impls/pseudo/posindep.c

/*
       Code for Timestepping with implicit backwards Euler.
*/
#include <petsc/private/tsimpl.h>                /*I   "petscts.h"   I*/

typedef struct {
  Vec update;       /* work vector where new solution is formed */
  Vec func;         /* work vector where F(t[i],u[i]) is stored */
  Vec xdot;         /* work vector for time derivative of state */

  /* information used for Pseudo-timestepping */

  PetscErrorCode (*dt)(TS,PetscReal*,void*);              /* compute next timestep, and related context */
  void *dtctx;
  PetscErrorCode (*verify)(TS,Vec,void*,PetscReal*,PetscBool*);  /* verify previous timestep and related context */
  void *verifyctx;

  PetscReal fnorm_initial,fnorm;                   /* original and current norm of F(u) */
  PetscReal fnorm_previous;

  PetscReal dt_initial;                     /* initial time-step */
  PetscReal dt_increment;                   /* scaling that dt is incremented each time-step */
  PetscReal dt_max;                         /* maximum time step */
  PetscBool increment_dt_from_initial_dt;
  PetscReal fatol,frtol;
} TS_Pseudo;

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

/*@C
    TSPseudoComputeTimeStep - Computes the next timestep for a currently running
    pseudo-timestepping process.

    Collective on TS

    Input Parameter:
.   ts - timestep context

    Output Parameter:
.   dt - newly computed timestep

    Level: developer

    Notes:
    The routine to be called here to compute the timestep should be
    set by calling TSPseudoSetTimeStep().

.seealso: TSPseudoTimeStepDefault(), TSPseudoSetTimeStep()
@*/
PetscErrorCode  TSPseudoComputeTimeStep(TS ts,PetscReal *dt)
{
  TS_Pseudo      *pseudo = (TS_Pseudo*)ts->data;

  PetscFunctionBegin;
  PetscCall(PetscLogEventBegin(TS_PseudoComputeTimeStep,ts,0,0,0));
  PetscCall((*pseudo->dt)(ts,dt,pseudo->dtctx));
  PetscCall(PetscLogEventEnd(TS_PseudoComputeTimeStep,ts,0,0,0));
  PetscFunctionReturn(0);
}

/* ------------------------------------------------------------------------------*/
/*@C
   TSPseudoVerifyTimeStepDefault - Default code to verify the quality of the last timestep.

   Collective on TS

   Input Parameters:
+  ts - the timestep context
.  dtctx - unused timestep context
-  update - latest solution vector

   Output Parameters:
+  newdt - the timestep to use for the next step
-  flag - flag indicating whether the last time step was acceptable

   Level: advanced

   Note:
   This routine always returns a flag of 1, indicating an acceptable
   timestep.

.seealso: TSPseudoSetVerifyTimeStep(), TSPseudoVerifyTimeStep()
@*/
PetscErrorCode  TSPseudoVerifyTimeStepDefault(TS ts,Vec update,void *dtctx,PetscReal *newdt,PetscBool  *flag)
{
  PetscFunctionBegin;
  *flag = PETSC_TRUE;
  PetscFunctionReturn(0);
}

/*@
    TSPseudoVerifyTimeStep - Verifies whether the last timestep was acceptable.

    Collective on TS

    Input Parameters:
+   ts - timestep context
-   update - latest solution vector

    Output Parameters:
+   dt - newly computed timestep (if it had to shrink)
-   flag - indicates if current timestep was ok

    Level: advanced

    Notes:
    The routine to be called here to compute the timestep should be
    set by calling TSPseudoSetVerifyTimeStep().

.seealso: TSPseudoSetVerifyTimeStep(), TSPseudoVerifyTimeStepDefault()
@*/
PetscErrorCode  TSPseudoVerifyTimeStep(TS ts,Vec update,PetscReal *dt,PetscBool *flag)
{
  TS_Pseudo      *pseudo = (TS_Pseudo*)ts->data;

  PetscFunctionBegin;
  *flag = PETSC_TRUE;
  if (pseudo->verify) {
    PetscCall((*pseudo->verify)(ts,update,pseudo->verifyctx,dt,flag));
  }
  PetscFunctionReturn(0);
}

