/*
  Code for timestepping with BDF methods
*/
#include <petsc/private/tsimpl.h>  /*I "petscts.h" I*/

static PetscBool  cited = PETSC_FALSE;
static const char citation[] =
  "@book{Brenan1995,\n"
  "  title     = {Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations},\n"
  "  author    = {Brenan, K. and Campbell, S. and Petzold, L.},\n"
  "  publisher = {Society for Industrial and Applied Mathematics},\n"
  "  year      = {1995},\n"
  "  doi       = {10.1137/1.9781611971224},\n}\n";

typedef struct {
  PetscInt  k,n;
  PetscReal time[6+2];
  Vec       work[6+2];
  PetscReal shift;
  Vec       vec_dot;
  Vec       vec_lte;

  PetscInt     order;
  PetscBool    adapt;
  TSStepStatus status;
} TS_BDF;


PETSC_STATIC_INLINE void LagrangeBasisVals(PetscInt n,PetscReal t,const PetscReal T[],PetscScalar L[])
{
  PetscInt k,j;
  for (k=0; k<n; k++)
    for (L[k]=1, j=0; j<n; j++)
      if (j != k)
        L[k] *= (t - T[j])/(T[k] - T[j]);
}

PETSC_STATIC_INLINE void LagrangeBasisDers(PetscInt n,PetscReal t,const PetscReal T[],PetscScalar dL[])
{
  PetscInt  k,j,i;
  for (k=0; k<n; k++)
    for (dL[k]=0, j=0; j<n; j++)
      if (j != k) {
        PetscReal L = 1/(T[k] - T[j]);
        for (i=0; i<n; i++)
          if (i != j && i != k)
            L *= (t - T[i])/(T[k] - T[i]);
        dL[k] += L;
      }
}

static PetscErrorCode TSBDF_Advance(TS ts,PetscReal t,Vec X)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscInt       i,n = (PetscInt)(sizeof(bdf->work)/sizeof(Vec));
  Vec            tail = bdf->work[n-1];
  PetscErrorCode ierr;

  PetscFunctionBegin;
  for (i=n-1; i>=2; i--) {
    bdf->time[i] = bdf->time[i-1];
    bdf->work[i] = bdf->work[i-1];
  }
  bdf->n       = PetscMin(bdf->n+1,n-1);
  bdf->time[1] = t;
  bdf->work[1] = tail;
  ierr = VecCopy(X,tail);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSBDF_VecDot(TS ts,PetscInt order,PetscReal t,Vec X,Vec Xdot,PetscReal *shift)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscInt       i,n = order+1;
  PetscReal      time[7];
  Vec            vecs[7];
  PetscScalar    alpha[7];
  PetscErrorCode ierr;

  PetscFunctionBegin;
  n = PetscMax(n,2);
  time[0] = t; for (i=1; i<n; i++) time[i] = bdf->time[i];
  vecs[0] = X; for (i=1; i<n; i++) vecs[i] = bdf->work[i];
  LagrangeBasisDers(n,t,time,alpha);
  ierr = VecZeroEntries(Xdot);CHKERRQ(ierr);
  ierr = VecMAXPY(Xdot,n,alpha,vecs);CHKERRQ(ierr);
  if (shift) *shift = PetscRealPart(alpha[0]);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSBDF_VecLTE(TS ts,PetscInt order,Vec lte)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscInt       i,n = order+1;
  PetscReal      *time = bdf->time;
  Vec            *vecs = bdf->work;
  PetscScalar    a[8],b[8],alpha[8];
  PetscErrorCode ierr;

