xref: /petsc/src/ts/impls/arkimex/arkimex.c (revision 24655328f080896bc75518026e6e8bd491ef3aa6)

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

  Notes:
  The general system is written as

  F(t,U,Udot) = G(t,U)

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

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

static TSARKIMEXType TSARKIMEXDefault = TSARKIMEX3;
static PetscBool     TSARKIMEXRegisterAllCalled;
static PetscBool     TSARKIMEXPackageInitialized;
static PetscInt      explicit_stage_time_id;

typedef struct _ARKTableau *ARKTableau;
struct _ARKTableau {
  char      *name;
  PetscInt  order;                /* Classical approximation order of the method */
  PetscInt  s;                    /* Number of stages */
  PetscBool stiffly_accurate;     /* The implicit part is stiffly accurate*/
  PetscBool FSAL_implicit;        /* The implicit part is FSAL*/
  PetscBool explicit_first_stage; /* The implicit part has an explicit first stage*/
  PetscInt  pinterp;              /* Interpolation order */
  PetscReal *At,*bt,*ct;          /* Stiff tableau */
  PetscReal *A,*b,*c;             /* Non-stiff tableau */
  PetscReal *bembedt,*bembed;     /* Embedded formula of order one less (order-1) */
  PetscReal *binterpt,*binterp;   /* Dense output formula */
  PetscReal ccfl;                 /* Placeholder for CFL coefficient relative to forward Euler */
};
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          Ydot0;            /* Holds the slope from the previous step in FSAL case */
  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    scoeff;           /* shift = scoeff/dt */
  PetscReal    stage_time;
  PetscBool    imex;
  TSStepStatus status;
} TS_ARKIMEX;
/*MC
     TSARKIMEXARS122 - Second order ARK IMEX scheme.

     This method has one explicit stage and one implicit stage.

     References:
     U. Ascher, S. Ruuth, R. J. Spitheri, Implicit-explicit Runge-Kutta methods for time dependent Partial Differential Equations. Appl. Numer. Math. 25, (1997), pp. 151–167.

     Level: advanced

.seealso: TSARKIMEX
M*/
/*MC
     TSARKIMEXA2 - Second order ARK IMEX scheme with A-stable implicit part.

     This method has an explicit stage and one implicit stage, and has an A-stable implicit scheme. This method was provided by Emil Constantinescu.

     Level: advanced

.seealso: TSARKIMEX
M*/
/*MC
     TSARKIMEXL2 - Second order ARK IMEX scheme with L-stable implicit part.

     This method has two implicit stages, and L-stable implicit scheme.

    References:
     L. Pareschi, G. Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxations. Journal of Scientific Computing Volume: 25, Issue: 1, October, 2005, pp. 129-155

     Level: advanced

.seealso: TSARKIMEX
M*/
/*MC
     TSARKIMEX1BEE - First order Backward Euler represented as an ARK IMEX scheme with extrapolation as error estimator. This is a 3-stage method.

     This method is aimed at starting the integration of implicit DAEs when explicit first-stage ARK methods are used.

     Level: advanced

.seealso: TSARKIMEX
M*/
/*MC
     TSARKIMEX2C - Second order ARK IMEX scheme with L-stable implicit part.

     This method has one explicit stage and two implicit stages. The implicit part is the same as in TSARKIMEX2D and TSARKIMEX2E, but the explicit part has a larger stability region on the negative real axis. This method was provided by Emil Constantinescu.

     Level: advanced

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

     This method has one explicit stage and two implicit stages. The stability function is independent of the explicit part in the infinity limit of the implict component. This method was provided by Emil Constantinescu.

     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
     TSARKIMEXPRSSP2 - Second order SSP ARK IMEX scheme.

     This method has three implicit stages.

     References:
     L. Pareschi, G. Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxations. Journal of Scientific Computing Volume: 25, Issue: 1, October, 2005, pp. 129-155

     This method is referred to as SSP2-(3,3,2) in http://arxiv.org/abs/1110.4375

     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
     TSARKIMEXARS443 - Third order ARK IMEX scheme.

     This method has one explicit stage and four implicit stages.

     References:
     U. Ascher, S. Ruuth, R. J. Spitheri, Implicit-explicit Runge-Kutta methods for time dependent Partial Differential Equations. Appl. Numer. Math. 25, (1997), pp. 151–167.

     This method is referred to as ARS(4,4,3) in http://arxiv.org/abs/1110.4375

     Level: advanced

.seealso: TSARKIMEX
M*/
/*MC
     TSARKIMEXBPR3 - Third order ARK IMEX scheme.

     This method has one explicit stage and four implicit stages.

