xref: /petsc/src/ts/impls/arkimex/arkimex.c (revision f1b97656f5355c1550c55f6f45aadd5df55e69a0)

/*
  Code for timestepping with additive Runge-Kutta IMEX method

  Notes:
  The general system is written as

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

  where F represents the stiff part of the physics and G represents the non-stiff part.

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

static const TSARKIMEXType TSARKIMEXDefault = TSARKIMEX2E;
static PetscBool TSARKIMEXRegisterAllCalled;
static PetscBool TSARKIMEXPackageInitialized;

typedef struct _ARKTableau *ARKTableau;
struct _ARKTableau {
  char *name;
  PetscInt order;               /* Classical approximation order of the method */
  PetscInt s;                   /* Number of stages */
  PetscInt pinterp;             /* Interpolation order */
  PetscReal *At,*bt,*ct;        /* Stiff tableau */
  PetscReal *A,*b,*c;           /* Non-stiff tableau */
  PetscReal *binterpt,*binterp; /* Dense output formula */
};
typedef struct _ARKTableauLink *ARKTableauLink;
struct _ARKTableauLink {
  struct _ARKTableau tab;
  ARKTableauLink next;
};
static ARKTableauLink ARKTableauList;

typedef struct {
  ARKTableau  tableau;
  Vec         *Y;               /* States computed during the step */
  Vec         *YdotI;           /* Time derivatives for the stiff part */
  Vec         *YdotRHS;         /* Function evaluations for the non-stiff part */
  Vec         Ydot;             /* Work vector holding Ydot during residual evaluation */
  Vec         Work;             /* Generic work vector */
  Vec         Z;                /* Ydot = shift(Y-Z) */
  PetscScalar *work;            /* Scalar work */
  PetscReal   shift;
  PetscReal   stage_time;
  PetscBool   imex;
} TS_ARKIMEX;

/*MC
     TSARKIMEX2D - Second order ARK IMEX scheme with L-stable implicit part.

     This method has one explicit stage and two implicit stages.

     Level: advanced

.seealso: TSARKIMEX
M*/
/*MC
     TSARKIMEX2E - Second order ARK IMEX scheme with L-stable implicit part.

     This method has one explicit stage and two implicit stages. It is is an optimal method developed by Emil Constantinescu.

     Level: advanced

.seealso: TSARKIMEX
M*/
/*MC
     TSARKIMEX3 - Third order ARK IMEX scheme with L-stable implicit part.

     This method has one explicit stage and three implicit stages.

     References:
     Kennedy and Carpenter 2003.

     Level: advanced

.seealso: TSARKIMEX
M*/
/*MC
     TSARKIMEX4 - Fourth order ARK IMEX scheme with L-stable implicit part.

     This method has one explicit stage and four implicit stages.

     References:
     Kennedy and Carpenter 2003.

     Level: advanced

.seealso: TSARKIMEX
M*/
/*MC
     TSARKIMEX5 - Fifth order ARK IMEX scheme with L-stable implicit part.

     This method has one explicit stage and five implicit stages.

     References:
     Kennedy and Carpenter 2003.

     Level: advanced

.seealso: TSARKIMEX
M*/

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

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

  Level: advanced

.keywords: TS, TSARKIMEX, register, all

.seealso:  TSARKIMEXRegisterDestroy()
@*/
PetscErrorCode TSARKIMEXRegisterAll(void)
{
  PetscErrorCode ierr;

