van der Pol model formulated for IMEX-RK. More...

#include <VanDerPol_IMEX_ImplicitModel_decl.hpp>

Inheritance diagram for Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >:

Public Member Functions
	VanDerPol_IMEX_ImplicitModel (Teuchos::RCP< Teuchos::ParameterList > pList=Teuchos::null)

Private functions overridden from ModelEvaluatorDefaultBase.
int	dim_
	Number of state unknowns (2)

int	Np_
	Number of parameter vectors (1)

int	np_
	Number of parameters in this vector (1)

int	Ng_
	Number of observation functions (0)

int	ng_
	Number of elements in this observation function (0)

bool	haveIC_
	false => no nominal values are provided (default=true)

bool	acceptModelParams_
	Changes inArgs to require parameters.

bool	useDfDpAsTangent_
	Treat DfDp OutArg as tangent (df/dx*dx/dp+df/dp)

bool	isInitialized_

Thyra::ModelEvaluatorBase::InArgs< Scalar >	inArgs_

Thyra::ModelEvaluatorBase::OutArgs< Scalar >	outArgs_

Thyra::ModelEvaluatorBase::InArgs< Scalar >	nominalValues_

Teuchos::RCP< const Thyra::VectorSpaceBase< Scalar > >	x_space_

Teuchos::RCP< const Thyra::VectorSpaceBase< Scalar > >	f_space_

Teuchos::RCP< const Thyra::VectorSpaceBase< Scalar > >	p_space_

Teuchos::RCP< const Thyra::VectorSpaceBase< Scalar > >	dxdp_space_

Teuchos::RCP< const Thyra::VectorSpaceBase< Scalar > >	g_space_

Scalar	epsilon_
	This is a model parameter.

Scalar	t0_ic_
	initial time

Scalar	x0_ic_
	initial condition for x0

Scalar	x1_ic_
	initial condition for x1

Thyra::ModelEvaluatorBase::OutArgs< Scalar >	createOutArgsImpl () const

void	evalModelImpl (const Thyra::ModelEvaluatorBase::InArgs< Scalar > &inArgs_bar, const Thyra::ModelEvaluatorBase::OutArgs< Scalar > &outArgs_bar) const

Public functions overridden from ModelEvaluator.
Teuchos::RCP< const Thyra::VectorSpaceBase< Scalar > >	get_x_space () const

Teuchos::RCP< const Thyra::VectorSpaceBase< Scalar > >	get_f_space () const

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

Teuchos::RCP< Thyra::LinearOpWithSolveBase< Scalar > >	create_W () const

Teuchos::RCP< Thyra::LinearOpBase< Scalar > >	create_W_op () const

Teuchos::RCP< const Thyra::LinearOpWithSolveFactoryBase< Scalar > >	get_W_factory () const

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

Teuchos::RCP< const Thyra::VectorSpaceBase< Scalar > >	get_p_space (int l) const

Teuchos::RCP< const Teuchos::Array< std::string > >	get_p_names (int l) const

Teuchos::RCP< const Thyra::VectorSpaceBase< Scalar > >	get_g_space (int j) const

Public functions overridden from ParameterListAcceptor.
void	setParameterList (Teuchos::RCP< Teuchos::ParameterList > const &paramList)

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

void	setupInOutArgs_ () const

Detailed Description

template<class Scalar>
class Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >

van der Pol model formulated for IMEX-RK.

This is a canonical equation of a nonlinear oscillator (Hairer, Norsett, and Wanner, pp. 111-115, and Hairer and Wanner, pp. 4-5) for an electrical circuit. In implicit ODE form, $\mathcal{F}(\dot{x},x,t) = 0$ , the scaled problem can be written as

$\begin{eqnarray*} \dot{x}_0(t) - x_1(t) & = & 0 \\ \dot{x}_1(t) - [(1-x_0^2)x_1-x_0]/\epsilon & = & 0 \end{eqnarray*}$

where the initial conditions are

$\begin{eqnarray*} x_0(t_0=0) & = & 2 \\ x_1(t_0=0) & = & 0 \end{eqnarray*}$

and the initial time derivatives are

$\begin{eqnarray*} \dot{x}_0(t_0=0) & = & x_1(t_0=0) = 0 \\ \dot{x}_1(t_0=0) & = & [(1-x_0^2)x_1-x_0]/\epsilon = -2/\epsilon \end{eqnarray*}$

For an IMEX-RK time stepper, we need to rewrite this in the following form

$\begin{eqnarray*} M(x,t)\, \dot{x} + G(x,t) + F(x,t) & = & 0, \\ \mathcal{G}(\dot{x},x,t) + F(x,t) & = & 0, \end{eqnarray*}$

where $\mathcal{G}(\dot{x},x,t) = M(x,t)\, \dot{x} + G(x,t)$ , is the mass matrix, is the operator representing the "slow" physics (and evolved explicitly), and is the operator representing the "fast" physics. For the van der Pol problem, we can separate the terms as follows

$x = \left\{\begin{array}{c} x_0 \\ x_1 \end{array}\right\},\; F(x,t) = \left\{\begin{array}{c} -x_1 \\ x_0/\epsilon\end{array}\right\}, \mbox{ and } G(x,t) = \left\{\begin{array}{c} 0 \\ -(1-x_0^2)x_1/\epsilon \end{array}\right\}$

where is the identity matrix.