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

static PetscErrorCode TSStep_Pseudo(TS ts)
{
  TS_Pseudo           *pseudo = (TS_Pseudo*)ts->data;
  PetscInt            nits,lits,reject;
  PetscBool           stepok;
  PetscReal           next_time_step = ts->time_step;

  PetscFunctionBegin;
  if (ts->steps == 0) pseudo->dt_initial = ts->time_step;
  PetscCall(VecCopy(ts->vec_sol,pseudo->update));
  PetscCall(TSPseudoComputeTimeStep(ts,&next_time_step));
  for (reject=0; reject<ts->max_reject; reject++,ts->reject++) {
    ts->time_step = next_time_step;
    PetscCall(TSPreStage(ts,ts->ptime+ts->time_step));
    PetscCall(SNESSolve(ts->snes,NULL,pseudo->update));
    PetscCall(SNESGetIterationNumber(ts->snes,&nits));
    PetscCall(SNESGetLinearSolveIterations(ts->snes,&lits));
    ts->snes_its += nits; ts->ksp_its += lits;
    PetscCall(TSPostStage(ts,ts->ptime+ts->time_step,0,&(pseudo->update)));
    PetscCall(TSAdaptCheckStage(ts->adapt,ts,ts->ptime+ts->time_step,pseudo->update,&stepok));
    if (!stepok) {next_time_step = ts->time_step; continue;}
    pseudo->fnorm = -1; /* The current norm is no longer valid, monitor must recompute it. */
    PetscCall(TSPseudoVerifyTimeStep(ts,pseudo->update,&next_time_step,&stepok));
    if (stepok) break;
  }
  if (reject >= ts->max_reject) {
    ts->reason = TS_DIVERGED_STEP_REJECTED;
    PetscCall(PetscInfo(ts,"Step=%D, step rejections %D greater than current TS allowed, stopping solve\n",ts->steps,reject));
    PetscFunctionReturn(0);
  }

  PetscCall(VecCopy(pseudo->update,ts->vec_sol));
  ts->ptime += ts->time_step;
  ts->time_step = next_time_step;

  if (pseudo->fnorm < 0) {
    PetscCall(VecZeroEntries(pseudo->xdot));
    PetscCall(TSComputeIFunction(ts,ts->ptime,ts->vec_sol,pseudo->xdot,pseudo->func,PETSC_FALSE));
    PetscCall(VecNorm(pseudo->func,NORM_2,&pseudo->fnorm));
  }
  if (pseudo->fnorm < pseudo->fatol) {
    ts->reason = TS_CONVERGED_PSEUDO_FATOL;
    PetscCall(PetscInfo(ts,"Step=%D, converged since fnorm %g < fatol %g\n",ts->steps,pseudo->fnorm,pseudo->frtol));
    PetscFunctionReturn(0);
  }
  if (pseudo->fnorm/pseudo->fnorm_initial < pseudo->frtol) {
    ts->reason = TS_CONVERGED_PSEUDO_FRTOL;
    PetscCall(PetscInfo(ts,"Step=%D, converged since fnorm %g / fnorm_initial %g < frtol %g\n",ts->steps,pseudo->fnorm,pseudo->fnorm_initial,pseudo->fatol));
    PetscFunctionReturn(0);
  }
  PetscFunctionReturn(0);
}

/*------------------------------------------------------------*/
static PetscErrorCode TSReset_Pseudo(TS ts)
{
  TS_Pseudo      *pseudo = (TS_Pseudo*)ts->data;

  PetscFunctionBegin;
  PetscCall(VecDestroy(&pseudo->update));
  PetscCall(VecDestroy(&pseudo->func));
  PetscCall(VecDestroy(&pseudo->xdot));
  PetscFunctionReturn(0);
}

static PetscErrorCode TSDestroy_Pseudo(TS ts)
{
  PetscFunctionBegin;
  PetscCall(TSReset_Pseudo(ts));
  PetscCall(PetscFree(ts->data));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSPseudoSetVerifyTimeStep_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSPseudoSetTimeStepIncrement_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSPseudoSetMaxTimeStep_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSPseudoIncrementDtFromInitialDt_C",NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSPseudoSetTimeStep_C",NULL));
  PetscFunctionReturn(0);
}