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

  {
    const PetscReal
      A[3][3] = {{0,0,0},
                 {0.41421356237309504880,0,0},
                 {0.75,0.25,0}},
      At[3][3] = {{0,0,0},
                  {0.12132034355964257320,0.29289321881345247560,0},
                  {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}},
      binterpt[3][2] = {{1,-0.5},{0,0},{0,0.5}};
    ierr = TSARKIMEXRegister(TSARKIMEX2D,2,3,&At[0][0],PETSC_NULL,PETSC_NULL,&A[0][0],PETSC_NULL,PETSC_NULL,2,binterpt[0],PETSC_NULL);CHKERRQ(ierr);
  }
  {                             /* Optimal for linear implicit part */
    const PetscReal s2 = PetscSqrtReal((PetscReal)2.0),
      A[3][3] = {{0,0,0},
                 {2-s2,0,0},
                 {(3-2*s2)/6,(3+2*s2)/6,0}},
      At[3][3] = {{0,0,0},
                  {1-1/s2,1-1/s2,0},
                  {1/(2*s2),1/(2*s2),1-1/s2}},
      binterpt[3][2] = {{1,-0.5},{0,0},{0,0.5}};
    ierr = TSARKIMEXRegister(TSARKIMEX2E,2,3,&At[0][0],PETSC_NULL,PETSC_NULL,&A[0][0],PETSC_NULL,PETSC_NULL,2,binterpt[0],PETSC_NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[4][4] = {{0,0,0,0},
                 {1767732205903./2027836641118.,0,0,0},
                 {5535828885825./10492691773637.,788022342437./10882634858940.,0,0},
                 {6485989280629./16251701735622.,-4246266847089./9704473918619.,10755448449292./10357097424841.,0}},
      At[4][4] = {{0,0,0,0},
                  {1767732205903./4055673282236.,1767732205903./4055673282236.,0,0},
                  {2746238789719./10658868560708.,-640167445237./6845629431997.,1767732205903./4055673282236.,0},
                  {1471266399579./7840856788654.,-4482444167858./7529755066697.,11266239266428./11593286722821.,1767732205903./4055673282236.}},
      binterpt[4][2] = {{4655552711362./22874653954995., -215264564351./13552729205753.},
                        {-18682724506714./9892148508045.,17870216137069./13817060693119.},
                        {34259539580243./13192909600954.,-28141676662227./17317692491321.},
                        {584795268549./6622622206610.,   2508943948391./7218656332882.}};
    ierr = TSARKIMEXRegister(TSARKIMEX3,3,4,&At[0][0],PETSC_NULL,PETSC_NULL,&A[0][0],PETSC_NULL,PETSC_NULL,2,binterpt[0],PETSC_NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[6][6] = {{0,0,0,0,0,0},
                 {1./2,0,0,0,0,0},
                 {13861./62500.,6889./62500.,0,0,0,0},
                 {-116923316275./2393684061468.,-2731218467317./15368042101831.,9408046702089./11113171139209.,0,0,0},
                 {-451086348788./2902428689909.,-2682348792572./7519795681897.,12662868775082./11960479115383.,3355817975965./11060851509271.,0,0},
                 {647845179188./3216320057751.,73281519250./8382639484533.,552539513391./3454668386233.,3354512671639./8306763924573.,4040./17871.,0}},
      At[6][6] = {{0,0,0,0,0,0},
                  {1./4,1./4,0,0,0,0},
                  {8611./62500.,-1743./31250.,1./4,0,0,0},
                  {5012029./34652500.,-654441./2922500.,174375./388108.,1./4,0,0},
                  {15267082809./155376265600.,-71443401./120774400.,730878875./902184768.,2285395./8070912.,1./4,0},
                  {82889./524892.,0,15625./83664.,69875./102672.,-2260./8211,1./4}},
      binterpt[6][3] = {{6943876665148./7220017795957.,-54480133./30881146.,6818779379841./7100303317025.},
                        {0,0,0},
                        {7640104374378./9702883013639.,-11436875./14766696.,2173542590792./12501825683035.},
                        {-20649996744609./7521556579894.,174696575./18121608.,-31592104683404./5083833661969.},
                        {8854892464581./2390941311638.,-12120380./966161.,61146701046299./7138195549469.},
                        {-11397109935349./6675773540249.,3843./706.,-17219254887155./4939391667607.}};
    ierr = TSARKIMEXRegister(TSARKIMEX4,4,6,&At[0][0],PETSC_NULL,PETSC_NULL,&A[0][0],PETSC_NULL,PETSC_NULL,3,binterpt[0],PETSC_NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[8][8] = {{0,0,0,0,0,0,0,0},
                 {41./100,0,0,0,0,0,0,0},
                 {367902744464./2072280473677.,677623207551./8224143866563.,0,0,0,0,0,0},
                 {1268023523408./10340822734521.,0,1029933939417./13636558850479.,0,0,0,0,0},
                 {14463281900351./6315353703477.,0,66114435211212./5879490589093.,-54053170152839./4284798021562.,0,0,0,0},
                 {14090043504691./34967701212078.,0,15191511035443./11219624916014.,-18461159152457./12425892160975.,-281667163811./9011619295870.,0,0,0},
                 {19230459214898./13134317526959.,0,21275331358303./2942455364971.,-38145345988419./4862620318723.,-1./8,-1./8,0,0},
                 {-19977161125411./11928030595625.,0,-40795976796054./6384907823539.,177454434618887./12078138498510.,782672205425./8267701900261.,-69563011059811./9646580694205.,7356628210526./4942186776405.,0}},
      At[8][8] = {{0,0,0,0,0,0,0,0},
                  {41./200.,41./200.,0,0,0,0,0,0},
                  {41./400.,-567603406766./11931857230679.,41./200.,0,0,0,0,0},
                  {683785636431./9252920307686.,0,-110385047103./1367015193373.,41./200.,0,0,0,0},
                  {3016520224154./10081342136671.,0,30586259806659./12414158314087.,-22760509404356./11113319521817.,41./200.,0,0,0},
                  {218866479029./1489978393911.,0,638256894668./5436446318841.,-1179710474555./5321154724896.,-60928119172./8023461067671.,41./200.,0,0},
                  {1020004230633./5715676835656.,0,25762820946817./25263940353407.,-2161375909145./9755907335909.,-211217309593./5846859502534.,-4269925059573./7827059040749.,41./200,0},
                  {-872700587467./9133579230613.,0,0,22348218063261./9555858737531.,-1143369518992./8141816002931.,-39379526789629./19018526304540.,32727382324388./42900044865799.,41./200.}},
      binterpt[8][3] = {{-17674230611817./10670229744614. ,  43486358583215./12773830924787. , -9257016797708./5021505065439.},
                        {0                               ,  0                              , 0                            },
                        {0                               ,  0                              , 0                            },
                        {65168852399939./7868540260826.  ,  -91478233927265./11067650958493., 26096422576131./11239449250142.},
                        {15494834004392./5936557850923.  ,  -79368583304911./10890268929626., 92396832856987./20362823103730.},
                        {-99329723586156./26959484932159.,  -12239297817655./9152339842473. , 30029262896817./10175596800299.},
                        {-19024464361622./5461577185407. ,  115839755401235./10719374521269., -26136350496073./3983972220547.},
                        {-6511271360970./6095937251113.  ,  5843115559534./2180450260947.   , -5289405421727./3760307252460. }};
    ierr = TSARKIMEXRegister(TSARKIMEX5,5,8,&At[0][0],PETSC_NULL,PETSC_NULL,&A[0][0],PETSC_NULL,PETSC_NULL,3,binterpt[0],PETSC_NULL);CHKERRQ(ierr);
  }

