Step Control Strategy for first-order time integration. More...

#include <Rythmos_FirstOrderErrorStepControlStrategy_decl.hpp>

Inheritance diagram for Rythmos::FirstOrderErrorStepControlStrategy< Scalar >:

Overridden from ParameterListAcceptor
void	setParameterList (RCP< Teuchos::ParameterList > const &paramList)
RCP< Teuchos::ParameterList >	getNonconstParameterList ()
RCP< Teuchos::ParameterList >	unsetParameterList ()
RCP< const Teuchos::ParameterList >	getValidParameters () const
void	initialize (const StepperBase< Scalar > &stepper)

Overridden from StepControlStrategyBase
void	setRequestedStepSize (const StepperBase< Scalar > &stepper, const Scalar &stepSize, const StepSizeType &stepSizeType)
void	nextStepSize (const StepperBase< Scalar > &stepper, Scalar stepSize, StepSizeType stepSizeType, int *order)
void	setCorrection (const StepperBase< Scalar > &stepper, const RCP< const Thyra::VectorBase< Scalar > > &soln, const RCP< const Thyra::VectorBase< Scalar > > &ee, int solveStatus)
bool	acceptStep (const StepperBase< Scalar > &stepper, Scalar *LETValue)
void	completeStep (const StepperBase< Scalar > &stepper)
AttemptedStepStatusFlag	rejectStep (const StepperBase< Scalar > &stepper)
StepControlStrategyState	getCurrentState ()
int	getMaxOrder () const
void	setStepControlData (const StepperBase< Scalar > &stepper)
bool	supportsCloning () const
RCP< StepControlStrategyBase< Scalar > >	cloneStepControlStrategyAlgorithm () const

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

Detailed Description

template<class Scalar>
class Rythmos::FirstOrderErrorStepControlStrategy< Scalar >

Step Control Strategy for first-order time integration.

This step-control strategy assumes a first-order predictor (Forward Euler) for a first-order time integrator (Backward Euler) and tries to maintain a constant error based on relative and absolute tolerances by adjusting the step size. Although specifically built from Forward and Backward Euler, one could use this for other first-order schemes. See Grasho and Sani, "Incompressible Flow and the Finite Element Method", Volume One, p. 268.

Theory

To control the error through step-size selection, Forward Euler is used as a predictor

$\‍[ x^P_n = x_{n-1} + \Delta t f(x_{n-1}, t_{n-1}) \‍]$

with the local truncation error (LTE) of

$\begin{eqnarray*} d_n &=& x^P_n - x(t_n) &=& -\frac{\Delta t^2}{2} \ddot{x}_n + O(\Delta t^3) \end{eqnarray*}$

Similarly for Backward Euler

$\‍[ x_n = x_{n-1} + \Delta t f(x_n, t_n) \‍]$

with the local truncation error (LTE) of

$\begin{eqnarray*} d_n &=& x_n - x(t_n) &=& \frac{\Delta t^2}{2} \ddot{x}_n + O(\Delta t^3) \end{eqnarray*}$

Using the Forward Euler as an estimate of the LTE for the Backward Euler, eliminating $\ddot{x}_n$ , and forming $ d_n = x_n - x(t_n) $ in terms of $ (x_n - x^P_n) $ , we get

$\‍[ d_n = x_n - x(t_n) = (x_n - x^P_n)/2 \‍]$

which gives us an estimate of the LTE, $ d_n $ in term of the difference between the Backward Euler solution, $ x_n $ , and the predicted Forward Euler solution, $ x^P_n $ .

To select the next step size, we would like the LTE to be meet a user-supplied tolerance,

$\‍[ d_{n+1} \leq \epsilon_r ||x_n|| + \epsilon_a \‍]$

where $\epsilon_r$ and $\epsilon_a$ are the relative and absolute tolerances, and $||\cdot||$ is the norm. Normalizing this with respect to the current step, which allows us the relate the current step size to the next step size, and using the LTE, we find

$\‍[ \frac{d_{n+1}}{d_n} = \frac{\epsilon_r ||x_n|| + \epsilon_a}{d_n} = \frac{\Delta t^2_{n+1} \ddot{x}_{n+1}}{\Delta t^2_n \ddot{x}_n} = \left(\frac{\Delta t_{n+1}}{\Delta t_n}\right)^2 + O(\Delta t) \‍]$

leading to the value for the next step size

$\‍[ \Delta t_{n+1} = \Delta t_n \left(\frac{\epsilon_r ||x_n|| + \epsilon_a}{d_n}\right)^{1/2} \‍]$

Definition at line 101 of file Rythmos_FirstOrderErrorStepControlStrategy_decl.hpp.

Member Function Documentation

◆ setRequestedStepSize()

template<class Scalar>

void Rythmos::FirstOrderErrorStepControlStrategy< Scalar >::setRequestedStepSize	(	const StepperBase< Scalar > &	stepper,
		const Scalar &	stepSize,
		const StepSizeType &	stepSizeType )

virtual

Implements Rythmos::StepControlStrategyBase< Scalar >.

Definition at line 201 of file Rythmos_FirstOrderErrorStepControlStrategy_def.hpp.

◆ nextStepSize()

template<class Scalar>

void Rythmos::FirstOrderErrorStepControlStrategy< Scalar >::nextStepSize	(	const StepperBase< Scalar > &	stepper,
		Scalar *	stepSize,
		StepSizeType *	stepSizeType,
		int *	order )

virtual

Implements Rythmos::StepControlStrategyBase< Scalar >.

Definition at line 236 of file Rythmos_FirstOrderErrorStepControlStrategy_def.hpp.

◆ setCorrection()

template<class Scalar>

void Rythmos::FirstOrderErrorStepControlStrategy< Scalar >::setCorrection	(	const StepperBase< Scalar > &	stepper,
		const RCP< const Thyra::VectorBase< Scalar > > &	soln,
		const RCP< const Thyra::VectorBase< Scalar > > &	ee,
		int	solveStatus )