TimeAdvanceBDFVariableStep - Backward differencing formula time discretization with non-uniform time step. More...

#include <TimeAdvanceBDFVariableStep.hpp>

Collaboration diagram for TimeAdvanceBDFVariableStep< FEVectorType >:

Private Attributes
UInt	M_order
	Order of the BDF derivative/extrapolation: the time-derivative coefficients vector has size n+1, the extrapolation vector has size n. More...

container_Type	M_alpha
	Coefficients $\alpha_i$ of the time bdf discretization. More...

container_Type	M_beta
	Coefficients $\beta_i$ of the extrapolation. More...

container_Type	M_timeStep
	container_Type $\delta_t$ of time intervals More...

container_Type	M_timeStepBack

feVectorPtrContainer_Type	M_unknowns
	Last n state vectors. More...

feVectorPtrContainer_Type	M_unknownsBack

Public Types
typedef ScalarVector	container_Type

typedef FEVectorType	feVector_Type

typedef std::shared_ptr< feVector_Type >	feVectorPtr_Type

typedef std::vector< feVectorPtr_Type >	feVectorPtrContainer_Type

Constructor & Destructor
	TimeAdvanceBDFVariableStep ()
	Empty constructor. More...

	~TimeAdvanceBDFVariableStep ()
	Destructor. More...

Methods
void	setup (const UInt order)
	Setup the TimeAdvanceBDFVariableStep with order n. More...

void	setInitialCondition (feVector_Type u0, Real const timeStep, bool startup=0)
	Initialize all the entries of the unknown vector to be derived with the vector u0 (duplicated if startup=0) More...

void	setInitialCondition (std::vector< feVector_Type > u0s, Real const timeStep)
	Initialize all the entries of the unknown vector to be derived with a set of vectors u0s. More...

template<typename FunctionType , typename FESpaceType >
void	setInitialCondition (const FunctionType &u0Function, feVector_Type &u0Vector, FESpaceType &feSpace, Real t0, Real timeStep)
	Initialize all the entries of the unknonwn vectors with a given function. More...

feVector_Type	rhsContribution () const
	Returns the right hand side $\bar{p}$ of the time derivative formula divided by dt. More...

feVector_Type	rhsContributionTimeStep (Real timeStep=1) const
	Returns the right hand side $\bar{p}$ of the time derivative formula divided by timeStep (backward compatibility version, will be discontinued) More...

feVector_Type	extrapolateSolution () const
	Compute the polynomial extrapolation approximation of order n-1 of u^{n+1} defined by the n stored state vectors. More...

void	shiftRight (feVector_Type const &uCurrent)
	Update the vectors of the previous time steps by shifting on the right the old values. More...

void	shiftRight (feVector_Type const &uCurrent, Real timeStepNew)
	Update the vectors of the previous time steps by shifting on the right the old values. More...

void	storeSolution ()
	Save the current vector M_unknowns and the current vector M_timeStep. More...

void	restoreSolution ()
	Restore the vector M_unknowns and the vector M_timeStep with the ones saved with store_unk() More...

void	restoreSolution (Real timeStep)
	It is equivalent to do : storeSolution() + setTimeStep() More...

void	showMe () const
	Show informations about the BDF. More...

Set Methods
void	setTimeStep (Real timeStep)
	Replace the current time-step with timeStep and computes the coefficients a_i and beta_i as functions of M_timeStep. More...

Get Methods
const Real &	coefficientDerivative (UInt i) const
	Return the i-th coefficient of the time derivative alpha_i. More...

Real	coefficientDerivativeOverTimeStep (UInt i) const
	Return the i-th coefficient of the time derivative alpha_i divided by dt. More...

const Real &	coefficientExtrapolation (UInt i) const
	Return the i-th coefficient of the time extrapolation beta_i. More...

const container_Type &	timeStepVector () const
	Return the vector of the time steps, ordered starting from the most recent one. More...

const feVectorPtrContainer_Type &	stateVector () const
	Return a vector with the last n state vectors. More...

const feVectorPtrContainer_Type &	stateVectorBack () const

void	computeCoefficient ()
	Private methods. More...

Detailed Description

template<typename FEVectorType = VectorEpetra>
class LifeV::TimeAdvanceBDFVariableStep< FEVectorType >

TimeAdvanceBDFVariableStep - Backward differencing formula time discretization with non-uniform time step.

A differential equation of the form

$ M u' = A u + f $

is discretized in time as

$M p'(t_{k+1}) = A u_{k+1} + f_{k+1}$

where p denotes the polynomial of order n in t that interpolates $ (t_i,u_i) $ for i = k-n+1,...,k+1.