  PetscFunctionReturn(0);
}

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

   Not Collective

   Level: advanced

.keywords: TSARKIMEX, register, destroy
.seealso: TSARKIMEXRegister(), TSARKIMEXRegisterAll(), TSARKIMEXRegisterDynamic()
@*/
PetscErrorCode TSARKIMEXRegisterDestroy(void)
{
  PetscErrorCode ierr;
  ARKTableauLink link;

  PetscFunctionBegin;
  while ((link = ARKTableauList)) {
    ARKTableau t = &link->tab;
    ARKTableauList = link->next;
    ierr = PetscFree6(t->At,t->bt,t->ct,t->A,t->b,t->c);CHKERRQ(ierr);
    ierr = PetscFree2(t->binterpt,t->binterp);CHKERRQ(ierr);
    ierr = PetscFree(t->name);CHKERRQ(ierr);
    ierr = PetscFree(link);CHKERRQ(ierr);
  }
  TSARKIMEXRegisterAllCalled = PETSC_FALSE;
  PetscFunctionReturn(0);
}

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

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

  Level: developer

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

  PetscFunctionBegin;
  if (TSARKIMEXPackageInitialized) PetscFunctionReturn(0);
  TSARKIMEXPackageInitialized = PETSC_TRUE;
  ierr = TSARKIMEXRegisterAll();CHKERRQ(ierr);
  ierr = PetscRegisterFinalize(TSARKIMEXFinalizePackage);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

  Level: developer

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

  PetscFunctionBegin;
  TSARKIMEXPackageInitialized = PETSC_FALSE;
  ierr = TSARKIMEXRegisterDestroy();CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSARKIMEXRegister"
/*@C
   TSARKIMEXRegister - register an ARK IMEX 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
.  At - Butcher table of stage coefficients for stiff part (dimension s*s, row-major)
.  bt - Butcher table for completing the stiff part of the step (dimension s; PETSC_NULL to use the last row of At)
.  ct - Abscissa of each stiff stage (dimension s, PETSC_NULL to use row sums of At)
.  A - Non-stiff stage coefficients (dimension s*s, row-major)
.  b - Non-stiff step completion table (dimension s; PETSC_NULL to use last row of At)
.  c - Non-stiff abscissa (dimension s; PETSC_NULL to use row sums of A)
.  pinterp - Order of the interpolation scheme, equal to the number of columns of binterpt and binterp
.  binterpt - Coefficients of the interpolation formula for the stiff part (dimension s*pinterp)
-  binterp - Coefficients of the interpolation formula for the non-stiff part (dimension s*pinterp; PETSC_NULL to reuse binterpt)

