xref: /petsc/src/ts/impls/multirate/mprk.c (revision 9d6e09e94d6628fa1ac345b00002cb4d5a8e9bf4)

/*
  Code for time stepping with the Partitioned Runge-Kutta method

  Notes:
  1) The general system is written as
     Udot = F(t,U) for nonsplit RHS multi-rate RK,
     user should give the indexes for both slow and fast components;
  2) The general system is written as
     Usdot = Fs(t,Us,Uf)
     Ufdot = Ff(t,Us,Uf) for split RHS multi-rate RK,
     user should partioned RHS by themselves and also provide the indexes for both slow and fast components.
  3) To correct The confusing terminology in the paper, we use 'slow method', 'slow buffer method' and 'fast method' to denote the methods applied to 'slow region', 'slow buffer region' and 'fast region' respectively. The 'slow method' in the original paper actually means the 'slow buffer method'.
  4) Why does the buffer region have to be inside the slow region? The buffer region is treated with a slow method essentially. Applying the slow method to a region with a fast characteristic time scale is apparently not a good choice.

  Reference:
  Emil M. Constantinescu, Adrian Sandu, Multirate Timestepping Methods for Hyperbolic Conservation Laws, Journal of Scientific Computing 2007

*/

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

static TSMPRKType TSMPRKDefault = TSMPRK2A22;
static PetscBool TSMPRKRegisterAllCalled;
static PetscBool TSMPRKPackageInitialized;

typedef struct _MPRKTableau *MPRKTableau;
struct _MPRKTableau {
  char      *name;
  PetscInt  order;                          /* Classical approximation order of the method i */
  PetscInt  sbase;                          /* Number of stages in the base method*/
  PetscInt  s;                              /* Number of stages */
  PetscInt  np;                             /* Number of partitions */
  PetscReal *Af,*bf,*cf;                    /* Tableau for fast components */
  PetscReal *Amb,*bmb,*cmb;                 /* Tableau for medium components */
  PetscInt  *rmb;                           /* Array of flags for repeated stages in medium method */
  PetscReal *Asb,*bsb,*csb;                 /* Tableau for slow components */
  PetscInt  *rsb;                           /* Array of flags for repeated staged in slow method*/
};
typedef struct _MPRKTableauLink *MPRKTableauLink;
struct _MPRKTableauLink {
  struct _MPRKTableau tab;
  MPRKTableauLink     next;
};
static MPRKTableauLink MPRKTableauList;

typedef struct {
  MPRKTableau         tableau;
  TSMPRKMultirateType mprkmtype;
  Vec                 *Y;                          /* States computed during the step                           */
  Vec                 *YdotRHS;
  Vec                 *YdotRHS_slow;               /* Function evaluations by slow tableau for slow components  */
  Vec                 *YdotRHS_slowbuffer;         /* Function evaluations by slow tableau for slow components  */
  Vec                 *YdotRHS_medium;             /* Function evaluations by slow tableau for slow components  */
  Vec                 *YdotRHS_mediumbuffer;       /* Function evaluations by slow tableau for slow components  */
  Vec                 *YdotRHS_fast;               /* Function evaluations by fast tableau for fast components  */
  PetscScalar         *work_slow;                  /* Scalar work_slow by slow tableau                          */
  PetscScalar         *work_slowbuffer;            /* Scalar work_slow by slow tableau                          */
  PetscScalar         *work_medium;                /* Scalar work_slow by medium tableau                        */
  PetscScalar         *work_mediumbuffer;          /* Scalar work_slow by medium tableau                        */
  PetscScalar         *work_fast;                  /* Scalar work_fast by fast tableau                          */
  PetscReal           stage_time;
  TSStepStatus        status;
  PetscReal           ptime;
  PetscReal           time_step;
  IS                  is_slow,is_slowbuffer,is_medium,is_mediumbuffer,is_fast;
  TS                  subts_slow,subts_slowbuffer,subts_medium,subts_mediumbuffer,subts_fast;
} TS_MPRK;

static PetscErrorCode TSMPRKGenerateTableau2(PetscInt ratio,PetscInt s,const PetscReal Abase[],const PetscReal bbase[],PetscReal A1[],PetscReal b1[],PetscReal A2[],PetscReal b2[])
{
  PetscInt i,j,k,l;

  PetscFunctionBegin;
  for (k=0; k<ratio; k++) {
    /* diagonal blocks */
    for (i=0; i<s; i++)
      for (j=0; j<s; j++) {
        A1[(k*s+i)*ratio*s+k*s+j] = Abase[i*s+j];
        A2[(k*s+i)*ratio*s+k*s+j] = Abase[i*s+j]/ratio;
      }
    /* off diagonal blocks */
    for (l=0; l<k; l++)
      for (i=0; i<s; i++)
        for (j=0; j<s; j++)
          A2[(k*s+i)*ratio*s+l*s+j] = bbase[j]/ratio;
    for (j=0; j<s; j++) {
      b1[k*s+j] = bbase[j]/ratio;
      b2[k*s+j] = bbase[j]/ratio;
    }
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TSMPRKGenerateTableau3(PetscInt ratio,PetscInt s,const PetscReal Abase[],const PetscReal bbase[],PetscReal A1[],PetscReal b1[],PetscReal A2[],PetscReal b2[],PetscReal A3[],PetscReal b3[])
{
  PetscInt i,j,k,l,m,n;

  PetscFunctionBegin;
  for (k=0; k<ratio; k++) { /* diagonal blocks of size ratio*s by ratio*s */
    for (l=0; l<ratio; l++) /* diagonal sub-blocks of size s by s */
      for (i=0; i<s; i++)
        for (j=0; j<s; j++) {
          A1[((k*ratio+l)*s+i)*ratio*ratio*s+(k*ratio+l)*s+j] = Abase[i*s+j];
          A2[((k*ratio+l)*s+i)*ratio*ratio*s+(k*ratio+l)*s+j] = Abase[i*s+j]/ratio;
          A3[((k*ratio+l)*s+i)*ratio*ratio*s+(k*ratio+l)*s+j] = Abase[i*s+j]/ratio/ratio;
        }
    for (l=0; l<k; l++) /* off-diagonal blocks of size ratio*s by ratio*s */
      for (m=0; m<ratio; m++)
        for (n=0; n<ratio; n++)
          for (i=0; i<s; i++)
            for (j=0; j<s; j++) {
               A2[((k*ratio+m)*s+i)*ratio*ratio*s+(l*ratio+n)*s+j] = bbase[j]/ratio/ratio;
               A3[((k*ratio+m)*s+i)*ratio*ratio*s+(l*ratio+n)*s+j] = bbase[j]/ratio/ratio;
            }
    for (m=0; m<ratio; m++)
      for (n=0; n<m; n++) /* off-diagonal sub-blocks of size s by s in the diagonal blocks */
          for (i=0; i<s; i++)
            for (j=0; j<s; j++)
               A3[((k*ratio+m)*s+i)*ratio*ratio*s+(k*ratio+n)*s+j] = bbase[j]/ratio/ratio;
    for (n=0; n<ratio; n++)
      for (j=0; j<s; j++) {
        b1[(k*ratio+n)*s+j] = bbase[j]/ratio/ratio;
        b2[(k*ratio+n)*s+j] = bbase[j]/ratio/ratio;
        b3[(k*ratio+n)*s+j] = bbase[j]/ratio/ratio;
      }
  }
  PetscFunctionReturn(0);
}

/*MC
     TSMPRK2A22 - Second Order Partitioned Runge Kutta scheme based on RK2A.

     This method has four stages for slow and fast parts. The refinement factor of the stepsize is 2.
     r = 2, np = 2
     Options database:
.     -ts_mprk_type 2a22

     Level: advanced

.seealso: TSMPRK, TSMPRKType, TSMPRKSetType()
M*/
/*MC
     TSMPRK2A23 - Second Order Partitioned Runge Kutta scheme based on RK2A.

     This method has eight stages for slow and medium and fast parts. The refinement factor of the stepsize is 2.
     r = 2, np = 3
     Options database:
.     -ts_mprk_type 2a23

     Level: advanced

.seealso: TSMPRK, TSMPRKType, TSMPRKSetType()
M*/
/*MC
     TSMPRK2A32 - Second Order Partitioned Runge Kutta scheme based on RK2A.

     This method has four stages for slow and fast parts. The refinement factor of the stepsize is 3.
     r = 3, np = 2
     Options database:
.     -ts_mprk_type 2a32

     Level: advanced

.seealso: TSMPRK, TSMPRKType, TSMPRKSetType()
M*/
/*MC
     TSMPRK2A33 - Second Order Partitioned Runge Kutta scheme based on RK2A.

     This method has eight stages for slow and medium and fast parts. The refinement factor of the stepsize is 3.
     r = 3, np = 3
     Options database:
.     -ts_mprk_type 2a33

     Level: advanced

.seealso: TSMPRK, TSMPRKType, TSMPRKSetType()
M*/
/*MC
     TSMPRK3P2M - Third Order Partitioned Runge Kutta scheme.

     This method has eight stages for both slow and fast parts.

     Options database:
.     -ts_mprk_type pm3  (put here temporarily)

     Level: advanced

.seealso: TSMPRK, TSMPRKType, TSMPRKSetType()
M*/
/*MC
     TSMPRKP2 - Second Order Partitioned Runge Kutta scheme.

     This method has five stages for both slow and fast parts.