     References:
     This method is referred to as ARK3 in http://arxiv.org/abs/1110.4375

     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] = {{0.0,0.0,0.0},
                 {0.0,0.0,0.0},
                 {0.0,0.5,0.0}},
      At[3][3] = {{1.0,0.0,0.0},
                  {0.0,0.5,0.0},
                  {0.0,0.5,0.5}},
      b[3]       = {0.0,0.5,0.5},
      bembedt[3] = {1.0,0.0,0.0};
    ierr = TSARKIMEXRegister(TSARKIMEX1BEE,2,3,&At[0][0],b,NULL,&A[0][0],b,NULL,bembedt,bembedt,1,b,NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[2][2] = {{0.0,0.0},
                 {0.5,0.0}},
      At[2][2] = {{0.0,0.0},
                  {0.0,0.5}},
      b[2]       = {0.0,1.0},
      bembedt[2] = {0.5,0.5};
    /* binterpt[2][2] = {{1.0,-1.0},{0.0,1.0}};  second order dense output has poor stability properties and hence it is not currently in use*/
    ierr = TSARKIMEXRegister(TSARKIMEXARS122,2,2,&At[0][0],b,NULL,&A[0][0],b,NULL,bembedt,bembedt,1,b,NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[2][2] = {{0.0,0.0},
                 {1.0,0.0}},
      At[2][2] = {{0.0,0.0},
                  {0.5,0.5}},
      b[2]       = {0.5,0.5},
      bembedt[2] = {0.0,1.0};
    /* binterpt[2][2] = {{1.0,-0.5},{0.0,0.5}}  second order dense output has poor stability properties and hence it is not currently in use*/
    ierr = TSARKIMEXRegister(TSARKIMEXA2,2,2,&At[0][0],b,NULL,&A[0][0],b,NULL,bembedt,bembedt,1,b,NULL);CHKERRQ(ierr);
  }
  {
    /* const PetscReal us2 = 1.0-1.0/PetscSqrtReal((PetscReal)2.0);    Direct evaluation: 0.2928932188134524755992. Used below to ensure all values are available at compile time   */
    const PetscReal
      A[2][2] = {{0.0,0.0},
                 {1.0,0.0}},
      At[2][2] = {{0.2928932188134524755992,0.0},
                  {1.0-2.0*0.2928932188134524755992,0.2928932188134524755992}},
      b[2]       = {0.5,0.5},
      bembedt[2] = {0.0,1.0},
      binterpt[2][2] = {{  (0.2928932188134524755992-1.0)/(2.0*0.2928932188134524755992-1.0),-1/(2.0*(1.0-2.0*0.2928932188134524755992))},
                        {1-(0.2928932188134524755992-1.0)/(2.0*0.2928932188134524755992-1.0),-1/(2.0*(1.0-2.0*0.2928932188134524755992))}},
      binterp[2][2] = {{1.0,-0.5},{0.0,0.5}};
    ierr = TSARKIMEXRegister(TSARKIMEXL2,2,2,&At[0][0],b,NULL,&A[0][0],b,NULL,bembedt,bembedt,2,binterpt[0],binterp[0]);CHKERRQ(ierr);
  }
  {
    /* const PetscReal s2 = PetscSqrtReal((PetscReal)2.0),  Direct evaluation: 1.414213562373095048802. Used below to ensure all values are available at compile time   */
    const PetscReal
      A[3][3] = {{0,0,0},
                 {2-1.414213562373095048802,0,0},
                 {0.5,0.5,0}},
      At[3][3] = {{0,0,0},
                  {1-1/1.414213562373095048802,1-1/1.414213562373095048802,0},
                  {1/(2*1.414213562373095048802),1/(2*1.414213562373095048802),1-1/1.414213562373095048802}},
      bembedt[3] = {(4.-1.414213562373095048802)/8.,(4.-1.414213562373095048802)/8.,1/(2.*1.414213562373095048802)},
      binterpt[3][2] = {{1.0/1.414213562373095048802,-1.0/(2.0*1.414213562373095048802)},
                        {1.0/1.414213562373095048802,-1.0/(2.0*1.414213562373095048802)},
                        {1.0-1.414213562373095048802,1.0/1.414213562373095048802}};
    ierr = TSARKIMEXRegister(TSARKIMEX2C,2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembedt,2,binterpt[0],NULL);CHKERRQ(ierr);
  }
  {
    /* const PetscReal s2 = PetscSqrtReal((PetscReal)2.0),  Direct evaluation: 1.414213562373095048802. Used below to ensure all values are available at compile time   */
    const PetscReal
      A[3][3] = {{0,0,0},
                 {2-1.414213562373095048802,0,0},
                 {0.75,0.25,0}},
      At[3][3] = {{0,0,0},
                  {1-1/1.414213562373095048802,1-1/1.414213562373095048802,0},
                  {1/(2*1.414213562373095048802),1/(2*1.414213562373095048802),1-1/1.414213562373095048802}},
      bembedt[3] = {(4.-1.414213562373095048802)/8.,(4.-1.414213562373095048802)/8.,1/(2.*1.414213562373095048802)},
      binterpt[3][2] =  {{1.0/1.414213562373095048802,-1.0/(2.0*1.414213562373095048802)},
