implicit/alpha/alpha1.c

/*
  Code for timestepping with implicit generalized-\alpha method
  for first order systems.
*/
#include <petsc/private/tsimpl.h> /*I   "petscts.h"   I*/

static PetscBool  cited      = PETSC_FALSE;
static const char citation[] = "@article{Jansen2000,\n"
                               "  title   = {A generalized-$\\alpha$ method for integrating the filtered {N}avier--{S}tokes equations with a stabilized finite element method},\n"
                               "  author  = {Kenneth E. Jansen and Christian H. Whiting and Gregory M. Hulbert},\n"
                               "  journal = {Computer Methods in Applied Mechanics and Engineering},\n"
                               "  volume  = {190},\n"
                               "  number  = {3--4},\n"
                               "  pages   = {305--319},\n"
                               "  year    = {2000},\n"
                               "  issn    = {0045-7825},\n"
                               "  doi     = {http://dx.doi.org/10.1016/S0045-7825(00)00203-6}\n}\n";

typedef struct {
  PetscReal stage_time;
  PetscReal shift_V;
  PetscReal scale_F;
  Vec       X0, Xa, X1;
  Vec       V0, Va, V1;

  PetscReal Alpha_m;
  PetscReal Alpha_f;
  PetscReal Gamma;
  PetscInt  order;

  Vec vec_sol_prev;
  Vec vec_lte_work;

  TSStepStatus status;
} TS_Alpha;

/* We need to transfer X0 which will be copied into sol_prev */
static PetscErrorCode TSResizeRegister_Alpha(TS ts, PetscBool reg)
{
  TS_Alpha  *th     = (TS_Alpha *)ts->data;
  const char name[] = "ts:alpha:X0";

  PetscFunctionBegin;
  if (reg && th->vec_sol_prev) {
    PetscCall(TSResizeRegisterVec(ts, name, th->X0));
  } else if (!reg) {
    PetscCall(TSResizeRetrieveVec(ts, name, &th->X0));
    PetscCall(PetscObjectReference((PetscObject)th->X0));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSAlpha_StageTime(TS ts)
{
  TS_Alpha *th      = (TS_Alpha *)ts->data;
  PetscReal t       = ts->ptime;
  PetscReal dt      = ts->time_step;
  PetscReal Alpha_m = th->Alpha_m;
  PetscReal Alpha_f = th->Alpha_f;
  PetscReal Gamma   = th->Gamma;

  PetscFunctionBegin;
  th->stage_time = t + Alpha_f * dt;
  th->shift_V    = Alpha_m / (Alpha_f * Gamma * dt);
  th->scale_F    = 1 / Alpha_f;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSAlpha_StageVecs(TS ts, Vec X)
{
  TS_Alpha *th = (TS_Alpha *)ts->data;
  Vec       X1 = X, V1 = th->V1;
  Vec       Xa = th->Xa, Va = th->Va;
  Vec       X0 = th->X0, V0 = th->V0;
  PetscReal dt      = ts->time_step;
  PetscReal Alpha_m = th->Alpha_m;
  PetscReal Alpha_f = th->Alpha_f;
  PetscReal Gamma   = th->Gamma;

  PetscFunctionBegin;
  /* V1 = 1/(Gamma*dT)*(X1-X0) + (1-1/Gamma)*V0 */
  PetscCall(VecWAXPY(V1, -1.0, X0, X1));
  PetscCall(VecAXPBY(V1, 1 - 1 / Gamma, 1 / (Gamma * dt), V0));
  /* Xa = X0 + Alpha_f*(X1-X0) */
  PetscCall(VecWAXPY(Xa, -1.0, X0, X1));
  PetscCall(VecAYPX(Xa, Alpha_f, X0));
  /* Va = V0 + Alpha_m*(V1-V0) */
  PetscCall(VecWAXPY(Va, -1.0, V0, V1));
  PetscCall(VecAYPX(Va, Alpha_m, V0));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSAlpha_SNESSolve(TS ts, Vec b, Vec x)
{
  PetscInt nits, lits;