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

/*
    Compute Xdot = (X^{n+1}-X^n)/dt) = 0
*/
static PetscErrorCode TSPseudoGetXdot(TS ts,Vec X,Vec *Xdot)
{
  TS_Pseudo         *pseudo = (TS_Pseudo*)ts->data;
  const PetscScalar mdt     = 1.0/ts->time_step,*xnp1,*xn;
  PetscScalar       *xdot;
  PetscInt          i,n;

  PetscFunctionBegin;
  *Xdot = NULL;
  PetscCall(VecGetArrayRead(ts->vec_sol,&xn));
  PetscCall(VecGetArrayRead(X,&xnp1));
  PetscCall(VecGetArray(pseudo->xdot,&xdot));
  PetscCall(VecGetLocalSize(X,&n));
  for (i=0; i<n; i++) xdot[i] = mdt*(xnp1[i] - xn[i]);
  PetscCall(VecRestoreArrayRead(ts->vec_sol,&xn));
  PetscCall(VecRestoreArrayRead(X,&xnp1));
  PetscCall(VecRestoreArray(pseudo->xdot,&xdot));
  *Xdot = pseudo->xdot;
  PetscFunctionReturn(0);
}

/*
    The transient residual is

        F(U^{n+1},(U^{n+1}-U^n)/dt) = 0

    or for ODE,

        (U^{n+1} - U^{n})/dt - F(U^{n+1}) = 0

    This is the function that must be evaluated for transient simulation and for
    finite difference Jacobians.  On the first Newton step, this algorithm uses
    a guess of U^{n+1} = U^n in which case the transient term vanishes and the
    residual is actually the steady state residual.  Pseudotransient
    continuation as described in the literature is a linearly implicit
    algorithm, it only takes this one Newton step with the steady state
    residual, and then advances to the next time step.
*/
static PetscErrorCode SNESTSFormFunction_Pseudo(SNES snes,Vec X,Vec Y,TS ts)
{
  Vec            Xdot;

  PetscFunctionBegin;
  PetscCall(TSPseudoGetXdot(ts,X,&Xdot));
  PetscCall(TSComputeIFunction(ts,ts->ptime+ts->time_step,X,Xdot,Y,PETSC_FALSE));
  PetscFunctionReturn(0);
}

/*
   This constructs the Jacobian needed for SNES.  For DAE, this is

       dF(X,Xdot)/dX + shift*dF(X,Xdot)/dXdot

    and for ODE:

       J = I/dt - J_{Frhs}   where J_{Frhs} is the given Jacobian of Frhs.
*/
static PetscErrorCode SNESTSFormJacobian_Pseudo(SNES snes,Vec X,Mat AA,Mat BB,TS ts)
{
  Vec            Xdot;

  PetscFunctionBegin;
  PetscCall(TSPseudoGetXdot(ts,X,&Xdot));
  PetscCall(TSComputeIJacobian(ts,ts->ptime+ts->time_step,X,Xdot,1./ts->time_step,AA,BB,PETSC_FALSE));
  PetscFunctionReturn(0);
}

static PetscErrorCode TSSetUp_Pseudo(TS ts)
{
  TS_Pseudo      *pseudo = (TS_Pseudo*)ts->data;

  PetscFunctionBegin;
  PetscCall(VecDuplicate(ts->vec_sol,&pseudo->update));
  PetscCall(VecDuplicate(ts->vec_sol,&pseudo->func));
  PetscCall(VecDuplicate(ts->vec_sol,&pseudo->xdot));
  PetscFunctionReturn(0);
}
/*------------------------------------------------------------*/

static PetscErrorCode TSPseudoMonitorDefault(TS ts,PetscInt step,PetscReal ptime,Vec v,void *dummy)
{
  TS_Pseudo      *pseudo = (TS_Pseudo*)ts->data;
  PetscViewer    viewer = (PetscViewer) dummy;