                         {1.0/1.414213562373095048802,-1.0/(2.0*1.414213562373095048802)},
                         {1.0-1.414213562373095048802,1.0/1.414213562373095048802}};
    ierr = TSARKIMEXRegister(TSARKIMEX2D,2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembedt,2,binterpt[0],NULL);CHKERRQ(ierr);
  }
  {                             /* Optimal for linear implicit part */
    /* const PetscReal s2 = PetscSqrtReal((PetscReal)2.0),  Direct evaluation: 1.414213562373095048802. Used below to ensure all values are available at compile time   */
    const PetscReal
      A[3][3] = {{0,0,0},
                 {2-1.414213562373095048802,0,0},
                 {(3-2*1.414213562373095048802)/6,(3+2*1.414213562373095048802)/6,0}},
      At[3][3] = {{0,0,0},
                  {1-1/1.414213562373095048802,1-1/1.414213562373095048802,0},
                  {1/(2*1.414213562373095048802),1/(2*1.414213562373095048802),1-1/1.414213562373095048802}},
      bembedt[3] = {(4.-1.414213562373095048802)/8.,(4.-1.414213562373095048802)/8.,1/(2.*1.414213562373095048802)},
      binterpt[3][2] =  {{1.0/1.414213562373095048802,-1.0/(2.0*1.414213562373095048802)},
                         {1.0/1.414213562373095048802,-1.0/(2.0*1.414213562373095048802)},
                         {1.0-1.414213562373095048802,1.0/1.414213562373095048802}};
    ierr = TSARKIMEXRegister(TSARKIMEX2E,2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembedt,2,binterpt[0],NULL);CHKERRQ(ierr);
  }
  {                             /* Optimal for linear implicit part */
    const PetscReal
      A[3][3] = {{0,0,0},
                 {0.5,0,0},
                 {0.5,0.5,0}},
      At[3][3] = {{0.25,0,0},
                  {0,0.25,0},
                  {1./3,1./3,1./3}};
    ierr = TSARKIMEXRegister(TSARKIMEXPRSSP2,2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,NULL,NULL,0,NULL,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.}},
      bembedt[4]     = {2756255671327./12835298489170.,-10771552573575./22201958757719.,9247589265047./10645013368117.,2193209047091./5459859503100.},
      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],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembedt,2,binterpt[0],NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[5][5] = {{0,0,0,0,0},
                 {1./2,0,0,0,0},
                 {11./18,1./18,0,0,0},
                 {5./6,-5./6,.5,0,0},
                 {1./4,7./4,3./4,-7./4,0}},
      At[5][5] = {{0,0,0,0,0},
                  {0,1./2,0,0,0},
                  {0,1./6,1./2,0,0},
                  {0,-1./2,1./2,1./2,0},
                  {0,3./2,-3./2,1./2,1./2}},
    *bembedt = NULL;
    ierr = TSARKIMEXRegister(TSARKIMEXARS443,3,5,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembedt,0,NULL,NULL);CHKERRQ(ierr);
  }
  {
    const PetscReal
      A[5][5] = {{0,0,0,0,0},
                 {1,0,0,0,0},
                 {4./9,2./9,0,0,0},
                 {1./4,0,3./4,0,0},
                 {1./4,0,3./5,0,0}},
      At[5][5] = {{0,0,0,0,0},
                  {.5,.5,0,0,0},
                  {5./18,-1./9,.5,0,0},
                  {.5,0,0,.5,0},
                  {.25,0,.75,-.5,.5}},
    *bembedt = NULL;
    ierr = TSARKIMEXRegister(TSARKIMEXBPR3,3,5,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembedt,0,NULL,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}},
      bembedt[6]     = {4586570599./29645900160.,0,178811875./945068544.,814220225./1159782912.,-3700637./11593932.,61727./225920.},
      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],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembedt,3,binterpt[0],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.}},
      bembedt[8]     = {-975461918565./9796059967033.,0,0,78070527104295./32432590147079.,-548382580838./3424219808633.,-33438840321285./15594753105479.,3629800801594./4656183773603.,4035322873751./18575991585200.},
      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],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembedt,3,binterpt[0],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()
@*/
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->bembedt,t->bembed);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.