     Options database:
.     -ts_mprk_type p2

     Level: advanced

.seealso: TSMPRK, TSMPRKType, TSMPRKSetType()
M*/
/*MC
     TSMPRKP3 - Third Order Partitioned Runge Kutta scheme.

     This method has ten stages for both slow and fast parts.

     Options database:
.     -ts_mprk_type p3

     Level: advanced

.seealso: TSMPRK, TSMPRKType, TSMPRKSetType()
M*/

/*@C
  TSMPRKRegisterAll - Registers all of the Partirioned Runge-Kutta explicit methods in TSMPRK

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

  Level: advanced

.keywords: TS, TSMPRK, register, all

.seealso:  TSMPRKRegisterDestroy()
@*/
PetscErrorCode TSMPRKRegisterAll(void)
{
  PetscErrorCode ierr;

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

#define RC PetscRealConstant
  {
    const PetscReal
      Abase[2][2] = {{0,0},
                     {RC(1.0),0}},
      bbase[2] = {RC(0.5),RC(0.5)};
    PetscReal
      Asb[4][4] = {{0}},Af[4][4] = {{0}},bsb[4] = {0},bf[4] = {0};
    PetscInt
      rsb[4] = {0,0,1,2};
    ierr = TSMPRKGenerateTableau2(2,2,&Abase[0][0],bbase,&Asb[0][0],bsb,&Af[0][0],bf);CHKERRQ(ierr);
    ierr = TSMPRKRegister(TSMPRK2A22,2,2,2,1,&Asb[0][0],bsb,NULL,rsb,NULL,NULL,NULL,NULL,&Af[0][0],bf,NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      Abase[2][2] = {{0,0},
                     {RC(1.0),0}},
      bbase[2]    = {RC(0.5),RC(0.5)};
    PetscReal
      Asb[8][8] = {{0}},Amb[8][8] = {{0}},Af[8][8] = {{0}},bsb[8] ={0},bmb[8] = {0},bf[8] = {0};
    PetscInt
      rsb[8] = {0,0,1,2,1,2,1,2},rmb[8] = {0,0,1,2,0,0,5,6};
    ierr = TSMPRKGenerateTableau3(2,2,&Abase[0][0],bbase,&Asb[0][0],bsb,&Amb[0][0],bmb,&Af[0][0],bf);CHKERRQ(ierr);
    ierr = TSMPRKRegister(TSMPRK2A23,2,2,2,2,&Asb[0][0],bsb,NULL,rsb,&Amb[0][0],bmb,NULL,rmb,&Af[0][0],bf,NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      Abase[2][2] = {{0,0},
                     {RC(1.0),0}},
      bbase[2]    = {RC(0.5),RC(0.5)};
    PetscReal
      Asb[6][6] = {{0}},Af[6][6] = {{0}},bsb[6] = {0},bf[6] = {0};
    PetscInt
      rsb[6] = {0,0,1,2,1,2};
    ierr = TSMPRKGenerateTableau2(3,2,&Abase[0][0],bbase,&Asb[0][0],bsb,&Af[0][0],bf);CHKERRQ(ierr);
    ierr = TSMPRKRegister(TSMPRK2A32,2,2,3,1,&Asb[0][0],bsb,NULL,rsb,NULL,NULL,NULL,NULL,&Af[0][0],bf,NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      Abase[2][2] = {{0,0},
                     {RC(1.0),0}},
      bbase[2]    = {RC(0.5),RC(0.5)};
    PetscReal
      Asb[18][18] = {{0}},Amb[18][18] = {{0}},Af[18][18] = {{0}},bsb[18] ={0},bmb[18] = {0},bf[18] = {0};
    PetscInt
      rsb[18] = {0,0,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2},rmb[18] = {0,0,1,2,1,2,0,0,7,8,7,8,0,0,13,14,13,14};
    ierr = TSMPRKGenerateTableau3(3,2,&Abase[0][0],bbase,&Asb[0][0],bsb,&Amb[0][0],bmb,&Af[0][0],bf);CHKERRQ(ierr);
    ierr = TSMPRKRegister(TSMPRK2A33,2,2,3,3,&Asb[0][0],bsb,NULL,rsb,&Amb[0][0],bmb,NULL,rmb,&Af[0][0],bf,NULL);CHKERRQ(ierr);
  }
/*
    PetscReal
      Asb[8][8] = {{Abase[0][0],Abase[0][1],0,0,0,0,0,0},
                   {Abase[1][0],Abase[1][1],0,0,0,0,0,0},
                   {0,0,Abase[0][0],Abase[0][1],0,0,0,0},
                   {0,0,Abase[1][0],Abase[1][1],0,0,0,0},
                   {0,0,0,0,Abase[0][0],Abase[0][1],0,0},
                   {0,0,0,0,Abase[1][0],Abase[1][1],0,0},
                   {0,0,0,0,0,0,Abase[0][0],Abase[0][1]},
                   {0,0,0,0,0,0,Abase[1][0],Abase[1][1]}},
      Amb[8][8] = {{Abase[0][0]/m,Abase[0][1]/m,0,0,0,0,0,0},
                   {Abase[1][0]/m,Abase[1][1]/m,0,0,0,0,0,0},
                   {0,0,Abase[0][0]/m,Abase[0][1]/m,0,0,0,0},
                   {0,0,Abase[1][0]/m,Abase[1][1]/m,0,0,0,0},
                   {bbase[0]/m,bbase[1]/m,bbase[0]/m,bbase[1]/m,Abase[0][0]/m,Abase[0][1]/m,0,0},
                   {bbase[0]/m,bbase[1]/m,bbase[0]/m,bbase[1]/m,Abase[1][0]/m,Abase[1][1]/m,0,0},
                   {bbase[0]/m,bbase[1]/m,bbase[0]/m,bbase[1]/m,0,0,Abase[0][0]/m,Abase[0][1]/m},
                   {bbase[0]/m,bbase[1]/m,bbase[0]/m,bbase[1]/m,0,0,Abase[1][0]/m,Abase[1][1]/m}},
      Af[8][8] = {{Abase[0][0]/m/m,Abase[0][1]/m/m,0,0,0,0,0,0},
                   {Abase[1][0]/m/m,Abase[1][1]/m/m,0,0,0,0,0,0},
                   {0,0,Abase[0][0]/m/m,Abase[0][1]/m/m,0,0,0,0},
                   {0,0,Abase[1][0]/m/m,Abase[1][1]/m/m,0,0,0,0},
                   {bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m/m,bbase[1]/m/m,Abase[0][0]/m/m,Abase[0][1]/m/m,0,0},
                   {bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m/m,bbase[1]/m/m,Abase[1][0]/m/m,Abase[1][1]/m/m,0,0},
                   {bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m/m,bbase[1]/m/m,Abase[0][0]/m,Abase[0][1]/m},
                   {bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m/m,bbase[1]/m/m,Abase[1][0]/m,Abase[1][1]/m}},
      bsb[8]    = {bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m/m,bbase[1]/m/m},
      bmb[8]    = {bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m,bbase[1]/m/m},
      bf[8]     = {bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m/m,bbase[1]/m/m,bbase[0]/m/m,bbase[1]/m,bbase[0]/m/m,bbase[1]/m/m},
*/
  /*{
      const PetscReal
        As[8][8] = {{0,0,0,0,0,0,0,0},
                    {RC(1.0)/RC(2.0),0,0,0,0,0,0,0},
                    {RC(-1.0)/RC(6.0),RC(2.0)/RC(3.0),0,0,0,0,0,0},
                    {RC(1.0)/RC(3.0),RC(-1.0)/RC(3.0),RC(1.0),0,0,0,0,0},
                    {0,0,0,0,0,0,0,0},
                    {0,0,0,0,RC(1.0)/RC(2.0),0,0,0},
                    {0,0,0,0,RC(-1.0)/RC(6.0),RC(2.0)/RC(3.0),0,0},
                    {0,0,0,0,RC(1.0)/RC(3.0),RC(-1.0)/RC(3.0),RC(1.0),0}},
         A[8][8] = {{0,0,0,0,0,0,0,0},
                    {RC(1.0)/RC(4.0),0,0,0,0,0,0,0},
                    {RC(-1.0)/RC(12.0),RC(1.0)/RC(3.0),0,0,0,0,0,0},
                    {RC(1.0)/RC(6.0),RC(-1.0)/RC(6.0),RC(1.0)/RC(2.0),0,0,0,0,0},
                    {RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),0,0,0,0},
                    {RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),RC(1.0)/RC(4.0),0,0,0},
                    {RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),RC(-1.0)/RC(12.0),RC(1.0)/RC(3.0),0,0},
                    {RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(-1.0)/RC(6.0),RC(1.0)/RC(2.0),0}},
          bs[8] = {RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0)},
           b[8] = {RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0)};
           ierr = TSMPRKRegister(TSMPRKPM3,3,8,&As[0][0],bs,NULL,&A[0][0],b,NULL);CHKERRQ(ierr);
  }*/