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

   Level: advanced

.keywords: TS, register

.seealso: TSARKIMEX
@*/
PetscErrorCode TSARKIMEXRegister(const TSARKIMEXType name,PetscInt order,PetscInt s,
                                 const PetscReal At[],const PetscReal bt[],const PetscReal ct[],
                                 const PetscReal A[],const PetscReal b[],const PetscReal c[],
                                 PetscInt pinterp,const PetscReal binterpt[],const PetscReal binterp[])
{
  PetscErrorCode ierr;
  ARKTableauLink link;
  ARKTableau t;
  PetscInt i,j;

  PetscFunctionBegin;
  ierr = PetscMalloc(sizeof(*link),&link);CHKERRQ(ierr);
  ierr = PetscMemzero(link,sizeof(*link));CHKERRQ(ierr);
  t = &link->tab;
  ierr = PetscStrallocpy(name,&t->name);CHKERRQ(ierr);
  t->order = order;
  t->s = s;
  ierr = PetscMalloc6(s*s,PetscReal,&t->At,s,PetscReal,&t->bt,s,PetscReal,&t->ct,s*s,PetscReal,&t->A,s,PetscReal,&t->b,s,PetscReal,&t->c);CHKERRQ(ierr);
  ierr = PetscMemcpy(t->At,At,s*s*sizeof(At[0]));CHKERRQ(ierr);
  ierr = PetscMemcpy(t->A,A,s*s*sizeof(A[0]));CHKERRQ(ierr);
  if (bt) {ierr = PetscMemcpy(t->bt,bt,s*sizeof(bt[0]));CHKERRQ(ierr);}
  else for (i=0; i<s; i++) t->bt[i] = At[(s-1)*s+i];
  if (b) {ierr = PetscMemcpy(t->b,b,s*sizeof(b[0]));CHKERRQ(ierr);}
  else for (i=0; i<s; i++) t->b[i] = At[(s-1)*s+i];
  if (ct) {ierr = PetscMemcpy(t->ct,ct,s*sizeof(ct[0]));CHKERRQ(ierr);}
  else for (i=0; i<s; i++) for (j=0,t->ct[i]=0; j<s; j++) t->ct[i] += At[i*s+j];
  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->pinterp = pinterp;
  ierr = PetscMalloc2(s*pinterp,PetscReal,&t->binterpt,s*pinterp,PetscReal,&t->binterp);CHKERRQ(ierr);
  ierr = PetscMemcpy(t->binterpt,binterpt,s*pinterp*sizeof(binterpt[0]));CHKERRQ(ierr);
  ierr = PetscMemcpy(t->binterp,binterp?binterp:binterpt,s*pinterp*sizeof(binterpt[0]));CHKERRQ(ierr);
  link->next = ARKTableauList;
  ARKTableauList = link;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSStep_ARKIMEX"
static PetscErrorCode TSStep_ARKIMEX(TS ts)
{
  TS_ARKIMEX          *ark = (TS_ARKIMEX*)ts->data;
  ARKTableau          tab  = ark->tableau;
  const PetscInt      s    = tab->s;
  const PetscReal     *At  = tab->At,*A = tab->A,*bt = tab->bt,*b = tab->b,*ct = tab->ct,*c = tab->c;
  PetscScalar         *w   = ark->work;
  Vec                 *Y   = ark->Y,*YdotI = ark->YdotI,*YdotRHS = ark->YdotRHS,Ydot = ark->Ydot,W = ark->Work,Z = ark->Z;
  SNES                snes;
  SNESConvergedReason snesreason;
  PetscInt            i,j,its,lits;
  PetscReal           next_time_step;
  PetscReal           h,t;
  PetscErrorCode      ierr;

