BDF2 (Backward-Difference-Formula-2) time stepper. More...

#include <Tempus_StepperBDF2_decl.hpp>

Inheritance diagram for Tempus::StepperBDF2< Scalar >:

Public Member Functions
	StepperBDF2 ()
	Default constructor. More...

	StepperBDF2 (const Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > &appModel, const Teuchos::RCP< Thyra::NonlinearSolverBase< Scalar > > &solver, const Teuchos::RCP< Stepper< Scalar > > &startUpStepper, bool useFSAL, std::string ICConsistency, bool ICConsistencyCheck, bool zeroInitialGuess, const Teuchos::RCP< StepperBDF2AppAction< Scalar > > &stepperBDF2AppAction)

virtual Scalar	getAlpha (const Scalar dt) const
	Return alpha = d(xDot)/dx. More...

virtual Scalar	getAlpha (const Scalar dt, const Scalar dtOld) const

virtual Scalar	getBeta (const Scalar) const
	Return beta = d(x)/dx. More...

virtual void	computeStartUp (const Teuchos::RCP< SolutionHistory< Scalar > > &solutionHistory)
	Compute the first time step given the supplied startup stepper. More...

Teuchos::RCP< const Teuchos::ParameterList >	getValidParameters () const

virtual bool	isValidSetup (Teuchos::FancyOStream &out) const

Public Member Functions inherited from Tempus::StepperImplicit< Scalar >
Teuchos::RCP < Teuchos::ParameterList >	getValidParametersBasicImplicit () const

void	setStepperImplicitValues (Teuchos::RCP< Teuchos::ParameterList > pl)
	Set StepperImplicit member data from the ParameterList. More...

void	setStepperSolverValues (Teuchos::RCP< Teuchos::ParameterList > pl)
	Set solver from ParameterList. More...

void	setSolverName (std::string i)
	Set the Solver Name. More...

std::string	getSolverName () const
	Get the Solver Name. More...

virtual Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > >	getModel () const override

virtual Teuchos::RCP< const WrapperModelEvaluator< Scalar > >	getWrapperModel ()

virtual void	setDefaultSolver ()

virtual void	setSolver (Teuchos::RCP< Thyra::NonlinearSolverBase< Scalar > > solver) override
	Set solver. More...

virtual Teuchos::RCP < Thyra::NonlinearSolverBase < Scalar > >	getSolver () const override
	Get solver. More...

const Thyra::SolveStatus< Scalar >	solveImplicitODE (const Teuchos::RCP< Thyra::VectorBase< Scalar > > &x, const Teuchos::RCP< Thyra::VectorBase< Scalar > > &xDot, const Scalar time, const Teuchos::RCP< ImplicitODEParameters< Scalar > > &p, const Teuchos::RCP< Thyra::VectorBase< Scalar > > &y=Teuchos::null, const int index=-1)
	Solve implicit ODE, f(x, xDot, t, p) = 0. More...

void	evaluateImplicitODE (Teuchos::RCP< Thyra::VectorBase< Scalar > > &f, const Teuchos::RCP< Thyra::VectorBase< Scalar > > &x, const Teuchos::RCP< Thyra::VectorBase< Scalar > > &xDot, const Scalar time, const Teuchos::RCP< ImplicitODEParameters< Scalar > > &p)
	Evaluate implicit ODE residual, f(x, xDot, t, p). More...

virtual void	setInitialGuess (Teuchos::RCP< const Thyra::VectorBase< Scalar > > initialGuess) override
	Pass initial guess to Newton solver (only relevant for implicit solvers) More...

virtual void	setZeroInitialGuess (bool zIG)

virtual bool	getZeroInitialGuess () const

virtual Scalar	getInitTimeStep (const Teuchos::RCP< SolutionHistory< Scalar > > &) const override

Public Member Functions inherited from Tempus::Stepper< Scalar >
void	setStepperValues (const Teuchos::RCP< Teuchos::ParameterList > pl)
	Set Stepper member data from ParameterList. More...

