xref: /petsc/src/ts/impls/glee/glee.c (revision dd8e379b23d2ef935d8131fb74f7eb73fef09263)

/*
  Code for time stepping with the General Linear with Error Estimation method

  Notes:
  The general system is written as

  Udot = F(t,U)

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

static PetscBool  cited      = PETSC_FALSE;
static const char citation[] = "@ARTICLE{Constantinescu_TR2016b,\n"
                               " author = {Constantinescu, E.M.},\n"
                               " title = {Estimating Global Errors in Time Stepping},\n"
                               " journal = {ArXiv e-prints},\n"
                               " year = 2016,\n"
                               " adsurl = {http://adsabs.harvard.edu/abs/2015arXiv150305166C}\n}\n";

static TSGLEEType TSGLEEDefaultType = TSGLEE35;
static PetscBool  TSGLEERegisterAllCalled;
static PetscBool  TSGLEEPackageInitialized;
static PetscInt   explicit_stage_time_id;

typedef struct _GLEETableau *GLEETableau;
struct _GLEETableau {
  char      *name;
  PetscInt   order;                     /* Classical approximation order of the method i*/
  PetscInt   s;                         /* Number of stages */
  PetscInt   r;                         /* Number of steps */
  PetscReal  gamma;                     /* LTE ratio */
  PetscReal *A, *B, *U, *V, *S, *F, *c; /* Tableau */
  PetscReal *Fembed;                    /* Embedded final method coefficients */
  PetscReal *Ferror;                    /* Coefficients for computing error   */
  PetscReal *Serror;                    /* Coefficients for initializing the error   */
  PetscInt   pinterp;                   /* Interpolation order */
  PetscReal *binterp;                   /* Interpolation coefficients */
  PetscReal  ccfl;                      /* Placeholder for CFL coefficient relative to forward Euler  */
};
typedef struct _GLEETableauLink *GLEETableauLink;
struct _GLEETableauLink {
  struct _GLEETableau tab;
  GLEETableauLink     next;
};
static GLEETableauLink GLEETableauList;

typedef struct {
  GLEETableau  tableau;
  Vec         *Y;         /* Solution vector (along with auxiliary solution y~ or eps) */
  Vec         *X;         /* Temporary solution vector */
  Vec         *YStage;    /* Stage values */
  Vec         *YdotStage; /* Stage right-hand side */
  Vec          W;         /* Right-hand-side for implicit stage solve */
  Vec          Ydot;      /* Work vector holding Ydot during residual evaluation */
  Vec          yGErr;     /* Vector holding the global error after a step is completed */
  PetscScalar *swork;     /* Scalar work (size of the number of stages)*/
  PetscScalar *rwork;     /* Scalar work (size of the number of steps)*/
  PetscReal    scoeff;    /* shift = scoeff/dt */
  PetscReal    stage_time;
  TSStepStatus status;
} TS_GLEE;

/*MC
     TSGLEE23 - Second order three stage GLEE method

     This method has three stages.
     s = 3, r = 2

     Level: advanced

.seealso: [](ch_ts), `TSGLEE`
M*/
/*MC
     TSGLEE24 - Second order four stage GLEE method

     This method has four stages.
     s = 4, r = 2

     Level: advanced

.seealso: [](ch_ts), `TSGLEE`
M*/
/*MC
     TSGLEE25i - Second order five stage GLEE method

     This method has five stages.
     s = 5, r = 2

     Level: advanced

.seealso: [](ch_ts), `TSGLEE`
M*/
/*MC
     TSGLEE35  - Third order five stage GLEE method

     This method has five stages.
     s = 5, r = 2

     Level: advanced

.seealso: [](ch_ts), `TSGLEE`
M*/
/*MC
     TSGLEEEXRK2A  - Second order six stage GLEE method

     This method has six stages.
     s = 6, r = 2

     Level: advanced

.seealso: [](ch_ts), `TSGLEE`
M*/
/*MC
     TSGLEERK32G1  - Third order eight stage GLEE method

     This method has eight stages.
     s = 8, r = 2

     Level: advanced

.seealso: [](ch_ts), `TSGLEE`
M*/
/*MC
     TSGLEERK285EX  - Second order nine stage GLEE method

     This method has nine stages.
     s = 9, r = 2

     Level: advanced

.seealso: [](ch_ts), `TSGLEE`
M*/

/*@C
  TSGLEERegisterAll - Registers all of the General Linear with Error Estimation methods in `TSGLEE`

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

  Level: advanced

.seealso: [](ch_ts), `TSGLEERegisterDestroy()`
@*/
PetscErrorCode TSGLEERegisterAll(void)
{
  PetscFunctionBegin;
  if (TSGLEERegisterAllCalled) PetscFunctionReturn(PETSC_SUCCESS);
  TSGLEERegisterAllCalled = PETSC_TRUE;