  Level: developer

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

  PetscFunctionBegin;
  if (TSARKIMEXPackageInitialized) PetscFunctionReturn(0);
  TSARKIMEXPackageInitialized = PETSC_TRUE;
  ierr = TSARKIMEXRegisterAll();CHKERRQ(ierr);
  ierr = PetscObjectComposedDataRegister(&explicit_stage_time_id);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; NULL to use the last row of At)
.  ct - Abscissa of each stiff stage (dimension s, 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; NULL to use last row of At)
.  c - Non-stiff abscissa (dimension s; NULL to use row sums of A)
.  bembedt - Stiff part of completion table for embedded method (dimension s; NULL if not available)
.  bembed - Non-stiff part of completion table for embedded method (dimension s; NULL to use bembedt if provided)
.  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; 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(TSARKIMEXType name,PetscInt order,PetscInt s,
                                 const PetscReal At[],const PetscReal bt[],const PetscReal ct[],
                                 const PetscReal A[],const PetscReal b[],const PetscReal c[],
                                 const PetscReal bembedt[],const PetscReal bembed[],
                                 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->stiffly_accurate = PETSC_TRUE;
  for (i=0; i<s; i++) if (t->At[(s-1)*s+i] != t->bt[i]) t->stiffly_accurate = PETSC_FALSE;
  t->explicit_first_stage = PETSC_TRUE;
  for (i=0; i<s; i++) if (t->At[i] != 0.0) t->explicit_first_stage = PETSC_FALSE;
  /*def of FSAL can be made more precise*/
  t->FSAL_implicit = (PetscBool)(t->explicit_first_stage && t->stiffly_accurate);
  if (bembedt) {
    ierr = PetscMalloc2(s,PetscReal,&t->bembedt,s,PetscReal,&t->bembed);CHKERRQ(ierr);
    ierr = PetscMemcpy(t->bembedt,bembedt,s*sizeof(bembedt[0]));CHKERRQ(ierr);
    ierr = PetscMemcpy(t->bembed,bembed ? bembed : bembedt,s*sizeof(bembed[0]));CHKERRQ(ierr);
  }

  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__ "TSEvaluateStep_ARKIMEX"
/*
 The step completion formula is

 x1 = x0 - h bt^T YdotI + h b^T YdotRHS

 This function can be called before or after ts->vec_sol has been updated.
 Suppose we have a completion formula (bt,b) and an embedded formula (bet,be) of different order.
 We can write

 x1e = x0 - h bet^T YdotI + h be^T YdotRHS
     = x1 + h bt^T YdotI - h b^T YdotRHS - h bet^T YdotI + h be^T YdotRHS
     = x1 - h (bet - bt)^T YdotI + h (be - b)^T YdotRHS

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

  PetscFunctionBegin;
  switch (ark->status) {
  case TS_STEP_INCOMPLETE:
  case TS_STEP_PENDING:
    h = ts->time_step; break;
  case TS_STEP_COMPLETE:
    h = ts->time_step_prev; break;
  default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
  }
  if (order == tab->order) {
    if (ark->status == TS_STEP_INCOMPLETE) {
      if (!ark->imex && tab->stiffly_accurate) { /* Only the stiffly accurate implicit formula is used */
        ierr = VecCopy(ark->Y[s-1],X);CHKERRQ(ierr);
      } else { /* Use the standard completion formula (bt,b) */
        ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
        for (j=0; j<s; j++) w[j] = h*tab->bt[j];
        ierr = VecMAXPY(X,s,w,ark->YdotI);CHKERRQ(ierr);
        if (ark->imex) { /* Method is IMEX, complete the explicit formula */
          for (j=0; j<s; j++) w[j] = h*tab->b[j];
          ierr = VecMAXPY(X,s,w,ark->YdotRHS);CHKERRQ(ierr);
        }
      }
    } else {ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);}
    if (done) *done = PETSC_TRUE;
    PetscFunctionReturn(0);
  } else if (order == tab->order-1) {
    if (!tab->bembedt) goto unavailable;
    if (ark->status == TS_STEP_INCOMPLETE) { /* Complete with the embedded method (bet,be) */
      ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
      for (j=0; j<s; j++) w[j] = h*tab->bembedt[j];
      ierr = VecMAXPY(X,s,w,ark->YdotI);CHKERRQ(ierr);
      for (j=0; j<s; j++) w[j] = h*tab->bembed[j];
      ierr = VecMAXPY(X,s,w,ark->YdotRHS);CHKERRQ(ierr);
    } else {                    /* Rollback and re-complete using (bet-be,be-b) */
      ierr = VecCopy(ts->vec_sol,X);CHKERRQ(ierr);
      for (j=0; j<s; j++) w[j] = h*(tab->bembedt[j] - tab->bt[j]);
      ierr = VecMAXPY(X,tab->s,w,ark->YdotI);CHKERRQ(ierr);
      for (j=0; j<s; j++) w[j] = h*(tab->bembed[j] - tab->b[j]);
      ierr = VecMAXPY(X,s,w,ark->YdotRHS);CHKERRQ(ierr);
    }
    if (done) *done = PETSC_TRUE;
    PetscFunctionReturn(0);
  }
unavailable:
  if (done) *done = PETSC_FALSE;
  else SETERRQ3(PetscObjectComm((PetscObject)ts),PETSC_ERR_SUP,"ARKIMEX '%s' of order %D cannot evaluate step at order %D",tab->name,tab->order,order);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSRollBack_ARKIMEX"
static PetscErrorCode TSRollBack_ARKIMEX(TS ts)
{
  TS_ARKIMEX      *ark = (TS_ARKIMEX*)ts->data;
  ARKTableau      tab  = ark->tableau;
  const PetscInt  s    = tab->s;
  const PetscReal *bt = tab->bt,*b = tab->b;
  PetscScalar     *w   = ark->work;
  Vec             *YdotI = ark->YdotI,*YdotRHS = ark->YdotRHS;
  PetscInt        j;
  PetscReal       h=ts->time_step;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  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);
  ark->status   = TS_STEP_INCOMPLETE;
  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,*ct = tab->ct,*c = tab->c;
  PetscScalar     *w   = ark->work;
  Vec             *Y   = ark->Y,*YdotI = ark->YdotI,*YdotRHS = ark->YdotRHS,Ydot = ark->Ydot,Ydot0 = ark->Ydot0,W = ark->Work,Z = ark->Z;
  TSAdapt         adapt;
  SNES            snes;
  PetscInt        i,j,its,lits,reject,next_scheme;
  PetscReal       t;
  PetscReal       next_time_step;
  PetscBool       accept;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  if (ts->equation_type >= TS_EQ_IMPLICIT && tab->explicit_first_stage) {
    PetscReal valid_time;
    PetscBool isvalid;
    ierr = PetscObjectComposedDataGetReal((PetscObject)ts->vec_sol,
                                          explicit_stage_time_id,
                                          valid_time,
                                          isvalid);
    CHKERRQ(ierr);
    if (!isvalid || valid_time != ts->ptime) {
      TS        ts_start;
      SNES      snes_start;
      DM        dm;
      PetscReal atol;
      Vec       vatol;
      PetscReal rtol;
      Vec       vrtol;