  PetscFunctionBegin;
  PetscCall(SNESSolve(ts->snes, b, x));
  PetscCall(SNESGetIterationNumber(ts->snes, &nits));
  PetscCall(SNESGetLinearSolveIterations(ts->snes, &lits));
  ts->snes_its += nits;
  ts->ksp_its += lits;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
  Compute a consistent initial state for the generalized-alpha method.
  - Solve two successive backward Euler steps with halved time step.
  - Compute the initial time derivative using backward differences.
  - If using adaptivity, estimate the LTE of the initial step.
*/
static PetscErrorCode TSAlpha_Restart(TS ts, PetscBool *initok)
{
  TS_Alpha *th = (TS_Alpha *)ts->data;
  PetscReal time_step;
  PetscReal alpha_m, alpha_f, gamma;
  Vec       X0 = ts->vec_sol, X1, X2 = th->X1;
  PetscBool stageok;

  PetscFunctionBegin;
  PetscCall(VecDuplicate(X0, &X1));

  /* Setup backward Euler with halved time step */
  PetscCall(TSAlphaGetParams(ts, &alpha_m, &alpha_f, &gamma));
  PetscCall(TSAlphaSetParams(ts, 1, 1, 1));
  PetscCall(TSGetTimeStep(ts, &time_step));
  ts->time_step = time_step / 2;
  PetscCall(TSAlpha_StageTime(ts));
  th->stage_time = ts->ptime;
  PetscCall(VecZeroEntries(th->V0));

  /* First BE step, (t0,X0) -> (t1,X1) */
  th->stage_time += ts->time_step;
  PetscCall(VecCopy(X0, th->X0));
  PetscCall(TSPreStage(ts, th->stage_time));
  PetscCall(VecCopy(th->X0, X1));
  PetscCall(TSAlpha_SNESSolve(ts, NULL, X1));
  PetscCall(TSPostStage(ts, th->stage_time, 0, &X1));
  PetscCall(TSAdaptCheckStage(ts->adapt, ts, th->stage_time, X1, &stageok));
  if (!stageok) goto finally;

  /* Second BE step, (t1,X1) -> (t2,X2) */
  th->stage_time += ts->time_step;
  PetscCall(VecCopy(X1, th->X0));
  PetscCall(TSPreStage(ts, th->stage_time));
  PetscCall(VecCopy(th->X0, X2));
  PetscCall(TSAlpha_SNESSolve(ts, NULL, X2));
  PetscCall(TSPostStage(ts, th->stage_time, 0, &X2));
  PetscCall(TSAdaptCheckStage(ts->adapt, ts, th->stage_time, X2, &stageok));
  if (!stageok) goto finally;

  /* Compute V0 ~ dX/dt at t0 with backward differences */
  PetscCall(VecZeroEntries(th->V0));
  PetscCall(VecAXPY(th->V0, -3 / ts->time_step, X0));
  PetscCall(VecAXPY(th->V0, +4 / ts->time_step, X1));
  PetscCall(VecAXPY(th->V0, -1 / ts->time_step, X2));

  /* Rough, lower-order estimate LTE of the initial step */
  if (th->vec_lte_work) {
    PetscCall(VecZeroEntries(th->vec_lte_work));
    PetscCall(VecAXPY(th->vec_lte_work, +2, X2));
    PetscCall(VecAXPY(th->vec_lte_work, -4, X1));
    PetscCall(VecAXPY(th->vec_lte_work, +2, X0));
  }

finally:
  /* Revert TSAlpha to the initial state (t0,X0) */
  if (initok) *initok = stageok;
  PetscCall(TSSetTimeStep(ts, time_step));
  PetscCall(TSAlphaSetParams(ts, alpha_m, alpha_f, gamma));
  PetscCall(VecCopy(ts->vec_sol, th->X0));

  PetscCall(VecDestroy(&X1));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSStep_Alpha(TS ts)
{
  TS_Alpha *th         = (TS_Alpha *)ts->data;
  PetscInt  rejections = 0;
  PetscBool stageok, accept = PETSC_TRUE;
  PetscReal next_time_step = ts->time_step;

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

  if (!ts->steprollback) {
    if (th->vec_sol_prev) PetscCall(VecCopy(th->X0, th->vec_sol_prev));
    PetscCall(VecCopy(ts->vec_sol, th->X0));
    PetscCall(VecCopy(th->V1, th->V0));
  }

  th->status = TS_STEP_INCOMPLETE;
  while (!ts->reason && th->status != TS_STEP_COMPLETE) {
    if (ts->steprestart) {
      PetscCall(TSAlpha_Restart(ts, &stageok));
      if (!stageok) goto reject_step;
    }