  PetscFunctionBegin;
  ierr = TSGetSNES(ts,&snes);CHKERRQ(ierr);
  next_time_step = ts->time_step;
  h = ts->time_step;
  t = ts->ptime;

  for (i=0; i<s; i++) {
    if (At[i*s+i] == 0) {           /* This stage is explicit */
      ierr = VecCopy(ts->vec_sol,Y[i]);CHKERRQ(ierr);
      for (j=0; j<i; j++) w[j] = -h*At[i*s+j];
      ierr = VecMAXPY(Y[i],i,w,YdotI);CHKERRQ(ierr);
      for (j=0; j<i; j++) w[j] = h*A[i*s+j];
      ierr = VecMAXPY(Y[i],i,w,YdotRHS);CHKERRQ(ierr);
    } else {
      ark->stage_time = t + h*ct[i];
      ark->shift = 1./(h*At[i*s+i]);
      /* Affine part */
      ierr = VecZeroEntries(W);CHKERRQ(ierr);
      for (j=0; j<i; j++) w[j] = h*A[i*s+j];
      ierr = VecMAXPY(W,i,w,YdotRHS);CHKERRQ(ierr);
      /* Ydot = shift*(Y-Z) */
      ierr = VecCopy(ts->vec_sol,Z);CHKERRQ(ierr);
      for (j=0; j<i; j++) w[j] = -h*At[i*s+j];
      ierr = VecMAXPY(Z,i,w,YdotI);CHKERRQ(ierr);
      /* Initial guess taken from last stage */
      ierr = VecCopy(i>0?Y[i-1]:ts->vec_sol,Y[i]);CHKERRQ(ierr);
      ierr = SNESSolve(snes,W,Y[i]);CHKERRQ(ierr);
      ierr = SNESGetIterationNumber(snes,&its);CHKERRQ(ierr);
      ierr = SNESGetLinearSolveIterations(snes,&lits);CHKERRQ(ierr);
      ierr = SNESGetConvergedReason(ts->snes,&snesreason);CHKERRQ(ierr);
      ts->nonlinear_its += its; ts->linear_its += lits;
      if (snesreason < 0 && ts->max_snes_failures > 0 && ++ts->num_snes_failures >= ts->max_snes_failures) {
        ts->reason = TS_DIVERGED_NONLINEAR_SOLVE;
        ierr = PetscInfo2(ts,"Step=%D, nonlinear solve solve failures %D greater than current TS allowed, stopping solve\n",ts->steps,ts->num_snes_failures);CHKERRQ(ierr);
        PetscFunctionReturn(0);
      }
    }
    ierr = VecZeroEntries(Ydot);CHKERRQ(ierr);
    ierr = TSComputeIFunction(ts,t+h*ct[i],Y[i],Ydot,YdotI[i],ark->imex);CHKERRQ(ierr);
    if (ark->imex) {
      ierr = TSComputeRHSFunction(ts,t+h*c[i],Y[i],YdotRHS[i]);CHKERRQ(ierr);
    } else {
      ierr = VecZeroEntries(YdotRHS[i]);CHKERRQ(ierr);
    }
  }
  for (j=0; j<s; j++) w[j] = -h*bt[j];
  ierr = VecMAXPY(ts->vec_sol,s,w,YdotI);CHKERRQ(ierr);
  for (j=0; j<s; j++) w[j] = h*b[j];
  ierr = VecMAXPY(ts->vec_sol,s,w,YdotRHS);CHKERRQ(ierr);