      ierr = TSCreate(PetscObjectComm((PetscObject)ts),&ts_start);CHKERRQ(ierr);
      ierr = TSGetSNES(ts,&snes_start);CHKERRQ(ierr);
      ierr = TSSetSNES(ts_start,snes_start);CHKERRQ(ierr);
      ierr = TSGetDM(ts,&dm);CHKERRQ(ierr);
      ierr = TSSetDM(ts_start,dm);CHKERRQ(ierr);

      ts_start->adapt=ts->adapt;
      PetscObjectReference((PetscObject)ts_start->adapt);

      ierr = TSSetSolution(ts_start,ts->vec_sol);CHKERRQ(ierr);
      ierr = TSSetTime(ts_start,ts->ptime);CHKERRQ(ierr);
      ierr = TSSetDuration(ts_start,1,ts->ptime+ts->time_step);CHKERRQ(ierr);
      ierr = TSSetTimeStep(ts_start,ts->time_step);CHKERRQ(ierr);
      ierr = TSSetType(ts_start,TSARKIMEX);CHKERRQ(ierr);
      ierr = TSARKIMEXSetFullyImplicit(ts_start,PETSC_TRUE);CHKERRQ(ierr);
      ierr = TSARKIMEXSetType(ts_start,TSARKIMEX1BEE);CHKERRQ(ierr);
      ierr = TSSetEquationType(ts_start,ts->equation_type);CHKERRQ(ierr);
      ierr = TSGetTolerances(ts,&atol,&vatol,&rtol,&vrtol);CHKERRQ(ierr);
      ierr = TSSetTolerances(ts_start,atol,vatol,rtol,vrtol);CHKERRQ(ierr);
      ierr = TSSolve(ts_start,ts->vec_sol);CHKERRQ(ierr);
      ierr = TSGetTime(ts_start,&ts->ptime);CHKERRQ(ierr);

      ts->time_step = ts_start->time_step;
      ts->steps++;
      ierr = VecCopy(((TS_ARKIMEX*)ts_start->data)->Ydot0,Ydot0);CHKERRQ(ierr);
      ts_start->snes=NULL;
      ierr = TSSetSNES(ts,snes_start);CHKERRQ(ierr);
      ierr = SNESDestroy(&snes_start);CHKERRQ(ierr);
      ierr = TSDestroy(&ts_start);CHKERRQ(ierr);
    }
  }

  ierr           = TSGetSNES(ts,&snes);CHKERRQ(ierr);
  t              = ts->ptime;
  next_time_step = ts->time_step;
  accept         = PETSC_TRUE;
  ark->status    = TS_STEP_INCOMPLETE;


