Implementation of Rythmos::Stepper for explicit Taylor polynomial time integration of ODEs. More...

#include <Rythmos_ExplicitTaylorPolynomialStepper.hpp>

Inheritance diagram for Rythmos::ExplicitTaylorPolynomialStepper< Scalar >:

Public Types
typedef Teuchos::ScalarTraits < Scalar >::magnitudeType	ScalarMag
	Typename of magnitude of scalars.
Public Member Functions
	ExplicitTaylorPolynomialStepper ()
	Constructor.
	~ExplicitTaylorPolynomialStepper ()
	Destructor.
RCP< const Thyra::VectorSpaceBase< Scalar > >	get_x_space () const
	Return the space for `x` and `x_dot`
void	setModel (const RCP< const Thyra::ModelEvaluator< Scalar > > &model)
	Set model.
void	setNonconstModel (const RCP< Thyra::ModelEvaluator< Scalar > > &model)
	Set model.
RCP< const Thyra::ModelEvaluator< Scalar > >	getModel () const

RCP< Thyra::ModelEvaluator < Scalar > >	getNonconstModel ()

void	setInitialCondition (const Thyra::ModelEvaluatorBase::InArgs< Scalar > &initialCondition)

Thyra::ModelEvaluatorBase::InArgs < Scalar >	getInitialCondition () const

Scalar	takeStep (Scalar dt, StepSizeType flag)
	Take a time step of magnitude `dt`.
const StepStatus< Scalar >	getStepStatus () const

void	setParameterList (RCP< Teuchos::ParameterList > const &paramList)
	Redefined from Teuchos::ParameterListAcceptor.
RCP< Teuchos::ParameterList >	getNonconstParameterList ()

RCP< Teuchos::ParameterList >	unsetParameterList ()

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

std::string	description () const

void	describe (Teuchos::FancyOStream &out, const Teuchos::EVerbosityLevel verbLevel=Teuchos::Describable::verbLevel_default) const

void	addPoints (const Array< Scalar > &time_vec, const Array< RCP< const Thyra::VectorBase< Scalar > > > &x_vec, const Array< RCP< const Thyra::VectorBase< Scalar > > > &xdot_vec)
void	getPoints (const Array< Scalar > &time_vec, Array< RCP< const Thyra::VectorBase< Scalar > > > x_vec, Array< RCP< const Thyra::VectorBase< Scalar > > > xdot_vec, Array< ScalarMag > *accuracy_vec) const
	Get values from buffer.
void	setRange (const TimeRange< Scalar > &range, const InterpolationBufferBase< Scalar > &IB)
	Fill data in from another interpolation buffer.
TimeRange< Scalar >	getTimeRange () const

void	getNodes (Array< Scalar > *time_vec) const
	Get interpolation nodes.
void	removeNodes (Array< Scalar > &time_vec)
	Remove interpolation nodes.
int	getOrder () const
	Get order of interpolation.

Detailed Description

template<class Scalar>
class Rythmos::ExplicitTaylorPolynomialStepper< Scalar >

Implementation of Rythmos::Stepper for explicit Taylor polynomial time integration of ODEs.

Let

$\frac{dx}{dt} = f(x,t), \quad x(t_0) = a$

be an ODE initial-value problem. This class implements a single time step of an explicit Taylor polynomial time integration method for computing numerical solutions to the IVP. The method consists of computing a local truncated Taylor series solution to the ODE (section Computing the Taylor Polynomial), estimating a step size within the radius of convergence of the Taylor series (section Computing a Step Size) and then summing the polynomial at that step to compute the next point in the numerical integration (section Summing the Polynomial). The algorithmic parameters to the method are controlled through the params argument to the constructor which are described in section Parameters.

Computing the Taylor Polynomial

Let

$x(t) = \sum_{k=0}^\infty x_k (t-t_0)^k$

be a power series solution to the IVP above. Then $f(x(t))$ can be expaned in a power series along the solution curve $x(t)$ :

$f(x(t),t) = \sum_{k=0}^\infty f_k (t-t_0)^k$

where

$f_k = \left.\frac{1}{k!}\frac{d^k}{dt^k} f(x(t),t)\right|_{t=t_0}.$

By differentiating the power series for $x(t)$ to compute a power series for $dx/dt$ and then comparing coefficients in the equation $dx/dt=f(x(t),t)$ we find the following recurrence relationship for the Taylor coefficients $\{x_k\}$ :

$x_{k+1} = \frac{1}{k+1} f_k, \quad k=0,1,\dots$

where each coefficient $f_k$ is a function only of $x_0,\dots,x_k$ and can be efficiently computed using the Taylor polynomial mode of automatic differentation. This allows the Taylor coefficients to be iteratively computed starting with the initial point $x_0$ up to some fixed degree $d$ to yield a local truncated Taylor polynomial approximating the solution to the IVP.

Computing a Step Size

With the truncated Taylor polynomial solution $\tilde{x}(t) = \sum_{k=0}^d x_k (t-t_0)^k$ in hand, a step size is chosen by estimating the truncation error in the polynomial solution and forcing this error to be less than some prescribed tolerance. Let

$\rho = \max_{d/2\leq k\leq d} (1+\|x_k\|_\infty)^{1/k}$

so $\|x_k\|_\infty\leq\rho^k$ for $d/2\leq k \leq d$ . Assume $\|x_k\|\leq\rho^k$ for $k>d$ as well, then for any $h<1/\rho$ it can be shown that the truncation error is bounded by

$\frac{(\rho h)^{d+1}}{1-\rho h}.$

A step size $h$ is then given by

$h = \exp\left(\frac{1}{d+1}\log\varepsilon-\log\rho\right)$

for some error tolerance $\varepsilon$ given an error of approximatly $\varepsilon$ .