  {
#define GAMMA 0.5
    /* y-eps form */
    const PetscInt  p = 1, s = 3, r = 2;
    const PetscReal A[3][3] =
      {
        {1.0, 0,   0  },
        {0,   0.5, 0  },
        {0,   0.5, 0.5}
    },
                    B[2][3] = {{1.0, 0, 0}, {-2.0, 1.0, 1.0}}, U[3][2] = {{1.0, 0}, {1.0, 0.5}, {1.0, 0.5}}, V[2][2] = {{1, 0}, {0, 1}}, S[2] = {1, 0}, F[2] = {1, 0}, Fembed[2] = {1, 1 - GAMMA}, Ferror[2] = {0, 1}, Serror[2] = {1, 0};
    PetscCall(TSGLEERegister(TSGLEEi1, p, s, r, GAMMA, &A[0][0], &B[0][0], &U[0][0], &V[0][0], S, F, NULL, Fembed, Ferror, Serror, 0, NULL));
  }
  {
#undef GAMMA
#define GAMMA 0.0
    /* y-eps form */
    const PetscInt  p = 2, s = 3, r = 2;
    const PetscReal A[3][3] =
      {
        {0,    0,    0},
        {1,    0,    0},
        {0.25, 0.25, 0}
    },
                    B[2][3] = {{1.0 / 12.0, 1.0 / 12.0, 5.0 / 6.0}, {1.0 / 12.0, 1.0 / 12.0, -1.0 / 6.0}}, U[3][2] = {{1, 0}, {1, 10}, {1, -1}}, V[2][2] = {{1, 0}, {0, 1}}, S[2] = {1, 0}, F[2] = {1, 0}, Fembed[2] = {1, 1 - GAMMA}, Ferror[2] = {0, 1}, Serror[2] = {1, 0};
    PetscCall(TSGLEERegister(TSGLEE23, p, s, r, GAMMA, &A[0][0], &B[0][0], &U[0][0], &V[0][0], S, F, NULL, Fembed, Ferror, Serror, 0, NULL));
  }
  {
#undef GAMMA
#define GAMMA 0.0
    /* y-y~ form */
    const PetscInt  p = 2, s = 4, r = 2;
    const PetscReal A[4][4] =
      {
        {0,            0,            0,            0},
        {0.75,         0,            0,            0},
        {0.25,         29.0 / 60.0,  0,            0},
        {-21.0 / 44.0, 145.0 / 44.0, -20.0 / 11.0, 0}
    },
                    B[2][4] = {{109.0 / 275.0, 58.0 / 75.0, -37.0 / 110.0, 1.0 / 6.0}, {3.0 / 11.0, 0, 75.0 / 88.0, -1.0 / 8.0}}, U[4][2] = {{0, 1}, {75.0 / 58.0, -17.0 / 58.0}, {0, 1}, {0, 1}}, V[2][2] = {{1, 0}, {0, 1}}, S[2] = {1, 1}, F[2] = {1, 0}, Fembed[2] = {0, 1}, Ferror[2] = {-1.0 / (1.0 - GAMMA), 1.0 / (1.0 - GAMMA)}, Serror[2] = {1.0 - GAMMA, 1.0};
    PetscCall(TSGLEERegister(TSGLEE24, p, s, r, GAMMA, &A[0][0], &B[0][0], &U[0][0], &V[0][0], S, F, NULL, Fembed, Ferror, Serror, 0, NULL));
  }
  {
#undef GAMMA
#define GAMMA 0.0
    /* y-y~ form */
    const PetscInt  p = 2, s = 5, r = 2;
    const PetscReal A[5][5] =
      {
        {0,                       0,                       0,                       0,                      0},
        {-0.94079244066783383269, 0,                       0,                       0,                      0},
        {0.64228187778301907108,  0.10915356933958500042,  0,                       0,                      0},
        {-0.51764297742287450812, 0.74414270351096040738,  -0.71404164927824538121, 0,                      0},
        {-0.44696561556825969206, -0.76768425657590196518, 0.20111608138142987881,  0.93828186737840469796, 0}
    },
                    B[2][5] = {{-0.029309178948150356153, -0.49671981884013874923, 0.34275801517650053274, 0.32941112623949194988, 0.85385985637229662276}, {0.78133219686062535272, 0.074238691892675897635, 0.57957363498384957966, -0.24638502829674959968, -0.18875949544040123033}}, U[5][2] = {{0.16911424754448327735, 0.83088575245551672265}, {0.53638465733199574340, 0.46361534266800425660}, {0.39901579167169582526, 0.60098420832830417474}, {0.87689005530618575480, 0.12310994469381424520}, {0.99056100455550913009, 0.0094389954444908699092}}, V[2][2] = {{1, 0}, {0, 1}}, S[2] = {1, 1}, F[2] = {1, 0}, Fembed[2] = {0, 1}, Ferror[2] = {-1.0 / (1.0 - GAMMA), 1.0 / (1.0 - GAMMA)}, Serror[2] = {1.0 - GAMMA, 1.0};
    PetscCall(TSGLEERegister(TSGLEE25I, p, s, r, GAMMA, &A[0][0], &B[0][0], &U[0][0], &V[0][0], S, F, NULL, Fembed, Ferror, Serror, 0, NULL));
  }
  {
#undef GAMMA
#define GAMMA 0.0
    /* y-y~ form */
    const PetscInt  p = 3, s = 5, r = 2;
    const PetscReal A[5][5] =
      {
        {0,                                                0,                                                 0,                                                 0,                                               0},
        {-2169604947363702313.0 / 24313474998937147335.0,  0,                                                 0,                                                 0,                                               0},
        {46526746497697123895.0 / 94116917485856474137.0,  -10297879244026594958.0 / 49199457603717988219.0,  0,                                                 0,                                               0},
        {23364788935845982499.0 / 87425311444725389446.0,  -79205144337496116638.0 / 148994349441340815519.0, 40051189859317443782.0 / 36487615018004984309.0,   0,                                               0},
        {42089522664062539205.0 / 124911313006412840286.0, -15074384760342762939.0 / 137927286865289746282.0, -62274678522253371016.0 / 125918573676298591413.0, 13755475729852471739.0 / 79257927066651693390.0, 0}
    },
                    B[2][5] = {{61546696837458703723.0 / 56982519523786160813.0, -55810892792806293355.0 / 206957624151308356511.0, 24061048952676379087.0 / 158739347956038723465.0, 3577972206874351339.0 / 7599733370677197135.0, -59449832954780563947.0 / 137360038685338563670.0}, {-9738262186984159168.0 / 99299082461487742983.0, -32797097931948613195.0 / 61521565616362163366.0, 42895514606418420631.0 / 71714201188501437336.0, 22608567633166065068.0 / 55371917805607957003.0, 94655809487476459565.0 / 151517167160302729021.0}}, U[5][2] = {{70820309139834661559.0 / 80863923579509469826.0, 10043614439674808267.0 / 80863923579509469826.0}, {161694774978034105510.0 / 106187653640211060371.0, -55507121337823045139.0 / 106187653640211060371.0}, {78486094644566264568.0 / 88171030896733822981.0, 9684936252167558413.0 / 88171030896733822981.0}, {65394922146334854435.0 / 84570853840405479554.0, 19175931694070625119.0 / 84570853840405479554.0}, {8607282770183754108.0 / 108658046436496925911.0, 100050763666313171803.0 / 108658046436496925911.0}}, V[2][2] = {{1, 0}, {0, 1}}, S[2] = {1, 1}, F[2] = {1, 0}, Fembed[2] = {0, 1}, Ferror[2] = {-1.0 / (1.0 - GAMMA), 1.0 / (1.0 - GAMMA)}, Serror[2] = {1.0 - GAMMA, 1.0};