  ts->ptime += ts->time_step;
  ts->time_step = next_time_step;
  ts->steps++;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSInterpolate_ARKIMEX"
static PetscErrorCode TSInterpolate_ARKIMEX(TS ts,PetscReal itime,Vec X)
{
  TS_ARKIMEX *ark = (TS_ARKIMEX*)ts->data;
  PetscInt s = ark->tableau->s,pinterp = ark->tableau->pinterp,i,j;
  PetscReal tt,t = (itime - ts->ptime)/ts->time_step + 1; /* In the interval [0,1] */
  PetscScalar *bt,*b;
  const PetscReal *Bt = ark->tableau->binterpt,*B = ark->tableau->binterp;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!Bt || !B) SETERRQ1(((PetscObject)ts)->comm,PETSC_ERR_SUP,"TSARKIMEX %s does not have an interpolation formula",ark->tableau->name);
  ierr = PetscMalloc2(s,PetscScalar,&bt,s,PetscScalar,&b);CHKERRQ(ierr);
  for (i=0; i<s; i++) bt[i] = b[i] = 0;
  for (j=0,tt=t; j<pinterp; j++,tt*=t) {
    for (i=0; i<s; i++) {
      bt[i] += ts->time_step * Bt[i*pinterp+j] * tt * -1.0;
      b[i]  += ts->time_step * B[i*pinterp+j] * tt;
    }
  }
  if (ark->tableau->At[0*s+0] != 0.0) SETERRQ(((PetscObject)ts)->comm,PETSC_ERR_SUP,"First stage not explicit so starting stage not saved");
  ierr = VecCopy(ark->Y[0],X);CHKERRQ(ierr);
  ierr = VecMAXPY(X,s,bt,ark->YdotI);CHKERRQ(ierr);
  ierr = VecMAXPY(X,s,b,ark->YdotRHS);CHKERRQ(ierr);
  ierr = PetscFree2(bt,b);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*------------------------------------------------------------*/
#undef __FUNCT__
#define __FUNCT__ "TSReset_ARKIMEX"
static PetscErrorCode TSReset_ARKIMEX(TS ts)
{
  TS_ARKIMEX      *ark = (TS_ARKIMEX*)ts->data;
  PetscInt        s;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  if (!ark->tableau) PetscFunctionReturn(0);
   s = ark->tableau->s;
  ierr = VecDestroyVecs(s,&ark->Y);CHKERRQ(ierr);
  ierr = VecDestroyVecs(s,&ark->YdotI);CHKERRQ(ierr);
  ierr = VecDestroyVecs(s,&ark->YdotRHS);CHKERRQ(ierr);
  ierr = VecDestroy(&ark->Ydot);CHKERRQ(ierr);
  ierr = VecDestroy(&ark->Work);CHKERRQ(ierr);
  ierr = VecDestroy(&ark->Z);CHKERRQ(ierr);
  ierr = PetscFree(ark->work);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  ierr = TSReset_ARKIMEX(ts);CHKERRQ(ierr);
  ierr = PetscFree(ts->data);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunctionDynamic((PetscObject)ts,"TSARKIMEXGetType_C","",PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunctionDynamic((PetscObject)ts,"TSARKIMEXSetType_C","",PETSC_NULL);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*
  This defines the nonlinear equation that is to be solved with SNES
  G(U) = F[t0+Theta*dt, U, (U-U0)*shift] = 0
*/
#undef __FUNCT__
#define __FUNCT__ "SNESTSFormFunction_ARKIMEX"
static PetscErrorCode SNESTSFormFunction_ARKIMEX(SNES snes,Vec X,Vec F,TS ts)
{
  TS_ARKIMEX     *ark = (TS_ARKIMEX*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = VecAXPBYPCZ(ark->Ydot,-ark->shift,ark->shift,0,ark->Z,X);CHKERRQ(ierr); /* Ydot = shift*(X-Z) */
  ierr = TSComputeIFunction(ts,ark->stage_time,X,ark->Ydot,F,ark->imex);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  /* ark->Ydot has already been computed in SNESTSFormFunction_ARKIMEX (SNES guarantees this) */
  ierr = TSComputeIJacobian(ts,ark->stage_time,X,ark->Ydot,ark->shift,A,B,str,PETSC_TRUE);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSSetUp_ARKIMEX"
static PetscErrorCode TSSetUp_ARKIMEX(TS ts)
{
  TS_ARKIMEX     *ark = (TS_ARKIMEX*)ts->data;
  ARKTableau     tab  = ark->tableau;
  PetscInt       s = tab->s;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!ark->tableau) {
    ierr = TSARKIMEXSetType(ts,TSARKIMEXDefault);CHKERRQ(ierr);
  }
  ierr = VecDuplicateVecs(ts->vec_sol,s,&ark->Y);CHKERRQ(ierr);
  ierr = VecDuplicateVecs(ts->vec_sol,s,&ark->YdotI);CHKERRQ(ierr);
  ierr = VecDuplicateVecs(ts->vec_sol,s,&ark->YdotRHS);CHKERRQ(ierr);
  ierr = VecDuplicate(ts->vec_sol,&ark->Ydot);CHKERRQ(ierr);
  ierr = VecDuplicate(ts->vec_sol,&ark->Work);CHKERRQ(ierr);
  ierr = VecDuplicate(ts->vec_sol,&ark->Z);CHKERRQ(ierr);
  ierr = PetscMalloc(s*sizeof(ark->work[0]),&ark->work);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}
/*------------------------------------------------------------*/

#undef __FUNCT__
#define __FUNCT__ "TSSetFromOptions_ARKIMEX"
static PetscErrorCode TSSetFromOptions_ARKIMEX(TS ts)
{
  TS_ARKIMEX     *ark = (TS_ARKIMEX*)ts->data;
  PetscErrorCode ierr;
  char           arktype[256];