  {
    const PetscReal
      Asb[5][5] = {{0,0,0,0,0},
                   {RC(1.0)/RC(2.0),0,0,0,0},
                   {RC(1.0)/RC(2.0),0,0,0,0},
                   {RC(1.0),0,0,0,0},
                   {RC(1.0),0,0,0,0}},
      Af[5][5]  = {{0,0,0,0,0},
                   {RC(1.0)/RC(2.0),0,0,0,0},
                   {RC(1.0)/RC(4.0),RC(1.0)/RC(4.0),0,0,0},
                   {RC(1.0)/RC(4.0),RC(1.0)/RC(4.0),RC(1.0)/RC(2.0),0,0},
                   {RC(1.0)/RC(4.0),RC(1.0)/RC(4.0),RC(1.0)/RC(4.0),RC(1.0)/RC(4.0),0}},
      bsb[5]    = {RC(1.0)/RC(2.0),0,0,0,RC(1.0)/RC(2.0)},
      bf[5]     = {RC(1.0)/RC(4.0),RC(1.0)/RC(4.0),RC(1.0)/RC(4.0),RC(1.0)/RC(4.0),0};
    const PetscInt
      rsb[5]    = {0,0,2,0,4};
    ierr = TSMPRKRegister(TSMPRKP2,2,5,1,1,&Asb[0][0],bsb,NULL,rsb,NULL,NULL,NULL,NULL,&Af[0][0],bf,NULL);CHKERRQ(ierr);
  }

  {
    const PetscReal
      Asb[10][10] = {{0,0,0,0,0,0,0,0,0,0},
                     {RC(1.0)/RC(4.0),0,0,0,0,0,0,0,0,0},
                     {RC(1.0)/RC(4.0),0,0,0,0,0,0,0,0,0},
                     {RC(1.0)/RC(2.0),0,0,0,0,0,0,0,0,0},
                     {RC(1.0)/RC(2.0),0,0,0,0,0,0,0,0,0},
                     {RC(-1.0)/RC(6.0),0,0,0,RC(2.0)/RC(3.0),0,0,0,0,0},
                     {RC(1.0)/RC(12.0),0,0,0,RC(1.0)/RC(6.0),RC(1.0)/RC(2.0),0,0,0,0},
                     {RC(1.0)/RC(12.0),0,0,0,RC(1.0)/RC(6.0),RC(1.0)/RC(2.0),0,0,0,0},
                     {RC(1.0)/RC(3.0),0,0,0,RC(-1.0)/RC(3.0),RC(1.0),0,0,0,0},
                     {RC(1.0)/RC(3.0),0,0,0,RC(-1.0)/RC(3.0),RC(1.0),0,0,0,0}},
      Af[10][10]  = {{0,0,0,0,0,0,0,0,0,0},
                     {RC(1.0)/RC(4.0),0,0,0,0,0,0,0,0,0},
                     {RC(-1.0)/RC(12.0),RC(1.0)/RC(3.0),0,0,0,0,0,0,0,0},
                     {RC(1.0)/RC(6.0),RC(-1.0)/RC(6.0),RC(1.0)/RC(2.0),0,0,0,0,0,0,0},
                     {RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),0,0,0,0,0,0},
                     {RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),0,0,0,0,0,0},
                     {RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),0,RC(1.0)/RC(4.0),0,0,0,0},
                     {RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),0,RC(-1.0)/RC(12.0),RC(1.0)/RC(3.0),0,0,0},
                     {RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),0,RC(1.0)/RC(6.0),RC(-1.0)/RC(6.0),RC(1.0)/RC(2.0),0,0},
                     {RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),0,RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),0}},
      bsb[10]     = {RC(1.0)/RC(6.0),0,0,0,RC(1.0)/RC(3.0),RC(1.0)/RC(3.0),0,0,0,RC(1.0)/RC(6.0)},
      bf[10]      = {RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),0,RC(1.0)/RC(12.0),RC(1.0)/RC(6.0),RC(1.0)/RC(6.0),RC(1.0)/RC(12.0),0};
    const PetscInt
      rsb[10]     = {0,0,2,0,4,0,0,7,0,9};
    ierr = TSMPRKRegister(TSMPRKP3,3,5,2,1,&Asb[0][0],bsb,NULL,rsb,NULL,NULL,NULL,NULL,&Af[0][0],bf,NULL);CHKERRQ(ierr);
  }
#undef RC
  PetscFunctionReturn(0);
}

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

   Not Collective

   Level: advanced

.keywords: TSMPRK, register, destroy
.seealso: TSMPRKRegister(), TSMPRKRegisterAll()
@*/
PetscErrorCode TSMPRKRegisterDestroy(void)
{
  PetscErrorCode ierr;
  MPRKTableauLink link;

  PetscFunctionBegin;
  while ((link = MPRKTableauList)) {
    MPRKTableau t = &link->tab;
    MPRKTableauList = link->next;
    ierr = PetscFree3(t->Asb,t->bsb,t->csb);CHKERRQ(ierr);
    ierr = PetscFree3(t->Amb,t->bmb,t->cmb);CHKERRQ(ierr);
    ierr = PetscFree3(t->Af,t->bf,t->cf);CHKERRQ(ierr);
    ierr = PetscFree2(t->rsb,t->rmb);CHKERRQ(ierr);
    ierr = PetscFree (t->name);CHKERRQ(ierr);
    ierr = PetscFree (link);CHKERRQ(ierr);
  }
  TSMPRKRegisterAllCalled = PETSC_FALSE;
  PetscFunctionReturn(0);
}

/*@C
  TSMPRKInitializePackage - This function initializes everything in the TSMPRK package. It is called
  from PetscDLLibraryRegister() when using dynamic libraries, and on the first call to TSCreate_MPRK()
  when using static libraries.

  Level: developer

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

  PetscFunctionBegin;
  if (TSMPRKPackageInitialized) PetscFunctionReturn(0);
  TSMPRKPackageInitialized = PETSC_TRUE;
  ierr = TSMPRKRegisterAll();CHKERRQ(ierr);
  ierr = PetscRegisterFinalize(TSMPRKFinalizePackage);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

  Level: developer

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

  PetscFunctionBegin;
  TSMPRKPackageInitialized = PETSC_FALSE;
  ierr = TSMPRKRegisterDestroy();CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@C
   TSMPRKRegister - register a MPRK scheme by providing the entries in the Butcher tableau

   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 in the base methods
.  ratio1 - stepsize ratio at 1st level (e.g. slow/medium)
.  ratio2 - stepsize ratio at 2nd level (e.g. medium/fast)
.  Af - stage coefficients for fast components(dimension s*s, row-major)
.  bf - step completion table for fast components(dimension s)
.  cf - abscissa for fast components(dimension s)
.  As - stage coefficients for slow components(dimension s*s, row-major)
.  bs - step completion table for slow components(dimension s)
-  cs - abscissa for slow components(dimension s)

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

   Level: advanced

.keywords: TS, register

.seealso: TSMPRK
@*/
PetscErrorCode TSMPRKRegister(TSMPRKType name,PetscInt order,
                              PetscInt sbase,PetscInt ratio1,PetscInt ratio2,
                              const PetscReal Asb[],const PetscReal bsb[],const PetscReal csb[],const PetscInt rsb[],
                              const PetscReal Amb[],const PetscReal bmb[],const PetscReal cmb[],const PetscInt rmb[],
                              const PetscReal Af[],const PetscReal bf[],const PetscReal cf[])
{
  MPRKTableauLink link;
  MPRKTableau     t;
  PetscInt        s,i,j;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  PetscValidCharPointer(name,1);
  PetscValidRealPointer(Asb,4);
  if (bsb) PetscValidRealPointer(bsb,7);
  if (csb) PetscValidRealPointer(csb,8);
  if (rsb) PetscValidRealPointer(rsb,9);
  if (Amb) PetscValidRealPointer(Amb,10);
  if (bmb) PetscValidRealPointer(bmb,11);
  if (cmb) PetscValidRealPointer(cmb,12);
  if (rmb) PetscValidRealPointer(rmb,13);
  PetscValidRealPointer(Af,14);
  if (bf) PetscValidRealPointer(bf,15);
  if (cf) PetscValidRealPointer(cf,16);

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

  ierr = PetscStrallocpy(name,&t->name);CHKERRQ(ierr);
  s = sbase*ratio1*ratio2; /*  this is the dimension of the matrices below */
  t->order = order;
  t->sbase = sbase;
  t->s  = s;
  t->np = 2;

  ierr = PetscMalloc3(s*s,&t->Af,s,&t->bf,s,&t->cf);CHKERRQ(ierr);
  ierr = PetscMemcpy(t->Af,Af,s*s*sizeof(Af[0]));CHKERRQ(ierr);
  if (bf) {
    ierr = PetscMemcpy(t->bf,bf,s*sizeof(bf[0]));CHKERRQ(ierr);
  } else
    for (i=0; i<s; i++) t->bf[i] = Af[(s-1)*s+i];
  if (cf) {
    ierr = PetscMemcpy(t->cf,cf,s*sizeof(cf[0]));CHKERRQ(ierr);
  } else {
    for (i=0; i<s; i++)
      for (j=0,t->cf[i]=0; j<s; j++)
        t->cf[i] += Af[i*s+j];
  }

  if (Amb) {
    t->np = 3;
    ierr = PetscMalloc3(s*s,&t->Amb,s,&t->bmb,s,&t->cmb);CHKERRQ(ierr);
    ierr = PetscCalloc1(s,&t->rmb);CHKERRQ(ierr);
    ierr = PetscMemcpy(t->Amb,Amb,s*s*sizeof(Amb[0]));CHKERRQ(ierr);
    if (bmb) {
      ierr = PetscMemcpy(t->bmb,bmb,s*sizeof(bmb[0]));CHKERRQ(ierr);
    } else {
      for (i=0; i<s; i++) t->bmb[i] = Amb[(s-1)*s+i];
    }
    if (cmb) {
      ierr = PetscMemcpy(t->cmb,cmb,s*sizeof(cmb[0]));CHKERRQ(ierr);
    } else {
      for (i=0; i<s; i++)
        for (j=0,t->cmb[i]=0; j<s; j++)
          t->cmb[i] += Amb[i*s+j];
    }
    if (rmb) {
      ierr = PetscMemcpy(t->rmb,rmb,s*sizeof(rmb[0]));CHKERRQ(ierr);
    }
  }

