slow);CHKERRQ(ierr);
  ierr = ISGetIndices(ts->isf,&is_fast);CHKERRQ(ierr);
  ierr = ISRestoreIndices(ts->isf,&is_fast);CHKERRQ(ierr);
  ierr = ISRestoreIndices(ts->iss,&is_slow);CHKERRQ(ierr);
  for (i=0; i<s; i++) {
    rk->stage_time = t + h*c[i];
    ierr = TSPreStage(ts,rk->stage_time);CHKERRQ(ierr);
    ierr = VecCopy(ts->vec_sol,Y[i]);CHKERRQ(ierr);
    for (j=0; j<i; j++) w[j] = h*A[i*s+j];
    ierr = VecMAXPY(Y[i],i,w,YdotRHS);CHKERRQ(ierr);
    ierr = TSPostStage(ts,rk->stage_time,i,Y); CHKERRQ(ierr);
    /* compute the stage RHS */
    ierr = TSComputeRHSFunction(ts,t+h*c[i],Y[i],YdotRHS[i]);CHKERRQ(ierr);
  }
  ierr = VecCopy(ts->vec_sol,prev_sol);CHKERRQ(ierr);
  rk->slow = 0;
  ierr = TSEvaluateStep_RKMULTIRATE(ts,tab->order,ts->vec_sol,NULL);CHKERRQ(ierr);
  for (k=0; k<rk->dtratio; k++){
    for (i=0; i<s; i++) {
      rk->stage_time = t + h/rk->dtratio*c[i];
      ierr = TSPreStage(ts,rk->stage_time);CHKERRQ(ierr);
      ierr = VecCopy(prev_sol,Y[i]);CHKERRQ(ierr);
      ierr = VecCopy(prev_sol,stage_slow);CHKERRQ(ierr);
      ierr = VecCopy(prev_sol,stage_fast);CHKERRQ(ierr);
      /* guarantee the stage RHS for slow components do not change while updating the stage RHS for fast components */
      ierr = VecCopy(YdotRHS[i],YdotRHS_prev);CHKERRQ(ierr);
      /* update the fast components stage value */
      for (j=0; j<i; j++) w[j] = h/rk->dtratio*A[i*s+j];
      ierr = VecMAXPY(stage_fast,i,w,YdotRHS);CHKERRQ(ierr);
      /* update the slow components stage value */
      ierr = VecCopy(prev_sol,Y[0]);CHKERRQ(ierr);
      ierr = TSInterpolate_RK(ts,t+k*h/rk->dtratio+h/rk->dtratio*c[i],stage_slow);CHKERRQ(ierr);
      /* combine the update fast components stage value and slow components stage value to Y[i] */
      ierr = VecGetArray(Y[i],&y_ptr);CHKERRQ(ierr);
      ierr = VecGetArray(stage_slow,&slowstage_ptr);CHKERRQ(ierr);
      ierr = VecGetArray(stage_fast,&faststage_ptr);CHKERRQ(ierr);
      for (j=0; j<len_slow; j++) y_ptr[is_slow[j]] = slowstage_ptr[is_slow[j]];
      for (j=0; j<len_fast; j++) y_ptr[is_fast[j]] = faststage_ptr[is_fast[j]];
      ierr = VecRestoreArray(Y[i],&y_ptr);CHKERRQ(ierr);
      ierr = VecRestoreArray(stage_slow,&slowstage_ptr);CHKERRQ(ierr);
      ierr = VecRestoreArray(stage_fast,&faststage_ptr);CHKERRQ(ierr);
      ierr = TSPostStage(ts,rk->stage_time,i,Y); CHKERRQ(ierr);
      /* compute the stage RHS */
      ierr = TSComputeRHSFunction(ts,t+k*h/rk->dtratio+h/rk->dtratio*c[i],Y[i],YdotRHS[i]);CHKERRQ(ierr);
      /* re-assign the value of slow components in YdotRHS[i] to the calculated value before working on smaller time step-size */
      ierr = VecGetArray(YdotRHS[i],&ydotrhs_ptr);CHKERRQ(ierr);
      ierr = VecGetArray(YdotRHS_prev,&ydotrhsprev_ptr);CHKERRQ(ierr);
      for (j=0; j<len_slow; j++) ydotrhs_ptr[is_slow[j]] = ydotrhsprev_ptr[is_slow[j]];
      ierr = VecRestoreArray(YdotRHS[i],&ydotrhs_ptr);CHKERRQ(ierr);
      ierr = VecRestoreArray(YdotRHS_prev,&ydotrhsprev_ptr);CHKERRQ(ierr);
    }
    rk->slow = 1;
    ierr = TSEvaluateStep_RKMULTIRATE(ts,tab->order,ts->vec_sol,NULL);CHKERRQ(ierr);
    /* update the intial value for fast components */
    ierr = VecGetSubVector(prev_sol,ts->isf,&presol_fast);CHKERRQ(ierr);
    ierr = VecGetSubVector(ts->vec_sol,ts->isf,&newslo_fast);CHKERRQ(ierr);
    ierr = VecCopy(newslo_fast,presol_fast);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(prev_sol,ts->isf,&presol_fast);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(ts->vec_sol,ts->isf,&newslo_fast);CHKERRQ(ierr);
  }
  ts->ptime += ts->time_step;
  ts->time_step = next_time_step;
  rk->slow = 0;
  rk->status = TS_STEP_COMPLETE;
  /* free memory of work vectors */
  ierr = VecDestroy(&stage_fast);CHKERRQ(ierr);
  ierr = VecDestroy(&stage_slow);CHKERRQ(ierr);
  ierr = VecDestroy(&YdotRHS_prev);CHKERRQ(ierr);
  ierr = VecDestroy(&prev_sol);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*
 This is for partitioned RHS multirate RK method
 The step completion formula is

