Rythmos::ExplicitTaylorPolynomialStepper

Rythmos::ExplicitTaylorPolynomialStepper< Scalar > Class Template Reference

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 Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > &model)
	Set model.
Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > >	getModel () const

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

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

Teuchos::RCP< Teuchos::ParameterList >	unsetParameterList ()

std::string	description () const

std::ostream &	describe (std::ostream &out, const Teuchos::EVerbosityLevel verbLevel, const std::string leadingIndent, const std::string indentSpacer) const

void	addPoints (const Array< Scalar > &time_vec, const Array< Teuchos::RCP< const Thyra::VectorBase< Scalar > > > &x_vec, const Array< Teuchos::RCP< const Thyra::VectorBase< Scalar > > > &xdot_vec)
void	getPoints (const Array< Scalar > &time_vec, Array< Teuchos::RCP< const Thyra::VectorBase< Scalar > > > x_vec, Array< Teuchos::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 140 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 145 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

Constructor & Destructor Documentation

template<class Scalar>

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

Constructor.

Definition at line 373 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

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

Destructor.

Definition at line 378 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 710 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

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

Set model.

Implements Rythmos::StepperBase< Scalar >.

Definition at line 383 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

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

Implements Rythmos::StepperBase< Scalar >.

Definition at line 398 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 405 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

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

Implements Rythmos::StepperBase< Scalar >.

Definition at line 513 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

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

Redefined from Teuchos::ParameterListAcceptor .

Implements Teuchos::ParameterListAcceptor.

Definition at line 529 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

Teuchos::RCP< Teuchos::ParameterList > Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::getParameterList ( ) [virtual]

Implements Teuchos::ParameterListAcceptor.

Definition at line 560 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

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

Implements Teuchos::ParameterListAcceptor.

Definition at line 567 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

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

Reimplemented from Teuchos::Describable.

Definition at line 575 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

std::ostream & Rythmos::ExplicitTaylorPolynomialStepper< Scalar >::describe	(	std::ostream &	out,
		const Teuchos::EVerbosityLevel	verbLevel,
		const std::string	leadingIndent,
		const std::string	indentSpacer
	)			const

Definition at line 582 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

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

Redefined from InterpolationBufferBase Add points to buffer

Implements Rythmos::InterpolationBufferBase< Scalar >.

Definition at line 617 of file Rythmos_ExplicitTaylorPolynomialStepper.hpp.

template<class Scalar>

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