Teuchos::RCP < Teuchos::ParameterList >	getValidParametersBasic () const
	Add basic parameters to Steppers ParameterList. More...

virtual bool	isInitialized ()
	True if stepper's member data is initialized. More...

virtual void	checkInitialized ()
	Check initialization, and error out on failure. More...

void	setStepperName (std::string s)
	Set the stepper name. More...

std::string	getStepperName () const
	Get the stepper name. More...

std::string	getStepperType () const
	Get the stepper type. The stepper type is used as an identifier for the stepper, and can only be set by the derived Stepper class. More...

virtual void	setUseFSAL (bool a)

void	setUseFSALTrueOnly (bool a)

void	setUseFSALFalseOnly (bool a)

bool	getUseFSAL () const

void	setICConsistency (std::string s)

std::string	getICConsistency () const

void	setICConsistencyCheck (bool c)

bool	getICConsistencyCheck () const

virtual Teuchos::RCP < Thyra::VectorBase< Scalar > >	getStepperX ()
	Get Stepper x. More...

virtual Teuchos::RCP < Thyra::VectorBase< Scalar > >	getStepperXDot ()
	Get Stepper xDot. More...

virtual Teuchos::RCP < Thyra::VectorBase< Scalar > >	getStepperXDotDot ()
	Get Stepper xDotDot. More...

virtual Teuchos::RCP < Thyra::VectorBase< Scalar > >	getStepperXDotDot (Teuchos::RCP< SolutionState< Scalar > > state)
	Get xDotDot from SolutionState or Stepper storage. More...

virtual std::string	description () const

Public Member Functions inherited from Teuchos::Describable
void	describe (std::ostream &out, const EVerbosityLevel verbLevel=verbLevel_default) const

virtual	~Describable ()

	LabeledObject ()

virtual	~LabeledObject ()

virtual void	setObjectLabel (const std::string &objectLabel)

virtual std::string	getObjectLabel () const

DescribableStreamManipulatorState	describe (const Describable &describable, const EVerbosityLevel verbLevel=Describable::verbLevel_default)

std::ostream &	operator<< (std::ostream &os, const DescribableStreamManipulatorState &d)

Public Member Functions inherited from Teuchos::VerboseObject< Stepper< Scalar > >
	VerboseObject (const EVerbosityLevel verbLevel=VERB_DEFAULT, const RCP< FancyOStream > &oStream=Teuchos::null)

virtual const VerboseObject &	setVerbLevel (const EVerbosityLevel verbLevel) const

virtual const VerboseObject &	setOverridingVerbLevel (const EVerbosityLevel verbLevel) const

virtual EVerbosityLevel	getVerbLevel () const

TEUCHOSPARAMETERLIST_LIB_DLL_EXPORT RCP< const ParameterList >	getValidVerboseObjectSublist ()

TEUCHOSPARAMETERLIST_LIB_DLL_EXPORT void	setupVerboseObjectSublist (ParameterList *paramList)

TEUCHOSPARAMETERLIST_LIB_DLL_EXPORT void	readVerboseObjectSublist (ParameterList paramList, RCP< FancyOStream > oStream, EVerbosityLevel *verbLevel)

void	readVerboseObjectSublist (ParameterList paramList, VerboseObject< ObjectType > verboseObject)

Public Member Functions inherited from Teuchos::VerboseObjectBase
virtual	~VerboseObjectBase ()

	VerboseObjectBase (const RCP< FancyOStream > &oStream=Teuchos::null)

virtual const VerboseObjectBase &	setOStream (const RCP< FancyOStream > &oStream) const

virtual const VerboseObjectBase &	setOverridingOStream (const RCP< FancyOStream > &oStream) const

virtual VerboseObjectBase &	setLinePrefix (const std::string &linePrefix)

virtual RCP< FancyOStream >	getOStream () const

virtual RCP< FancyOStream >	getOverridingOStream () const

virtual std::string	getLinePrefix () const

virtual OSTab	getOSTab (const int tabs=1, const std::string &linePrefix="") const

Private Attributes
Teuchos::RCP< Stepper< Scalar > >	startUpStepper_

Teuchos::RCP < StepperBDF2AppAction< Scalar > >	stepperBDF2AppAction_

Scalar	order_ = Scalar(2.0)

Basic stepper methods
virtual void	setModel (const Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > &appModel)
	Set the model. More...