    PetscCall(TSGLEERegister(TSGLEE35, p, s, r, GAMMA, &A[0][0], &B[0][0], &U[0][0], &V[0][0], S, F, NULL, Fembed, Ferror, Serror, 0, NULL));
  }
  {
#undef GAMMA
#define GAMMA 0.25
    /* y-eps form */
    const PetscInt  p = 2, s = 6, r = 2;
    const PetscReal A[6][6] =
      {
        {0, 0, 0,    0,    0,   0},
        {1, 0, 0,    0,    0,   0},
        {0, 0, 0,    0,    0,   0},
        {0, 0, 0.5,  0,    0,   0},
        {0, 0, 0.25, 0.25, 0,   0},
        {0, 0, 0.25, 0.25, 0.5, 0}
    },
                    B[2][6] = {{0.5, 0.5, 0, 0, 0, 0}, {-2.0 / 3.0, -2.0 / 3.0, 1.0 / 3.0, 1.0 / 3.0, 1.0 / 3.0, 1.0 / 3.0}}, U[6][2] = {{1, 0}, {1, 0}, {1, 0.75}, {1, 0.75}, {1, 0.75}, {1, 0.75}}, V[2][2] = {{1, 0}, {0, 1}}, S[2] = {1, 0}, F[2] = {1, 0}, Fembed[2] = {1, 1 - GAMMA}, Ferror[2] = {0, 1}, Serror[2] = {1, 0};
    PetscCall(TSGLEERegister(TSGLEEEXRK2A, p, s, r, GAMMA, &A[0][0], &B[0][0], &U[0][0], &V[0][0], S, F, NULL, Fembed, Ferror, Serror, 0, NULL));
  }
  {
#undef GAMMA
#define GAMMA 0.0
    /* y-eps form */
    const PetscInt  p = 3, s = 8, r = 2;
    const PetscReal A[8][8] =
      {
        {0,           0,          0,          0,          0,         0,         0,         0},
        {0.5,         0,          0,          0,          0,         0,         0,         0},
        {-1,          2,          0,          0,          0,         0,         0,         0},
        {1.0 / 6.0,   2.0 / 3.0,  1.0 / 6.0,  0,          0,         0,         0,         0},
        {0,           0,          0,          0,          0,         0,         0,         0},
        {-7.0 / 24.0, 1.0 / 3.0,  1.0 / 12.0, -1.0 / 8.0, 0.5,       0,         0,         0},
        {7.0 / 6.0,   -4.0 / 3.0, -1.0 / 3.0, 0.5,        -1.0,      2.0,       0,         0},
        {0,           0,          0,          0,          1.0 / 6.0, 2.0 / 3.0, 1.0 / 6.0, 0}
    },
                    B[2][8] = {{1.0 / 6.0, 2.0 / 3.0, 1.0 / 6.0, 0, 0, 0, 0, 0}, {-1.0 / 6.0, -2.0 / 3.0, -1.0 / 6.0, 0, 1.0 / 6.0, 2.0 / 3.0, 1.0 / 6.0, 0}}, U[8][2] = {{1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 1}, {1, 1}, {1, 1}, {1, 1}}, V[2][2] = {{1, 0}, {0, 1}}, S[2] = {1, 0}, F[2] = {1, 0}, Fembed[2] = {1, 1 - GAMMA}, Ferror[2] = {0, 1}, Serror[2] = {1, 0};
    PetscCall(TSGLEERegister(TSGLEERK32G1, p, s, r, GAMMA, &A[0][0], &B[0][0], &U[0][0], &V[0][0], S, F, NULL, Fembed, Ferror, Serror, 0, NULL));
  }
  {
#undef GAMMA
#define GAMMA 0.25
    /* y-eps form */
    const PetscInt  p = 2, s = 9, r = 2;
    const PetscReal A[9][9] =
      {
        {0,                    0,                    0, 0,                     0,                    0,                    0,                     0,                    0},
        {0.585786437626904966, 0,                    0, 0,                     0,                    0,                    0,                     0,                    0},
        {0.149999999999999994, 0.849999999999999978, 0, 0,                     0,                    0,                    0,                     0,                    0},
        {0,                    0,                    0, 0,                     0,                    0,                    0,                     0,                    0},
        {0,                    0,                    0, 0.292893218813452483,  0,                    0,                    0,                     0,                    0},
        {0,                    0,                    0, 0.0749999999999999972, 0.424999999999999989, 0,                    0,                     0,                    0},
        {0,                    0,                    0, 0.176776695296636893,  0.176776695296636893, 0.146446609406726241, 0,                     0,                    0},
        {0,                    0,                    0, 0.176776695296636893,  0.176776695296636893, 0.146446609406726241, 0.292893218813452483,  0,                    0},
        {0,                    0,                    0, 0.176776695296636893,  0.176776695296636893, 0.146446609406726241, 0.0749999999999999972, 0.424999999999999989, 0}
    },
                    B[2][9] = {{0.353553390593273786, 0.353553390593273786, 0.292893218813452483, 0, 0, 0, 0, 0, 0}, {-0.471404520791031678, -0.471404520791031678, -0.390524291751269959, 0.235702260395515839, 0.235702260395515839, 0.195262145875634979, 0.235702260395515839, 0.235702260395515839, 0.195262145875634979}}, U[9][2] = {{1, 0}, {1, 0}, {1, 0}, {1, 0.75}, {1, 0.75}, {1, 0.75}, {1, 0.75}, {1, 0.75}, {1, 0.75}}, V[2][2] = {{1, 0}, {0, 1}}, S[2] = {1, 0}, F[2] = {1, 0}, Fembed[2] = {1, 1 - GAMMA}, Ferror[2] = {0, 1}, Serror[2] = {1, 0};
    PetscCall(TSGLEERegister(TSGLEERK285EX, p, s, r, GAMMA, &A[0][0], &B[0][0], &U[0][0], &V[0][0], S, F, NULL, Fembed, Ferror, Serror, 0, NULL));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  TSGLEERegisterDestroy - Frees the list of schemes that were registered by `TSGLEERegister()`.