    PetscCall(TSAlpha_StageTime(ts));
    PetscCall(VecCopy(th->X0, th->X1));
    PetscCall(TSPreStage(ts, th->stage_time));
    PetscCall(TSAlpha_SNESSolve(ts, NULL, th->X1));
    PetscCall(TSPostStage(ts, th->stage_time, 0, &th->Xa));
    PetscCall(TSAdaptCheckStage(ts->adapt, ts, th->stage_time, th->Xa, &stageok));
    if (!stageok) goto reject_step;

    th->status = TS_STEP_PENDING;
    PetscCall(VecCopy(th->X1, ts->vec_sol));
    PetscCall(TSAdaptChoose(ts->adapt, ts, ts->time_step, NULL, &next_time_step, &accept));
    th->status = accept ? TS_STEP_COMPLETE : TS_STEP_INCOMPLETE;
    if (!accept) {
      PetscCall(VecCopy(th->X0, ts->vec_sol));
      ts->time_step = next_time_step;
      goto reject_step;
    }

    ts->ptime += ts->time_step;
    ts->time_step = next_time_step;
    break;

  reject_step:
    ts->reject++;
    accept = PETSC_FALSE;
    if (!ts->reason && ++rejections > ts->max_reject && ts->max_reject >= 0) {
      ts->reason = TS_DIVERGED_STEP_REJECTED;
      PetscCall(PetscInfo(ts, "Step=%" PetscInt_FMT ", step rejections %" PetscInt_FMT " greater than current TS allowed, stopping solve\n", ts->steps, rejections));
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSEvaluateWLTE_Alpha(TS ts, NormType wnormtype, PetscInt *order, PetscReal *wlte)
{
  TS_Alpha *th = (TS_Alpha *)ts->data;
  Vec       X  = th->X1;           /* X = solution */
  Vec       Y  = th->vec_lte_work; /* Y = X + LTE  */
  PetscReal wltea, wlter;

  PetscFunctionBegin;
  if (!th->vec_sol_prev) {
    *wlte = -1;
    PetscFunctionReturn(PETSC_SUCCESS);
  }
  if (!th->vec_lte_work) {
    *wlte = -1;
    PetscFunctionReturn(PETSC_SUCCESS);
  }
  if (ts->steprestart) {
    /* th->vec_lte_work is set to the LTE in TSAlpha_Restart() */
    PetscCall(VecAXPY(Y, 1, X));
  } else {
    /* Compute LTE using backward differences with non-constant time step */
    PetscReal   h = ts->time_step, h_prev = ts->ptime - ts->ptime_prev;
    PetscReal   a = 1 + h_prev / h;
    PetscScalar scal[3];
    Vec         vecs[3];
    scal[0] = +1 / a;
    scal[1] = -1 / (a - 1);
    scal[2] = +1 / (a * (a - 1));
    vecs[0] = th->X1;
    vecs[1] = th->X0;
    vecs[2] = th->vec_sol_prev;
    PetscCall(VecCopy(X, Y));
    PetscCall(VecMAXPY(Y, 3, scal, vecs));
  }
  PetscCall(TSErrorWeightedNorm(ts, X, Y, wnormtype, wlte, &wltea, &wlter));
  if (order) *order = 2;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSRollBack_Alpha(TS ts)
{
  TS_Alpha *th = (TS_Alpha *)ts->data;

  PetscFunctionBegin;
  PetscCall(VecCopy(th->X0, ts->vec_sol));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSInterpolate_Alpha(TS ts, PetscReal t, Vec X)
{
  TS_Alpha *th = (TS_Alpha *)ts->data;
  PetscReal dt = t - ts->ptime;

  PetscFunctionBegin;
  PetscCall(VecCopy(ts->vec_sol, X));
  PetscCall(VecAXPY(X, th->Gamma * dt, th->V1));
  PetscCall(VecAXPY(X, (1 - th->Gamma) * dt, th->V0));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode SNESTSFormFunction_Alpha(PETSC_UNUSED SNES snes, Vec X, Vec F, TS ts)
{
  TS_Alpha *th = (TS_Alpha *)ts->data;
  PetscReal ta = th->stage_time;
  Vec       Xa = th->Xa, Va = th->Va;