  ierr = PetscMalloc3(s*s,&t->Asb,s,&t->bsb,s,&t->csb);CHKERRQ(ierr);
  ierr = PetscMemcpy(t->Asb,Asb,s*s*sizeof(Asb[0]));CHKERRQ(ierr);
  ierr = PetscCalloc1(s,&t->rsb);CHKERRQ(ierr);
  if (bsb) {
    ierr = PetscMemcpy(t->bsb,bsb,s*sizeof(bsb[0]));CHKERRQ(ierr);
  } else
    for (i=0; i<s; i++) t->bsb[i] = Asb[(s-1)*s+i];
  if (csb) {
    ierr = PetscMemcpy(t->csb,csb,s*sizeof(csb[0]));CHKERRQ(ierr);
  } else {
    for (i=0; i<s; i++)
      for (j=0,t->csb[i]=0; j<s; j++)
        t->csb[i] += Asb[i*s+j];
  }
  if (rsb) {
    ierr = PetscMemcpy(t->rsb,rsb,s*sizeof(rsb[0]));CHKERRQ(ierr);
  }
  link->next = MPRKTableauList;
  MPRKTableauList = link;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSMPRKSetSplits(TS ts)
{
  TS_MPRK        *mprk = (TS_MPRK*)ts->data;
  MPRKTableau    tab = mprk->tableau;
  DM             dm,subdm,newdm;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = TSRHSSplitGetSubTS(ts,"slow",&mprk->subts_slow);CHKERRQ(ierr);
  ierr = TSRHSSplitGetSubTS(ts,"fast",&mprk->subts_fast);CHKERRQ(ierr);
  if (!mprk->subts_slow || !mprk->subts_fast) SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_USER,"Must set up the RHSFunctions for 'slow' and 'fast' components using TSRHSSplitSetRHSFunction() or calling TSSetRHSFunction() for each sub-TS");

  /* Only copy the DM */
  ierr = TSGetDM(ts,&dm);CHKERRQ(ierr);

  ierr = TSRHSSplitGetSubTS(ts,"slowbuffer",&mprk->subts_slowbuffer);CHKERRQ(ierr);
  if (!mprk->subts_slowbuffer) {
    mprk->subts_slowbuffer = mprk->subts_slow;
    mprk->subts_slow       = NULL;
  }
  if (mprk->subts_slow) {
    ierr = DMClone(dm,&newdm);CHKERRQ(ierr);
    ierr = TSGetDM(mprk->subts_slow,&subdm);CHKERRQ(ierr);
    ierr = DMCopyDMTS(subdm,newdm);CHKERRQ(ierr);
    ierr = DMCopyDMSNES(subdm,newdm);CHKERRQ(ierr);
    ierr = TSSetDM(mprk->subts_slow,newdm);CHKERRQ(ierr);
    ierr = DMDestroy(&newdm);CHKERRQ(ierr);
  }
  ierr = DMClone(dm,&newdm);CHKERRQ(ierr);
  ierr = TSGetDM(mprk->subts_slowbuffer,&subdm);CHKERRQ(ierr);
  ierr = DMCopyDMTS(subdm,newdm);CHKERRQ(ierr);
  ierr = DMCopyDMSNES(subdm,newdm);CHKERRQ(ierr);
  ierr = TSSetDM(mprk->subts_slowbuffer,newdm);CHKERRQ(ierr);
  ierr = DMDestroy(&newdm);CHKERRQ(ierr);

  ierr = DMClone(dm,&newdm);CHKERRQ(ierr);
  ierr = TSGetDM(mprk->subts_fast,&subdm);CHKERRQ(ierr);
  ierr = DMCopyDMTS(subdm,newdm);CHKERRQ(ierr);
  ierr = DMCopyDMSNES(subdm,newdm);CHKERRQ(ierr);
  ierr = TSSetDM(mprk->subts_fast,newdm);CHKERRQ(ierr);
  ierr = DMDestroy(&newdm);CHKERRQ(ierr);

  if (tab->np == 3) {
    ierr = TSRHSSplitGetSubTS(ts,"medium",&mprk->subts_medium);CHKERRQ(ierr);
    ierr = TSRHSSplitGetSubTS(ts,"mediumbuffer",&mprk->subts_mediumbuffer);CHKERRQ(ierr);
    if (mprk->subts_medium && !mprk->subts_mediumbuffer) {
      mprk->subts_mediumbuffer = mprk->subts_medium;
      mprk->subts_medium       = NULL;
    }
    if (mprk->subts_medium) {
      ierr = DMClone(dm,&newdm);CHKERRQ(ierr);
      ierr = TSGetDM(mprk->subts_medium,&subdm);CHKERRQ(ierr);
      ierr = DMCopyDMTS(subdm,newdm);CHKERRQ(ierr);
      ierr = DMCopyDMSNES(subdm,newdm);CHKERRQ(ierr);
      ierr = TSSetDM(mprk->subts_medium,newdm);CHKERRQ(ierr);
      ierr = DMDestroy(&newdm);CHKERRQ(ierr);
    }
    ierr = DMClone(dm,&newdm);CHKERRQ(ierr);
    ierr = TSGetDM(mprk->subts_mediumbuffer,&subdm);CHKERRQ(ierr);
    ierr = DMCopyDMTS(subdm,newdm);CHKERRQ(ierr);
    ierr = DMCopyDMSNES(subdm,newdm);CHKERRQ(ierr);
    ierr = TSSetDM(mprk->subts_mediumbuffer,newdm);CHKERRQ(ierr);
    ierr = DMDestroy(&newdm);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

/*
 This if for nonsplit RHS MPRK
 The step completion formula is

 x1 = x0 + h b^T YdotRHS

*/
static PetscErrorCode TSEvaluateStep_MPRK(TS ts,PetscInt order,Vec X,PetscBool *done)
{
  TS_MPRK        *mprk = (TS_MPRK*)ts->data;
  MPRKTableau    tab = mprk->tableau;
  PetscScalar    *wf = mprk->work_fast;
  PetscReal      h = ts->time_step;
  PetscInt       s = tab->s,j;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  for (j=0; j<s; j++) wf[j] = h*tab->bf[j];
  ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
  ierr = VecMAXPY(X,s,wf,mprk->YdotRHS);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSStep_MPRK(TS ts)
{
  TS_MPRK         *mprk = (TS_MPRK*)ts->data;
  Vec             *Y = mprk->Y,*YdotRHS = mprk->YdotRHS,*YdotRHS_fast = mprk->YdotRHS_fast,*YdotRHS_slow = mprk->YdotRHS_slow,*YdotRHS_slowbuffer = mprk->YdotRHS_slowbuffer;
  Vec             Yslow,Yslowbuffer,Yfast;
  MPRKTableau     tab = mprk->tableau;
  const PetscInt  s = tab->s;
  const PetscReal *Af = tab->Af,*cf = tab->cf,*Asb = tab->Asb,*csb = tab->csb;
  PetscScalar     *wf = mprk->work_fast,*ws = mprk->work_slow,*wsb = mprk->work_slowbuffer;
  PetscInt        i,j,computedstages;
  PetscReal       next_time_step = ts->time_step,t = ts->ptime,h = ts->time_step;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  for (i=0; i<s; i++) {
    mprk->stage_time = t + h*cf[i];
    ierr = TSPreStage(ts,mprk->stage_time);CHKERRQ(ierr);
    ierr = VecCopy(ts->vec_sol, Y[i]);CHKERRQ(ierr);

    /* slow buffer region */
    for (j=0; j<i; j++) wsb[j] = h*Asb[i*s+j];
    for (j=0; j<i; j++) {
      ierr = VecGetSubVector(YdotRHS[j],mprk->is_slowbuffer,&YdotRHS_slowbuffer[j]);CHKERRQ(ierr);
    }
    ierr = VecGetSubVector(Y[i],mprk->is_slowbuffer,&Yslowbuffer);CHKERRQ(ierr);
    ierr = VecMAXPY(Yslowbuffer,i,wsb,mprk->YdotRHS_slowbuffer);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(Y[i],mprk->is_slowbuffer,&Yslowbuffer);CHKERRQ(ierr);
    for (j=0; j<i; j++) {
      ierr = VecRestoreSubVector(YdotRHS[j],mprk->is_slowbuffer,&YdotRHS_slowbuffer[j]);CHKERRQ(ierr);
    }
    /* slow region */
    if (mprk->is_slow) {
      computedstages = 0;
      for (j=0; j<i; j++) {
        if (tab->rsb[j]) ws[tab->rsb[j]-1] += h*tab->Asb[i*s+j];
        else ws[computedstages++] = h*tab->Asb[i*s+j];
      }
      for (j=0; j<computedstages; j++) {
        ierr = VecGetSubVector(YdotRHS[j],mprk->is_slow,&YdotRHS_slow[j]);CHKERRQ(ierr);
      }
      ierr = VecGetSubVector(Y[i],mprk->is_slow,&Yslow);CHKERRQ(ierr);
      ierr = VecMAXPY(Yslow,computedstages,ws,mprk->YdotRHS_slow);CHKERRQ(ierr);
      ierr = VecRestoreSubVector(Y[i],mprk->is_slow,&Yslow);CHKERRQ(ierr);
      for (j=0; j<computedstages; j++) {
        ierr = VecRestoreSubVector(YdotRHS[j],mprk->is_slow,&YdotRHS_slow[j]);CHKERRQ(ierr);
      }
    }