  for (reject=0; reject<ts->max_reject && !ts->reason; reject++,ts->reject++) {
    PetscReal h = ts->time_step;
    ierr = TSPreStep(ts);CHKERRQ(ierr);
    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->scoeff     = 1./At[i*s+i];
        ierr            = TSPreStage(ts,ark->stage_time);CHKERRQ(ierr);
        /* 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);
        ierr = VecScale(W, ark->scoeff/h);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          = (ts->ops->snesfunction)(snes,Y[i],W,ts);CHKERRQ(ierr);
        ierr          = SNESGetIterationNumber(snes,&its);CHKERRQ(ierr);
        ierr          = SNESGetLinearSolveIterations(snes,&lits);CHKERRQ(ierr);
        ts->snes_its += its; ts->ksp_its += lits;
        ierr          = TSGetAdapt(ts,&adapt);CHKERRQ(ierr);
        ierr          = TSAdaptCheckStage(adapt,ts,&accept);CHKERRQ(ierr);
        if (!accept) goto reject_step;
      }
      if (ts->equation_type>=TS_EQ_IMPLICIT) {
        if (i==0 && tab->explicit_first_stage) {
          ierr = VecCopy(Ydot0,YdotI[0]);CHKERRQ(ierr);
        } else {
          ierr = VecAXPBYPCZ(YdotI[i],-ark->scoeff/h,ark->scoeff/h,0,Z,Y[i]);CHKERRQ(ierr); /* Ydot = shift*(X-Z) */
        }
      } else {
        ierr = VecZeroEntries(Ydot);CHKERRQ(ierr);
        ierr = TSComputeIFunction(ts,t+h*ct[i],Y[i],Ydot,YdotI[i],ark->imex);CHKERRQ(ierr);
        ierr = VecScale(YdotI[i], -1.0);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);
        }
      }
    }
    ierr = TSEvaluateStep(ts,tab->order,ts->vec_sol,NULL);CHKERRQ(ierr);
    ark->status = TS_STEP_PENDING;

    /* Register only the current method as a candidate because we're not supporting multiple candidates yet. */
    ierr = TSGetAdapt(ts,&adapt);CHKERRQ(ierr);
    ierr = TSAdaptCandidatesClear(adapt);CHKERRQ(ierr);
    ierr = TSAdaptCandidateAdd(adapt,tab->name,tab->order,1,tab->ccfl,1.*tab->s,PETSC_TRUE);CHKERRQ(ierr);
    ierr = TSAdaptChoose(adapt,ts,ts->time_step,&next_scheme,&next_time_step,&accept);CHKERRQ(ierr);
    if (accept) {
      /* ignore next_scheme for now */
      ts->ptime    += ts->time_step;
      ts->time_step = next_time_step;
      ts->steps++;
      if (ts->equation_type>=TS_EQ_IMPLICIT) { /* save the initial slope for the next step*/
        ierr = VecCopy(YdotI[s-1],Ydot0);CHKERRQ(ierr);
      }
      ark->status = TS_STEP_COMPLETE;
      if (tab->explicit_first_stage) {
        ierr = PetscObjectComposedDataSetReal((PetscObject)ts->vec_sol,explicit_stage_time_id,ts->ptime);CHKERRQ(ierr);
      }

      break;
    } else {                    /* Roll back the current step */
      ts->ptime += next_time_step; /* This will be undone in rollback */
      ierr = TSRollBack(ts);CHKERRQ(ierr);
    }
reject_step: continue;
  }
  if (ark->status != TS_STEP_COMPLETE && !ts->reason) ts->reason = TS_DIVERGED_STEP_REJECTED;
  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       h;
  PetscReal       tt,t;
  PetscScalar     *bt,*b;
  const PetscReal *Bt = ark->tableau->binterpt,*B = ark->tableau->binterp;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  if (!Bt || !B) SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_SUP,"TSARKIMEX %s does not have an interpolation formula",ark->tableau->name);
  switch (ark->status) {
  case TS_STEP_INCOMPLETE:
  case TS_STEP_PENDING:
    h = ts->time_step;
    t = (itime - ts->ptime)/h;
    break;
  case TS_STEP_COMPLETE:
    h = ts->time_step_prev;
    t = (itime - ts->ptime)/h + 1; /* In the interval [0,1] */
    break;
  default: SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_PLIB,"Invalid TSStepStatus");
  }
  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] += h * Bt[i*pinterp+j] * tt * -1.0;
      b[i]  += h * B[i*pinterp+j] * tt;
    }
  }
  if (ark->tableau->At[0*s+0] != 0.0) SETERRQ(PetscObjectComm((PetscObject)ts),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->Ydot0);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 = PetscObjectComposeFunction((PetscObject)ts,"TSARKIMEXGetType_C",NULL);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSARKIMEXSetType_C",NULL);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSARKIMEXSetFullyImplicit_C",NULL);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}


#undef __FUNCT__
#define __FUNCT__ "TSARKIMEXGetVecs"
static PetscErrorCode TSARKIMEXGetVecs(TS ts,DM dm,Vec *Z,Vec *Ydot)
{
  TS_ARKIMEX     *ax = (TS_ARKIMEX*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (Z) {
    if (dm && dm != ts->dm) {
      ierr = DMGetNamedGlobalVector(dm,"TSARKIMEX_Z",Z);CHKERRQ(ierr);
    } else *Z = ax->Z;
  }
  if (Ydot) {
    if (dm && dm != ts->dm) {
      ierr = DMGetNamedGlobalVector(dm,"TSARKIMEX_Ydot",Ydot);CHKERRQ(ierr);
    } else *Ydot = ax->Ydot;
  }
  PetscFunctionReturn(0);
}