 x1 = x0 + h b^T YdotRHS

*/
static PetscErrorCode TSEvaluateStep_RKPMULTIRATE(TS ts,PetscInt order,Vec X,PetscBool *done)
{
  TS_RK           *rk = (TS_RK*)ts->data;
  RKTableau       tab  = rk->tableau;
  Vec             Xslow,Xfast;                  /* subvectors of X which store slow components and fast components respectively */
  PetscScalar     *w = rk->work;
  PetscReal       h = ts->time_step;
  PetscInt        s = tab->s,j;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
  if (!rk->slow){
    for (j=0; j<s; j++) w[j] = h*tab->b[j];
    ierr = VecGetSubVector(ts->vec_sol,ts->iss,&Xslow);CHKERRQ(ierr);
    ierr = VecMAXPY(Xslow,s,w,rk->YdotRHS_slow);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(ts->vec_sol,ts->iss,&Xslow);CHKERRQ(ierr);;
    } else {;
    for (j=0; j<s; j++) w[j] = h/rk->dtratio*tab->b[j];
    ierr = VecGetSubVector(X,ts->isf,&Xfast);CHKERRQ(ierr);
    ierr = VecMAXPY(Xfast,s,w,rk->YdotRHS_fast);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(X,ts->isf,&Xfast);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TSStep_RKPMULTIRATE(TS ts)
{
  TS_RK             *rk = (TS_RK*)ts->data;
  RKTableau         tab  = rk->tableau;
  Vec               *Y = rk->Y;
  Vec               *YdotRHS_fast = rk->YdotRHS_fast,*YdotRHS_slow = rk->YdotRHS_slow;
  Vec               Yslow,Yfast;                         /* subvectors store the stges of slow components and fast components respectively                           */
  Vec               Yfast_prev,Yfast_new,prev_sol;       /* subvectors store the previous and new solution of fast components and vector store the previous solution */
  const PetscInt    s = tab->s;
  const PetscReal   *A = tab->A,*c = tab->c;
  PetscScalar       *w = rk->work;
  PetscInt          i,j,k;
  PetscReal         next_time_step = ts->time_step,t = ts->ptime,h = ts->time_step;
  PetscErrorCode    ierr;