  PetscFunctionBegin;
  PetscCall(TSAlpha_StageVecs(ts, X));
  /* F = Function(ta,Xa,Va) */
  PetscCall(TSComputeIFunction(ts, ta, Xa, Va, F, PETSC_FALSE));
  PetscCall(VecScale(F, th->scale_F));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode SNESTSFormJacobian_Alpha(PETSC_UNUSED SNES snes, PETSC_UNUSED Vec X, Mat J, Mat P, TS ts)
{
  TS_Alpha *th = (TS_Alpha *)ts->data;
  PetscReal ta = th->stage_time;
  Vec       Xa = th->Xa, Va = th->Va;
  PetscReal dVdX = th->shift_V;

  PetscFunctionBegin;
  /* J,P = Jacobian(ta,Xa,Va) */
  PetscCall(TSComputeIJacobian(ts, ta, Xa, Va, dVdX, J, P, PETSC_FALSE));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSReset_Alpha(TS ts)
{
  TS_Alpha *th = (TS_Alpha *)ts->data;

  PetscFunctionBegin;
  PetscCall(VecDestroy(&th->X0));
  PetscCall(VecDestroy(&th->Xa));
  PetscCall(VecDestroy(&th->X1));
  PetscCall(VecDestroy(&th->V0));
  PetscCall(VecDestroy(&th->Va));
  PetscCall(VecDestroy(&th->V1));
  PetscCall(VecDestroy(&th->vec_sol_prev));
  PetscCall(VecDestroy(&th->vec_lte_work));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSDestroy_Alpha(TS ts)
{
  PetscFunctionBegin;
  PetscCall(TSReset_Alpha(ts));
  PetscCall(PetscFree(ts->data));

  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSAlphaSetRadius_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSAlphaSetParams_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSAlphaGetParams_C", NULL));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSSetUp_Alpha(TS ts)
{
  TS_Alpha *th = (TS_Alpha *)ts->data;
  PetscBool match;

  PetscFunctionBegin;
  if (!th->X0) PetscCall(VecDuplicate(ts->vec_sol, &th->X0));
  PetscCall(VecDuplicate(ts->vec_sol, &th->Xa));
  PetscCall(VecDuplicate(ts->vec_sol, &th->X1));
  PetscCall(VecDuplicate(ts->vec_sol, &th->V0));
  PetscCall(VecDuplicate(ts->vec_sol, &th->Va));
  PetscCall(VecDuplicate(ts->vec_sol, &th->V1));

  PetscCall(TSGetAdapt(ts, &ts->adapt));
  PetscCall(TSAdaptCandidatesClear(ts->adapt));
  PetscCall(PetscObjectTypeCompare((PetscObject)ts->adapt, TSADAPTNONE, &match));
  if (!match) {
    PetscCall(VecDuplicate(ts->vec_sol, &th->vec_sol_prev));
    PetscCall(VecDuplicate(ts->vec_sol, &th->vec_lte_work));
  }

  PetscCall(TSGetSNES(ts, &ts->snes));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSSetFromOptions_Alpha(TS ts, PetscOptionItems *PetscOptionsObject)
{
  TS_Alpha *th = (TS_Alpha *)ts->data;

  PetscFunctionBegin;
  PetscOptionsHeadBegin(PetscOptionsObject, "Generalized-Alpha ODE solver options");
  {
    PetscBool flg;
    PetscReal radius = 1;
    PetscCall(PetscOptionsReal("-ts_alpha_radius", "Spectral radius (high-frequency dissipation)", "TSAlphaSetRadius", radius, &radius, &flg));
    if (flg) PetscCall(TSAlphaSetRadius(ts, radius));
    PetscCall(PetscOptionsReal("-ts_alpha_alpha_m", "Algorithmic parameter alpha_m", "TSAlphaSetParams", th->Alpha_m, &th->Alpha_m, NULL));
    PetscCall(PetscOptionsReal("-ts_alpha_alpha_f", "Algorithmic parameter alpha_f", "TSAlphaSetParams", th->Alpha_f, &th->Alpha_f, NULL));
    PetscCall(PetscOptionsReal("-ts_alpha_gamma", "Algorithmic parameter gamma", "TSAlphaSetParams", th->Gamma, &th->Gamma, NULL));
    PetscCall(TSAlphaSetParams(ts, th->Alpha_m, th->Alpha_f, th->Gamma));
  }
  PetscOptionsHeadEnd();
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSView_Alpha(TS ts, PetscViewer viewer)
{
  TS_Alpha *th = (TS_Alpha *)ts->data;
  PetscBool iascii;