The approximative time derivative $p'(t_{k+1})$ is a linear combination of state vectors $ u_i $ :

$p'(t_{k+1}) = \frac{1}{\Delta t} (\alpha_0 u_{k+1} - \sum_{i=0}^n \alpha_i u_{k+1-i} )$

Thus we have

$\frac{\alpha_0}{\Delta t} M u_{k+1} = A u_{k+1} + f + M \bar{p}$

with

$\bar{p} = \frac{1}{\Delta t} \sum_{i=1}^n \alpha_i u_{k+1-i}$

This class stores the n last state vectors in order to be able to calculate $\bar{p}$ . It also provides $\alpha_i$ and can extrapolate the the new state from the n last states with a polynomial of order n-1:

$u_{k+1} \approx \sum_{i=0}^{n-1} \beta_i u_{k-i}$

The class allows to change the time-steps, recomputing the coefficients. to change $\Delta t$ at each time-step, use shift_right(uCurrent, timeStepNew) instead of shift_right(uCurrent).

Other methods that could be helpful while adapting in time are: set_deltat, store_unk and restore_unk

Definition at line 97 of file TimeAdvanceBDFVariableStep.hpp.

Member Typedef Documentation

◆ container_Type

typedef ScalarVector container_Type

Definition at line 103 of file TimeAdvanceBDFVariableStep.hpp.

◆ feVector_Type

typedef FEVectorType feVector_Type

Definition at line 104 of file TimeAdvanceBDFVariableStep.hpp.

◆ feVectorPtr_Type

typedef std::shared_ptr< feVector_Type > feVectorPtr_Type

Definition at line 105 of file TimeAdvanceBDFVariableStep.hpp.

◆ feVectorPtrContainer_Type

typedef std::vector< feVectorPtr_Type > feVectorPtrContainer_Type

Definition at line 106 of file TimeAdvanceBDFVariableStep.hpp.

Constructor & Destructor Documentation

◆ TimeAdvanceBDFVariableStep()

TimeAdvanceBDFVariableStep ( )

Empty constructor.

Definition at line 332 of file TimeAdvanceBDFVariableStep.hpp.

◆ ~TimeAdvanceBDFVariableStep()

~TimeAdvanceBDFVariableStep ( )

Destructor.

Definition at line 342 of file TimeAdvanceBDFVariableStep.hpp.

Member Function Documentation

◆ setup()

void setup ( const UInt order )

Setup the TimeAdvanceBDFVariableStep with order n.

Parameters

order order of the BDF

Definition at line 349 of file TimeAdvanceBDFVariableStep.hpp.

◆ setInitialCondition() [1/3]

void setInitialCondition	(	feVector_Type	u0,
		Real const	timeStep,
		bool	startup = `0`
	)

Initialize all the entries of the unknown vector to be derived with the vector u0 (duplicated if startup=0)

If startup = 0, it initializes all the entries of the unknown vectors with a given function The array of initial conditions needed by the selected BDF is initialized as follows: unk=[ u0(t0), u0(t0-dt), u0(t0-2*dt), ...] When startUp = true, only M_unknown[0] is initialized (with u0). At the first step, class Bfd computes the coefficients of BDF1, at the second step, the coefficient of BDF2, and so on, until it reaches the wanted method. Second order of accuracy is obtained using this procedure to startup BDF2. Using the startup procedure with BDF3, the accuracy order is limited to the second order; anyway it is better to use the startup procedure instead of taking u{-2} = u_{-1} = u_0.

Parameters

u0	initial condition vector
timeStep	time step
startup

Definition at line 370 of file TimeAdvanceBDFVariableStep.hpp.

◆ setInitialCondition() [2/3]

void setInitialCondition	(	std::vector< feVector_Type >	u0s,
		Real const	timeStep
	)

Initialize all the entries of the unknown vector to be derived with a set of vectors u0s.

Parameters

u0s	initial condition vectors
timeStep	time step

Definition at line 395 of file TimeAdvanceBDFVariableStep.hpp.

◆ setInitialCondition() [3/3]

void setInitialCondition	(	const FunctionType &	u0Function,
		feVector_Type &	u0Vector,
		FESpaceType &	feSpace,
		Real	t0,
		Real	timeStep
	)

Initialize all the entries of the unknonwn vectors with a given function.

The array of initial conditions needed by the selected BDF is initialized as follows: M_unknown=[ u0Function(t0), u0Function(t0-dt), u0Function(t0-2*dt), ...] For the space dependence of the initial conditions we need informations on:

Parameters

u0Function	the function we want to interpolate
u0Vector	corrected mapped vector, it contains the u0Function(t0) in output
feSpace	finite element space
t0	initial time (t0) and the
timeStep	time step (for solutions before the initial instant)

Based on the NavierStokesHandler::initialize by M. Fernandez

Definition at line 426 of file TimeAdvanceBDFVariableStep.hpp.

◆ rhsContribution()

TimeAdvanceBDFVariableStep< FEVectorType >::feVector_Type rhsContribution ( ) const

Returns the right hand side $\bar{p}$ of the time derivative formula divided by dt.

Returns: a feVector_Type, by default feVector_Type = VectorEpetra

Definition at line 448 of file TimeAdvanceBDFVariableStep.hpp.