  Not Collective

  Level: advanced

.seealso: [](ch_ts), `TSGLEERegister()`, `TSGLEERegisterAll()`
@*/
PetscErrorCode TSGLEERegisterDestroy(void)
{
  GLEETableauLink link;

  PetscFunctionBegin;
  while ((link = GLEETableauList)) {
    GLEETableau t   = &link->tab;
    GLEETableauList = link->next;
    PetscCall(PetscFree5(t->A, t->B, t->U, t->V, t->c));
    PetscCall(PetscFree2(t->S, t->F));
    PetscCall(PetscFree(t->Fembed));
    PetscCall(PetscFree(t->Ferror));
    PetscCall(PetscFree(t->Serror));
    PetscCall(PetscFree(t->binterp));
    PetscCall(PetscFree(t->name));
    PetscCall(PetscFree(link));
  }
  TSGLEERegisterAllCalled = PETSC_FALSE;
  PetscFunctionReturn(PETSC_SUCCESS);
}

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

  Level: developer

.seealso: [](ch_ts), `PetscInitialize()`
@*/
PetscErrorCode TSGLEEInitializePackage(void)
{
  PetscFunctionBegin;
  if (TSGLEEPackageInitialized) PetscFunctionReturn(PETSC_SUCCESS);
  TSGLEEPackageInitialized = PETSC_TRUE;
  PetscCall(TSGLEERegisterAll());
  PetscCall(PetscObjectComposedDataRegister(&explicit_stage_time_id));
  PetscCall(PetscRegisterFinalize(TSGLEEFinalizePackage));
  PetscFunctionReturn(PETSC_SUCCESS);
}

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

  Level: developer

.seealso: [](ch_ts), `PetscFinalize()`
@*/
PetscErrorCode TSGLEEFinalizePackage(void)
{
  PetscFunctionBegin;
  TSGLEEPackageInitialized = PETSC_FALSE;
  PetscCall(TSGLEERegisterDestroy());
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  TSGLEERegister - register a new `TSGLEE` 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   - order of method
. s       - number of stages
. r       - number of steps
. gamma   - LTE ratio
. A       - stage coefficients (dimension s*s, row-major)
. B       - step completion coefficients (dimension r*s, row-major)
. U       - method coefficients (dimension s*r, row-major)
. V       - method coefficients (dimension r*r, row-major)
. S       - starting coefficients
. F       - finishing coefficients
. c       - abscissa (dimension s; NULL to use row sums of A)
. Fembed  - step completion coefficients for embedded method
. Ferror  - error computation coefficients
. Serror  - error initialization coefficients
. pinterp - order of interpolation (0 if unavailable)
- binterp - array of interpolation coefficients (NULL if unavailable)

  Level: advanced

  Note:
  Several `TSGLEE` methods are provided, this function is only needed to create new methods.

.seealso: [](ch_ts), `TSGLEE`
@*/
PetscErrorCode TSGLEERegister(TSGLEEType name, PetscInt order, PetscInt s, PetscInt r, PetscReal gamma, const PetscReal A[], const PetscReal B[], const PetscReal U[], const PetscReal V[], const PetscReal S[], const PetscReal F[], const PetscReal c[], const PetscReal Fembed[], const PetscReal Ferror[], const PetscReal Serror[], PetscInt pinterp, const PetscReal binterp[])
{
  GLEETableauLink link;
  GLEETableau     t;
  PetscInt        i, j;

  PetscFunctionBegin;
  PetscCall(TSGLEEInitializePackage());
  PetscCall(PetscNew(&link));
  t = &link->tab;
  PetscCall(PetscStrallocpy(name, &t->name));
  t->order = order;
  t->s     = s;
  t->r     = r;
  t->gamma = gamma;
  PetscCall(PetscMalloc5(s * s, &t->A, r * r, &t->V, s, &t->c, r * s, &t->B, s * r, &t->U));
  PetscCall(PetscMalloc2(r, &t->S, r, &t->F));
  PetscCall(PetscArraycpy(t->A, A, s * s));
  PetscCall(PetscArraycpy(t->B, B, r * s));
  PetscCall(PetscArraycpy(t->U, U, s * r));
  PetscCall(PetscArraycpy(t->V, V, r * r));
  PetscCall(PetscArraycpy(t->S, S, r));
  PetscCall(PetscArraycpy(t->F, F, r));
  if (c) {
    PetscCall(PetscArraycpy(t->c, c, s));
  } else {
    for (i = 0; i < s; i++)
      for (j = 0, t->c[i] = 0; j < s; j++) t->c[i] += A[i * s + j];
  }
  PetscCall(PetscMalloc1(r, &t->Fembed));
  PetscCall(PetscMalloc1(r, &t->Ferror));
  PetscCall(PetscMalloc1(r, &t->Serror));
  PetscCall(PetscArraycpy(t->Fembed, Fembed, r));
  PetscCall(PetscArraycpy(t->Ferror, Ferror, r));
  PetscCall(PetscArraycpy(t->Serror, Serror, r));
  t->pinterp = pinterp;
  PetscCall(PetscMalloc1(s * pinterp, &t->binterp));
  PetscCall(PetscArraycpy(t->binterp, binterp, s * pinterp));

  link->next      = GLEETableauList;
  GLEETableauList = link;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSEvaluateStep_GLEE(TS ts, PetscInt order, Vec X, PetscBool *done)
{
  TS_GLEE     *glee = (TS_GLEE *)ts->data;
  GLEETableau  tab  = glee->tableau;
  PetscReal    h, *B = tab->B, *V = tab->V, *F = tab->F, *Fembed = tab->Fembed;
  PetscInt     s = tab->s, r = tab->r, i, j;
  Vec         *Y = glee->Y, *YdotStage = glee->YdotStage;
  PetscScalar *ws = glee->swork, *wr = glee->rwork;

