implicit/alpha/alpha2.c

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

static PetscBool  cited      = PETSC_FALSE;
static const char citation[] = "@article{Chung1993,\n"
                               "  title   = {A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation: The Generalized-$\\alpha$ Method},\n"
                               "  author  = {J. Chung, G. M. Hubert},\n"
                               "  journal = {ASME Journal of Applied Mechanics},\n"
                               "  volume  = {60},\n"
                               "  number  = {2},\n"
                               "  pages   = {371--375},\n"
                               "  year    = {1993},\n"
                               "  issn    = {0021-8936},\n"
                               "  doi     = {http://dx.doi.org/10.1115/1.2900803}\n}\n";

typedef struct {
  PetscReal stage_time;
  PetscReal shift_V;
  PetscReal shift_A;
  PetscReal scale_F;
  Vec       X0, Xa, X1;
  Vec       V0, Va, V1;
  Vec       A0, Aa, A1;

  Vec vec_dot;

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

  Vec vec_sol_prev;
  Vec vec_dot_prev;
  Vec vec_lte_work[2];

  TSStepStatus status;
} TS_Alpha;

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;
  PetscReal Beta    = th->Beta;

  PetscFunctionBegin;
  th->stage_time = t + Alpha_f * dt;
  th->shift_V    = Gamma / (dt * Beta);
  th->shift_A    = Alpha_m / (Alpha_f * dt * dt * Beta);
  th->scale_F    = 1 / Alpha_f;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSAlpha_StageVecs(TS ts, Vec X)
{
  TS_Alpha *th = (TS_Alpha *)ts->data;
  Vec       X1 = X, V1 = th->V1, A1 = th->A1;
  Vec       Xa = th->Xa, Va = th->Va, Aa = th->Aa;
  Vec       X0 = th->X0, V0 = th->V0, A0 = th->A0;
  PetscReal dt      = ts->time_step;
  PetscReal Alpha_m = th->Alpha_m;
  PetscReal Alpha_f = th->Alpha_f;
  PetscReal Gamma   = th->Gamma;
  PetscReal Beta    = th->Beta;

  PetscFunctionBegin;
  /* A1 = ... */
  PetscCall(VecWAXPY(A1, -1.0, X0, X1));
  PetscCall(VecAXPY(A1, -dt, V0));
  PetscCall(VecAXPBY(A1, -(1 - 2 * Beta) / (2 * Beta), 1 / (dt * dt * Beta), A0));
  /* V1 = ... */
  PetscCall(VecWAXPY(V1, (1.0 - Gamma) / Gamma, A0, A1));
  PetscCall(VecAYPX(V1, dt * Gamma, V0));
  /* Xa = X0 + Alpha_f*(X1-X0) */
  PetscCall(VecWAXPY(Xa, -1.0, X0, X1));
  PetscCall(VecAYPX(Xa, Alpha_f, X0));
  /* Va = V0 + Alpha_f*(V1-V0) */
  PetscCall(VecWAXPY(Va, -1.0, V0, V1));
  PetscCall(VecAYPX(Va, Alpha_f, V0));
  /* Aa = A0 + Alpha_m*(A1-A0) */
  PetscCall(VecWAXPY(Aa, -1.0, A0, A1));
  PetscCall(VecAYPX(Aa, Alpha_m, A0));
  PetscFunctionReturn(0);
}

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(0);
}

/*
  Compute a consistent initial state for the generalized-alpha method.
  - Solve two successive backward Euler steps with halved time step.
  - Compute the initial second 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, beta;
  Vec       X0 = ts->vec_sol, X1, X2 = th->X1;
  Vec       V0 = ts->vec_dot, V1, V2 = th->V1;
  PetscBool stageok;

  PetscFunctionBegin;
  PetscCall(VecDuplicate(X0, &X1));
  PetscCall(VecDuplicate(V0, &V1));