  PetscFunctionBegin;
  rk->status = TS_STEP_INCOMPLETE;
  for (i=0; i<s; i++) {
    rk->stage_time = t + h*c[i];
    ierr = TSPreStage(ts,rk->stage_time);CHKERRQ(ierr);
    ierr = VecCopy(ts->vec_sol,Y[i]);CHKERRQ(ierr);
    /*ierr = VecCopy(ts->vec_sol,Yc[i]);CHKERRQ(ierr);*/
    ierr = VecGetSubVector(Y[i],ts->iss,&Yslow);CHKERRQ(ierr);
    ierr = VecGetSubVector(Y[i],ts->isf,&Yfast);CHKERRQ(ierr);
    for (j=0; j<i; j++) w[j] = h*A[i*s+j];
    ierr = VecMAXPY(Yslow,i,w,YdotRHS_slow);CHKERRQ(ierr);
    ierr = VecMAXPY(Yfast,i,w,YdotRHS_fast);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(Y[i],ts->iss,&Yslow);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(Y[i],ts->isf,&Yfast);CHKERRQ(ierr);
    ierr = TSPostStage(ts,rk->stage_time,i,Y); CHKERRQ(ierr);
    /* compute the stage RHS for both slow and fast components */
    ierr = TSComputeRHSFunctionslow(ts,t+h*c[i],Y[i],YdotRHS_slow[i]);CHKERRQ(ierr);
    ierr = TSComputeRHSFunctionfast(ts,t+h*c[i],Y[i],YdotRHS_fast[i]);CHKERRQ(ierr);
  }
  /* store the value of slow components at previous time  */
  ierr = VecDuplicate(ts->vec_sol,&prev_sol);CHKERRQ(ierr);
  ierr = VecCopy(ts->vec_sol,prev_sol);CHKERRQ(ierr);
  rk->slow = 0;
  ierr = TSEvaluateStep_RKPMULTIRATE(ts,tab->order,ts->vec_sol,NULL);CHKERRQ(ierr);
  for (k=0; k<rk->dtratio; k++){
    for (i=0; i<s; i++) {
    rk->stage_time = t + h/rk->dtratio*c[i];
    ierr = TSPreStage(ts,rk->stage_time);CHKERRQ(ierr);
    ierr = VecCopy(prev_sol,Y[i]);CHKERRQ(ierr);
    /* stage value for fast components */
    for (j=0; j<i; j++) w[j] = h/rk->dtratio*A[i*s+j];
    ierr = VecGetSubVector(Y[i],ts->isf,&Yfast);CHKERRQ(ierr);
    ierr = VecMAXPY(Yfast,i,w,YdotRHS_fast);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(Y[i],ts->isf,&Yfast);CHKERRQ(ierr);
    /* stage value for slow components */
    ierr = VecCopy(prev_sol,Y[0]);CHKERRQ(ierr);
    ierr = VecGetSubVector(Y[i],ts->iss,&Yslow);CHKERRQ(ierr);
    ierr = TSInterpolate_PARTITIONEDMRK(ts,t+k*h/rk->dtratio+h/rk->dtratio*c[i],Yslow);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(Y[i],ts->iss,&Yslow);CHKERRQ(ierr);
    ierr = TSPostStage(ts,rk->stage_time,i,Y); CHKERRQ(ierr);
    /* compute the stage RHS for fast components */
    ierr = TSComputeRHSFunctionfast(ts,t+k*h/rk->dtratio+h/rk->dtratio*c[i],Y[i],YdotRHS_fast[i]);CHKERRQ(ierr);
   }
    /* update the value of fast components whenusing fast time step */
    rk->slow = 1;
    ierr = TSEvaluateStep_RKPMULTIRATE(ts,tab->order,ts->vec_sol,NULL);CHKERRQ(ierr);
    ierr = VecGetSubVector(prev_sol,ts->isf,&Yfast_prev);CHKERRQ(ierr);
    ierr = VecGetSubVector(ts->vec_sol,ts->isf,&Yfast_new);CHKERRQ(ierr);
    ierr = VecCopy(Yfast_new,Yfast_prev);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(prev_sol,ts->isf,&Yfast_prev);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(ts->vec_sol,ts->isf,&Yfast_new);CHKERRQ(ierr);
  }
  ts->ptime += ts->time_step;
  ts->time_step = next_time_step;
  rk->slow = 0;
  rk->status = TS_STEP_COMPLETE;
  ierr = VecDestroy(&prev_sol);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSAdjointSetUp_RK(TS ts)
{
  TS_RK          *rk  = (TS_RK*)ts->data;
  RKTableau      tab = rk->tableau;
  PetscInt       s   = tab->s;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (ts->adjointsetupcalled++) PetscFunctionReturn(0);
  ierr = VecDuplicateVecs(ts->vecs_sensi[0],s*ts->numcost,&rk->VecDeltaLam);CHKERRQ(ierr);
  if(ts->vecs_sensip) {
    ierr = VecDuplicateVecs(ts->vecs_sensip[0],s*ts->numcost,&rk->VecDeltaMu);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TSAdjointStep_RK(TS ts)
{
  TS_RK            *rk  = (TS_RK*)ts->data;
  RKTableau        tab  = rk->tableau;
  const PetscInt   s    = tab->s;
  const PetscReal  *A = tab->A,*b = tab->b,*c = tab->c;
  PetscScalar      *w    = rk->work;
  Vec              *Y    = rk->Y,*VecDeltaLam = rk->VecDeltaLam,*VecDeltaMu = rk->VecDeltaMu;
  PetscInt         i,j,nadj;
  PetscReal        t = ts->ptime;
  PetscErrorCode   ierr;
  PetscReal        h = ts->time_step;