  PetscFunctionBegin;
  switch (glee->status) {
  case TS_STEP_INCOMPLETE:
  case TS_STEP_PENDING:
    h = ts->time_step;
    break;
  case TS_STEP_COMPLETE:
    h = ts->ptime - ts->ptime_prev;
    break;
  default:
    SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_PLIB, "Invalid TSStepStatus");
  }

  if (order == tab->order) {
    /* Note: Irrespective of whether status is TS_STEP_INCOMPLETE
             or TS_STEP_COMPLETE, glee->X has the solution at the
             beginning of the time step. So no need to roll-back.
    */
    if (glee->status == TS_STEP_INCOMPLETE) {
      for (i = 0; i < r; i++) {
        PetscCall(VecZeroEntries(Y[i]));
        for (j = 0; j < r; j++) wr[j] = V[i * r + j];
        PetscCall(VecMAXPY(Y[i], r, wr, glee->X));
        for (j = 0; j < s; j++) ws[j] = h * B[i * s + j];
        PetscCall(VecMAXPY(Y[i], s, ws, YdotStage));
      }
      PetscCall(VecZeroEntries(X));
      for (j = 0; j < r; j++) wr[j] = F[j];
      PetscCall(VecMAXPY(X, r, wr, Y));
    } else PetscCall(VecCopy(ts->vec_sol, X));
    PetscFunctionReturn(PETSC_SUCCESS);

  } else if (order == tab->order - 1) {
    /* Complete with the embedded method (Fembed) */
    for (i = 0; i < r; i++) {
      PetscCall(VecZeroEntries(Y[i]));
      for (j = 0; j < r; j++) wr[j] = V[i * r + j];
      PetscCall(VecMAXPY(Y[i], r, wr, glee->X));
      for (j = 0; j < s; j++) ws[j] = h * B[i * s + j];
      PetscCall(VecMAXPY(Y[i], s, ws, YdotStage));
    }
    PetscCall(VecZeroEntries(X));
    for (j = 0; j < r; j++) wr[j] = Fembed[j];
    PetscCall(VecMAXPY(X, r, wr, Y));

    if (done) *done = PETSC_TRUE;
    PetscFunctionReturn(PETSC_SUCCESS);
  }
  if (done) *done = PETSC_FALSE;
  else SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "GLEE '%s' of order %" PetscInt_FMT " cannot evaluate step at order %" PetscInt_FMT, tab->name, tab->order, order);
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSStep_GLEE(TS ts)
{
  TS_GLEE       *glee = (TS_GLEE *)ts->data;
  GLEETableau    tab  = glee->tableau;
  const PetscInt s = tab->s, r = tab->r;
  PetscReal     *A = tab->A, *U = tab->U, *F = tab->F, *c = tab->c;
  Vec           *Y = glee->Y, *X = glee->X, *YStage = glee->YStage, *YdotStage = glee->YdotStage, W = glee->W;
  SNES           snes;
  PetscScalar   *ws = glee->swork, *wr = glee->rwork;
  TSAdapt        adapt;
  PetscInt       i, j, reject, next_scheme, its, lits;
  PetscReal      next_time_step;
  PetscReal      t;
  PetscBool      accept;

  PetscFunctionBegin;
  PetscCall(PetscCitationsRegister(citation, &cited));

  for (i = 0; i < r; i++) PetscCall(VecCopy(Y[i], X[i]));

  PetscCall(TSGetSNES(ts, &snes));
  next_time_step = ts->time_step;
  t              = ts->ptime;
  accept         = PETSC_TRUE;
  glee->status   = TS_STEP_INCOMPLETE;

  for (reject = 0; reject < ts->max_reject && !ts->reason; reject++, ts->reject++) {
    PetscReal h = ts->time_step;
    PetscCall(TSPreStep(ts));

    for (i = 0; i < s; i++) {
      glee->stage_time = t + h * c[i];
      PetscCall(TSPreStage(ts, glee->stage_time));

      if (A[i * s + i] == 0) { /* Explicit stage */
        PetscCall(VecZeroEntries(YStage[i]));
        for (j = 0; j < r; j++) wr[j] = U[i * r + j];
        PetscCall(VecMAXPY(YStage[i], r, wr, X));
        for (j = 0; j < i; j++) ws[j] = h * A[i * s + j];
        PetscCall(VecMAXPY(YStage[i], i, ws, YdotStage));
      } else { /* Implicit stage */
        glee->scoeff = 1.0 / A[i * s + i];
        /* compute right-hand side */
        PetscCall(VecZeroEntries(W));
        for (j = 0; j < r; j++) wr[j] = U[i * r + j];
        PetscCall(VecMAXPY(W, r, wr, X));
        for (j = 0; j < i; j++) ws[j] = h * A[i * s + j];
        PetscCall(VecMAXPY(W, i, ws, YdotStage));
        PetscCall(VecScale(W, glee->scoeff / h));
        /* set initial guess */
        PetscCall(VecCopy(i > 0 ? YStage[i - 1] : ts->vec_sol, YStage[i]));
        /* solve for this stage */
        PetscCall(SNESSolve(snes, W, YStage[i]));
        PetscCall(SNESGetIterationNumber(snes, &its));
        PetscCall(SNESGetLinearSolveIterations(snes, &lits));
        ts->snes_its += its;
        ts->ksp_its += lits;
      }
      PetscCall(TSGetAdapt(ts, &adapt));
      PetscCall(TSAdaptCheckStage(adapt, ts, glee->stage_time, YStage[i], &accept));
      if (!accept) goto reject_step;
      PetscCall(TSPostStage(ts, glee->stage_time, i, YStage));
      PetscCall(TSComputeRHSFunction(ts, t + h * c[i], YStage[i], YdotStage[i]));
    }
    PetscCall(TSEvaluateStep(ts, tab->order, ts->vec_sol, NULL));
    glee->status = TS_STEP_PENDING;

    /* Register only the current method as a candidate because we're not supporting multiple candidates yet. */
    PetscCall(TSGetAdapt(ts, &adapt));
    PetscCall(TSAdaptCandidatesClear(adapt));
    PetscCall(TSAdaptCandidateAdd(adapt, tab->name, tab->order, 1, tab->ccfl, (PetscReal)tab->s, PETSC_TRUE));
    PetscCall(TSAdaptChoose(adapt, ts, ts->time_step, &next_scheme, &next_time_step, &accept));
    if (accept) {
      /* ignore next_scheme for now */
      ts->ptime += ts->time_step;
      ts->time_step = next_time_step;
      glee->status  = TS_STEP_COMPLETE;
      /* compute and store the global error */
      /* Note: this is not needed if TSAdaptGLEE is not used */
      PetscCall(TSGetTimeError(ts, 0, &glee->yGErr));
      PetscCall(PetscObjectComposedDataSetReal((PetscObject)ts->vec_sol, explicit_stage_time_id, ts->ptime));
      break;
    } else { /* Roll back the current step */
      for (j = 0; j < r; j++) wr[j] = F[j];
      PetscCall(VecMAXPY(ts->vec_sol, r, wr, X));
      ts->time_step = next_time_step;
      glee->status  = TS_STEP_INCOMPLETE;
    }
  reject_step:
    continue;
  }
  if (glee->status != TS_STEP_COMPLETE && !ts->reason) ts->reason = TS_DIVERGED_STEP_REJECTED;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSInterpolate_GLEE(TS ts, PetscReal itime, Vec X)
{
  TS_GLEE         *glee = (TS_GLEE *)ts->data;
  PetscInt         s = glee->tableau->s, pinterp = glee->tableau->pinterp, i, j;
  PetscReal        h, tt, t;
  PetscScalar     *b;
  const PetscReal *B = glee->tableau->binterp;