Thus the explicit van der Pol model (VanDerPol_IMEX_ExplicitModel) formulated for IMEX is

$\begin{eqnarray*} \mathcal{F}_0 & = & \dot{x}_0(t) - x_1(t) = 0 \\ \mathcal{F}_1 & = & \dot{x}_1(t) + x_0/\epsilon = 0 \end{eqnarray*}$

and the implicit van der Pol model (VanDerPol_IMEX_ImplicitModel) formulated for IMEX is

$\begin{eqnarray*} \mathcal{G}_0 & = & \dot{x}_0(t) = 0 \\ \mathcal{G}_1 & = & \dot{x}_1(t) - (1-x_0^2)x_1/\epsilon = 0 \end{eqnarray*}$

Recalling the defintion of the iteration matrix, ,

$W_{ij} \equiv \frac{d\mathcal{G}_i}{dx_j} = \alpha \frac{\partial\mathcal{G}_i}{\partial \dot{x}_j} + \beta \frac{\partial\mathcal{G}_i}{\partial x_j}$

where

$\alpha = \left\{ \begin{array}{cl} \frac{\partial\dot{x}_i}{\partial x_j} & \mbox{ if } i = j \\ 0 & \mbox{ if } i \neq j \end{array} \right. \;\;\;\; \mbox{ and } \;\;\;\; \beta = 1$

we can write for the implicit van der Pol model (VanDerPol_IMEX_ImplicitModel)

$\begin{eqnarray*} W_{00} = \alpha \frac{\partial\mathcal{G}_0}{\partial \dot{x}_0} + \beta \frac{\partial\mathcal{G}_0}{\partial x_0} & = & \alpha \\ W_{01} = \alpha \frac{\partial\mathcal{G}_0}{\partial \dot{x}_1} + \beta \frac{\partial\mathcal{G}_0}{\partial x_1} & = & 0 \\ W_{10} = \alpha \frac{\partial\mathcal{G}_1}{\partial \dot{x}_0} + \beta \frac{\partial\mathcal{G}_1}{\partial x_0} & = & 2 \beta x_0 x_1/\epsilon \\ W_{11} = \alpha \frac{\partial\mathcal{G}_1}{\partial \dot{x}_1} + \beta \frac{\partial\mathcal{G}_1}{\partial x_1} & = & \alpha + \beta (x^2_0 - 1)/\epsilon \\ \end{eqnarray*}$

Definition at line 108 of file VanDerPol_IMEX_ImplicitModel_decl.hpp.

Constructor & Destructor Documentation

◆ VanDerPol_IMEX_ImplicitModel()

template<class Scalar >

Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::VanDerPol_IMEX_ImplicitModel ( Teuchos::RCP< Teuchos::ParameterList > pList = Teuchos::null )

Definition at line 30 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

Member Function Documentation

◆ get_x_space()

template<class Scalar >

Teuchos::RCP< const Thyra::VectorSpaceBase< Scalar > > Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::get_x_space ( ) const

Definition at line 61 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ get_f_space()

template<class Scalar >

Teuchos::RCP< const Thyra::VectorSpaceBase< Scalar > > Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::get_f_space ( ) const

Definition at line 68 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ getNominalValues()

template<class Scalar >

Thyra::ModelEvaluatorBase::InArgs< Scalar > Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::getNominalValues ( ) const

Definition at line 75 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ create_W()

template<class Scalar >

Teuchos::RCP< Thyra::LinearOpWithSolveBase< Scalar > > Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::create_W ( ) const

Definition at line 84 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ create_W_op()

template<class Scalar >

Teuchos::RCP< Thyra::LinearOpBase< Scalar > > Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::create_W_op ( ) const

Definition at line 123 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ get_W_factory()

template<class Scalar >

Teuchos::RCP< const Thyra::LinearOpWithSolveFactoryBase< Scalar > > Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::get_W_factory ( ) const

Definition at line 132 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ createInArgs()

template<class Scalar >

Thyra::ModelEvaluatorBase::InArgs< Scalar > Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::createInArgs ( ) const

Definition at line 141 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ get_p_space()

template<class Scalar >

Teuchos::RCP< const Thyra::VectorSpaceBase< Scalar > > Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::get_p_space ( int l ) const

Definition at line 285 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ get_p_names()

template<class Scalar >

Teuchos::RCP< const Teuchos::Array< std::string > > Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::get_p_names ( int l ) const

Definition at line 300 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ get_g_space()

template<class Scalar >

Teuchos::RCP< const Thyra::VectorSpaceBase< Scalar > > Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::get_g_space ( int j ) const

Definition at line 320 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ setParameterList()

template<class Scalar >

void Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::setParameterList ( Teuchos::RCP< Teuchos::ParameterList > const & paramList )

Definition at line 397 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ getValidParameters()

template<class Scalar >

Teuchos::RCP< const Teuchos::ParameterList > Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::getValidParameters ( ) const

Definition at line 426 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ setupInOutArgs_()

template<class Scalar >

void Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::setupInOutArgs_ ( ) const

private

Definition at line 327 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ createOutArgsImpl()

template<class Scalar >

Thyra::ModelEvaluatorBase::OutArgs< Scalar > Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::createOutArgsImpl ( ) const

private

Definition at line 151 of file VanDerPol_IMEX_ImplicitModel_impl.hpp.

◆ evalModelImpl()

template<class Scalar >

void Tempus_Test::VanDerPol_IMEX_ImplicitModel< Scalar >::evalModelImpl	(	const Thyra::ModelEvaluatorBase::InArgs< Scalar > &	inArgs_bar,
		const Thyra::ModelEvaluatorBase::OutArgs< Scalar > &	outArgs_bar
	)		const