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &iascii));
  if (iascii) PetscCall(PetscViewerASCIIPrintf(viewer, "  Alpha_m=%g, Alpha_f=%g, Gamma=%g\n", (double)th->Alpha_m, (double)th->Alpha_f, (double)th->Gamma));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSAlphaSetRadius_Alpha(TS ts, PetscReal radius)
{
  PetscReal alpha_m, alpha_f, gamma;

  PetscFunctionBegin;
  PetscCheck(radius >= 0 && radius <= 1, PetscObjectComm((PetscObject)ts), PETSC_ERR_ARG_OUTOFRANGE, "Radius %g not in range [0,1]", (double)radius);
  alpha_m = (PetscReal)0.5 * (3 - radius) / (1 + radius);
  alpha_f = 1 / (1 + radius);
  gamma   = (PetscReal)0.5 + alpha_m - alpha_f;
  PetscCall(TSAlphaSetParams(ts, alpha_m, alpha_f, gamma));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSAlphaSetParams_Alpha(TS ts, PetscReal alpha_m, PetscReal alpha_f, PetscReal gamma)
{
  TS_Alpha *th  = (TS_Alpha *)ts->data;
  PetscReal tol = 100 * PETSC_MACHINE_EPSILON;
  PetscReal res = ((PetscReal)0.5 + alpha_m - alpha_f) - gamma;

  PetscFunctionBegin;
  th->Alpha_m = alpha_m;
  th->Alpha_f = alpha_f;
  th->Gamma   = gamma;
  th->order   = (PetscAbsReal(res) < tol) ? 2 : 1;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TSAlphaGetParams_Alpha(TS ts, PetscReal *alpha_m, PetscReal *alpha_f, PetscReal *gamma)
{
  TS_Alpha *th = (TS_Alpha *)ts->data;

  PetscFunctionBegin;
  if (alpha_m) *alpha_m = th->Alpha_m;
  if (alpha_f) *alpha_f = th->Alpha_f;
  if (gamma) *gamma = th->Gamma;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*MC
  TSALPHA - ODE/DAE solver using the implicit Generalized-Alpha method {cite}`jansen_2000` {cite}`chung1993` for first-order systems

  Level: beginner

.seealso: [](ch_ts), `TS`, `TSCreate()`, `TSSetType()`, `TSAlphaSetRadius()`, `TSAlphaSetParams()`
M*/
PETSC_EXTERN PetscErrorCode TSCreate_Alpha(TS ts)
{
  TS_Alpha *th;

  PetscFunctionBegin;
  ts->ops->reset          = TSReset_Alpha;
  ts->ops->destroy        = TSDestroy_Alpha;
  ts->ops->view           = TSView_Alpha;
  ts->ops->setup          = TSSetUp_Alpha;
  ts->ops->setfromoptions = TSSetFromOptions_Alpha;
  ts->ops->step           = TSStep_Alpha;
  ts->ops->evaluatewlte   = TSEvaluateWLTE_Alpha;
  ts->ops->rollback       = TSRollBack_Alpha;
  ts->ops->interpolate    = TSInterpolate_Alpha;
  ts->ops->resizeregister = TSResizeRegister_Alpha;
  ts->ops->snesfunction   = SNESTSFormFunction_Alpha;
  ts->ops->snesjacobian   = SNESTSFormJacobian_Alpha;
  ts->default_adapt_type  = TSADAPTNONE;

  ts->usessnes = PETSC_TRUE;

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

  th->Alpha_m = 0.5;
  th->Alpha_f = 0.5;
  th->Gamma   = 0.5;
  th->order   = 2;

  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSAlphaSetRadius_C", TSAlphaSetRadius_Alpha));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSAlphaSetParams_C", TSAlphaSetParams_Alpha));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSAlphaGetParams_C", TSAlphaGetParams_Alpha));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  TSAlphaSetRadius - sets the desired spectral radius of the method for `TSALPHA`
  (i.e. high-frequency numerical damping)

  Logically Collective

  Input Parameters:
+ ts     - timestepping context
- radius - the desired spectral radius