  /* Setup backward Euler with halved time step */
  PetscCall(TSAlpha2GetParams(ts, &alpha_m, &alpha_f, &gamma, &beta));
  PetscCall(TSAlpha2SetParams(ts, 1, 1, 1, 0.5));
  PetscCall(TSGetTimeStep(ts, &time_step));
  ts->time_step = time_step / 2;
  PetscCall(TSAlpha_StageTime(ts));
  th->stage_time = ts->ptime;
  PetscCall(VecZeroEntries(th->A0));

  /* First BE step, (t0,X0,V0) -> (t1,X1,V1) */
  th->stage_time += ts->time_step;
  PetscCall(VecCopy(X0, th->X0));
  PetscCall(VecCopy(V0, th->V0));
  PetscCall(TSPreStage(ts, th->stage_time));
  PetscCall(VecCopy(th->X0, X1));
  PetscCall(TSAlpha_SNESSolve(ts, NULL, X1));
  PetscCall(VecCopy(th->V1, V1));
  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,V1) -> (t2,X2,V2) */
  th->stage_time += ts->time_step;
  PetscCall(VecCopy(X1, th->X0));
  PetscCall(VecCopy(V1, th->V0));
  PetscCall(TSPreStage(ts, th->stage_time));
  PetscCall(VecCopy(th->X0, X2));
  PetscCall(TSAlpha_SNESSolve(ts, NULL, X2));
  PetscCall(VecCopy(th->V1, V2));
  PetscCall(TSPostStage(ts, th->stage_time, 0, &X2));
  PetscCall(TSAdaptCheckStage(ts->adapt, ts, th->stage_time, X1, &stageok));
  if (!stageok) goto finally;

  /* Compute A0 ~ dV/dt at t0 with backward differences */
  PetscCall(VecZeroEntries(th->A0));
  PetscCall(VecAXPY(th->A0, -3 / ts->time_step, V0));
  PetscCall(VecAXPY(th->A0, +4 / ts->time_step, V1));
  PetscCall(VecAXPY(th->A0, -1 / ts->time_step, V2));

  /* Rough, lower-order estimate LTE of the initial step */
  if (th->vec_lte_work[0]) {
    PetscCall(VecZeroEntries(th->vec_lte_work[0]));
    PetscCall(VecAXPY(th->vec_lte_work[0], +2, X2));
    PetscCall(VecAXPY(th->vec_lte_work[0], -4, X1));
    PetscCall(VecAXPY(th->vec_lte_work[0], +2, X0));
  }
  if (th->vec_lte_work[1]) {
    PetscCall(VecZeroEntries(th->vec_lte_work[1]));
    PetscCall(VecAXPY(th->vec_lte_work[1], +2, V2));
    PetscCall(VecAXPY(th->vec_lte_work[1], -4, V1));
    PetscCall(VecAXPY(th->vec_lte_work[1], +2, V0));
  }

finally:
  /* Revert TSAlpha to the initial state (t0,X0,V0) */
  if (initok) *initok = stageok;
  PetscCall(TSSetTimeStep(ts, time_step));
  PetscCall(TSAlpha2SetParams(ts, alpha_m, alpha_f, gamma, beta));
  PetscCall(VecCopy(ts->vec_sol, th->X0));
  PetscCall(VecCopy(ts->vec_dot, th->V0));

  PetscCall(VecDestroy(&X1));
  PetscCall(VecDestroy(&V1));
  PetscFunctionReturn(0);
}

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));
    if (th->vec_dot_prev) PetscCall(VecCopy(th->V0, th->vec_dot_prev));
    PetscCall(VecCopy(ts->vec_sol, th->X0));
    PetscCall(VecCopy(ts->vec_dot, th->V0));
    PetscCall(VecCopy(th->A1, th->A0));
  }

  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(VecCopy(th->V1, ts->vec_dot));
    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));
      PetscCall(VecCopy(th->V0, ts->vec_dot));
      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(0);
}

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       V  = th->V1;              /* V = solution */
  Vec       Y  = th->vec_lte_work[0]; /* Y = X + LTE  */
  Vec       Z  = th->vec_lte_work[1]; /* Z = V + LTE  */
  PetscReal enormX, enormV, enormXa, enormVa, enormXr, enormVr;