virtual void	setAppAction (Teuchos::RCP< StepperBDF2AppAction< Scalar > > appAction)

virtual Teuchos::RCP < StepperBDF2AppAction< Scalar > >	getAppAction () const

void	setStartUpStepper (std::string startupStepperType)
	Set the stepper to use in first step. More...

void	setStartUpStepper (Teuchos::RCP< Stepper< Scalar > > startupStepper)

virtual void	initialize ()
	Initialize during construction and after changing input parameters. More...

virtual void	setInitialConditions (const Teuchos::RCP< SolutionHistory< Scalar > > &solutionHistory)
	Set the initial conditions and make them consistent. More...

virtual void	takeStep (const Teuchos::RCP< SolutionHistory< Scalar > > &solutionHistory)
	Take the specified timestep, dt, and return true if successful. More...

virtual Teuchos::RCP < Tempus::StepperState< Scalar > >	getDefaultStepperState ()
	Get a default (initial) StepperState. More...

virtual Scalar	getOrder () const

virtual Scalar	getOrderMin () const

virtual Scalar	getOrderMax () const

virtual bool	isExplicit () const

virtual bool	isImplicit () const

virtual bool	isExplicitImplicit () const

virtual bool	isOneStepMethod () const

virtual bool	isMultiStepMethod () const

virtual OrderODE	getOrderODE () const

Overridden from Teuchos::Describable
virtual void	describe (Teuchos::FancyOStream &out, const Teuchos::EVerbosityLevel verbLevel) const

Additional Inherited Members
Static Public Member Functions inherited from Teuchos::VerboseObject< Stepper< Scalar > >
static void	setDefaultVerbLevel (const EVerbosityLevel defaultVerbLevel)

static EVerbosityLevel	getDefaultVerbLevel ()

Static Public Member Functions inherited from Teuchos::VerboseObjectBase
static void	setDefaultOStream (const RCP< FancyOStream > &defaultOStream)

static RCP< FancyOStream >	getDefaultOStream ()

Static Public Attributes inherited from Teuchos::Describable
static const EVerbosityLevel	verbLevel_default

Protected Member Functions inherited from Tempus::Stepper< Scalar >
virtual void	setStepperX (Teuchos::RCP< Thyra::VectorBase< Scalar > > x)
	Set x for Stepper storage. More...

virtual void	setStepperXDot (Teuchos::RCP< Thyra::VectorBase< Scalar > > xDot)
	Set xDot for Stepper storage. More...

virtual void	setStepperXDotDot (Teuchos::RCP< Thyra::VectorBase< Scalar > > xDotDot)
	Set x for Stepper storage. More...

void	setStepperType (std::string s)
	Set the stepper type. More...

Protected Member Functions inherited from Teuchos::VerboseObject< Stepper< Scalar > >
void	initializeVerboseObject (const EVerbosityLevel verbLevel=VERB_DEFAULT, const RCP< FancyOStream > &oStream=Teuchos::null)

Protected Member Functions inherited from Teuchos::VerboseObjectBase
void	initializeVerboseObjectBase (const RCP< FancyOStream > &oStream=Teuchos::null)

virtual void	informUpdatedVerbosityState () const

Protected Attributes inherited from Tempus::StepperImplicit< Scalar >
Teuchos::RCP < WrapperModelEvaluator < Scalar > >	wrapperModel_

Teuchos::RCP < Thyra::NonlinearSolverBase < Scalar > >	solver_

Teuchos::RCP< const Thyra::VectorBase< Scalar > >	initialGuess_

bool	zeroInitialGuess_

std::string	solverName_

Protected Attributes inherited from Tempus::Stepper< Scalar >
bool	useFSAL_ = false
	Use First-Same-As-Last (FSAL) principle. More...

bool	isInitialized_
	True if stepper's member data is initialized. More...

Detailed Description

template<class Scalar>
class Tempus::StepperBDF2< Scalar >

BDF2 (Backward-Difference-Formula-2) time stepper.

For the implicit ODE system, $\mathcal{F}(\dot{x},x,t) = 0$ , the solution, $\dot{x}$ and $x$ , is determined using a solver (e.g., a non-linear solver, like NOX). This stepper allows for a variable time-step, $\Delta t$ . It is a 2-step method.

