BDF2 Class Reference

An order 2 BDF-Method with fixed-point iteration and variable step size. More...

Inheritance diagram for BDF2:

Collaboration diagram for BDF2:

Detailed Description

An order 2 BDF-Method with fixed-point iteration and variable step size.

Derived from

• Roussel C. and Roussel M. (2003) "Generic Object-Oriented Differential Equation Integrators", C/C++ User Journal, Nov. 2003, http://www.ddj.com/cpp/184401724?pgno=8 and
• Eckert S., Baaser H., Gross D. and Scherf O. (2004) "A BDF2 integration method with step size control for elasto-plasticity", Computational Mechanics 34, 377 - 386, DOI: 10.1007/s00466-004-0581-1

Most important function: Integrate

Public Member Functions
Integrate
void	integrate_until (cmf::math::Time t_max, cmf::math::Time dt=Time(), bool reset=false)
	Integrates the vector of state variables until t_max.

Protected Attributes
Accuracy parameters
const real	Epsilon =1e-9
	Tolerable error.

Gear functions: The following functions evaluate the Gear formulas of specified order, loading the results into compareStates.
int	max_order
	Maximum order, can be 1 (implicit euler) or 2 (BDF2)
ptrdiff_t	get_error_position () const
	Returns the position of the biggest error.
void	reset () override
	Resets any saved history (for multistep methods)
	BDF2 (real epsilon=1e-9, cmf::math::Time tStepMin=cmf::math::timespan(10))
	Constructs a new BDF2 integrator.
	BDF2 (const cmf::math::state_list &states, real epsilon=1e-9, cmf::math::Time tStepMin=cmf::math::timespan(10))
	Constructs a new Gears_var_Step.
	BDF2 (const Integrator &templ)
	Constructs a new BDF2 integrator.
Integrator *	copy () const override
	Polymorphic copy constructor.
std::string	to_string () const override
	Desrcibes the iterator.
int	integrate (cmf::math::Time MaxTime, cmf::math::Time TimeStep) override
	Integrates the vector of state variables.

model time
cmf::math::Time	get_t () const
	Returns the current model time.
void	set_t (cmf::math::Time val)
	Sets the current model time.
cmf::math::Time	get_dt () const
	Returns the last time step.
cmf::math::Time	m_t
	Actual time of the solver.
cmf::math::Time	m_dt
	last time step of the solver

Constructor & Destructor Documentation

BDF2() [1/3]

BDF2 ( real epsilon = 1e-9, cmf::math::Time tStepMin = cmf::math::timespan(10) )

explicit

Constructs a new BDF2 integrator.

Parameters

epsilon	relative error tolerance per time step (default=1e-9)
tStepMin	minimum time step (default=10s)

BDF2() [2/3]

BDF2 ( const cmf::math::state_list & states, real epsilon = 1e-9, cmf::math::Time tStepMin = cmf::math::timespan(10) )

explicit

Constructs a new Gears_var_Step.

Parameters

states	States to be added to the solver
epsilon	relative error tolerance per time step (default=1e-9)
tStepMin	minimum time step (default=10s)

BDF2() [3/3]

BDF2 ( const Integrator & templ )

explicit

Constructs a new BDF2 integrator.

Parameters

templ

Template to be used to construct a BDF2 method

Member Function Documentation

integrate()

int integrate ( cmf::math::Time MaxTime, cmf::math::Time TimeStep )

overridevirtual

Integrates the vector of state variables.

• if \( t_{max}-t_{now}\leq 2h_n \) then \( h_{n+1}= t_{max}-t_{now} \) else \( h_{n+1}= 2h_n \)
• Uses pastStatesArray to store \( y_n,y_{n-1} \)
• if more than one step is taken start iterating:
  • \( y_{n+1}^i=\frac{1}{1+2\rho }\left( (\rho +1)^{2}y_{n}-\rho ^{2}y_{n-1}+(\rho +1)h\frac{dy}{dt}(y_{n+1}^{i-1},t_{n+1} \right), y_{n+1}^0=y_n \) with \( \rho = \frac{h_{n+1}}{h_n} \)
• until \( y_{n+1}^{i} - y_{n+1}^{i-1} \leq \ \|\epsilon + \epsilon y_{n+1}\|_{\infty} \)
• If too many iterations are needed, or the error is rising, repeat iteration with \( h_{n+1} = \frac{h_{n+1}}{2} \)

  Parameters

  MaxTime	To stop the model (if running in a model framework) at time steps of value exchange e.g. full hours, the next value exchange time can be given
  TimeStep	Takes the proposed timestep, ignored by this solver

Implements Integrator.

integrate_until()

void integrate_until ( cmf::math::Time t_max, cmf::math::Time dt = Time(), bool reset = false )

inherited

Integrates the vector of state variables until t_max.

Parameters

t_max	Time, the solver should run to
dt	Time step (may be omitted)
reset	If true, solver is reseted before integration starts