  PetscFunctionBegin;
  rk->status = TS_STEP_INCOMPLETE;
  for (i=s-1; i>=0; i--) {
    Mat       J;
    PetscReal stage_time = t + h*(1.0-c[i]);
    PetscBool zero = PETSC_FALSE;

    ierr = TSGetRHSJacobian(ts,&J,NULL,NULL,NULL);CHKERRQ(ierr);
    ierr = TSComputeRHSJacobian(ts,stage_time,Y[i],J,J);CHKERRQ(ierr);
    if (ts->vec_costintegral) {
      ierr = TSAdjointComputeDRDYFunction(ts,stage_time,Y[i],ts->vecs_drdy);CHKERRQ(ierr);
    }
    /* Stage values of mu */
    if (ts->vecs_sensip) {
      ierr = TSAdjointComputeRHSJacobian(ts,stage_time,Y[i],ts->Jacp);CHKERRQ(ierr);
      if (ts->vec_costintegral) {
        ierr = TSAdjointComputeDRDPFunction(ts,stage_time,Y[i],ts->vecs_drdp);CHKERRQ(ierr);
      }
    }

    if (b[i] == 0 && i == s-1) zero = PETSC_TRUE;
    for (nadj=0; nadj<ts->numcost; nadj++) {
      DM  dm;
      Vec VecSensiTemp;

      ierr = TSGetDM(ts,&dm);CHKERRQ(ierr);
      ierr = DMGetGlobalVector(dm,&VecSensiTemp);CHKERRQ(ierr);
      /* Stage values of lambda */
      if (!zero) {
        ierr = VecCopy(ts->vecs_sensi[nadj],VecSensiTemp);CHKERRQ(ierr);
        ierr = VecScale(VecSensiTemp,-h*b[i]);CHKERRQ(ierr);
        for (j=i+1; j<s; j++) w[j-i-1]  = -h*A[j*s+i];
        ierr = VecMAXPY(VecSensiTemp,s-i-1,w,&VecDeltaLam[nadj*s+i+1]);CHKERRQ(ierr);
        ierr = MatMultTranspose(J,VecSensiTemp,VecDeltaLam[nadj*s+i]);CHKERRQ(ierr);
      } else {
        ierr = VecSet(VecDeltaLam[nadj*s+i],0);CHKERRQ(ierr);
      }
      if (ts->vec_costintegral) {
        ierr = VecAXPY(VecDeltaLam[nadj*s+i],-h*b[i],ts->vecs_drdy[nadj]);CHKERRQ(ierr);
      }