  PetscFunctionBegin;
  LagrangeBasisDers(n+0,time[0],time,a); a[n] =0;
  LagrangeBasisDers(n+1,time[0],time,b);
  for (i=0; i<n+1; i++) alpha[i] = (a[i]-b[i])/a[0];
  ierr = VecZeroEntries(lte);CHKERRQ(ierr);
  ierr = VecMAXPY(lte,n+1,alpha,vecs);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSBDF_Extrapolate(TS ts,PetscInt order,PetscReal t,Vec X)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscInt       n = order+1;
  PetscReal      *time = bdf->time+1;
  Vec            *vecs = bdf->work+1;
  PetscScalar    alpha[7];
  PetscErrorCode ierr;

  PetscFunctionBegin;
  n = PetscMin(n,bdf->n);
  LagrangeBasisVals(n,t,time,alpha);
  ierr = VecZeroEntries(X);CHKERRQ(ierr);
  ierr = VecMAXPY(X,n,alpha,vecs);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSBDF_Interpolate(TS ts,PetscInt order,PetscReal t,Vec X)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscInt       n = order+1;
  PetscReal      *time = bdf->time;
  Vec            *vecs = bdf->work;
  PetscScalar    alpha[7];
  PetscErrorCode ierr;

  PetscFunctionBegin;
  LagrangeBasisVals(n,t,time,alpha);
  ierr = VecZeroEntries(X);CHKERRQ(ierr);
  ierr = VecMAXPY(X,n,alpha,vecs);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  ierr = SNESSolve(ts->snes,b,x);CHKERRQ(ierr);
  ierr = SNESGetIterationNumber(ts->snes,&nits);CHKERRQ(ierr);
  ierr = SNESGetLinearSolveIterations(ts->snes,&lits);CHKERRQ(ierr);
  ts->snes_its += nits; ts->ksp_its += lits;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSBDF_Restart(TS ts,PetscBool *accept)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  bdf->k = 1; bdf->n = 0;
  ierr = TSBDF_Advance(ts,ts->ptime,ts->vec_sol);CHKERRQ(ierr);

  bdf->time[0] = ts->ptime + ts->time_step/2;
  ierr = VecCopy(bdf->work[1],bdf->work[0]);CHKERRQ(ierr);
  ierr = TSPreStage(ts,bdf->time[0]);CHKERRQ(ierr);
  ierr = TS_SNESSolve(ts,NULL,bdf->work[0]);CHKERRQ(ierr);
  ierr = TSPostStage(ts,bdf->time[0],0,&bdf->work[0]);CHKERRQ(ierr);
  ierr = TSAdaptCheckStage(ts->adapt,ts,bdf->time[0],bdf->work[0],accept);CHKERRQ(ierr);
  if (!*accept) PetscFunctionReturn(0);

  bdf->k = PetscMin(2,bdf->order); bdf->n++;
  ierr = VecCopy(bdf->work[0],bdf->work[2]);CHKERRQ(ierr);
  bdf->time[2] = bdf->time[0];
  PetscFunctionReturn(0);
}

static const char *const BDF_SchemeName[] = {"", "1", "2", "3", "4", "5", "6"};

static PetscErrorCode TSStep_BDF(TS ts)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscInt       rejections = 0;
  PetscBool      stageok,accept = PETSC_TRUE;
  PetscReal      next_time_step = ts->time_step;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscCitationsRegister(citation,&cited);CHKERRQ(ierr);

  if (!ts->steprollback && !ts->steprestart) {
    bdf->k = PetscMin(bdf->k+1,bdf->order);
    ierr = TSBDF_Advance(ts,ts->ptime,ts->vec_sol);CHKERRQ(ierr);
  }

  bdf->status = TS_STEP_INCOMPLETE;
  while (!ts->reason && bdf->status != TS_STEP_COMPLETE) {

    if (ts->steprestart) {
      ierr = TSBDF_Restart(ts,&stageok);CHKERRQ(ierr);
      if (!stageok) goto reject_step;
    }