  Options Database Key:
. -ts_alpha_radius <radius> - set alpha radius

  Level: intermediate

  Notes:
  The algorithmic parameters $\alpha_m$ and $\alpha_f$ of the generalized-$\alpha$ method can
  be computed in terms of a specified spectral radius $\rho$ in [0, 1] for infinite time step
  in order to control high-frequency numerical damping\:

  $$
  \begin{align*}
  \alpha_m = 0.5*(3-\rho)/(1+\rho) \\
  \alpha_f = 1/(1+\rho)
  \end{align*}
  $$

.seealso: [](ch_ts), `TS`, `TSALPHA`, `TSAlphaSetParams()`, `TSAlphaGetParams()`
@*/
PetscErrorCode TSAlphaSetRadius(TS ts, PetscReal radius)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts, TS_CLASSID, 1);
  PetscValidLogicalCollectiveReal(ts, radius, 2);
  PetscCheck(radius >= 0 && radius <= 1, ((PetscObject)ts)->comm, PETSC_ERR_ARG_OUTOFRANGE, "Radius %g not in range [0,1]", (double)radius);
  PetscTryMethod(ts, "TSAlphaSetRadius_C", (TS, PetscReal), (ts, radius));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  TSAlphaSetParams - sets the algorithmic parameters for `TSALPHA`

  Logically Collective

  Input Parameters:
+ ts      - timestepping context
. alpha_m - algorithmic parameter
. alpha_f - algorithmic parameter
- gamma   - algorithmic parameter

  Options Database Keys:
+ -ts_alpha_alpha_m <alpha_m> - set alpha_m
. -ts_alpha_alpha_f <alpha_f> - set alpha_f
- -ts_alpha_gamma   <gamma>   - set gamma

  Level: advanced

  Note:
  Second-order accuracy can be obtained so long as\:  $\gamma = 0.5 + \alpha_m - \alpha_f$

  Unconditional stability requires\: $\alpha_m >= \alpha_f >= 0.5$

  Backward Euler method is recovered with\: $\alpha_m = \alpha_f = \gamma = 1$

  Use of this function is normally only required to hack `TSALPHA` to use a modified
  integration scheme. Users should call `TSAlphaSetRadius()` to set the desired spectral radius
  of the methods (i.e. high-frequency damping) in order so select optimal values for these
  parameters.

.seealso: [](ch_ts), `TS`, `TSALPHA`, `TSAlphaSetRadius()`, `TSAlphaGetParams()`
@*/
PetscErrorCode TSAlphaSetParams(TS ts, PetscReal alpha_m, PetscReal alpha_f, PetscReal gamma)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts, TS_CLASSID, 1);
  PetscValidLogicalCollectiveReal(ts, alpha_m, 2);
  PetscValidLogicalCollectiveReal(ts, alpha_f, 3);
  PetscValidLogicalCollectiveReal(ts, gamma, 4);
  PetscTryMethod(ts, "TSAlphaSetParams_C", (TS, PetscReal, PetscReal, PetscReal), (ts, alpha_m, alpha_f, gamma));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  TSAlphaGetParams - gets the algorithmic parameters for `TSALPHA`

  Not Collective

  Input Parameter:
. ts - timestepping context

  Output Parameters:
+ alpha_m - algorithmic parameter
. alpha_f - algorithmic parameter
- gamma   - algorithmic parameter

  Level: advanced

  Note:
  Use of this function is normally only required to hack `TSALPHA` to use a modified
  integration scheme. Users should call `TSAlphaSetRadius()` to set the high-frequency damping
  (i.e. spectral radius of the method) in order so select optimal values for these parameters.

.seealso: [](ch_ts), `TS`, `TSALPHA`, `TSAlphaSetRadius()`, `TSAlphaSetParams()`
@*/
PetscErrorCode TSAlphaGetParams(TS ts, PetscReal *alpha_m, PetscReal *alpha_f, PetscReal *gamma)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts, TS_CLASSID, 1);
  if (alpha_m) PetscAssertPointer(alpha_m, 2);
  if (alpha_f) PetscAssertPointer(alpha_f, 3);
  if (gamma) PetscAssertPointer(gamma, 4);
  PetscUseMethod(ts, "TSAlphaGetParams_C", (TS, PetscReal *, PetscReal *, PetscReal *), (ts, alpha_m, alpha_f, gamma));
  PetscFunctionReturn(PETSC_SUCCESS);
}