    /* fast region */
    for (j=0; j<i; j++) wf[j] = h*Af[i*s+j];
    for (j=0; j<i; j++) {
      ierr = VecGetSubVector(YdotRHS[j],mprk->is_fast,&YdotRHS_fast[j]);CHKERRQ(ierr);
    }
    ierr = VecGetSubVector(Y[i],mprk->is_fast,&Yfast);CHKERRQ(ierr);
    ierr = VecMAXPY(Yfast,i,wf,mprk->YdotRHS_fast);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(Y[i],mprk->is_fast,&Yfast);CHKERRQ(ierr);
    for (j=0; j<i; j++) {
      ierr = VecRestoreSubVector(YdotRHS[j],mprk->is_fast,&YdotRHS_fast[j]);CHKERRQ(ierr);
    }
    if (tab->np == 3) {
      Vec         *YdotRHS_medium = mprk->YdotRHS_medium,*YdotRHS_mediumbuffer = mprk->YdotRHS_mediumbuffer;
      Vec         Ymedium,Ymediumbuffer;
      PetscScalar *wm = mprk->work_medium,*wmb = mprk->work_mediumbuffer;
      /* medium region */
      if (mprk->is_medium) {
        computedstages = 0;
        for (j=0; j<i; j++) {
          if (tab->rmb[j]) wm[computedstages-tab->sbase+(tab->rmb[j]-1)%tab->sbase] += h*tab->Amb[i*s+j];
          else wm[computedstages++] = h*tab->Amb[i*s+j];
        }
        for (j=0; j<computedstages; j++) {
          ierr = VecGetSubVector(YdotRHS[j],mprk->is_medium,&YdotRHS_medium[j]);CHKERRQ(ierr);
        }
        ierr = VecGetSubVector(Y[i],mprk->is_medium,&Ymedium);CHKERRQ(ierr);
        ierr = VecMAXPY(Ymedium,computedstages,wm,mprk->YdotRHS_medium);CHKERRQ(ierr);
        ierr = VecRestoreSubVector(Y[i],mprk->is_medium,&Ymedium);CHKERRQ(ierr);
        for (j=0; j<computedstages; j++) {
          ierr = VecRestoreSubVector(YdotRHS[j],mprk->is_medium,&YdotRHS_medium[j]);CHKERRQ(ierr);
        }
      }
      /* medium buffer region */
      for (j=0; j<i; j++) wmb[j] = h*tab->Amb[i*s+j];
      for (j=0; j<i; j++) {
        ierr = VecGetSubVector(YdotRHS[j],mprk->is_mediumbuffer,&YdotRHS_mediumbuffer[j]);CHKERRQ(ierr);
      }
      ierr = VecGetSubVector(Y[i],mprk->is_mediumbuffer,&Ymediumbuffer);CHKERRQ(ierr);
      ierr = VecMAXPY(Ymediumbuffer,i,wmb,mprk->YdotRHS_mediumbuffer);CHKERRQ(ierr);
      ierr = VecRestoreSubVector(Y[i],mprk->is_mediumbuffer,&Ymediumbuffer);CHKERRQ(ierr);
      for (j=0; j<i; j++) {
        ierr = VecRestoreSubVector(YdotRHS[j],mprk->is_mediumbuffer,&YdotRHS_mediumbuffer[j]);CHKERRQ(ierr);
      }
    }
    ierr = TSPostStage(ts,mprk->stage_time,i,Y); CHKERRQ(ierr);
    /* compute the stage RHS by fast and slow tableau respectively */
    ierr = TSComputeRHSFunction(ts,t+h*csb[i],Y[i],YdotRHS[i]);CHKERRQ(ierr);
  }
  ierr = TSEvaluateStep(ts,tab->order,ts->vec_sol,NULL);CHKERRQ(ierr);
  ts->ptime += ts->time_step;
  ts->time_step = next_time_step;
  PetscFunctionReturn(0);
}

/*
 This if for partitioned RHS MPRK
 The step completion formula is

 x1 = x0 + h b^T YdotRHS

*/
static PetscErrorCode TSEvaluateStep_MPRKSPLIT(TS ts,PetscInt order,Vec X,PetscBool *done)
{
  TS_MPRK        *mprk = (TS_MPRK*)ts->data;
  MPRKTableau    tab  = mprk->tableau;
  Vec            Xslow,Xfast,Xslowbuffer; /* subvectors for slow and fast componets in X respectively */
  PetscScalar    *wf = mprk->work_fast,*ws = mprk->work_slow,*wsb = mprk->work_slowbuffer;
  PetscReal      h = ts->time_step;
  PetscInt       s = tab->s,j,computedstages;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);

  /* slow region */
  if (mprk->is_slow) {
    computedstages = 0;
    for (j=0; j<s; j++) {
      if (tab->rsb[j]) ws[tab->rsb[j]-1] += h*tab->bsb[j];
      else ws[computedstages++] = h*tab->bsb[j];
    }
    ierr = VecGetSubVector(X,mprk->is_slow,&Xslow);CHKERRQ(ierr);
    ierr = VecMAXPY(Xslow,computedstages,ws,mprk->YdotRHS_slow);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(X,mprk->is_slow,&Xslow);CHKERRQ(ierr);
  }

  if (tab->np == 3 && mprk->is_medium) {
    computedstages = 0;
    for (j=0; j<s; j++) {
      if (tab->rmb[j]) wsb[computedstages-tab->sbase+(tab->rmb[j]-1)%tab->sbase] += h*tab->bsb[j];
      else wsb[computedstages++] = h*tab->bsb[j];
    }
    ierr = VecGetSubVector(X,mprk->is_slowbuffer,&Xslowbuffer);CHKERRQ(ierr);
    ierr = VecMAXPY(Xslowbuffer,computedstages,wsb,mprk->YdotRHS_slowbuffer);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(X,mprk->is_slowbuffer,&Xslowbuffer);CHKERRQ(ierr);
  } else {
    /* slow buffer region */
    for (j=0; j<s; j++) wsb[j] = h*tab->bsb[j];
    ierr = VecGetSubVector(X,mprk->is_slowbuffer,&Xslowbuffer);CHKERRQ(ierr);
    ierr = VecMAXPY(Xslowbuffer,s,wsb,mprk->YdotRHS_slowbuffer);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(X,mprk->is_slowbuffer,&Xslowbuffer);CHKERRQ(ierr);
  }
  if (tab->np == 3) {
    Vec         Xmedium,Xmediumbuffer;
    PetscScalar *wm = mprk->work_medium,*wmb = mprk->work_mediumbuffer;
    /* medium region and slow buffer region */
    if (mprk->is_medium) {
      computedstages = 0;
      for (j=0; j<s; j++) {
        if (tab->rmb[j]) wm[computedstages-tab->sbase+(tab->rmb[j]-1)%tab->sbase] += h*tab->bmb[j];
        else wm[computedstages++] = h*tab->bmb[j];
      }
      ierr = VecGetSubVector(X,mprk->is_medium,&Xmedium);CHKERRQ(ierr);
      ierr = VecMAXPY(Xmedium,computedstages,wm,mprk->YdotRHS_medium);CHKERRQ(ierr);
      ierr = VecRestoreSubVector(X,mprk->is_medium,&Xmedium);CHKERRQ(ierr);
    }
    /* medium buffer region */
    for (j=0; j<s; j++) wmb[j] = h*tab->bmb[j];
    ierr = VecGetSubVector(X,mprk->is_mediumbuffer,&Xmediumbuffer);CHKERRQ(ierr);
    ierr = VecMAXPY(Xmediumbuffer,s,wmb,mprk->YdotRHS_mediumbuffer);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(X,mprk->is_mediumbuffer,&Xmediumbuffer);CHKERRQ(ierr);
  }
  /* fast region */
  for (j=0; j<s; j++) wf[j] = h*tab->bf[j];
  ierr = VecGetSubVector(X,mprk->is_fast,&Xfast);CHKERRQ(ierr);
  ierr = VecMAXPY(Xfast,s,wf,mprk->YdotRHS_fast);CHKERRQ(ierr);
  ierr = VecRestoreSubVector(X,mprk->is_fast,&Xfast);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSStep_MPRKSPLIT(TS ts)
{
  TS_MPRK         *mprk = (TS_MPRK*)ts->data;
  MPRKTableau     tab = mprk->tableau;
  Vec             *Y = mprk->Y,*YdotRHS_fast = mprk->YdotRHS_fast,*YdotRHS_slow = mprk->YdotRHS_slow,*YdotRHS_slowbuffer = mprk->YdotRHS_slowbuffer;
  Vec             Yslow,Yslowbuffer,Yfast; /* subvectors for slow and fast components in Y[i] respectively */
  PetscInt        s = tab->s;
  const PetscReal *Af = tab->Af,*cf = tab->cf,*Asb = tab->Asb,*csb = tab->csb;
  PetscScalar     *wf = mprk->work_fast,*ws = mprk->work_slow,*wsb = mprk->work_slowbuffer;
  PetscInt        i,j,computedstages;
  PetscReal       next_time_step = ts->time_step,t = ts->ptime,h = ts->time_step;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  for (i=0; i<s; i++) {
    mprk->stage_time = t + h*cf[i];
    ierr = TSPreStage(ts,mprk->stage_time);CHKERRQ(ierr);
    /* calculate the stage value for fast and slow components respectively */
    ierr = VecCopy(ts->vec_sol,Y[i]);CHKERRQ(ierr);
    for (j=0; j<i; j++) wsb[j] = h*Asb[i*s+j];