    bdf->time[0] = ts->ptime + ts->time_step;
    ierr = TSBDF_Extrapolate(ts,bdf->k-(accept?0:1),bdf->time[0],bdf->work[0]);CHKERRQ(ierr);
    ierr = TSPreStage(ts,bdf->time[0]);CHKERRQ(ierr);
    ierr = TS_SNESSolve(ts,NULL,bdf->work[0]);CHKERRQ(ierr);
    ierr = TSPostStage(ts,bdf->time[0],0,&bdf->work[0]);CHKERRQ(ierr);
    ierr = TSAdaptCheckStage(ts->adapt,ts,bdf->time[0],bdf->work[0],&stageok);CHKERRQ(ierr);
    if (!stageok) goto reject_step;

    bdf->status = TS_STEP_PENDING;
    ierr = TSAdaptCandidatesClear(ts->adapt);CHKERRQ(ierr);
    ierr = TSAdaptCandidateAdd(ts->adapt,BDF_SchemeName[bdf->k],bdf->k,1,1.0,1.0,PETSC_TRUE);CHKERRQ(ierr);
    ierr = TSAdaptChoose(ts->adapt,ts,ts->time_step,NULL,&next_time_step,&accept);CHKERRQ(ierr);
    bdf->status = accept ? TS_STEP_COMPLETE : TS_STEP_INCOMPLETE;
    if (!accept) { ts->time_step = next_time_step; goto reject_step; }

    ierr = VecCopy(bdf->work[0],ts->vec_sol);CHKERRQ(ierr);
    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) {
      ierr = PetscInfo2(ts,"Step=%D, step rejections %D greater than current TS allowed, stopping solve\n",ts->steps,rejections);CHKERRQ(ierr);
      ts->reason = TS_DIVERGED_STEP_REJECTED;
    }
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TSInterpolate_BDF(TS ts,PetscReal t,Vec X)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = TSBDF_Interpolate(ts,bdf->k,t,X);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSEvaluateWLTE_BDF(TS ts,NormType wnormtype,PetscInt *order,PetscReal *wlte)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscInt       k = bdf->k;
  PetscReal      wltea,wlter;
  Vec            X = bdf->work[0], Y = bdf->vec_lte;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  k = PetscMin(k,bdf->n-1);
  ierr = TSBDF_VecLTE(ts,k,Y);CHKERRQ(ierr);
  ierr = VecAXPY(Y,1,X);CHKERRQ(ierr);
  ierr = TSErrorWeightedNorm(ts,X,Y,wnormtype,wlte,&wltea,&wlter);CHKERRQ(ierr);
  if (order) *order = k + 1;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSRollBack_BDF(TS ts)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = VecCopy(bdf->work[1],ts->vec_sol);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode SNESTSFormFunction_BDF(PETSC_UNUSED SNES snes,Vec X,Vec F,TS ts)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscInt       k = bdf->k;
  PetscReal      t = bdf->time[0];
  Vec            V = bdf->vec_dot;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = TSBDF_VecDot(ts,k,t,X,V,&bdf->shift);CHKERRQ(ierr);
  /* F = Function(t,X,V) */
  ierr = TSComputeIFunction(ts,t,X,V,F,PETSC_FALSE);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode SNESTSFormJacobian_BDF(PETSC_UNUSED SNES snes,
                                             PETSC_UNUSED Vec X,
                                             Mat J,Mat P,
                                             TS ts)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscReal      t = bdf->time[0];
  Vec            V = bdf->vec_dot;
  PetscReal      dVdX = bdf->shift;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  /* J,P = Jacobian(t,X,V) */
  ierr = TSComputeIJacobian(ts,t,X,V,dVdX,J,P,PETSC_FALSE);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSReset_BDF(TS ts)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  size_t         i,n = sizeof(bdf->work)/sizeof(Vec);
  PetscErrorCode ierr;