  PetscFunctionBegin;
  if (!th->vec_sol_prev) {
    *wlte = -1;
    PetscFunctionReturn(0);
  }
  if (!th->vec_dot_prev) {
    *wlte = -1;
    PetscFunctionReturn(0);
  }
  if (!th->vec_lte_work[0]) {
    *wlte = -1;
    PetscFunctionReturn(0);
  }
  if (!th->vec_lte_work[1]) {
    *wlte = -1;
    PetscFunctionReturn(0);
  }
  if (ts->steprestart) {
    /* th->vec_lte_prev is set to the LTE in TSAlpha_Restart() */
    PetscCall(VecAXPY(Y, 1, X));
    PetscCall(VecAXPY(Z, 1, V));
  } 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         vecX[3], vecV[3];
    scal[0] = +1 / a;
    scal[1] = -1 / (a - 1);
    scal[2] = +1 / (a * (a - 1));
    vecX[0] = th->X1;
    vecX[1] = th->X0;
    vecX[2] = th->vec_sol_prev;
    vecV[0] = th->V1;
    vecV[1] = th->V0;
    vecV[2] = th->vec_dot_prev;
    PetscCall(VecCopy(X, Y));
    PetscCall(VecMAXPY(Y, 3, scal, vecX));
    PetscCall(VecCopy(V, Z));
    PetscCall(VecMAXPY(Z, 3, scal, vecV));
  }
  /* XXX ts->atol and ts->vatol are not appropriate for computing enormV */
  PetscCall(TSErrorWeightedNorm(ts, X, Y, wnormtype, &enormX, &enormXa, &enormXr));
  PetscCall(TSErrorWeightedNorm(ts, V, Z, wnormtype, &enormV, &enormVa, &enormVr));
  if (wnormtype == NORM_2) *wlte = PetscSqrtReal(PetscSqr(enormX) / 2 + PetscSqr(enormV) / 2);
  else *wlte = PetscMax(enormX, enormV);
  if (order) *order = 2;
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  PetscCall(VecCopy(th->X0, ts->vec_sol));
  PetscCall(VecCopy(th->V0, ts->vec_dot));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  PetscCall(VecCopy(ts->vec_dot,V));
  PetscCall(VecAXPY(V,dt*(1-th->Gamma),th->A0));
  PetscCall(VecAXPY(V,dt*th->Gamma,th->A1));
  PetscCall(VecCopy(ts->vec_sol,X));
  PetscCall(VecAXPY(X,dt,V));
  PetscCall(VecAXPY(X,dt*dt*((PetscReal)0.5-th->Beta),th->A0));
  PetscCall(VecAXPY(X,dt*dt*th->Beta,th->A1));
  PetscFunctionReturn(0);
}
*/

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, Aa = th->Aa;

  PetscFunctionBegin;
  PetscCall(TSAlpha_StageVecs(ts, X));
  /* F = Function(ta,Xa,Va,Aa) */
  PetscCall(TSComputeI2Function(ts, ta, Xa, Va, Aa, F));
  PetscCall(VecScale(F, th->scale_F));
  PetscFunctionReturn(0);
}

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, Aa = th->Aa;
  PetscReal dVdX = th->shift_V, dAdX = th->shift_A;

  PetscFunctionBegin;
  /* J,P = Jacobian(ta,Xa,Va,Aa) */
  PetscCall(TSComputeI2Jacobian(ts, ta, Xa, Va, Aa, dVdX, dAdX, J, P));
  PetscFunctionReturn(0);
}

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->A0));
  PetscCall(VecDestroy(&th->Aa));
  PetscCall(VecDestroy(&th->A1));
  PetscCall(VecDestroy(&th->vec_sol_prev));
  PetscCall(VecDestroy(&th->vec_dot_prev));
  PetscCall(VecDestroy(&th->vec_lte_work[0]));
  PetscCall(VecDestroy(&th->vec_lte_work[1]));
  PetscFunctionReturn(0);
}