#undef __FUNCT__
#define __FUNCT__ "TSARKIMEXRestoreVecs"
static PetscErrorCode TSARKIMEXRestoreVecs(TS ts,DM dm,Vec *Z,Vec *Ydot)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (Z) {
    if (dm && dm != ts->dm) {
      ierr = DMRestoreNamedGlobalVector(dm,"TSARKIMEX_Z",Z);CHKERRQ(ierr);
    }
  }
  if (Ydot) {
    if (dm && dm != ts->dm) {
      ierr = DMRestoreNamedGlobalVector(dm,"TSARKIMEX_Ydot",Ydot);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;
  DM             dm,dmsave;
  Vec            Z,Ydot;
  PetscReal      shift = ark->scoeff / ts->time_step;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr   = SNESGetDM(snes,&dm);CHKERRQ(ierr);
  ierr   = TSARKIMEXGetVecs(ts,dm,&Z,&Ydot);CHKERRQ(ierr);
  ierr   = VecAXPBYPCZ(Ydot,-shift,shift,0,Z,X);CHKERRQ(ierr); /* Ydot = shift*(X-Z) */
  dmsave = ts->dm;
  ts->dm = dm;

  ierr = TSComputeIFunction(ts,ark->stage_time,X,Ydot,F,ark->imex);CHKERRQ(ierr);

  ts->dm = dmsave;
  ierr   = TSARKIMEXRestoreVecs(ts,dm,&Z,&Ydot);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;
  DM             dm,dmsave;
  Vec            Ydot;
  PetscReal      shift = ark->scoeff / ts->time_step;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = SNESGetDM(snes,&dm);CHKERRQ(ierr);
  ierr = TSARKIMEXGetVecs(ts,dm,NULL,&Ydot);CHKERRQ(ierr);
  /* ark->Ydot has already been computed in SNESTSFormFunction_ARKIMEX (SNES guarantees this) */
  dmsave = ts->dm;
  ts->dm = dm;

  ierr = TSComputeIJacobian(ts,ark->stage_time,X,Ydot,shift,A,B,str,ark->imex);CHKERRQ(ierr);

  ts->dm = dmsave;
  ierr   = TSARKIMEXRestoreVecs(ts,dm,NULL,&Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMCoarsenHook_TSARKIMEX"
static PetscErrorCode DMCoarsenHook_TSARKIMEX(DM fine,DM coarse,void *ctx)
{
  PetscFunctionBegin;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMRestrictHook_TSARKIMEX"
static PetscErrorCode DMRestrictHook_TSARKIMEX(DM fine,Mat restrct,Vec rscale,Mat inject,DM coarse,void *ctx)
{
  TS             ts = (TS)ctx;
  PetscErrorCode ierr;
  Vec            Z,Z_c;

  PetscFunctionBegin;
  ierr = TSARKIMEXGetVecs(ts,fine,&Z,NULL);CHKERRQ(ierr);
  ierr = TSARKIMEXGetVecs(ts,coarse,&Z_c,NULL);CHKERRQ(ierr);
  ierr = MatRestrict(restrct,Z,Z_c);CHKERRQ(ierr);
  ierr = VecPointwiseMult(Z_c,rscale,Z_c);CHKERRQ(ierr);
  ierr = TSARKIMEXRestoreVecs(ts,fine,&Z,NULL);CHKERRQ(ierr);
  ierr = TSARKIMEXRestoreVecs(ts,coarse,&Z_c,NULL);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}


#undef __FUNCT__
#define __FUNCT__ "DMSubDomainHook_TSARKIMEX"
static PetscErrorCode DMSubDomainHook_TSARKIMEX(DM dm,DM subdm,void *ctx)
{
  PetscFunctionBegin;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMSubDomainRestrictHook_TSARKIMEX"
static PetscErrorCode DMSubDomainRestrictHook_TSARKIMEX(DM dm,VecScatter gscat,VecScatter lscat,DM subdm,void *ctx)
{
  TS             ts = (TS)ctx;
  PetscErrorCode ierr;
  Vec            Z,Z_c;

  PetscFunctionBegin;
  ierr = TSARKIMEXGetVecs(ts,dm,&Z,NULL);CHKERRQ(ierr);
  ierr = TSARKIMEXGetVecs(ts,subdm,&Z_c,NULL);CHKERRQ(ierr);

  ierr = VecScatterBegin(gscat,Z,Z_c,INSERT_VALUES,SCATTER_FORWARD);CHKERRQ(ierr);
  ierr = VecScatterEnd(gscat,Z,Z_c,INSERT_VALUES,SCATTER_FORWARD);CHKERRQ(ierr);

  ierr = TSARKIMEXRestoreVecs(ts,dm,&Z,NULL);CHKERRQ(ierr);
  ierr = TSARKIMEXRestoreVecs(ts,subdm,&Z_c,NULL);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;
  PetscInt       s;
  PetscErrorCode ierr;
  DM             dm;

