Timestep strategy for a Nyström algorithm to solve second order problems in time with no first order time derivative. More...

#include <nystroem.hh>

Inheritance diagram for timestepping::Nystroem:

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Collaboration diagram for timestepping::Nystroem:

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List of all members.

Public Member Functions
	Nystroem (concepts::SolverFabric< Real > &fabric, concepts::Operator< Real > &D2, concepts::Operator< Real > &D0, timestepping::TimeVector &trhs, const concepts::Vector< Real > &Y0, const concepts::Vector< Real > &Z0, Real dt)
	Constructor.
Protected Member Functions
virtual std::ostream &	info (std::ostream &os) const
	Returns information in an output stream.
virtual void	next ()
	The overloaded member function next() has to calculate the new right hand side and to release the solution vector.
Protected Attributes
Real	dt_
	Time step size.
concepts::Operator< Real > *	liCo_
	Operator of the linear equation system which is solved by the friend class TimeStepping.
concepts::Vector< Real >	rhs_
	The right hand side vector of the linear equation system which is solved by the friend class TimeStepping.
concepts::Vector< Real >	sol_
	The solution vector of the linear equation system which is solved by the friend class TimeStepping.
std::auto_ptr < concepts::Operator< Real > >	solver_
	Solver for the linear system.
Real	t_
	Time of the actual solution.
Private Attributes
concepts::Operator< Real > &	D0_
	Space operator.
timestepping::TimeVector &	trhs_
	External driver function.
concepts::Vector< Real >	Z_
	Store velocity times timestep in Z.
Friends
class	TimeStepping

Detailed Description

Timestep strategy for a Nyström algorithm to solve second order problems in time with no first order time derivative.

$[ D_2 \partial_t^2 + D_0 ] y(x,t) = f(x,t)$

The scheme needs no parameters. It is explicit, has convergence order 2, is symplectic and conditionally stable. The stability criterion for the model problem $\partial_t^2 y = -\omega^2 y$ is $\Delta t < 2/\omega$ . For linear operators and f(x,t)=0, this scheme is identic to the leap frog scheme.

See also:: E. Hairer, S.P. Nørsett, G. Wanner Solving Ordinary Differential Equations I - Nonstiff Problems, Springer, Berlin, 1992.

Author:: Christoph Winkelmann, 2004

Definition at line 38 of file nystroem.hh.

Constructor & Destructor Documentation

timestepping::Nystroem::Nystroem	(	concepts::SolverFabric< Real > &	fabric,
		concepts::Operator< Real > &	D2,
		concepts::Operator< Real > &	D0,
		timestepping::TimeVector &	trhs,
		const concepts::Vector< Real > &	Y0,
		const concepts::Vector< Real > &	Z0,
		Real	dt
	)

Constructor.

Parameters:

fabric	Solver fabric for solving the occuring systems
D2	Space operator D₂
D0	Space operator D₀
trhs	Timedependent external driver f(x,t)
Y0	Initial condition y(x,0)
Z0	Initial condition d/dt y(x,0)
dt	Time step size