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

  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSAlpha2SetRadius_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSAlpha2SetParams_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSAlpha2GetParams_C", NULL));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  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(VecDuplicate(ts->vec_sol, &th->A0));
  PetscCall(VecDuplicate(ts->vec_sol, &th->Aa));
  PetscCall(VecDuplicate(ts->vec_sol, &th->A1));

  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_dot_prev));
    PetscCall(VecDuplicate(ts->vec_sol, &th->vec_lte_work[0]));
    PetscCall(VecDuplicate(ts->vec_sol, &th->vec_lte_work[1]));
  }

  PetscCall(TSGetSNES(ts, &ts->snes));
  PetscFunctionReturn(0);
}

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)", "TSAlpha2SetRadius", radius, &radius, &flg));
    if (flg) PetscCall(TSAlpha2SetRadius(ts, radius));
    PetscCall(PetscOptionsReal("-ts_alpha_alpha_m", "Algorithmic parameter alpha_m", "TSAlpha2SetParams", th->Alpha_m, &th->Alpha_m, NULL));
    PetscCall(PetscOptionsReal("-ts_alpha_alpha_f", "Algorithmic parameter alpha_f", "TSAlpha2SetParams", th->Alpha_f, &th->Alpha_f, NULL));
    PetscCall(PetscOptionsReal("-ts_alpha_gamma", "Algorithmic parameter gamma", "TSAlpha2SetParams", th->Gamma, &th->Gamma, NULL));
    PetscCall(PetscOptionsReal("-ts_alpha_beta", "Algorithmic parameter beta", "TSAlpha2SetParams", th->Beta, &th->Beta, NULL));
    PetscCall(TSAlpha2SetParams(ts, th->Alpha_m, th->Alpha_f, th->Gamma, th->Beta));
  }
  PetscOptionsHeadEnd();
  PetscFunctionReturn(0);
}

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, Beta=%g\n", (double)th->Alpha_m, (double)th->Alpha_f, (double)th->Gamma, (double)th->Beta));
  PetscFunctionReturn(0);
}

static PetscErrorCode TSAlpha2SetRadius_Alpha(TS ts, PetscReal radius)
{
  PetscReal alpha_m, alpha_f, gamma, beta;

  PetscFunctionBegin;
  PetscCheck(radius >= 0 && radius <= 1, PetscObjectComm((PetscObject)ts), PETSC_ERR_ARG_OUTOFRANGE, "Radius %g not in range [0,1]", (double)radius);
  alpha_m = (2 - radius) / (1 + radius);
  alpha_f = 1 / (1 + radius);
  gamma   = (PetscReal)0.5 + alpha_m - alpha_f;
  beta    = (PetscReal)0.5 * (1 + alpha_m - alpha_f);
  beta *= beta;
  PetscCall(TSAlpha2SetParams(ts, alpha_m, alpha_f, gamma, beta));
  PetscFunctionReturn(0);
}

static PetscErrorCode TSAlpha2SetParams_Alpha(TS ts, PetscReal alpha_m, PetscReal alpha_f, PetscReal gamma, PetscReal beta)
{
  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->Beta    = beta;
  th->order   = (PetscAbsReal(res) < tol) ? 2 : 1;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSAlpha2GetParams_Alpha(TS ts, PetscReal *alpha_m, PetscReal *alpha_f, PetscReal *gamma, PetscReal *beta)
{
  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;
  if (beta) *beta = th->Beta;
  PetscFunctionReturn(0);
}

/*MC
      TSALPHA2 - ODE/DAE solver using the implicit Generalized-Alpha method
                 for second-order systems

  Level: beginner

  References:
. * - J. Chung, G.M.Hubert. "A Time Integration Algorithm for Structural
  Dynamics with Improved Numerical Dissipation: The Generalized-alpha
  Method" ASME Journal of Applied Mechanics, 60, 371:375, 1993.