    /* slow buffer region */
    if (tab->np == 3 && mprk->is_medium) {
      if (tab->rmb[i]) {
        ierr = VecGetSubVector(Y[i],mprk->is_slowbuffer,&Yslowbuffer);CHKERRQ(ierr);
        ierr = VecISCopy(Y[tab->rmb[i]-1],mprk->is_slowbuffer,SCATTER_REVERSE,Yslowbuffer);CHKERRQ(ierr);
        ierr = VecRestoreSubVector(Y[i],mprk->is_slowbuffer,&Yslowbuffer);CHKERRQ(ierr);
      } else {
        PetscScalar *wm = mprk->work_medium;
        computedstages = 0;
        for (j=0; j<i; j++) {
          if (tab->rmb[j]) wm[computedstages-tab->sbase+(tab->rmb[j]-1)%tab->sbase] += wsb[j];
          else wm[computedstages++] = wsb[j];
        }
        ierr = VecGetSubVector(Y[i],mprk->is_slowbuffer,&Yslowbuffer);CHKERRQ(ierr);
        ierr = VecMAXPY(Yslowbuffer,computedstages,wm,YdotRHS_slowbuffer);CHKERRQ(ierr);
        ierr = VecRestoreSubVector(Y[i],mprk->is_slowbuffer,&Yslowbuffer);CHKERRQ(ierr);
      }
    } else {
      ierr = VecGetSubVector(Y[i],mprk->is_slowbuffer,&Yslowbuffer);CHKERRQ(ierr);
      ierr = VecMAXPY(Yslowbuffer,i,wsb,YdotRHS_slowbuffer);CHKERRQ(ierr);
      ierr = VecRestoreSubVector(Y[i],mprk->is_slowbuffer,&Yslowbuffer);CHKERRQ(ierr);
    }

    /* slow region */
    if (mprk->is_slow) {
      if (tab->rsb[i]) { /* repeat previous stage */
        ierr = VecGetSubVector(Y[i],mprk->is_slow,&Yslow);CHKERRQ(ierr);
        ierr = VecISCopy(Y[tab->rsb[i]-1],mprk->is_slow,SCATTER_REVERSE,Yslow);CHKERRQ(ierr);
        ierr = VecRestoreSubVector(Y[i],mprk->is_slow,&Yslow);CHKERRQ(ierr);
      } else {
        computedstages = 0;
        for (j=0; j<i; j++) {
          if (tab->rsb[j]) ws[tab->rsb[j]-1] += wsb[j];
          else ws[computedstages++] = wsb[j];
        }
        ierr = VecGetSubVector(Y[i],mprk->is_slow,&Yslow);CHKERRQ(ierr);
        ierr = VecMAXPY(Yslow,computedstages,ws,YdotRHS_slow);CHKERRQ(ierr);
        ierr = VecRestoreSubVector(Y[i],mprk->is_slow,&Yslow);CHKERRQ(ierr);
        /* only depends on the slow buffer region */
        ierr = TSComputeRHSFunction(mprk->subts_slow,t+h*csb[i],Y[i],YdotRHS_slow[computedstages]);CHKERRQ(ierr);
      }
    }

    /* fast region */
    for (j=0; j<i; j++) wf[j] = h*Af[i*s+j];
    ierr = VecGetSubVector(Y[i],mprk->is_fast,&Yfast);CHKERRQ(ierr);
    ierr = VecMAXPY(Yfast,i,wf,YdotRHS_fast);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(Y[i],mprk->is_fast,&Yfast);CHKERRQ(ierr);

    if (tab->np == 3) {
      Vec *YdotRHS_medium = mprk->YdotRHS_medium,*YdotRHS_mediumbuffer = mprk->YdotRHS_mediumbuffer;
      Vec Ymedium,Ymediumbuffer;
      const PetscReal *Amb = tab->Amb,*cmb = tab->cmb;
      PetscScalar *wm = mprk->work_medium,*wmb = mprk->work_mediumbuffer;

      for (j=0; j<i; j++) wmb[j] = h*Amb[i*s+j];
      /* medium buffer region */
      ierr = VecGetSubVector(Y[i],mprk->is_mediumbuffer,&Ymediumbuffer);CHKERRQ(ierr);
      ierr = VecMAXPY(Ymediumbuffer,i,wmb,YdotRHS_mediumbuffer);CHKERRQ(ierr);
      ierr = VecRestoreSubVector(Y[i],mprk->is_mediumbuffer,&Ymediumbuffer);CHKERRQ(ierr);
      /* medium region */
      if (mprk->is_medium) {
        if (tab->rmb[i]) { /* repeat previous stage */
          ierr = VecGetSubVector(Y[i],mprk->is_medium,&Ymedium);CHKERRQ(ierr);
          ierr = VecISCopy(Y[tab->rmb[i]-1],mprk->is_medium,SCATTER_REVERSE,Ymedium);CHKERRQ(ierr);
          ierr = VecRestoreSubVector(Y[i],mprk->is_medium,&Ymedium);CHKERRQ(ierr);
        } else {
          computedstages = 0;
          for (j=0; j<i; j++) {
            if (tab->rmb[j]) wm[computedstages-tab->sbase+(tab->rmb[j]-1)%tab->sbase] += wmb[j];
            else wm[computedstages++] = wmb[j];

          }
          ierr = VecGetSubVector(Y[i],mprk->is_medium,&Ymedium);CHKERRQ(ierr);
          ierr = VecMAXPY(Ymedium,computedstages,wm,YdotRHS_medium);CHKERRQ(ierr);
          ierr = VecRestoreSubVector(Y[i],mprk->is_medium,&Ymedium);CHKERRQ(ierr);
          /* only depends on the medium buffer region */
          ierr = TSComputeRHSFunction(mprk->subts_medium,t+h*cmb[i],Y[i],YdotRHS_medium[computedstages]);CHKERRQ(ierr);
          /* must be computed after all regions are updated in Y */
          ierr = TSComputeRHSFunction(mprk->subts_slowbuffer,t+h*csb[i],Y[i],YdotRHS_slowbuffer[computedstages]);CHKERRQ(ierr);
        }
      }
      /* must be computed after fast region and slow region are updated in Y */
      ierr = TSComputeRHSFunction(mprk->subts_mediumbuffer,t+h*cmb[i],Y[i],YdotRHS_mediumbuffer[i]);CHKERRQ(ierr);
    }
    if (!(tab->np == 3 && mprk->is_medium)) {
      ierr = TSComputeRHSFunction(mprk->subts_slowbuffer,t+h*csb[i],Y[i],YdotRHS_slowbuffer[i]);CHKERRQ(ierr);
    }
    ierr = TSComputeRHSFunction(mprk->subts_fast,t+h*cf[i],Y[i],YdotRHS_fast[i]);
  }

  ierr = TSEvaluateStep(ts,tab->order,ts->vec_sol,NULL);CHKERRQ(ierr);
  ts->ptime += ts->time_step;
  ts->time_step = next_time_step;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSMPRKTableauReset(TS ts)
{
  TS_MPRK        *mprk = (TS_MPRK*)ts->data;
  MPRKTableau    tab = mprk->tableau;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!tab) PetscFunctionReturn(0);
  ierr = PetscFree(mprk->work_fast);CHKERRQ(ierr);
  ierr = PetscFree(mprk->work_slow);CHKERRQ(ierr);
  ierr = PetscFree(mprk->work_slowbuffer);CHKERRQ(ierr);
  ierr = PetscFree(mprk->work_medium);CHKERRQ(ierr);
  ierr = PetscFree(mprk->work_mediumbuffer);CHKERRQ(ierr);
  ierr = VecDestroyVecs(tab->s,&mprk->Y);CHKERRQ(ierr);
  if (mprk->mprkmtype == TSMPRKNONSPLIT) {
    ierr = VecDestroyVecs(tab->s,&mprk->YdotRHS);CHKERRQ(ierr);
    if (mprk->is_slow) {
      ierr = PetscFree(mprk->YdotRHS_slow);CHKERRQ(ierr);
    }
    ierr = PetscFree(mprk->YdotRHS_slowbuffer);CHKERRQ(ierr);
    if (tab->np == 3) {
      if (mprk->is_medium) {
        ierr = PetscFree(mprk->YdotRHS_medium);CHKERRQ(ierr);
      }
      ierr = PetscFree(mprk->YdotRHS_mediumbuffer);CHKERRQ(ierr);
    }
    ierr = PetscFree(mprk->YdotRHS_fast);CHKERRQ(ierr);

  } else {
    ierr = VecDestroyVecs(tab->s,&mprk->YdotRHS_fast);CHKERRQ(ierr);
    ierr = VecDestroyVecs(tab->s,&mprk->YdotRHS_slow);CHKERRQ(ierr);
    ierr = VecDestroyVecs(tab->s,&mprk->YdotRHS_slowbuffer);CHKERRQ(ierr);
    ierr = VecDestroyVecs(tab->s,&mprk->YdotRHS_medium);CHKERRQ(ierr);
    ierr = VecDestroyVecs(tab->s,&mprk->YdotRHS_mediumbuffer);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TSReset_MPRK(TS ts)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = TSMPRKTableauReset(ts);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