  PetscFunctionBegin;
  if (pseudo->fnorm < 0) { /* The last computed norm is stale, recompute */
    PetscCall(VecZeroEntries(pseudo->xdot));
    PetscCall(TSComputeIFunction(ts,ts->ptime,ts->vec_sol,pseudo->xdot,pseudo->func,PETSC_FALSE));
    PetscCall(VecNorm(pseudo->func,NORM_2,&pseudo->fnorm));
  }
  PetscCall(PetscViewerASCIIAddTab(viewer,((PetscObject)ts)->tablevel));
  PetscCall(PetscViewerASCIIPrintf(viewer,"TS %D dt %g time %g fnorm %g\n",step,(double)ts->time_step,(double)ptime,(double)pseudo->fnorm));
  PetscCall(PetscViewerASCIISubtractTab(viewer,((PetscObject)ts)->tablevel));
  PetscFunctionReturn(0);
}

static PetscErrorCode TSSetFromOptions_Pseudo(PetscOptionItems *PetscOptionsObject,TS ts)
{
  TS_Pseudo      *pseudo = (TS_Pseudo*)ts->data;
  PetscBool      flg = PETSC_FALSE;
  PetscViewer    viewer;

  PetscFunctionBegin;
  PetscCall(PetscOptionsHead(PetscOptionsObject,"Pseudo-timestepping options"));
  PetscCall(PetscOptionsBool("-ts_monitor_pseudo","Monitor convergence","",flg,&flg,NULL));
  if (flg) {
    PetscCall(PetscViewerASCIIOpen(PetscObjectComm((PetscObject)ts),"stdout",&viewer));
    PetscCall(TSMonitorSet(ts,TSPseudoMonitorDefault,viewer,(PetscErrorCode (*)(void**))PetscViewerDestroy));
  }
  flg  = pseudo->increment_dt_from_initial_dt;
  PetscCall(PetscOptionsBool("-ts_pseudo_increment_dt_from_initial_dt","Increase dt as a ratio from original dt","TSPseudoIncrementDtFromInitialDt",flg,&flg,NULL));
  pseudo->increment_dt_from_initial_dt = flg;
  PetscCall(PetscOptionsReal("-ts_pseudo_increment","Ratio to increase dt","TSPseudoSetTimeStepIncrement",pseudo->dt_increment,&pseudo->dt_increment,NULL));
  PetscCall(PetscOptionsReal("-ts_pseudo_max_dt","Maximum value for dt","TSPseudoSetMaxTimeStep",pseudo->dt_max,&pseudo->dt_max,NULL));
  PetscCall(PetscOptionsReal("-ts_pseudo_fatol","Tolerance for norm of function","",pseudo->fatol,&pseudo->fatol,NULL));
  PetscCall(PetscOptionsReal("-ts_pseudo_frtol","Relative tolerance for norm of function","",pseudo->frtol,&pseudo->frtol,NULL));
  PetscCall(PetscOptionsTail());
  PetscFunctionReturn(0);
}

static PetscErrorCode TSView_Pseudo(TS ts,PetscViewer viewer)
{
  PetscBool      isascii;

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii));
  if (isascii) {
    TS_Pseudo *pseudo = (TS_Pseudo*) ts->data;
    PetscCall(PetscViewerASCIIPrintf(viewer,"  frtol - relative tolerance in function value %g\n",(double)pseudo->frtol));
    PetscCall(PetscViewerASCIIPrintf(viewer,"  fatol - absolute tolerance in function value %g\n",(double)pseudo->fatol));
    PetscCall(PetscViewerASCIIPrintf(viewer,"  dt_initial - initial timestep %g\n",(double)pseudo->dt_initial));
    PetscCall(PetscViewerASCIIPrintf(viewer,"  dt_increment - increase in timestep on successful step %g\n",(double)pseudo->dt_increment));
    PetscCall(PetscViewerASCIIPrintf(viewer,"  dt_max - maximum time %g\n",(double)pseudo->dt_max));
  }
  PetscFunctionReturn(0);
}

/* ----------------------------------------------------------------------------- */
/*@C
   TSPseudoSetVerifyTimeStep - Sets a user-defined routine to verify the quality of the
   last timestep.

   Logically Collective on TS

   Input Parameters:
+  ts - timestep context
.  dt - user-defined function to verify timestep
-  ctx - [optional] user-defined context for private data
         for the timestep verification routine (may be NULL)

   Level: advanced

   Calling sequence of func:
$  func (TS ts,Vec update,void *ctx,PetscReal *newdt,PetscBool  *flag);

+  update - latest solution vector
.  ctx - [optional] timestep context
.  newdt - the timestep to use for the next step
-  flag - flag indicating whether the last time step was acceptable

   Notes:
   The routine set here will be called by TSPseudoVerifyTimeStep()
   during the timestepping process.