Summing the Polynomial

With a step size $h$ computed,

$x^\ast = \sum_{k=0}^d x_k h^k$

is used as the next integration point where a new Taylor series is calculated. Local error per step can also be controlled by computing $\|dx^\ast/dt - f(x^\ast)\|_\infty$ . If this error is too large, the step size can be reduced to an appropriate size.

Parameters

This method recognizes the following algorithmic parameters that can be set in the params argument to the constructor:

"Initial Time" (Scalar) [Default = 0] Initial integration time
"Final Time" (Scalar) [Default = 1] Final integration time
"Local Error Tolerance" (Magnitude) [Default = 1.0e-10] Error tolerance on $\|dx^\ast/dt - f(x^\ast)\|_\infty$ as described above.
"Minimum Step Size" (Scalar) [Default = 1.0e-10] Minimum step size
"Maximum Step Size" (Scalar) [Default = 1.0] Maximum step size
"Taylor Polynomial Degree" (int) [Default = 40] Degree of local Taylor polynomial approximation.

Definition at line 163 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

Member Typedef Documentation

template<class Scalar>

typedef Teuchos::ScalarTraits<Scalar>::magnitudeType Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::ScalarMag

Typename of magnitude of scalars.

Reimplemented from Rythmos::InterpolationBufferBase< Scalar >.

Definition at line 168 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

Constructor & Destructor Documentation

template<class Scalar >

Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::ExplicitTaylorPolynomialStepper ( )

Constructor.

Definition at line 366 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar >

Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::~ExplicitTaylorPolynomialStepper ( )

Destructor.

Definition at line 374 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

Member Function Documentation

template<class Scalar >

RCP< const Thyra::VectorSpaceBase< Scalar > > Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::get_x_space ( ) const [virtual]

Return the space for x and x_dot

Implements Rythmos::InterpolationBufferBase< Scalar >.

Definition at line 865 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

void Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::setModel ( const RCP< const Thyra::ModelEvaluator< Scalar > > & model ) [virtual]

Set model.

Implements Rythmos::StepperBase< Scalar >.

Definition at line 408 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

void Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::setNonconstModel ( const RCP< Thyra::ModelEvaluator< Scalar > > & model ) [virtual]

Set model.

Implements Rythmos::StepperBase< Scalar >.

Definition at line 421 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar >

RCP< const Thyra::ModelEvaluator< Scalar > > Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::getModel ( ) const [virtual]

Implements Rythmos::StepperBase< Scalar >.

Definition at line 431 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar >

RCP< Thyra::ModelEvaluator< Scalar > > Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::getNonconstModel ( ) [virtual]

Implements Rythmos::StepperBase< Scalar >.

Definition at line 439 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

void Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::setInitialCondition ( const Thyra::ModelEvaluatorBase::InArgs< Scalar > & initialCondition ) [virtual]

Implements Rythmos::StepperBase< Scalar >.

Definition at line 446 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar >

Thyra::ModelEvaluatorBase::InArgs< Scalar > Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::getInitialCondition ( ) const [virtual]

Implements Rythmos::StepperBase< Scalar >.

Definition at line 469 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

Scalar Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::takeStep	(	Scalar	dt,
		StepSizeType	flag
	)		`[virtual]`

Take a time step of magnitude dt.

Implements Rythmos::StepperBase< Scalar >.

Definition at line 477 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar >

const StepStatus< Scalar > Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::getStepStatus ( ) const [virtual]

Implements Rythmos::StepperBase< Scalar >.

Definition at line 595 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar >

void Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::setParameterList ( RCP< Teuchos::ParameterList > const & paramList )

Redefined from Teuchos::ParameterListAcceptor.

Definition at line 628 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar >

RCP< Teuchos::ParameterList > Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::getNonconstParameterList ( )

Definition at line 663 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar >

RCP< Teuchos::ParameterList > Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::unsetParameterList ( )

Definition at line 671 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar >

RCP< const Teuchos::ParameterList > Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::getValidParameters ( ) const

Definition at line 681 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar >

std::string Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::description ( ) const

Definition at line 703 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar >

void Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::describe	(	Teuchos::FancyOStream &	out,
		const Teuchos::EVerbosityLevel	verbLevel = `Teuchos::Describable::verbLevel_default`
	)		const

Definition at line 711 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

void Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::addPoints	(	const Array< Scalar > &	time_vec,
		const Array< RCP< const Thyra::VectorBase< Scalar > > > &	x_vec,
		const Array< RCP< const Thyra::VectorBase< Scalar > > > &	xdot_vec
	)		`[virtual]`

Redefined from InterpolationBufferBase Add points to buffer

Implements Rythmos::InterpolationBufferBase< Scalar >.

Definition at line 743 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

void Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::getPoints	(	const Array< Scalar > &	time_vec,
		Array< RCP< const Thyra::VectorBase< Scalar > > > *	x_vec,
		Array< RCP< const Thyra::VectorBase< Scalar > > > *	xdot_vec,
		Array< ScalarMag > *	accuracy_vec
	)		const `[virtual]`