  PetscFunctionBegin;
  if (!ark->tableau) {
    ierr = TSARKIMEXSetType(ts,TSARKIMEXDefault);CHKERRQ(ierr);
  }
  tab  = ark->tableau;
  s    = tab->s;
  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->Ydot0);CHKERRQ(ierr);
  ierr = VecDuplicate(ts->vec_sol,&ark->Z);CHKERRQ(ierr);
  ierr = PetscMalloc(s*sizeof(ark->work[0]),&ark->work);CHKERRQ(ierr);
  ierr = TSGetDM(ts,&dm);CHKERRQ(ierr);
  if (dm) {
    ierr = DMCoarsenHookAdd(dm,DMCoarsenHook_TSARKIMEX,DMRestrictHook_TSARKIMEX,ts);CHKERRQ(ierr);
    ierr = DMSubDomainHookAdd(dm,DMSubDomainHook_TSARKIMEX,DMSubDomainRestrictHook_TSARKIMEX,ts);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,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;
  PetscInt       i;
  size_t         left,count;
  char           *p;

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

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

  PetscFunctionBegin;
  ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);CHKERRQ(ierr);
  if (iascii) {
    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,"Stiffly accurate: %s\n",tab->stiffly_accurate ? "yes" : "no");CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"Explicit first stage: %s\n",tab->explicit_first_stage ? "yes" : "no");CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"FSAL property: %s\n",tab->FSAL_implicit ? "yes" : "no");CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"  Nonstiff abscissa     c = %s\n",buf);CHKERRQ(ierr);
  }
  ierr = TSGetAdapt(ts,&adapt);CHKERRQ(ierr);
  ierr = TSAdaptView(adapt,viewer);CHKERRQ(ierr);
  ierr = SNESView(ts->snes,viewer);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "TSLoad_ARKIMEX"
static PetscErrorCode TSLoad_ARKIMEX(TS ts,PetscViewer viewer)
{
  PetscErrorCode ierr;
  SNES           snes;
  TSAdapt        tsadapt;

  PetscFunctionBegin;
  ierr = TSGetAdapt(ts,&tsadapt);CHKERRQ(ierr);
  ierr = TSAdaptLoad(tsadapt,viewer);CHKERRQ(ierr);
  ierr = TSGetSNES(ts,&snes);CHKERRQ(ierr);
  ierr = SNESLoad(snes,viewer);CHKERRQ(ierr);
  /* function and Jacobian context for SNES when used with TS is always ts object */
  ierr = SNESSetFunction(snes,NULL,NULL,ts);CHKERRQ(ierr);
  ierr = SNESSetJacobian(snes,NULL,NULL,NULL,ts);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(), TSARKIMEX, TSARKIMEX2D, TSARKIMEX2E, TSARKIMEXPRSSP2, TSARKIMEX3, TSARKIMEXBPR3, TSARKIMEXARS443, TSARKIMEX4, TSARKIMEX5
@*/
PetscErrorCode TSARKIMEXSetType(TS ts,TSARKIMEXType arktype)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  ierr = PetscTryMethod(ts,"TSARKIMEXSetType_C",(TS,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,TSARKIMEXType *arktype)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  ierr = PetscUseMethod(ts,"TSARKIMEXGetType_C",(TS,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);
}

#undef __FUNCT__
#define __FUNCT__ "TSARKIMEXGetType_ARKIMEX"
PetscErrorCode  TSARKIMEXGetType_ARKIMEX(TS ts,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,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(PetscObjectComm((PetscObject)ts),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);
}

/* ------------------------------------------------------------ */
/*MC
      TSARKIMEX - ODE and DAE 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 TSARKIMEX3, it can be changed with TSARKIMEXSetType() or -ts_arkimex_type

  Methods with an explicit stage can only be used with ODE in 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, TSARKIMEXPRSSP2, TSARKIMEXBPR3, TSARKIMEXType, TSARKIMEXRegister()

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

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

  ts->ops->reset          = TSReset_ARKIMEX;
  ts->ops->destroy        = TSDestroy_ARKIMEX;
  ts->ops->view           = TSView_ARKIMEX;
  ts->ops->load           = TSLoad_ARKIMEX;
  ts->ops->setup          = TSSetUp_ARKIMEX;
  ts->ops->step           = TSStep_ARKIMEX;
  ts->ops->interpolate    = TSInterpolate_ARKIMEX;
  ts->ops->evaluatestep   = TSEvaluateStep_ARKIMEX;
  ts->ops->rollback       = TSRollBack_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 = PetscObjectComposeFunction((PetscObject)ts,"TSARKIMEXGetType_C",TSARKIMEXGetType_ARKIMEX);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSARKIMEXSetType_C",TSARKIMEXSetType_ARKIMEX);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSARKIMEXSetFullyImplicit_C",TSARKIMEXSetFullyImplicit_ARKIMEX);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}