      /* Stage values of mu */
      if (ts->vecs_sensip) {
        if (!zero) {
          ierr = MatMultTranspose(ts->Jacp,VecSensiTemp,VecDeltaMu[nadj*s+i]);CHKERRQ(ierr);
        } else {
          ierr = VecSet(VecDeltaMu[nadj*s+i],0);CHKERRQ(ierr);
        }
        if (ts->vec_costintegral) {
          ierr = VecAXPY(VecDeltaMu[nadj*s+i],-h*b[i],ts->vecs_drdp[nadj]);CHKERRQ(ierr);
        }
      }
      ierr = DMRestoreGlobalVector(dm,&VecSensiTemp);CHKERRQ(ierr);
    }
  }

  for (j=0; j<s; j++) w[j] = 1.0;
  for (nadj=0; nadj<ts->numcost; nadj++) {
    ierr = VecMAXPY(ts->vecs_sensi[nadj],s,w,&VecDeltaLam[nadj*s]);CHKERRQ(ierr);
    if (ts->vecs_sensip) {
      ierr = VecMAXPY(ts->vecs_sensip[nadj],s,w,&VecDeltaMu[nadj*s]);CHKERRQ(ierr);
    }
  }
  rk->status = TS_STEP_COMPLETE;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSRKTableauReset(TS ts)
{
  TS_RK          *rk = (TS_RK*)ts->data;
  RKTableau      tab = rk->tableau;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!tab) PetscFunctionReturn(0);
  ierr = PetscFree(rk->work);CHKERRQ(ierr);
  ierr = VecDestroyVecs(tab->s,&rk->Y);CHKERRQ(ierr);
  ierr = VecDestroyVecs(tab->s,&rk->YdotRHS);CHKERRQ(ierr);
  ierr = VecDestroyVecs(tab->s,&rk->YdotRHS_fast);CHKERRQ(ierr);
  ierr = VecDestroyVecs(tab->s,&rk->YdotRHS_slow);CHKERRQ(ierr);
  ierr = VecDestroyVecs(tab->s*ts->numcost,&rk->VecDeltaLam);CHKERRQ(ierr);
  ierr = VecDestroyVecs(tab->s*ts->numcost,&rk->VecDeltaMu);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSReset_RK(TS ts)
{
  TS_RK         *rk = (TS_RK*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = TSRKTableauReset(ts);CHKERRQ(ierr);
  ierr = VecDestroy(&rk->VecCostIntegral0);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

static PetscErrorCode DMRestrictHook_TSRK(DM fine,Mat restrct,Vec rscale,Mat inject,DM coarse,void *ctx)
{
  PetscFunctionBegin;
  PetscFunctionReturn(0);
}


static PetscErrorCode DMSubDomainHook_TSRK(DM dm,DM subdm,void *ctx)
{
  PetscFunctionBegin;
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  PetscFunctionReturn(0);
}
/*
static PetscErrorCode RKSetAdjCoe(RKTableau tab)
{
  PetscReal *A,*b;
  PetscInt        s,i,j;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  s    = tab->s;
  ierr = PetscMalloc2(s*s,&A,s,&b);CHKERRQ(ierr);

  for (i=0; i<s; i++)
    for (j=0; j<s; j++) {
      A[i*s+j] = (tab->b[s-1-i]==0) ? 0: -tab->A[s-1-i+(s-1-j)*s] * tab->b[s-1-j] / tab->b[s-1-i];
      ierr = PetscPrintf(PETSC_COMM_WORLD,"Coefficients: A[%D][%D]=%.6f\n",i,j,A[i*s+j]);CHKERRQ(ierr);
    }

  for (i=0; i<s; i++) b[i] = (tab->b[s-1-i]==0)? 0: -tab->b[s-1-i];