  PetscFunctionBegin;
  for (i=0; i<n; i++) {ierr = VecDestroy(&bdf->work[i]);CHKERRQ(ierr);}
  ierr = VecDestroy(&bdf->vec_dot);CHKERRQ(ierr);
  ierr = VecDestroy(&bdf->vec_lte);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSDestroy_BDF(TS ts)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = TSReset_BDF(ts);CHKERRQ(ierr);
  ierr = PetscFree(ts->data);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSBDFSetOrder_C",NULL);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSBDFGetOrder_C",NULL);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSBDFUseAdapt_C",NULL);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSSetUp_BDF(TS ts)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  size_t         i,n = sizeof(bdf->work)/sizeof(Vec);
  PetscErrorCode ierr;

  PetscFunctionBegin;
  bdf->k = bdf->n = 0;
  for (i=0; i<n; i++) {ierr = VecDuplicate(ts->vec_sol,&bdf->work[i]);CHKERRQ(ierr);}
  ierr = VecDuplicate(ts->vec_sol,&bdf->vec_dot);CHKERRQ(ierr);
  ierr = VecDuplicate(ts->vec_sol,&bdf->vec_lte);CHKERRQ(ierr);

  ierr = TSGetAdapt(ts,&ts->adapt);CHKERRQ(ierr);
  ierr = TSAdaptCandidatesClear(ts->adapt);CHKERRQ(ierr);
  if (!bdf->adapt) {
    ierr = TSAdaptSetType(ts->adapt,TSADAPTNONE);CHKERRQ(ierr);
  } else {
    PetscReal low,high;
    ierr = TSAdaptBasicGetClip(ts->adapt,&low,&high);CHKERRQ(ierr);
    high = PetscMin(high,2.0);
    ierr = TSAdaptBasicSetClip(ts->adapt,low,high);CHKERRQ(ierr);
    if (ts->exact_final_time == TS_EXACTFINALTIME_UNSPECIFIED)
      ts->exact_final_time = TS_EXACTFINALTIME_MATCHSTEP;
  }

  ierr = TSGetSNES(ts,&ts->snes);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSSetFromOptions_BDF(PetscOptionItems *PetscOptionsObject,TS ts)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscOptionsHead(PetscOptionsObject,"BDF ODE solver options");CHKERRQ(ierr);
  {
    PetscBool flg;
    PetscInt  order = bdf->order;
    PetscBool adapt = bdf->adapt;
    ierr = PetscOptionsInt("-ts_bdf_order","Order of the BDF method","TSBDFSetOrder",order,&order,&flg);CHKERRQ(ierr);
    if (flg) {ierr = TSBDFSetOrder(ts,order);CHKERRQ(ierr);}
    ierr = PetscOptionsBool("-ts_bdf_adapt","Use time-step adaptivity with the BDF method","TSBDFUseAdapt",adapt,&adapt,&flg);CHKERRQ(ierr);
    if (flg) {ierr = TSBDFUseAdapt(ts,adapt);CHKERRQ(ierr);}
  }
  ierr = PetscOptionsTail();CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TSView_BDF(TS ts,PetscViewer viewer)
{
  TS_BDF         *bdf = (TS_BDF*)ts->data;
  PetscBool      iascii;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);CHKERRQ(ierr);
  if (iascii)    {
    ierr = PetscViewerASCIIPrintf(viewer,"  Order=%D\n",bdf->order);CHKERRQ(ierr);
  }
  if (ts->adapt) {ierr = TSAdaptView(ts->adapt,viewer);CHKERRQ(ierr);}
  if (ts->snes)  {ierr = SNESView(ts->snes,viewer);CHKERRQ(ierr);}
  PetscFunctionReturn(0);
}

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

static PetscErrorCode TSBDFSetOrder_BDF(TS ts,PetscInt order)
{
  TS_BDF *bdf = (TS_BDF*)ts->data;

  PetscFunctionBegin;
  if (order == bdf->order) PetscFunctionReturn(0);
  if (order < 1 || order > 6) SETERRQ1(PetscObjectComm((PetscObject)ts),PETSC_ERR_ARG_OUTOFRANGE,"BDF Order %D not implemented",order);
  bdf->order = order;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSBDFGetOrder_BDF(TS ts,PetscInt *order)
{
  TS_BDF *bdf = (TS_BDF*)ts->data;