.seealso: TSPseudoVerifyTimeStepDefault(), TSPseudoVerifyTimeStep()
@*/
PetscErrorCode  TSPseudoSetVerifyTimeStep(TS ts,PetscErrorCode (*dt)(TS,Vec,void*,PetscReal*,PetscBool*),void *ctx)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  PetscTryMethod(ts,"TSPseudoSetVerifyTimeStep_C",(TS,PetscErrorCode (*)(TS,Vec,void*,PetscReal*,PetscBool*),void*),(ts,dt,ctx));
  PetscFunctionReturn(0);
}

/*@
    TSPseudoSetTimeStepIncrement - Sets the scaling increment applied to
    dt when using the TSPseudoTimeStepDefault() routine.

   Logically Collective on TS

    Input Parameters:
+   ts - the timestep context
-   inc - the scaling factor >= 1.0

    Options Database Key:
.    -ts_pseudo_increment <increment> - set pseudo increment

    Level: advanced

.seealso: TSPseudoSetTimeStep(), TSPseudoTimeStepDefault()
@*/
PetscErrorCode  TSPseudoSetTimeStepIncrement(TS ts,PetscReal inc)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  PetscValidLogicalCollectiveReal(ts,inc,2);
  PetscTryMethod(ts,"TSPseudoSetTimeStepIncrement_C",(TS,PetscReal),(ts,inc));
  PetscFunctionReturn(0);
}

/*@
    TSPseudoSetMaxTimeStep - Sets the maximum time step
    when using the TSPseudoTimeStepDefault() routine.

   Logically Collective on TS

    Input Parameters:
+   ts - the timestep context
-   maxdt - the maximum time step, use a non-positive value to deactivate

    Options Database Key:
.    -ts_pseudo_max_dt <increment> - set pseudo max dt

    Level: advanced

.seealso: TSPseudoSetTimeStep(), TSPseudoTimeStepDefault()
@*/
PetscErrorCode  TSPseudoSetMaxTimeStep(TS ts,PetscReal maxdt)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  PetscValidLogicalCollectiveReal(ts,maxdt,2);
  PetscTryMethod(ts,"TSPseudoSetMaxTimeStep_C",(TS,PetscReal),(ts,maxdt));
  PetscFunctionReturn(0);
}

/*@
    TSPseudoIncrementDtFromInitialDt - Indicates that a new timestep
    is computed via the formula
$         dt = initial_dt*initial_fnorm/current_fnorm
      rather than the default update,
$         dt = current_dt*previous_fnorm/current_fnorm.

   Logically Collective on TS

    Input Parameter:
.   ts - the timestep context

    Options Database Key:
.    -ts_pseudo_increment_dt_from_initial_dt <true,false> - use the initial dt to determine increment

    Level: advanced

.seealso: TSPseudoSetTimeStep(), TSPseudoTimeStepDefault()
@*/
PetscErrorCode  TSPseudoIncrementDtFromInitialDt(TS ts)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  PetscTryMethod(ts,"TSPseudoIncrementDtFromInitialDt_C",(TS),(ts));
  PetscFunctionReturn(0);
}

/*@C
   TSPseudoSetTimeStep - Sets the user-defined routine to be
   called at each pseudo-timestep to update the timestep.

   Logically Collective on TS

   Input Parameters:
+  ts - timestep context
.  dt - function to compute timestep
-  ctx - [optional] user-defined context for private data
         required by the function (may be NULL)

   Level: intermediate

   Calling sequence of func:
$  func (TS ts,PetscReal *newdt,void *ctx);

+  newdt - the newly computed timestep
-  ctx - [optional] timestep context

   Notes:
   The routine set here will be called by TSPseudoComputeTimeStep()
   during the timestepping process.
   If not set then TSPseudoTimeStepDefault() is automatically used