  PetscFunctionBegin;
  ierr = PetscOptionsHead("ARKIMEX ODE solver options");CHKERRQ(ierr);
  {
    ARKTableauLink link;
    PetscInt count,choice;
    PetscBool flg;
    const char **namelist;
    ierr = PetscStrncpy(arktype,TSARKIMEXDefault,sizeof arktype);CHKERRQ(ierr);
    for (link=ARKTableauList,count=0; link; link=link->next,count++) ;
    ierr = PetscMalloc(count*sizeof(char*),&namelist);CHKERRQ(ierr);
    for (link=ARKTableauList,count=0; link; link=link->next,count++) namelist[count] = link->tab.name;
    ierr = PetscOptionsEList("-ts_arkimex_type","Family of ARK IMEX method","TSARKIMEXSetType",(const char*const*)namelist,count,arktype,&choice,&flg);CHKERRQ(ierr);
    ierr = TSARKIMEXSetType(ts,flg ? namelist[choice] : arktype);CHKERRQ(ierr);
    ierr = PetscFree(namelist);CHKERRQ(ierr);
    flg = (PetscBool)!ark->imex;
    ierr = PetscOptionsBool("-ts_arkimex_fully_implicit","Solve the problem fully implicitly","TSARKIMEXSetFullyImplicit",flg,&flg,PETSC_NULL);CHKERRQ(ierr);
    ark->imex = (PetscBool)!flg;
    ierr = SNESSetFromOptions(ts->snes);CHKERRQ(ierr);
  }
  ierr = PetscOptionsTail();CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

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

#undef __FUNCT__
#define __FUNCT__ "TSView_ARKIMEX"
static PetscErrorCode TSView_ARKIMEX(TS ts,PetscViewer viewer)
{
  TS_ARKIMEX     *ark = (TS_ARKIMEX*)ts->data;
  ARKTableau     tab = ark->tableau;
  PetscBool      iascii;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);CHKERRQ(ierr);
  if (iascii) {
    const TSARKIMEXType arktype;
    char buf[512];
    ierr = TSARKIMEXGetType(ts,&arktype);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  ARK IMEX %s\n",arktype);CHKERRQ(ierr);
    ierr = PetscFormatRealArray(buf,sizeof buf,"% 8.6f",tab->s,tab->ct);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  Stiff abscissa       ct = %s\n",buf);CHKERRQ(ierr);
    ierr = PetscFormatRealArray(buf,sizeof buf,"% 8.6f",tab->s,tab->c);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  Nonstiff abscissa     c = %s\n",buf);CHKERRQ(ierr);
  }
  ierr = SNESView(ts->snes,viewer);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSARKIMEXSetType"
/*@C
  TSARKIMEXSetType - Set the type of ARK IMEX scheme

  Logically collective

  Input Parameter:
+  ts - timestepping context
-  arktype - type of ARK-IMEX scheme

  Level: intermediate

.seealso: TSARKIMEXGetType()
@*/
PetscErrorCode TSARKIMEXSetType(TS ts,const TSARKIMEXType arktype)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  ierr = PetscTryMethod(ts,"TSARKIMEXSetType_C",(TS,const TSARKIMEXType),(ts,arktype));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSARKIMEXGetType"
/*@C
  TSARKIMEXGetType - Get the type of ARK IMEX scheme

  Logically collective

  Input Parameter:
.  ts - timestepping context

  Output Parameter:
.  arktype - type of ARK-IMEX scheme

  Level: intermediate

.seealso: TSARKIMEXGetType()
@*/
PetscErrorCode TSARKIMEXGetType(TS ts,const TSARKIMEXType *arktype)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  ierr = PetscUseMethod(ts,"TSARKIMEXGetType_C",(TS,const TSARKIMEXType*),(ts,arktype));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSARKIMEXSetFullyImplicit"
/*@C
  TSARKIMEXSetFullyImplicit - Solve both parts of the equation implicitly

  Logically collective

  Input Parameter:
+  ts - timestepping context
-  flg - PETSC_TRUE for fully implicit