  PetscFunctionBegin;
  PetscCheck(B, PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "TSGLEE %s does not have an interpolation formula", glee->tableau->name);
  switch (glee->status) {
  case TS_STEP_INCOMPLETE:
  case TS_STEP_PENDING:
    h = ts->time_step;
    t = (itime - ts->ptime) / h;
    break;
  case TS_STEP_COMPLETE:
    h = ts->ptime - ts->ptime_prev;
    t = (itime - ts->ptime) / h + 1; /* In the interval [0,1] */
    break;
  default:
    SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_PLIB, "Invalid TSStepStatus");
  }
  PetscCall(PetscMalloc1(s, &b));
  for (i = 0; i < s; i++) b[i] = 0;
  for (j = 0, tt = t; j < pinterp; j++, tt *= t) {
    for (i = 0; i < s; i++) b[i] += h * B[i * pinterp + j] * tt;
  }
  PetscCall(VecCopy(glee->YStage[0], X));
  PetscCall(VecMAXPY(X, s, b, glee->YdotStage));
  PetscCall(PetscFree(b));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*------------------------------------------------------------*/
static PetscErrorCode TSReset_GLEE(TS ts)
{
  TS_GLEE *glee = (TS_GLEE *)ts->data;
  PetscInt s, r;

  PetscFunctionBegin;
  if (!glee->tableau) PetscFunctionReturn(PETSC_SUCCESS);
  s = glee->tableau->s;
  r = glee->tableau->r;
  PetscCall(VecDestroyVecs(r, &glee->Y));
  PetscCall(VecDestroyVecs(r, &glee->X));
  PetscCall(VecDestroyVecs(s, &glee->YStage));
  PetscCall(VecDestroyVecs(s, &glee->YdotStage));
  PetscCall(VecDestroy(&glee->Ydot));
  PetscCall(VecDestroy(&glee->yGErr));
  PetscCall(VecDestroy(&glee->W));
  PetscCall(PetscFree2(glee->swork, glee->rwork));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSGLEEGetVecs(TS ts, DM dm, Vec *Ydot)
{
  TS_GLEE *glee = (TS_GLEE *)ts->data;

  PetscFunctionBegin;
  if (Ydot) {
    if (dm && dm != ts->dm) {
      PetscCall(DMGetNamedGlobalVector(dm, "TSGLEE_Ydot", Ydot));
    } else *Ydot = glee->Ydot;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSGLEERestoreVecs(TS ts, DM dm, Vec *Ydot)
{
  PetscFunctionBegin;
  if (Ydot) {
    if (dm && dm != ts->dm) PetscCall(DMRestoreNamedGlobalVector(dm, "TSGLEE_Ydot", Ydot));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
  This defines the nonlinear equation that is to be solved with SNES
*/
static PetscErrorCode SNESTSFormFunction_GLEE(SNES snes, Vec X, Vec F, TS ts)
{
  TS_GLEE  *glee = (TS_GLEE *)ts->data;
  DM        dm, dmsave;
  Vec       Ydot;
  PetscReal shift = glee->scoeff / ts->time_step;

  PetscFunctionBegin;
  PetscCall(SNESGetDM(snes, &dm));
  PetscCall(TSGLEEGetVecs(ts, dm, &Ydot));
  /* Set Ydot = shift*X */
  PetscCall(VecCopy(X, Ydot));
  PetscCall(VecScale(Ydot, shift));
  dmsave = ts->dm;
  ts->dm = dm;

  PetscCall(TSComputeIFunction(ts, glee->stage_time, X, Ydot, F, PETSC_FALSE));

  ts->dm = dmsave;
  PetscCall(TSGLEERestoreVecs(ts, dm, &Ydot));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode SNESTSFormJacobian_GLEE(SNES snes, Vec X, Mat A, Mat B, TS ts)
{
  TS_GLEE  *glee = (TS_GLEE *)ts->data;
  DM        dm, dmsave;
  Vec       Ydot;
  PetscReal shift = glee->scoeff / ts->time_step;

  PetscFunctionBegin;
  PetscCall(SNESGetDM(snes, &dm));
  PetscCall(TSGLEEGetVecs(ts, dm, &Ydot));
  /* glee->Ydot has already been computed in SNESTSFormFunction_GLEE (SNES guarantees this) */
  dmsave = ts->dm;
  ts->dm = dm;

  PetscCall(TSComputeIJacobian(ts, glee->stage_time, X, Ydot, shift, A, B, PETSC_FALSE));

  ts->dm = dmsave;
  PetscCall(TSGLEERestoreVecs(ts, dm, &Ydot));
  PetscFunctionReturn(PETSC_SUCCESS);
}

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

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

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

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

static PetscErrorCode TSSetUp_GLEE(TS ts)
{
  TS_GLEE    *glee = (TS_GLEE *)ts->data;
  GLEETableau tab;
  PetscInt    s, r;
  DM          dm;