  ierr  = PetscMemcpy(tab->A,A,s*s*sizeof(A[0]));CHKERRQ(ierr);
  ierr  = PetscMemcpy(tab->b,b,s*sizeof(b[0]));CHKERRQ(ierr);
  ierr  = PetscFree2(A,b);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}
*/

static PetscErrorCode TSRKTableauSetUp(TS ts)
{
  TS_RK          *rk  = (TS_RK*)ts->data;
  RKTableau      tab = rk->tableau;
  Vec            YdotRHS_fast,YdotRHS_slow;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscMalloc1(tab->s,&rk->work);CHKERRQ(ierr);
  ierr = VecDuplicateVecs(ts->vec_sol,tab->s,&rk->Y);CHKERRQ(ierr);
  ierr = VecDuplicateVecs(ts->vec_sol,tab->s,&rk->YdotRHS);CHKERRQ(ierr);
  ierr = VecGetSubVector(ts->vec_sol,ts->isf,&YdotRHS_fast);CHKERRQ(ierr);
  ierr = VecDuplicateVecs(YdotRHS_fast,tab->s,&rk->YdotRHS_fast);CHKERRQ(ierr);
  ierr = VecRestoreSubVector(ts->vec_sol,ts->isf,&YdotRHS_fast);CHKERRQ(ierr);
  ierr = VecGetSubVector(ts->vec_sol,ts->iss,&YdotRHS_slow);CHKERRQ(ierr);
  ierr = VecDuplicateVecs(YdotRHS_slow,tab->s,&rk->YdotRHS_slow);CHKERRQ(ierr);
  ierr = VecRestoreSubVector(ts->vec_sol,ts->iss,&YdotRHS_slow);CHKERRQ(ierr);

  PetscFunctionReturn(0);
}


static PetscErrorCode TSSetUp_RK(TS ts)
{
  TS_RK          *rk = (TS_RK*)ts->data;
  PetscErrorCode ierr;
  DM             dm;

  PetscFunctionBegin;
  ierr = TSCheckImplicitTerm(ts);CHKERRQ(ierr);
  ierr = TSRKTableauSetUp(ts);CHKERRQ(ierr);
  if (!rk->VecCostIntegral0 && ts->vec_costintegral && ts->costintegralfwd) { /* back up cost integral */
    ierr = VecDuplicate(ts->vec_costintegral,&rk->VecCostIntegral0);CHKERRQ(ierr);
  }
  ierr = TSGetDM(ts,&dm);CHKERRQ(ierr);
  ierr = DMCoarsenHookAdd(dm,DMCoarsenHook_TSRK,DMRestrictHook_TSRK,ts);CHKERRQ(ierr);
  ierr = DMSubDomainHookAdd(dm,DMSubDomainHook_TSRK,DMSubDomainRestrictHook_TSRK,ts);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*
  construct a database to choose single-step rk method, or combined rhs mutirate rk method or partitioned rhs rk method
*/
const char *const TSRKMultirateTypes[] = {"NONE","COMBINED","PARTITIONED","TSRKMultirateType","RKM_",0};

static PetscErrorCode TSSetFromOptions_RK(PetscOptionItems *PetscOptionsObject,TS ts)
{
  TS_RK          *rk = (TS_RK*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscOptionsHead(PetscOptionsObject,"RK ODE solver options");CHKERRQ(ierr);
  {
    RKTableauLink link;
    PetscInt      count,choice;
    PetscBool     flg;
    const char    **namelist;
    PetscInt      rkmtype = 0;