Algorithm The single-timestep algorithm for BDF2 is

$\begin{center} \parbox{5in}{ \rule{5in}{0.4pt} \\ {\bf Algorithm} BDF2 \\ \rule{5in}{0.4pt} \vspace{-15pt} \begin{enumerate} \setlength{\itemsep}{0pt} \setlength{\parskip}{0pt} \setlength{\parsep}{0pt} \item {\bf if ( "Startup", i.e., number of states $<$ 3) then} \item \quad {\bf Take timestep with startup Stepper.} \item \quad {\bf return} \item {\bf endif} \item {\it appAction.execute(solutionHistory, stepper, BEGIN\_STEP)} \item {\bf Get old states $x_{n-1}$ and $x_{n-2}$.} \item {\bf Set ODE parameters.} \item \quad {\bf Time derivative: } $\dot{x}_{n} = \frac{2\tau_n + \tau_{n-1}}{\tau_n + \tau_{n-1}} \left[ \frac{x_n-x_{n-1}}{\tau_n}\right] - \frac{\tau_n}{\tau_n + \tau_{n-1}} \left[ \frac{x_{n-1}-x_{n-2}}{\tau_{n-1}}\right]$ \item \quad {\bf Alpha: $\alpha = \frac{2\tau_n + \tau_{n-1}}{(\tau_n + \tau_{n-1})\tau_n}$} \item \quad {\bf Beta: $\beta = 1$} \item {\it appAction.execute(solutionHistory, stepper, BEFORE\_SOLVE)} \item {\bf Solve $\mathcal{F}(\dot{x}_n,x_n,t_n) = 0$ for $x_n$.} \item {\it appAction.execute(solutionHistory, stepper, AFTER\_SOLVE)} \item {\bf Update} $\dot{x}_{n} = \frac{2\tau_n + \tau_{n-1}}{\tau_n + \tau_{n-1}} \left[ \frac{x_n-x_{n-1}}{\tau_n}\right] - \frac{\tau_n}{\tau_n + \tau_{n-1}} \left[ \frac{x_{n-1}-x_{n-2}}{\tau_{n-1}}\right]$ \item {\it appAction.execute(solutionHistory, stepper, END\_STEP)} \end{enumerate} \vspace{-10pt} \rule{5in}{0.4pt} } \end{center}$

The startup stepper allows BDF2 to use user-specified Stepper for the first timestep in order to populate the SolutionHistory with past states. A one-step startup stepper is perfect for this situation, e.g., Backward Euler or RK4. The default startup stepper is 'DIRK 1 Stage Theta Method', which is second order accurate and allows an overall second-order solution.

The First-Same-As-Last (FSAL) principle is not needed for BDF2. The default is to set useFSAL=false, however useFSAL=true will also work but have no affect (i.e., no-op).

Iteration Matrix, . Recalling that the definition of the iteration matrix, $W$ , is

$W = \alpha \frac{\partial \mathcal{F}_n}{\partial \dot{x}_n} + \beta \frac{\partial \mathcal{F}_n}{\partial x_n},$

where $\alpha \equiv \frac{\partial \dot{x}_n(x_n) }{\partial x_n},$ and $\beta \equiv \frac{\partial x_n}{\partial x_n} = 1$ , and the time derivative for BDF2 is

$\dot{x}_n(x_n) = \frac{2\tau_n + \tau_{n-1}}{\tau_n + \tau_{n-1}} \left[ \frac{x_n-x_{n-1}}{\tau_n}\right] - \frac{\tau_n}{\tau_n + \tau_{n-1}} \left[ \frac{x_{n-1}-x_{n-2}}{\tau_{n-1}}\right],$

where $\Delta t_n = \tau_n = t_n - t_{n-1}$ , we can determine that $\alpha = \frac{2\tau_n + \tau_{n-1}}{(\tau_n + \tau_{n-1})\tau_n}$ and $\beta = 1$ , and therefore write

$W = \frac{2\tau_n + \tau_{n-1}}{(\tau_n + \tau_{n-1})\tau_n} \frac{\partial \mathcal{F}_n}{\partial \dot{x}_n} + \frac{\partial \mathcal{F}_n}{\partial x_n}.$

Definition at line 102 of file Tempus_StepperBDF2_decl.hpp.

Constructor & Destructor Documentation

template<class Scalar >

Tempus::StepperBDF2< Scalar >::StepperBDF2 ( )

Default constructor.