  Level: intermediate

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

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

EXTERN_C_BEGIN
#undef __FUNCT__
#define __FUNCT__ "TSARKIMEXGetType_ARKIMEX"
PetscErrorCode  TSARKIMEXGetType_ARKIMEX(TS ts,const TSARKIMEXType *arktype)
{
  TS_ARKIMEX *ark = (TS_ARKIMEX*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!ark->tableau) {ierr = TSARKIMEXSetType(ts,TSARKIMEXDefault);CHKERRQ(ierr);}
  *arktype = ark->tableau->name;
  PetscFunctionReturn(0);
}
#undef __FUNCT__
#define __FUNCT__ "TSARKIMEXSetType_ARKIMEX"
PetscErrorCode  TSARKIMEXSetType_ARKIMEX(TS ts,const TSARKIMEXType arktype)
{
  TS_ARKIMEX *ark = (TS_ARKIMEX*)ts->data;
  PetscErrorCode ierr;
  PetscBool match;
  ARKTableauLink link;

  PetscFunctionBegin;
  if (ark->tableau) {
    ierr = PetscStrcmp(ark->tableau->name,arktype,&match);CHKERRQ(ierr);
    if (match) PetscFunctionReturn(0);
  }
  for (link = ARKTableauList; link; link=link->next) {
    ierr = PetscStrcmp(link->tab.name,arktype,&match);CHKERRQ(ierr);
    if (match) {
      ierr = TSReset_ARKIMEX(ts);CHKERRQ(ierr);
      ark->tableau = &link->tab;
      PetscFunctionReturn(0);
    }
  }
  SETERRQ1(((PetscObject)ts)->comm,PETSC_ERR_ARG_UNKNOWN_TYPE,"Could not find '%s'",arktype);
  PetscFunctionReturn(0);
}
#undef __FUNCT__
#define __FUNCT__ "TSARKIMEXSetFullyImplicit_ARKIMEX"
PetscErrorCode  TSARKIMEXSetFullyImplicit_ARKIMEX(TS ts,PetscBool flg)
{
  TS_ARKIMEX *ark = (TS_ARKIMEX*)ts->data;

  PetscFunctionBegin;
  ark->imex = (PetscBool)!flg;
  PetscFunctionReturn(0);
}
EXTERN_C_END

/* ------------------------------------------------------------ */
/*MC
      TSARKIMEX - ODE solver using Additive Runge-Kutta IMEX schemes

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

  Notes:
  The default is TSARKIMEX2E, it can be changed with TSARKIMEXSetType() or -ts_arkimex_type

  This method currently only works with ODE, for which the stiff part G(t,X,Xdot) has the form Xdot + Ghat(t,X).

  Level: beginner

.seealso:  TSCreate(), TS, TSSetType(), TSARKIMEXSetType(), TSARKIMEXGetType(), TSARKIMEXSetFullyImplicit(), TSARKIMEX2D, TTSARKIMEX2E, TSARKIMEX3,
           TSARKIMEX4, TSARKIMEX5, TSARKIMEXType, TSARKIMEXRegister()

M*/
EXTERN_C_BEGIN
#undef __FUNCT__
#define __FUNCT__ "TSCreate_ARKIMEX"
PetscErrorCode  TSCreate_ARKIMEX(TS ts)
{
  TS_ARKIMEX       *th;
  PetscErrorCode ierr;

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

  ts->ops->reset          = TSReset_ARKIMEX;
  ts->ops->destroy        = TSDestroy_ARKIMEX;
  ts->ops->view           = TSView_ARKIMEX;
  ts->ops->setup          = TSSetUp_ARKIMEX;
  ts->ops->step           = TSStep_ARKIMEX;
  ts->ops->interpolate    = TSInterpolate_ARKIMEX;
  ts->ops->setfromoptions = TSSetFromOptions_ARKIMEX;
  ts->ops->snesfunction   = SNESTSFormFunction_ARKIMEX;
  ts->ops->snesjacobian   = SNESTSFormJacobian_ARKIMEX;

  ierr = PetscNewLog(ts,TS_ARKIMEX,&th);CHKERRQ(ierr);
  ts->data = (void*)th;
  th->imex = PETSC_TRUE;

  ierr = PetscObjectComposeFunctionDynamic((PetscObject)ts,"TSARKIMEXGetType_C","TSARKIMEXGetType_ARKIMEX",TSARKIMEXGetType_ARKIMEX);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunctionDynamic((PetscObject)ts,"TSARKIMEXSetType_C","TSARKIMEXSetType_ARKIMEX",TSARKIMEXSetType_ARKIMEX);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunctionDynamic((PetscObject)ts,"TSARKIMEXSetFullyImplicit_C","TSARKIMEXSetFullyImplicit_ARKIMEX",TSARKIMEXSetFullyImplicit_ARKIMEX);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}
EXTERN_C_END