    for (link=RKTableauList,count=0; link; link=link->next,count++) ;
    ierr = PetscMalloc1(count,(char***)&namelist);CHKERRQ(ierr);
    for (link=RKTableauList,count=0; link; link=link->next,count++) namelist[count] = link->tab.name;
    ierr = PetscOptionsEList("-ts_rk_multirate_type","Use Multirate or partioned RHS Multirate or single rate RK method","TSRKSetMultirateType",TSRKMultirateTypes,3,TSRKMultirateTypes[0],&rkmtype,&flg);CHKERRQ(ierr);
    if (flg) {ierr = TSRKSetMultirateType(ts,rkmtype);CHKERRQ(ierr);}
    ierr = PetscOptionsEList("-ts_rk_type","Family of RK method","TSRKSetType",(const char*const*)namelist,count,rk->tableau->name,&choice,&flg);CHKERRQ(ierr);
    if (flg) {ierr = TSRKSetType(ts,namelist[choice]);CHKERRQ(ierr);}
    ierr = PetscFree(namelist);CHKERRQ(ierr);
  }
  ierr = PetscOptionsTail();CHKERRQ(ierr);
  ierr = PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Multirate methods options","");CHKERRQ(ierr);
  ierr = PetscOptionsInt("-ts_rk_dtratio","time step ratio between slow and fast","",rk->dtratio,&rk->dtratio,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsEnd();CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSView_RK(TS ts,PetscViewer viewer)
{
  TS_RK          *rk = (TS_RK*)ts->data;
  PetscBool      iascii;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);CHKERRQ(ierr);
  if (iascii) {
    RKTableau tab  = rk->tableau;
    TSRKType  rktype;
    char      buf[512];
    ierr = TSRKGetType(ts,&rktype);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  RK type %s\n",rktype);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  Order: %D\n",tab->order);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  FSAL property: %s\n",tab->FSAL ? "yes" : "no");CHKERRQ(ierr);
    ierr = PetscFormatRealArray(buf,sizeof(buf),"% 8.6f",tab->s,tab->c);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  Abscissa c = %s\n",buf);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TSLoad_RK(TS ts,PetscViewer viewer)
{
  PetscErrorCode ierr;
  TSAdapt        adapt;

  PetscFunctionBegin;
  ierr = TSGetAdapt(ts,&adapt);CHKERRQ(ierr);
  ierr = TSAdaptLoad(adapt,viewer);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

  Logically collective

  Input Parameter:
+  ts - timestepping context
-  rktype - type of RK-scheme

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

  Level: intermediate

.seealso: TSRKGetType(), TSRK, TSRKType, TSRK1FE, TSRK2A, TSRK3, TSRK3BS, TSRK4, TSRK5F, TSRK5DP, TSRK5BS
@*/
PetscErrorCode TSRKSetType(TS ts,TSRKType rktype)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  PetscValidCharPointer(rktype,2);
  ierr = PetscTryMethod(ts,"TSRKSetType_C",(TS,TSRKType),(ts,rktype));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@C
  TSRKSetMultirateType - Set the type of RK Multirate scheme

  Logically collective

  Input Parameter:
+  ts - timestepping context
-  rkmtype - type of RKM-scheme

  Options Database:
.   -ts_rk_multiarte_type - <none,combined,partitioned>

  Level: intermediate
@*/
PetscErrorCode TSRKSetMultirateType(TS ts, TSRKMultirateType rkmtype)
{
  TS_RK *rk = (TS_RK*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  switch(rkmtype){
    case RKM_NONE:
      break;
    case RKM_COMBINED:
      ts->ops->step           = TSStep_RKMULTIRATE;
      ts->ops->evaluatestep   = TSEvaluateStep_RKMULTIRATE;
      rk->dtratio             = 2;
      ierr = TSRKSetType(ts,TSMRKDefault);CHKERRQ(ierr);
      break;
    case RKM_PARTITIONED:
      ts->ops->step           = TSStep_RKPMULTIRATE;
      ts->ops->evaluatestep   = TSEvaluateStep_RKPMULTIRATE;
      ts->ops->interpolate    = TSInterpolate_PARTITIONEDMRK;
      rk->dtratio             = 2;
      ierr = TSRKSetType(ts,TSMRKDefault);CHKERRQ(ierr);
      break;
    default :
      SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_ARG_UNKNOWN_TYPE,"Unknown type '%s'",rkmtype);
  }
  PetscFunctionReturn(0);
}