.seealso: TSPseudoTimeStepDefault(), TSPseudoComputeTimeStep()
@*/
PetscErrorCode  TSPseudoSetTimeStep(TS ts,PetscErrorCode (*dt)(TS,PetscReal*,void*),void *ctx)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(ts,TS_CLASSID,1);
  PetscTryMethod(ts,"TSPseudoSetTimeStep_C",(TS,PetscErrorCode (*)(TS,PetscReal*,void*),void*),(ts,dt,ctx));
  PetscFunctionReturn(0);
}

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

typedef PetscErrorCode (*FCN1)(TS,Vec,void*,PetscReal*,PetscBool*);  /* force argument to next function to not be extern C*/
static PetscErrorCode  TSPseudoSetVerifyTimeStep_Pseudo(TS ts,FCN1 dt,void *ctx)
{
  TS_Pseudo *pseudo = (TS_Pseudo*)ts->data;

  PetscFunctionBegin;
  pseudo->verify    = dt;
  pseudo->verifyctx = ctx;
  PetscFunctionReturn(0);
}

static PetscErrorCode  TSPseudoSetTimeStepIncrement_Pseudo(TS ts,PetscReal inc)
{
  TS_Pseudo *pseudo = (TS_Pseudo*)ts->data;

  PetscFunctionBegin;
  pseudo->dt_increment = inc;
  PetscFunctionReturn(0);
}

static PetscErrorCode  TSPseudoSetMaxTimeStep_Pseudo(TS ts,PetscReal maxdt)
{
  TS_Pseudo *pseudo = (TS_Pseudo*)ts->data;

  PetscFunctionBegin;
  pseudo->dt_max = maxdt;
  PetscFunctionReturn(0);
}

static PetscErrorCode  TSPseudoIncrementDtFromInitialDt_Pseudo(TS ts)
{
  TS_Pseudo *pseudo = (TS_Pseudo*)ts->data;

  PetscFunctionBegin;
  pseudo->increment_dt_from_initial_dt = PETSC_TRUE;
  PetscFunctionReturn(0);
}

typedef PetscErrorCode (*FCN2)(TS,PetscReal*,void*); /* force argument to next function to not be extern C*/
static PetscErrorCode  TSPseudoSetTimeStep_Pseudo(TS ts,FCN2 dt,void *ctx)
{
  TS_Pseudo *pseudo = (TS_Pseudo*)ts->data;

  PetscFunctionBegin;
  pseudo->dt    = dt;
  pseudo->dtctx = ctx;
  PetscFunctionReturn(0);
}

/* ----------------------------------------------------------------------------- */
/*MC
      TSPSEUDO - Solve steady state ODE and DAE problems with pseudo time stepping

  This method solves equations of the form

$    F(X,Xdot) = 0

  for steady state using the iteration

$    [G'] S = -F(X,0)
$    X += S

  where

$    G(Y) = F(Y,(Y-X)/dt)

  This is linearly-implicit Euler with the residual always evaluated "at steady
  state".  See note below.

  Options database keys:
+  -ts_pseudo_increment <real> - ratio of increase dt
.  -ts_pseudo_increment_dt_from_initial_dt <truth> - Increase dt as a ratio from original dt
.  -ts_pseudo_fatol <atol> - stop iterating when the function norm is less than atol
-  -ts_pseudo_frtol <rtol> - stop iterating when the function norm divided by the initial function norm is less than rtol

  Level: beginner

  References:
+  * - Todd S. Coffey and C. T. Kelley and David E. Keyes, Pseudotransient Continuation and Differential Algebraic Equations, 2003.
-  * - C. T. Kelley and David E. Keyes, Convergence analysis of Pseudotransient Continuation, 1998.

  Notes:
  The residual computed by this method includes the transient term (Xdot is computed instead of
  always being zero), but since the prediction from the last step is always the solution from the
  last step, on the first Newton iteration we have

$  Xdot = (Xpredicted - Xold)/dt = (Xold-Xold)/dt = 0

  Therefore, the linear system solved by the first Newton iteration is equivalent to the one
  described above and in the papers.  If the user chooses to perform multiple Newton iterations, the
  algorithm is no longer the one described in the referenced papers.