Requires the following calls before takeStep(): setModel() and initialize().

Definition at line 19 of file Tempus_StepperBDF2_impl.hpp.

template<class Scalar >

Tempus::StepperBDF2< Scalar >::StepperBDF2	(	const Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > &	appModel,
		const Teuchos::RCP< Thyra::NonlinearSolverBase< Scalar > > &	solver,
		const Teuchos::RCP< Stepper< Scalar > > &	startUpStepper,
		bool	useFSAL,
		std::string	ICConsistency,
		bool	ICConsistencyCheck,
		bool	zeroInitialGuess,
		const Teuchos::RCP< StepperBDF2AppAction< Scalar > > &	stepperBDF2AppAction
	)

Definition at line 34 of file Tempus_StepperBDF2_impl.hpp.

Member Function Documentation

template<class Scalar >

void Tempus::StepperBDF2< Scalar >::setModel ( const Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > & appModel )

virtual

Set the model.

Reimplemented from Tempus::StepperImplicit< Scalar >.

Definition at line 59 of file Tempus_StepperBDF2_impl.hpp.

template<class Scalar >

void Tempus::StepperBDF2< Scalar >::setAppAction ( Teuchos::RCP< StepperBDF2AppAction< Scalar > > appAction )

virtual

Definition at line 110 of file Tempus_StepperBDF2_impl.hpp.

template<class Scalar >

virtual Teuchos::RCP<StepperBDF2AppAction<Scalar> > Tempus::StepperBDF2< Scalar >::getAppAction ( ) const

inlinevirtual

Definition at line 126 of file Tempus_StepperBDF2_decl.hpp.

template<class Scalar >

void Tempus::StepperBDF2< Scalar >::setStartUpStepper ( std::string startupStepperType )

Set the stepper to use in first step.

Definition at line 73 of file Tempus_StepperBDF2_impl.hpp.

template<class Scalar >

void Tempus::StepperBDF2< Scalar >::setStartUpStepper ( Teuchos::RCP< Stepper< Scalar > > startupStepper )

Definition at line 89 of file Tempus_StepperBDF2_impl.hpp.

template<class Scalar >

void Tempus::StepperBDF2< Scalar >::initialize ( )

virtual

Initialize during construction and after changing input parameters.

Reimplemented from Tempus::Stepper< Scalar >.

Definition at line 124 of file Tempus_StepperBDF2_impl.hpp.

template<class Scalar >

void Tempus::StepperBDF2< Scalar >::setInitialConditions ( const Teuchos::RCP< SolutionHistory< Scalar > > & solutionHistory )

virtual

Set the initial conditions and make them consistent.

Reimplemented from Tempus::StepperImplicit< Scalar >.

Definition at line 130 of file Tempus_StepperBDF2_impl.hpp.

template<class Scalar >

void Tempus::StepperBDF2< Scalar >::takeStep ( const Teuchos::RCP< SolutionHistory< Scalar > > & solutionHistory )

virtual

Take the specified timestep, dt, and return true if successful.

Implements Tempus::Stepper< Scalar >.

Definition at line 147 of file Tempus_StepperBDF2_impl.hpp.

template<class Scalar >

Teuchos::RCP< Tempus::StepperState< Scalar > > Tempus::StepperBDF2< Scalar >::getDefaultStepperState ( )

virtual

Get a default (initial) StepperState.

Provide a StepperState to the SolutionState. This Stepper does not have any special state data, so just provide the base class StepperState with the Stepper description. This can be checked to ensure that the input StepperState can be used by this Stepper.

Implements Tempus::Stepper< Scalar >.

Definition at line 258 of file Tempus_StepperBDF2_impl.hpp.

template<class Scalar >

virtual Scalar Tempus::StepperBDF2< Scalar >::getOrder ( ) const

inlinevirtual

Implements Tempus::Stepper< Scalar >.

Definition at line 148 of file Tempus_StepperBDF2_decl.hpp.

template<class Scalar >

virtual Scalar Tempus::StepperBDF2< Scalar >::getOrderMin ( ) const

inlinevirtual

Implements Tempus::Stepper< Scalar >.

Definition at line 149 of file Tempus_StepperBDF2_decl.hpp.