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

  Logically collective

  Input Parameter:
.  ts - timestepping context

  Output Parameter:
.  rktype - type of RK-scheme

  Level: intermediate

.seealso: TSRKGetType()
@*/
PetscErrorCode TSRKGetType(TS ts,TSRKType *rktype)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  ierr = PetscUseMethod(ts,"TSRKGetType_C",(TS,TSRKType*),(ts,rktype));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  *rktype = rk->tableau->name;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSRKSetType_RK(TS ts,TSRKType rktype)
{
  TS_RK          *rk = (TS_RK*)ts->data;
  PetscErrorCode ierr;
  PetscBool      match;
  RKTableauLink  link;

  PetscFunctionBegin;
  if (rk->tableau) {
    ierr = PetscStrcmp(rk->tableau->name,rktype,&match);CHKERRQ(ierr);
    if (match) PetscFunctionReturn(0);
  }
  for (link = RKTableauList; link; link=link->next) {
    ierr = PetscStrcmp(link->tab.name,rktype,&match);CHKERRQ(ierr);
    if (match) {
      if (ts->setupcalled) {ierr = TSRKTableauReset(ts);CHKERRQ(ierr);}
      rk->tableau = &link->tab;
      if (ts->setupcalled) {ierr = TSRKTableauSetUp(ts);CHKERRQ(ierr);}
      ts->default_adapt_type = rk->tableau->bembed ? TSADAPTBASIC : TSADAPTNONE;
      PetscFunctionReturn(0);
    }
  }
  SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_ARG_UNKNOWN_TYPE,"Could not find '%s'",rktype);
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  if (ns) *ns = rk->tableau->s;
  if (Y)   *Y = rk->Y;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSDestroy_RK(TS ts)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = TSReset_RK(ts);CHKERRQ(ierr);
  if (ts->dm) {
    ierr = DMCoarsenHookRemove(ts->dm,DMCoarsenHook_TSRK,DMRestrictHook_TSRK,ts);CHKERRQ(ierr);
    ierr = DMSubDomainHookRemove(ts->dm,DMSubDomainHook_TSRK,DMSubDomainRestrictHook_TSRK,ts);CHKERRQ(ierr);
  }
  ierr = PetscFree(ts->data);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSRKGetType_C",NULL);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSRKSetType_C",NULL);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSRKSetMultirateType_C",NULL);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

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

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

  Level: beginner

.seealso:  TSCreate(), TS, TSSetType(), TSRKSetType(), TSRKGetType(), TSRKSetFullyImplicit(), TSRK2D, TTSRK2E, TSRK3,
           TSRK4, TSRK5, TSRKPRSSP2, TSRKBPR3, TSRKType, TSRKRegister(), TSRKSetMultirateType()

M*/
PETSC_EXTERN PetscErrorCode TSCreate_RK(TS ts)
{
  TS_RK          *rk;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = TSRKInitializePackage();CHKERRQ(ierr);

  ts->ops->reset          = TSReset_RK;
  ts->ops->destroy        = TSDestroy_RK;
  ts->ops->view           = TSView_RK;
  ts->ops->load           = TSLoad_RK;
  ts->ops->setup          = TSSetUp_RK;
  ts->ops->adjointsetup   = TSAdjointSetUp_RK;
  ts->ops->interpolate    = TSInterpolate_RK;
  ts->ops->step           = TSStep_RK;
  ts->ops->evaluatestep   = TSEvaluateStep_RK;
  ts->ops->rollback       = TSRollBack_RK;
  ts->ops->setfromoptions = TSSetFromOptions_RK;
  ts->ops->getstages      = TSGetStages_RK;
  ts->ops->adjointstep    = TSAdjointStep_RK;

  ts->ops->adjointintegral = TSAdjointCostIntegral_RK;
  ts->ops->forwardintegral = TSForwardCostIntegral_RK;

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

  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSRKGetType_C",TSRKGetType_RK);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSRKSetType_C",TSRKSetType_RK);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSRKSetMultirateType_C",TSRKSetMultirateType);CHKERRQ(ierr);

  ierr = TSRKSetType(ts,TSRKDefault);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}