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

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

static PetscErrorCode DMSubDomainRestrictHook_TSMPRK(DM dm,VecScatter gscat,VecScatter lscat,DM subdm,void *ctx)
{
  PetscFunctionBegin;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSMPRKTableauSetUp(TS ts)
{
  TS_MPRK        *mprk  = (TS_MPRK*)ts->data;
  MPRKTableau    tab = mprk->tableau;
  Vec            YdotRHS_slow,YdotRHS_slowbuffer,YdotRHS_medium,YdotRHS_mediumbuffer,YdotRHS_fast;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = VecDuplicateVecs(ts->vec_sol,tab->s,&mprk->Y);CHKERRQ(ierr);
  if (mprk->is_slow) {
    ierr = PetscMalloc1(tab->s,&mprk->work_slow);CHKERRQ(ierr);
  }
  ierr = PetscMalloc1(tab->s,&mprk->work_slowbuffer);CHKERRQ(ierr);
  if (tab->np == 3) {
    if (mprk->is_medium) {
      ierr = PetscMalloc1(tab->s,&mprk->work_medium);CHKERRQ(ierr);
    }
    ierr = PetscMalloc1(tab->s,&mprk->work_mediumbuffer);CHKERRQ(ierr);
  }
  ierr = PetscMalloc1(tab->s,&mprk->work_fast);CHKERRQ(ierr);

  if (mprk->mprkmtype == TSMPRKNONSPLIT) {
    ierr = VecDuplicateVecs(ts->vec_sol,tab->s,&mprk->YdotRHS);CHKERRQ(ierr);
    if (mprk->is_slow) {
      ierr = PetscMalloc1(tab->s,&mprk->YdotRHS_slow);CHKERRQ(ierr);
    }
    ierr = PetscMalloc1(tab->s,&mprk->YdotRHS_slowbuffer);CHKERRQ(ierr);
    if (tab->np == 3) {
      if (mprk->is_medium) {
        ierr = PetscMalloc1(tab->s,&mprk->YdotRHS_medium);CHKERRQ(ierr);
      }
      ierr = PetscMalloc1(tab->s,&mprk->YdotRHS_mediumbuffer);CHKERRQ(ierr);
    }
    ierr = PetscMalloc1(tab->s,&mprk->YdotRHS_fast);CHKERRQ(ierr);
  } else {
    if (mprk->is_slow) {
      ierr = VecGetSubVector(ts->vec_sol,mprk->is_slow,&YdotRHS_slow);CHKERRQ(ierr);
      ierr = VecDuplicateVecs(YdotRHS_slow,tab->s,&mprk->YdotRHS_slow);CHKERRQ(ierr);
      ierr = VecRestoreSubVector(ts->vec_sol,mprk->is_slow,&YdotRHS_slow);CHKERRQ(ierr);
    }
    ierr = VecGetSubVector(ts->vec_sol,mprk->is_slowbuffer,&YdotRHS_slowbuffer);CHKERRQ(ierr);
    ierr = VecDuplicateVecs(YdotRHS_slowbuffer,tab->s,&mprk->YdotRHS_slowbuffer);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(ts->vec_sol,mprk->is_slowbuffer,&YdotRHS_slowbuffer);CHKERRQ(ierr);
    if (tab->np == 3) {
      if (mprk->is_medium) {
        ierr = VecGetSubVector(ts->vec_sol,mprk->is_medium,&YdotRHS_medium);CHKERRQ(ierr);
        ierr = VecDuplicateVecs(YdotRHS_medium,tab->s,&mprk->YdotRHS_medium);CHKERRQ(ierr);
        ierr = VecRestoreSubVector(ts->vec_sol,mprk->is_medium,&YdotRHS_medium);CHKERRQ(ierr);
      }
      ierr = VecGetSubVector(ts->vec_sol,mprk->is_mediumbuffer,&YdotRHS_mediumbuffer);CHKERRQ(ierr);
      ierr = VecDuplicateVecs(YdotRHS_mediumbuffer,tab->s,&mprk->YdotRHS_mediumbuffer);CHKERRQ(ierr);
      ierr = VecRestoreSubVector(ts->vec_sol,mprk->is_mediumbuffer,&YdotRHS_mediumbuffer);CHKERRQ(ierr);
    }
    ierr = VecGetSubVector(ts->vec_sol,mprk->is_fast,&YdotRHS_fast);CHKERRQ(ierr);
    ierr = VecDuplicateVecs(YdotRHS_fast,tab->s,&mprk->YdotRHS_fast);CHKERRQ(ierr);
    ierr = VecRestoreSubVector(ts->vec_sol,mprk->is_fast,&YdotRHS_fast);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TSSetUp_MPRK(TS ts)
{
  TS_MPRK        *mprk = (TS_MPRK*)ts->data;
  MPRKTableau    tab = mprk->tableau;
  DM             dm;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = TSRHSSplitGetIS(ts,"slow",&mprk->is_slow);CHKERRQ(ierr);
  ierr = TSRHSSplitGetIS(ts,"fast",&mprk->is_fast);CHKERRQ(ierr);
  if (!mprk->is_slow || !mprk->is_fast) SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_USER,"Must set up RHSSplits with TSRHSSplitSetIS() using split names 'slow' and 'fast' respectively in order to use the method '%s'",tab->name);

  if (tab->np == 3) {
    ierr = TSRHSSplitGetIS(ts,"medium",&mprk->is_medium);CHKERRQ(ierr);
    if (!mprk->is_medium) SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_USER,"Must set up RHSSplits with TSRHSSplitSetIS() using split names 'slow' and 'medium' and 'fast' respectively in order to use the method '%s'",tab->name);
    ierr = TSRHSSplitGetIS(ts,"mediumbuffer",&mprk->is_mediumbuffer);CHKERRQ(ierr);
    if (!mprk->is_mediumbuffer) { /* let medium buffer cover whole medium region */
      mprk->is_mediumbuffer = mprk->is_medium;
      mprk->is_medium = NULL;
    }
  }

  /* If users do not provide buffer region settings, the solver will do them automatically, but with a performance penalty */
  ierr = TSRHSSplitGetIS(ts,"slowbuffer",&mprk->is_slowbuffer);CHKERRQ(ierr);
  if (!mprk->is_slowbuffer) { /* let slow buffer cover whole slow region */
    mprk->is_slowbuffer = mprk->is_slow;
    mprk->is_slow = NULL;
  }
/*
  if (!mprk->is_medium) {
    mprk->is_medium = mprk->is_fast;
    mprk->is_fast = NULL;
  } else {
    ierr = TSRHSSplitGetIS(ts,"mediumbuffer",&mprk->is_mediumbuffer);CHKERRQ(ierr);
  }
*/
  ierr = TSCheckImplicitTerm(ts);CHKERRQ(ierr);
  ierr = TSMPRKTableauSetUp(ts);CHKERRQ(ierr);
  ierr = TSGetDM(ts,&dm);CHKERRQ(ierr);
  ierr = DMCoarsenHookAdd(dm,DMCoarsenHook_TSMPRK,DMRestrictHook_TSMPRK,ts);CHKERRQ(ierr);
  ierr = DMSubDomainHookAdd(dm,DMSubDomainHook_TSMPRK,DMSubDomainRestrictHook_TSMPRK,ts);CHKERRQ(ierr);
  ierr = PetscTryMethod(ts,"TSMPRKSetSplits_C",(TS),(ts));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* construct a database to chose nonsplit RHS mutirate mprk method or split RHS MPRK method */
const char *const TSMPRKMultirateTypes[] = {"NONSPLIT","SPLIT","TSMPRKMultirateType","TSMPRK",0};

static PetscErrorCode TSSetFromOptions_MPRK(PetscOptionItems *PetscOptionsObject,TS ts)
{
  TS_MPRK        *mprk = (TS_MPRK*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscOptionsHead(PetscOptionsObject,"PRK ODE solver options");CHKERRQ(ierr);
  {
    MPRKTableauLink link;
    PetscInt        count,choice;
    PetscBool       flg;
    const char      **namelist;
    PetscInt        mprkmtype = 0;
    for (link=MPRKTableauList,count=0; link; link=link->next,count++) ;
    ierr = PetscMalloc1(count,(char***)&namelist);CHKERRQ(ierr);
    for (link=MPRKTableauList,count=0; link; link=link->next,count++) namelist[count] = link->tab.name;
    ierr = PetscOptionsEList("-ts_mprk_type","Family of MPRK method","TSMPRKSetType",(const char*const*)namelist,count,mprk->tableau->name,&choice,&flg);CHKERRQ(ierr);
    if (flg) {ierr = TSMPRKSetType(ts,namelist[choice]);CHKERRQ(ierr);}
    ierr = PetscFree(namelist);CHKERRQ(ierr);
    ierr = PetscOptionsEList("-ts_mprk_multirate_type","Use Combined RHS Multirate or Partioned RHS Multirat MPRK method","TSMPRKSetMultirateType",TSMPRKMultirateTypes,2,TSMPRKMultirateTypes[0],&mprkmtype,&flg);CHKERRQ(ierr);
     if (flg) {ierr = TSMPRKSetMultirateType(ts,mprkmtype);CHKERRQ(ierr);}
  }
  ierr = PetscOptionsTail();CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSView_MPRK(TS ts,PetscViewer viewer)
{
  TS_MPRK        *mprk = (TS_MPRK*)ts->data;
  PetscBool      iascii;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);CHKERRQ(ierr);
  if (iascii) {
    MPRKTableau tab  = mprk->tableau;
    TSMPRKType  mprktype;
    char        fbuf[512];
    char        sbuf[512];
    PetscInt    i;
    ierr = TSMPRKGetType(ts,&mprktype);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  MPRK type %s\n",mprktype);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  Order: %D\n",tab->order);CHKERRQ(ierr);