  PetscFunctionBegin;
  if (!glee->tableau) PetscCall(TSGLEESetType(ts, TSGLEEDefaultType));
  tab = glee->tableau;
  s   = tab->s;
  r   = tab->r;
  PetscCall(VecDuplicateVecs(ts->vec_sol, r, &glee->Y));
  PetscCall(VecDuplicateVecs(ts->vec_sol, r, &glee->X));
  PetscCall(VecDuplicateVecs(ts->vec_sol, s, &glee->YStage));
  PetscCall(VecDuplicateVecs(ts->vec_sol, s, &glee->YdotStage));
  PetscCall(VecDuplicate(ts->vec_sol, &glee->Ydot));
  PetscCall(VecDuplicate(ts->vec_sol, &glee->yGErr));
  PetscCall(VecZeroEntries(glee->yGErr));
  PetscCall(VecDuplicate(ts->vec_sol, &glee->W));
  PetscCall(PetscMalloc2(s, &glee->swork, r, &glee->rwork));
  PetscCall(TSGetDM(ts, &dm));
  PetscCall(DMCoarsenHookAdd(dm, DMCoarsenHook_TSGLEE, DMRestrictHook_TSGLEE, ts));
  PetscCall(DMSubDomainHookAdd(dm, DMSubDomainHook_TSGLEE, DMSubDomainRestrictHook_TSGLEE, ts));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSStartingMethod_GLEE(TS ts)
{
  TS_GLEE    *glee = (TS_GLEE *)ts->data;
  GLEETableau tab  = glee->tableau;
  PetscInt    r    = tab->r, i;
  PetscReal  *S    = tab->S;

  PetscFunctionBegin;
  for (i = 0; i < r; i++) {
    PetscCall(VecZeroEntries(glee->Y[i]));
    PetscCall(VecAXPY(glee->Y[i], S[i], ts->vec_sol));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*------------------------------------------------------------*/

static PetscErrorCode TSSetFromOptions_GLEE(TS ts, PetscOptionItems *PetscOptionsObject)
{
  char gleetype[256];

  PetscFunctionBegin;
  PetscOptionsHeadBegin(PetscOptionsObject, "GLEE ODE solver options");
  {
    GLEETableauLink link;
    PetscInt        count, choice;
    PetscBool       flg;
    const char    **namelist;

    PetscCall(PetscStrncpy(gleetype, TSGLEEDefaultType, sizeof(gleetype)));
    for (link = GLEETableauList, count = 0; link; link = link->next, count++);
    PetscCall(PetscMalloc1(count, (char ***)&namelist));
    for (link = GLEETableauList, count = 0; link; link = link->next, count++) namelist[count] = link->tab.name;
    PetscCall(PetscOptionsEList("-ts_glee_type", "Family of GLEE method", "TSGLEESetType", (const char *const *)namelist, count, gleetype, &choice, &flg));
    PetscCall(TSGLEESetType(ts, flg ? namelist[choice] : gleetype));
    PetscCall(PetscFree(namelist));
  }
  PetscOptionsHeadEnd();
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSView_GLEE(TS ts, PetscViewer viewer)
{
  TS_GLEE    *glee = (TS_GLEE *)ts->data;
  GLEETableau tab  = glee->tableau;
  PetscBool   iascii;

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &iascii));
  if (iascii) {
    TSGLEEType gleetype;
    char       buf[512];
    PetscCall(TSGLEEGetType(ts, &gleetype));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  GLEE type %s\n", gleetype));
    PetscCall(PetscFormatRealArray(buf, sizeof(buf), "% 8.6f", tab->s, tab->c));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  Abscissa     c = %s\n", buf));
    /* Note: print out r as well */
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSLoad_GLEE(TS ts, PetscViewer viewer)
{
  SNES    snes;
  TSAdapt tsadapt;

  PetscFunctionBegin;
  PetscCall(TSGetAdapt(ts, &tsadapt));
  PetscCall(TSAdaptLoad(tsadapt, viewer));
  PetscCall(TSGetSNES(ts, &snes));
  PetscCall(SNESLoad(snes, viewer));
  /* function and Jacobian context for SNES when used with TS is always ts object */
  PetscCall(SNESSetFunction(snes, NULL, NULL, ts));
  PetscCall(SNESSetJacobian(snes, NULL, NULL, NULL, ts));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  TSGLEESetType - Set the type of `TSGLEE` scheme

  Logically Collective

  Input Parameters:
+ ts       - timestepping context
- gleetype - type of `TSGLEE` scheme

  Level: intermediate

.seealso: [](ch_ts), `TSGLEEGetType()`, `TSGLEE`
@*/
PetscErrorCode TSGLEESetType(TS ts, TSGLEEType gleetype)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts, TS_CLASSID, 1);
  PetscAssertPointer(gleetype, 2);
  PetscTryMethod(ts, "TSGLEESetType_C", (TS, TSGLEEType), (ts, gleetype));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  TSGLEEGetType - Get the type of `TSGLEE` scheme

  Logically Collective

  Input Parameter:
. ts - timestepping context

  Output Parameter:
. gleetype - type of `TSGLEE` scheme

  Level: intermediate

.seealso: [](ch_ts), `TSGLEE`, `TSGLEESetType()`
@*/
PetscErrorCode TSGLEEGetType(TS ts, TSGLEEType *gleetype)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts, TS_CLASSID, 1);
  PetscUseMethod(ts, "TSGLEEGetType_C", (TS, TSGLEEType *), (ts, gleetype));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSGLEEGetType_GLEE(TS ts, TSGLEEType *gleetype)
{
  TS_GLEE *glee = (TS_GLEE *)ts->data;

  PetscFunctionBegin;
  if (!glee->tableau) PetscCall(TSGLEESetType(ts, TSGLEEDefaultType));
  *gleetype = glee->tableau->name;
  PetscFunctionReturn(PETSC_SUCCESS);
}
static PetscErrorCode TSGLEESetType_GLEE(TS ts, TSGLEEType gleetype)
{
  TS_GLEE        *glee = (TS_GLEE *)ts->data;
  PetscBool       match;
  GLEETableauLink link;

  PetscFunctionBegin;
  if (glee->tableau) {
    PetscCall(PetscStrcmp(glee->tableau->name, gleetype, &match));
    if (match) PetscFunctionReturn(PETSC_SUCCESS);
  }
  for (link = GLEETableauList; link; link = link->next) {
    PetscCall(PetscStrcmp(link->tab.name, gleetype, &match));
    if (match) {
      PetscCall(TSReset_GLEE(ts));
      glee->tableau = &link->tab;
      PetscFunctionReturn(PETSC_SUCCESS);
    }
  }
  SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_ARG_UNKNOWN_TYPE, "Could not find '%s'", gleetype);
}

static PetscErrorCode TSGetStages_GLEE(TS ts, PetscInt *ns, Vec **Y)
{
  TS_GLEE *glee = (TS_GLEE *)ts->data;

