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;
};

typedef struct {
  RKTableau    tableau;
  Vec          X0;
  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          *VecsDeltaLam;    /* Increment of the adjoint sensitivity w.r.t IC at stage                       */
  Vec          *VecsSensiTemp;
  Vec          VecDeltaMu;       /* Increment of the adjoint sensitivity w.r.t P at stage                        */
  Vec          *VecsDeltaLam2;   /* Increment of the 2nd-order adjoint sensitivity w.r.t IC at stage */
  Vec          VecDeltaMu2;      /* Increment of the 2nd-order adjoint sensitivity w.r.t P at stage */
  Vec          *VecsSensi2Temp;
  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                         */
  IS           is_fast,is_slow;
  TS           subts_fast,subts_slow,subts_current,ts_root;
  PetscBool    use_multirate;
  Mat          MatFwdSensip0;
  Mat          *MatsFwdStageSensip;
  Mat          *MatsFwdSensipTemp;
  Vec          VecDeltaFwdSensipCol; /* Working vector for holding one column of the sensitivity matrix */
} TS_RK;