.seealso: `TS`, `TSCreate()`, `TSSetType()`, `TSAlpha2SetRadius()`, `TSAlpha2SetParams()`
M*/
PETSC_EXTERN PetscErrorCode TSCreate_Alpha2(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->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->Beta    = 0.25;
  th->order   = 2;

  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSAlpha2SetRadius_C", TSAlpha2SetRadius_Alpha));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSAlpha2SetParams_C", TSAlpha2SetParams_Alpha));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts, "TSAlpha2GetParams_C", TSAlpha2GetParams_Alpha));
  PetscFunctionReturn(0);
}

/*@
  TSAlpha2SetRadius - sets the desired spectral radius of the method
                      (i.e. high-frequency numerical damping)

  Logically Collective on TS

  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:
    \alpha_m = (2-\rho)/(1+\rho)
    \alpha_f = 1/(1+\rho)

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

  Options Database:
.  -ts_alpha_radius <radius> - set the desired spectral radius

  Level: intermediate

.seealso: `TSAlpha2SetParams()`, `TSAlpha2GetParams()`
@*/
PetscErrorCode TSAlpha2SetRadius(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, "TSAlpha2SetRadius_C", (TS, PetscReal), (ts, radius));
  PetscFunctionReturn(0);
}

/*@
  TSAlpha2SetParams - sets the algorithmic parameters for TSALPHA2

  Logically Collective on TS

  Second-order accuracy can be obtained so long as:
    \gamma = 1/2 + alpha_m - alpha_f
    \beta  = 1/4 (1 + alpha_m - alpha_f)^2

  Unconditional stability requires:
    \alpha_m >= \alpha_f >= 1/2

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

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

  Note:
  Use of this function is normally only required to hack TSALPHA2 to
  use a modified integration scheme. Users should call
  TSAlpha2SetRadius() to set the desired spectral radius of the methods
  (i.e. high-frequency damping) in order so select optimal values for
  these parameters.

  Level: advanced

.seealso: `TSAlpha2SetRadius()`, `TSAlpha2GetParams()`
@*/
PetscErrorCode TSAlpha2SetParams(TS ts, PetscReal alpha_m, PetscReal alpha_f, PetscReal gamma, PetscReal beta)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts, TS_CLASSID, 1);
  PetscValidLogicalCollectiveReal(ts, alpha_m, 2);
  PetscValidLogicalCollectiveReal(ts, alpha_f, 3);
  PetscValidLogicalCollectiveReal(ts, gamma, 4);
  PetscValidLogicalCollectiveReal(ts, beta, 5);
  PetscTryMethod(ts, "TSAlpha2SetParams_C", (TS, PetscReal, PetscReal, PetscReal, PetscReal), (ts, alpha_m, alpha_f, gamma, beta));
  PetscFunctionReturn(0);
}

/*@
  TSAlpha2GetParams - gets the algorithmic parameters for TSALPHA2

  Not Collective

  Input Parameter:
. ts - timestepping context

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

  Note:
  Use of this function is normally only required to hack TSALPHA2 to
  use a modified integration scheme. Users should call
  TSAlpha2SetRadius() to set the high-frequency damping (i.e. spectral
  radius of the method) in order so select optimal values for these
  parameters.

  Level: advanced

.seealso: `TSAlpha2SetRadius()`, `TSAlpha2SetParams()`
@*/
PetscErrorCode TSAlpha2GetParams(TS ts, PetscReal *alpha_m, PetscReal *alpha_f, PetscReal *gamma, PetscReal *beta)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts, TS_CLASSID, 1);
  if (alpha_m) PetscValidRealPointer(alpha_m, 2);
  if (alpha_f) PetscValidRealPointer(alpha_f, 3);
  if (gamma) PetscValidRealPointer(gamma, 4);
  if (beta) PetscValidRealPointer(beta, 5);
  PetscUseMethod(ts, "TSAlpha2GetParams_C", (TS, PetscReal *, PetscReal *, PetscReal *, PetscReal *), (ts, alpha_m, alpha_f, gamma, beta));
  PetscFunctionReturn(0);
}