◆ rhsContributionTimeStep()

TimeAdvanceBDFVariableStep< FEVectorType >::feVector_Type rhsContributionTimeStep ( Real timeStep = 1 ) const

Returns the right hand side $\bar{p}$ of the time derivative formula divided by timeStep (backward compatibility version, will be discontinued)

Parameters

timeStep time step, 1 by default

Returns: a feVector_Type, by default feVector_Type = VectorEpetra

Definition at line 464 of file TimeAdvanceBDFVariableStep.hpp.

◆ extrapolateSolution()

TimeAdvanceBDFVariableStep< FEVectorType >::feVector_Type extrapolateSolution ( ) const

Compute the polynomial extrapolation approximation of order n-1 of u^{n+1} defined by the n stored state vectors.

Definition at line 480 of file TimeAdvanceBDFVariableStep.hpp.

◆ shiftRight() [1/2]

void shiftRight ( feVector_Type const & uCurrent )

Update the vectors of the previous time steps by shifting on the right the old values.

Parameters

uCurrent current (new) value of the state vector

Definition at line 496 of file TimeAdvanceBDFVariableStep.hpp.

◆ shiftRight() [2/2]

void shiftRight	(	feVector_Type const &	uCurrent,
		Real	timeStepNew
	)

Update the vectors of the previous time steps by shifting on the right the old values.

Set the the new time step and shift right the old time steps.

Parameters

uCurrent	current (new) value of the state vector
timeStepNew	new value of the time step

Definition at line 521 of file TimeAdvanceBDFVariableStep.hpp.

◆ storeSolution()

void storeSolution ( )

Save the current vector M_unknowns and the current vector M_timeStep.

Definition at line 548 of file TimeAdvanceBDFVariableStep.hpp.

◆ restoreSolution() [1/2]

void restoreSolution ( )

Restore the vector M_unknowns and the vector M_timeStep with the ones saved with store_unk()

Definition at line 563 of file TimeAdvanceBDFVariableStep.hpp.

◆ restoreSolution() [2/2]

void restoreSolution ( Real timeStep )

It is equivalent to do : storeSolution() + setTimeStep()

Parameters

timeStep new time step

Definition at line 580 of file TimeAdvanceBDFVariableStep.hpp.

◆ showMe()

void showMe ( ) const

Show informations about the BDF.

Definition at line 598 of file TimeAdvanceBDFVariableStep.hpp.

◆ setTimeStep()

void setTimeStep ( Real timeStep )

Replace the current time-step with timeStep and computes the coefficients a_i and beta_i as functions of M_timeStep.

Parameters

timeStep time step

Definition at line 617 of file TimeAdvanceBDFVariableStep.hpp.

◆ coefficientDerivative()

const Real & coefficientDerivative ( UInt i ) const

Return the i-th coefficient of the time derivative alpha_i.

Parameters

i	index of the coefficient

Definition at line 628 of file TimeAdvanceBDFVariableStep.hpp.

◆ coefficientDerivativeOverTimeStep()

Real coefficientDerivativeOverTimeStep ( UInt i ) const

Return the i-th coefficient of the time derivative alpha_i divided by dt.

Parameters

i	index of the coefficient

Definition at line 639 of file TimeAdvanceBDFVariableStep.hpp.

◆ coefficientExtrapolation()

const Real & coefficientExtrapolation ( UInt i ) const

Return the i-th coefficient of the time extrapolation beta_i.

Parameters

i	index of the coefficient

Definition at line 650 of file TimeAdvanceBDFVariableStep.hpp.

◆ timeStepVector()

const container_Type& timeStepVector ( ) const

inline

Return the vector of the time steps, ordered starting from the most recent one.

Definition at line 261 of file TimeAdvanceBDFVariableStep.hpp.

◆ stateVector()

const feVectorPtrContainer_Type& stateVector ( ) const

inline

Return a vector with the last n state vectors.

Definition at line 267 of file TimeAdvanceBDFVariableStep.hpp.

◆ stateVectorBack()

const feVectorPtrContainer_Type& stateVectorBack ( ) const

inline

Definition at line 272 of file TimeAdvanceBDFVariableStep.hpp.

◆ computeCoefficient()

void computeCoefficient ( )

private

Private methods.

Computes the coefficients a_i and beta_i as functions of _M_delta_t.

Arbitrary order, variable time step BDF coefficients. Reference: Comincioli pag. 618-619 Matlab implementation: function [alpha0, alpha, beta] = compute_coef(dts, order) assert( (length(dts) == order) ); rho = dts(1)./cumsum(dts);

alpha0 = sum(rho); alpha = zeros(order,1); beta = zeros(order,1);

for j = 1:order ind = setdiff([1:order],j); beta(j) = 1/prod(1-rho(ind)/rho(j)); alpha(j) = rho(j)*beta(j); end

end

Definition at line 662 of file TimeAdvanceBDFVariableStep.hpp.

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