  PetscFunctionBegin;
  *order = bdf->order;
  PetscFunctionReturn(0);
}

static PetscErrorCode TSBDFUseAdapt_BDF(TS ts,PetscBool use)
{
  TS_BDF *bdf = (TS_BDF*)ts->data;

  PetscFunctionBegin;
  if (use == bdf->adapt) PetscFunctionReturn(0);
  if (ts->setupcalled) SETERRQ(PetscObjectComm((PetscObject)ts),PETSC_ERR_ORDER,"Cannot change adaptivity after TSSetUp()");
  bdf->adapt = use;
  PetscFunctionReturn(0);
}

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

/*MC
      TSBDF - DAE solver using BDF methods

  Level: beginner

.seealso:  TS, TSCreate(), TSSetType()
M*/
PETSC_EXTERN PetscErrorCode TSCreate_BDF(TS ts)
{
  TS_BDF         *bdf;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ts->ops->reset          = TSReset_BDF;
  ts->ops->destroy        = TSDestroy_BDF;
  ts->ops->view           = TSView_BDF;
  ts->ops->setup          = TSSetUp_BDF;
  ts->ops->setfromoptions = TSSetFromOptions_BDF;
  ts->ops->step           = TSStep_BDF;
  ts->ops->evaluatewlte   = TSEvaluateWLTE_BDF;
  ts->ops->rollback       = TSRollBack_BDF;
  ts->ops->interpolate    = TSInterpolate_BDF;
  ts->ops->snesfunction   = SNESTSFormFunction_BDF;
  ts->ops->snesjacobian   = SNESTSFormJacobian_BDF;

  ierr = PetscNewLog(ts,&bdf);CHKERRQ(ierr);
  ts->data = (void*)bdf;

  bdf->order  = 2;
  bdf->adapt  = PETSC_FALSE;
  bdf->status = TS_STEP_COMPLETE;

  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSBDFSetOrder_C",TSBDFSetOrder_BDF);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSBDFGetOrder_C",TSBDFGetOrder_BDF);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)ts,"TSBDFUseAdapt_C",TSBDFUseAdapt_BDF);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

/*@
  TSBDFSetOrder - Set the order of the BDF method

  Logically Collective on TS

  Input Parameter:
+  ts - timestepping context
-  order - order of the method

  Options Database:
.  -ts_bdf_order <order>

  Level: intermediate

@*/
PetscErrorCode TSBDFSetOrder(TS ts,PetscInt order)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  PetscValidLogicalCollectiveInt(ts,order,2);
  ierr = PetscTryMethod(ts,"TSBDFSetOrder_C",(TS,PetscInt),(ts,order));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
  TSBDFGetOrder - Get the order of the BDF method

  Not Collective

  Input Parameter:
.  ts - timestepping context

  Output Parameter:
.  order - order of the method

  Level: intermediate

@*/
PetscErrorCode TSBDFGetOrder(TS ts,PetscInt *order)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  PetscValidIntPointer(order,2);
  ierr = PetscUseMethod(ts,"TSBDFGetOrder_C",(TS,PetscInt*),(ts,order));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
  TSBDFUseAdapt - Use time-step adaptivity with the BDF method

  Logically Collective on TS

  Input Parameter:
+  ts - timestepping context
-  use - flag to use adaptivity

  Options Database:
.  -ts_bdf_adapt

  Level: intermediate

.seealso: TSAdapt
@*/
PetscErrorCode TSBDFUseAdapt(TS ts,PetscBool use)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  PetscValidLogicalCollectiveBool(ts,use,2);
  ierr = PetscTryMethod(ts,"TSBDFUseAdapt_C",(TS,PetscBool),(ts,use));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}