.seealso:  TSCreate(), TS, TSSetType()

M*/
PETSC_EXTERN PetscErrorCode TSCreate_Pseudo(TS ts)
{
  TS_Pseudo      *pseudo;
  SNES           snes;
  SNESType       stype;

  PetscFunctionBegin;
  ts->ops->reset          = TSReset_Pseudo;
  ts->ops->destroy        = TSDestroy_Pseudo;
  ts->ops->view           = TSView_Pseudo;
  ts->ops->setup          = TSSetUp_Pseudo;
  ts->ops->step           = TSStep_Pseudo;
  ts->ops->setfromoptions = TSSetFromOptions_Pseudo;
  ts->ops->snesfunction   = SNESTSFormFunction_Pseudo;
  ts->ops->snesjacobian   = SNESTSFormJacobian_Pseudo;
  ts->default_adapt_type  = TSADAPTNONE;
  ts->usessnes            = PETSC_TRUE;

  PetscCall(TSGetSNES(ts,&snes));
  PetscCall(SNESGetType(snes,&stype));
  if (!stype) PetscCall(SNESSetType(snes,SNESKSPONLY));

  PetscCall(PetscNewLog(ts,&pseudo));
  ts->data = (void*)pseudo;

  pseudo->dt                           = TSPseudoTimeStepDefault;
  pseudo->dtctx                        = NULL;
  pseudo->dt_increment                 = 1.1;
  pseudo->increment_dt_from_initial_dt = PETSC_FALSE;
  pseudo->fnorm                        = -1;
  pseudo->fnorm_initial                = -1;
  pseudo->fnorm_previous               = -1;
 #if defined(PETSC_USE_REAL_SINGLE)
  pseudo->fatol                        = 1.e-25;
  pseudo->frtol                        = 1.e-5;
#else
  pseudo->fatol                        = 1.e-50;
  pseudo->frtol                        = 1.e-12;
#endif
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSPseudoSetVerifyTimeStep_C",TSPseudoSetVerifyTimeStep_Pseudo));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSPseudoSetTimeStepIncrement_C",TSPseudoSetTimeStepIncrement_Pseudo));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSPseudoSetMaxTimeStep_C",TSPseudoSetMaxTimeStep_Pseudo));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSPseudoIncrementDtFromInitialDt_C",TSPseudoIncrementDtFromInitialDt_Pseudo));
  PetscCall(PetscObjectComposeFunction((PetscObject)ts,"TSPseudoSetTimeStep_C",TSPseudoSetTimeStep_Pseudo));
  PetscFunctionReturn(0);
}

/*@C
   TSPseudoTimeStepDefault - Default code to compute pseudo-timestepping.
   Use with TSPseudoSetTimeStep().

   Collective on TS

   Input Parameters:
+  ts - the timestep context
-  dtctx - unused timestep context

   Output Parameter:
.  newdt - the timestep to use for the next step

   Level: advanced

.seealso: TSPseudoSetTimeStep(), TSPseudoComputeTimeStep()
@*/
PetscErrorCode  TSPseudoTimeStepDefault(TS ts,PetscReal *newdt,void *dtctx)
{
  TS_Pseudo      *pseudo = (TS_Pseudo*)ts->data;
  PetscReal      inc = pseudo->dt_increment;

  PetscFunctionBegin;
  PetscCall(VecZeroEntries(pseudo->xdot));
  PetscCall(TSComputeIFunction(ts,ts->ptime,ts->vec_sol,pseudo->xdot,pseudo->func,PETSC_FALSE));
  PetscCall(VecNorm(pseudo->func,NORM_2,&pseudo->fnorm));
  if (pseudo->fnorm_initial < 0) {
    /* first time through so compute initial function norm */
    pseudo->fnorm_initial  = pseudo->fnorm;
    pseudo->fnorm_previous = pseudo->fnorm;
  }
  if (pseudo->fnorm == 0.0)                      *newdt = 1.e12*inc*ts->time_step;
  else if (pseudo->increment_dt_from_initial_dt) *newdt = inc*pseudo->dt_initial*pseudo->fnorm_initial/pseudo->fnorm;
  else                                           *newdt = inc*ts->time_step*pseudo->fnorm_previous/pseudo->fnorm;
  if (pseudo->dt_max > 0) *newdt = PetscMin(*newdt,pseudo->dt_max);
  pseudo->fnorm_previous = pseudo->fnorm;
  PetscFunctionReturn(0);
}