    ierr = PetscFormatRealArray(fbuf,sizeof(fbuf),"% 8.6f",tab->s,tab->cf);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  Abscissa cf = %s\n",fbuf);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  Af = \n");CHKERRQ(ierr);
    for (i=0; i<tab->s; i++) {
      ierr = PetscFormatRealArray(fbuf,sizeof(fbuf),"% 8.6f",tab->s,&tab->Af[i*tab->s]);CHKERRQ(ierr);
      ierr = PetscViewerASCIIPrintf(viewer,"    %s\n",fbuf);CHKERRQ(ierr);
    }
    ierr = PetscFormatRealArray(fbuf,sizeof(fbuf),"% 8.6f",tab->s,tab->bf);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  bf = %s\n",fbuf);CHKERRQ(ierr);

    ierr = PetscFormatRealArray(sbuf,sizeof(sbuf),"% 8.6f",tab->s,tab->csb);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  Abscissa csb = %s\n",sbuf);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  Asb = \n");CHKERRQ(ierr);
    for (i=0; i<tab->s; i++) {
      ierr = PetscFormatRealArray(sbuf,sizeof(sbuf),"% 8.6f",tab->s,&tab->Asb[i*tab->s]);CHKERRQ(ierr);
      ierr = PetscViewerASCIIPrintf(viewer,"    %s\n",sbuf);CHKERRQ(ierr);
    }
    ierr = PetscFormatRealArray(sbuf,sizeof(sbuf),"% 8.6f",tab->s,tab->bsb);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  bsb = %s\n",sbuf);CHKERRQ(ierr);

    if (tab->np == 3) {
      char mbuf[512];
      ierr = PetscFormatRealArray(mbuf,sizeof(mbuf),"% 8.6f",tab->s,tab->cmb);CHKERRQ(ierr);
      ierr = PetscViewerASCIIPrintf(viewer,"  Abscissa cmb = %s\n",mbuf);CHKERRQ(ierr);
      ierr = PetscViewerASCIIPrintf(viewer,"  Amb = \n");CHKERRQ(ierr);
      for (i=0; i<tab->s; i++) {
        ierr = PetscFormatRealArray(mbuf,sizeof(mbuf),"% 8.6f",tab->s,&tab->Amb[i*tab->s]);CHKERRQ(ierr);
        ierr = PetscViewerASCIIPrintf(viewer,"    %s\n",mbuf);CHKERRQ(ierr);
      }
      ierr = PetscFormatRealArray(mbuf,sizeof(mbuf),"% 8.6f",tab->s,tab->bmb);CHKERRQ(ierr);
      ierr = PetscViewerASCIIPrintf(viewer,"  bmb = %s\n",mbuf);CHKERRQ(ierr);
    }
  }
  PetscFunctionReturn(0);
}

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

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

/*@C
  TSMPRKSetType - Set the type of MPRK scheme

  Logically collective

  Input Parameter:
+  ts - timestepping context
-  mprktype - type of MPRK-scheme

  Options Database:
.   -ts_mprk_type - <pm2,p2,p3>

  Level: intermediate

.seealso: TSMPRKGetType(), TSMPRK, TSMPRKType
@*/
PetscErrorCode TSMPRKSetType(TS ts,TSMPRKType mprktype)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  PetscValidCharPointer(mprktype,2);
  ierr = PetscTryMethod(ts,"TSMPRKSetType_C",(TS,TSMPRKType),(ts,mprktype));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@C
  TSMPRKGetType - Get the type of MPRK scheme

  Logically collective

  Input Parameter:
.  ts - timestepping context

  Output Parameter:
.  mprktype - type of MPRK-scheme

  Level: intermediate

.seealso: TSMPRKGetType()
@*/
PetscErrorCode TSMPRKGetType(TS ts,TSMPRKType *mprktype)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  ierr = PetscUseMethod(ts,"TSMPRKGetType_C",(TS,TSMPRKType*),(ts,mprktype));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@C
  TSMPRKSetMultirateType - Set the type of MPRK multirate scheme

  Logically collective

  Input Parameter:
+  ts - timestepping context
-  mprkmtype - type of the multirate configuration

  Options Database:
.   -ts_mprk_multirate_type - <nonsplit,split>

  Level: intermediate
@*/
PetscErrorCode TSMPRKSetMultirateType(TS ts, TSMPRKMultirateType mprkmtype)
{
  TS_MPRK        *mprk = (TS_MPRK*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  switch(mprkmtype){
    case TSMPRKNONSPLIT:
      ts->ops->step         = TSStep_MPRK;
      ts->ops->evaluatestep = TSEvaluateStep_MPRK;
      break;
    case TSMPRKSPLIT:
      ts->ops->step         = TSStep_MPRKSPLIT;
      ts->ops->evaluatestep = TSEvaluateStep_MPRKSPLIT;
      ierr = PetscObjectComposeFunction((PetscObject)ts,"TSMPRKSetSplits_C",TSMPRKSetSplits);CHKERRQ(ierr);
      break;
    default :
      SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_ARG_UNKNOWN_TYPE,"Unknown type '%s'",mprkmtype);
  }
  mprk->mprkmtype = mprkmtype;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSMPRKGetType_MPRK(TS ts,TSMPRKType *mprktype)
{
  TS_MPRK *mprk = (TS_MPRK*)ts->data;

  PetscFunctionBegin;
  *mprktype = mprk->tableau->name;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSMPRKSetType_MPRK(TS ts,TSMPRKType mprktype)
{
  TS_MPRK         *mprk = (TS_MPRK*)ts->data;
  PetscBool       match;
  MPRKTableauLink link;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  if (mprk->tableau) {
    ierr = PetscStrcmp(mprk->tableau->name,mprktype,&match);CHKERRQ(ierr);
    if (match) PetscFunctionReturn(0);
  }
  for (link = MPRKTableauList; link; link=link->next) {
    ierr = PetscStrcmp(link->tab.name,mprktype,&match);CHKERRQ(ierr);
    if (match) {
      if (ts->setupcalled) {ierr = TSMPRKTableauReset(ts);CHKERRQ(ierr);}
      mprk->tableau = &link->tab;
      if (ts->setupcalled) {ierr = TSMPRKTableauSetUp(ts);CHKERRQ(ierr);}
      PetscFunctionReturn(0);
    }
  }
  SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_ARG_UNKNOWN_TYPE,"Could not find '%s'",mprktype);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSGetStages_MPRK(TS ts,PetscInt *ns,Vec **Y)
{
  TS_MPRK *mprk = (TS_MPRK*)ts->data;

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

static PetscErrorCode TSDestroy_MPRK(TS ts)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = TSReset_MPRK(ts);CHKERRQ(ierr);
  if (ts->dm) {
    ierr = DMCoarsenHookRemove(ts->dm,DMCoarsenHook_TSMPRK,DMRestrictHook_TSMPRK,ts);CHKERRQ(ierr);
    ierr = DMSubDomainHookRemove(ts->dm,DMSubDomainHook_TSMPRK,DMSubDomainRestrictHook_TSMPRK,ts);CHKERRQ(ierr);
  }
  ierr = PetscFree(ts->data);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSMPRKGetType_C",NULL);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSMPRKSetType_C",NULL);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSMPRKSetMultirateType_C",NULL);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*MC
      TSMPRK - ODE solver using Partitioned Runge-Kutta schemes

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

  Notes:
  The default is TSMPRKPM2, it can be changed with TSRKSetType() or -ts_mprk_type

  Level: beginner

.seealso:  TSCreate(), TS, TSSetType(), TSMPRKSetType(), TSMPRKGetType(), TSMPRKType, TSMPRKRegister(), TSMPRKSetMultirateType()
           TSMPRKM2, TSMPRKM3, TSMPRKRFSMR3, TSMPRKRFSMR2

M*/
PETSC_EXTERN PetscErrorCode TSCreate_MPRK(TS ts)
{
  TS_MPRK        *mprk;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = TSMPRKInitializePackage();CHKERRQ(ierr);

  ts->ops->reset          = TSReset_MPRK;
  ts->ops->destroy        = TSDestroy_MPRK;
  ts->ops->view           = TSView_MPRK;
  ts->ops->load           = TSLoad_MPRK;
  ts->ops->setup          = TSSetUp_MPRK;
  ts->ops->step           = TSStep_MPRK;
  ts->ops->evaluatestep   = TSEvaluateStep_MPRK;
  ts->ops->setfromoptions = TSSetFromOptions_MPRK;
  ts->ops->getstages      = TSGetStages_MPRK;

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

  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSMPRKGetType_C",TSMPRKGetType_MPRK);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSMPRKSetType_C",TSMPRKSetType_MPRK);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSMPRKSetMultirateType_C",TSMPRKSetMultirateType);CHKERRQ(ierr);

  ierr = TSMPRKSetType(ts,TSMPRKDefault);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}