  PetscFunctionBegin;
  if (ns) *ns = glee->tableau->s;
  if (Y) *Y = glee->YStage;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSGetSolutionComponents_GLEE(TS ts, PetscInt *n, Vec *Y)
{
  TS_GLEE    *glee = (TS_GLEE *)ts->data;
  GLEETableau tab  = glee->tableau;

  PetscFunctionBegin;
  if (!Y) *n = tab->r;
  else {
    if ((*n >= 0) && (*n < tab->r)) {
      PetscCall(VecCopy(glee->Y[*n], *Y));
    } else SETERRQ(PetscObjectComm((PetscObject)ts), PETSC_ERR_ARG_OUTOFRANGE, "Second argument (%" PetscInt_FMT ") out of range[0,%" PetscInt_FMT "].", *n, tab->r - 1);
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSGetAuxSolution_GLEE(TS ts, Vec *X)
{
  TS_GLEE     *glee = (TS_GLEE *)ts->data;
  GLEETableau  tab  = glee->tableau;
  PetscReal   *F    = tab->Fembed;
  PetscInt     r    = tab->r;
  Vec         *Y    = glee->Y;
  PetscScalar *wr   = glee->rwork;
  PetscInt     i;

  PetscFunctionBegin;
  PetscCall(VecZeroEntries(*X));
  for (i = 0; i < r; i++) wr[i] = F[i];
  PetscCall(VecMAXPY(*X, r, wr, Y));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSGetTimeError_GLEE(TS ts, PetscInt n, Vec *X)
{
  TS_GLEE     *glee = (TS_GLEE *)ts->data;
  GLEETableau  tab  = glee->tableau;
  PetscReal   *F    = tab->Ferror;
  PetscInt     r    = tab->r;
  Vec         *Y    = glee->Y;
  PetscScalar *wr   = glee->rwork;
  PetscInt     i;

  PetscFunctionBegin;
  PetscCall(VecZeroEntries(*X));
  if (n == 0) {
    for (i = 0; i < r; i++) wr[i] = F[i];
    PetscCall(VecMAXPY(*X, r, wr, Y));
  } else if (n == -1) {
    *X = glee->yGErr;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSSetTimeError_GLEE(TS ts, Vec X)
{
  TS_GLEE    *glee = (TS_GLEE *)ts->data;
  GLEETableau tab  = glee->tableau;
  PetscReal  *S    = tab->Serror;
  PetscInt    r    = tab->r, i;
  Vec        *Y    = glee->Y;

  PetscFunctionBegin;
  PetscCheck(r == 2, PetscObjectComm((PetscObject)ts), PETSC_ERR_SUP, "TSSetTimeError_GLEE not supported for '%s' with r=%" PetscInt_FMT ".", tab->name, tab->r);
  for (i = 1; i < r; i++) {
    PetscCall(VecCopy(ts->vec_sol, Y[i]));
    PetscCall(VecAXPBY(Y[i], S[0], S[1], X));
    PetscCall(VecCopy(X, glee->yGErr));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSDestroy_GLEE(TS ts)
{
  PetscFunctionBegin;
  PetscCall(TSReset_GLEE(ts));
  if (ts->dm) {
    PetscCall(DMCoarsenHookRemove(ts->dm, DMCoarsenHook_TSGLEE, DMRestrictHook_TSGLEE, ts));
    PetscCall(DMSubDomainHookRemove(ts->dm, DMSubDomainHook_TSGLEE, DMSubDomainRestrictHook_TSGLEE, ts));
  }
  PetscCall(PetscFree(ts->data));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSGLEEGetType_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSGLEESetType_C", NULL));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* ------------------------------------------------------------ */
/*MC
      TSGLEE - ODE and DAE solver using General Linear with Error Estimation schemes

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

  Level: beginner

  Note:
  The default is `TSGLEE35`, it can be changed with `TSGLEESetType()` or -ts_glee_type

.seealso: [](ch_ts), `TSCreate()`, `TS`, `TSSetType()`, `TSGLEESetType()`, `TSGLEEGetType()`,
          `TSGLEE23`, `TTSGLEE24`, `TSGLEE35`, `TSGLEE25I`, `TSGLEEEXRK2A`,
          `TSGLEERK32G1`, `TSGLEERK285EX`, `TSGLEEType`, `TSGLEERegister()`, `TSType`
M*/
PETSC_EXTERN PetscErrorCode TSCreate_GLEE(TS ts)
{
  TS_GLEE *th;

  PetscFunctionBegin;
  PetscCall(TSGLEEInitializePackage());

  ts->ops->reset                 = TSReset_GLEE;
  ts->ops->destroy               = TSDestroy_GLEE;
  ts->ops->view                  = TSView_GLEE;
  ts->ops->load                  = TSLoad_GLEE;
  ts->ops->setup                 = TSSetUp_GLEE;
  ts->ops->step                  = TSStep_GLEE;
  ts->ops->interpolate           = TSInterpolate_GLEE;
  ts->ops->evaluatestep          = TSEvaluateStep_GLEE;
  ts->ops->setfromoptions        = TSSetFromOptions_GLEE;
  ts->ops->getstages             = TSGetStages_GLEE;
  ts->ops->snesfunction          = SNESTSFormFunction_GLEE;
  ts->ops->snesjacobian          = SNESTSFormJacobian_GLEE;
  ts->ops->getsolutioncomponents = TSGetSolutionComponents_GLEE;
  ts->ops->getauxsolution        = TSGetAuxSolution_GLEE;
  ts->ops->gettimeerror          = TSGetTimeError_GLEE;
  ts->ops->settimeerror          = TSSetTimeError_GLEE;
  ts->ops->startingmethod        = TSStartingMethod_GLEE;
  ts->default_adapt_type         = TSADAPTGLEE;

  ts->usessnes = PETSC_TRUE;

  PetscCall(PetscNew(&th));
  ts->data = (void *)th;

  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSGLEEGetType_C", TSGLEEGetType_GLEE));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSGLEESetType_C", TSGLEESetType_GLEE));
  PetscFunctionReturn(PETSC_SUCCESS);
}