# On energy conservation of the simplified Takahashi-Imada method

Ernst Hairer; Robert I. McLachlan; Robert D. Skeel

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

- Volume: 43, Issue: 4, page 631-644
- ISSN: 0764-583X

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topHairer, Ernst, McLachlan, Robert I., and Skeel, Robert D.. "On energy conservation of the simplified Takahashi-Imada method." ESAIM: Mathematical Modelling and Numerical Analysis 43.4 (2009): 631-644. <http://eudml.org/doc/250589>.

@article{Hairer2009,

abstract = {
In long-time numerical integration of Hamiltonian systems,
and especially in molecular dynamics simulation,
it is important that the energy is well conserved. For symplectic
integrators applied with sufficiently small step size, this
is guaranteed by the existence of a modified
Hamiltonian that is exactly conserved up to exponentially small
terms. This article is concerned with the simplified
Takahashi-Imada method, which is a modification
of the Störmer-Verlet method that is as easy to implement but
has improved accuracy. This integrator is symmetric and
volume-preserving, but no longer symplectic. We study its
long-time energy conservation and give theoretical
arguments, supported by numerical experiments, which
show the possibility of a drift in the energy (linear or like a random walk).
With respect to energy conservation, this article provides empirical
and theoretical data concerning the importance of using a symplectic
integrator.
},

author = {Hairer, Ernst, McLachlan, Robert I., Skeel, Robert D.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Symmetric and symplectic integrators; geometric
numerical integration; modified differential equation; energy conservation;
Hénon-Heiles problem; N-body problem in molecular dynamics.; symmetric and symplectic integrators; geometric numerical integration},

language = {eng},

month = {7},

number = {4},

pages = {631-644},

publisher = {EDP Sciences},

title = {On energy conservation of the simplified Takahashi-Imada method},

url = {http://eudml.org/doc/250589},

volume = {43},

year = {2009},

}

TY - JOUR

AU - Hairer, Ernst

AU - McLachlan, Robert I.

AU - Skeel, Robert D.

TI - On energy conservation of the simplified Takahashi-Imada method

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2009/7//

PB - EDP Sciences

VL - 43

IS - 4

SP - 631

EP - 644

AB -
In long-time numerical integration of Hamiltonian systems,
and especially in molecular dynamics simulation,
it is important that the energy is well conserved. For symplectic
integrators applied with sufficiently small step size, this
is guaranteed by the existence of a modified
Hamiltonian that is exactly conserved up to exponentially small
terms. This article is concerned with the simplified
Takahashi-Imada method, which is a modification
of the Störmer-Verlet method that is as easy to implement but
has improved accuracy. This integrator is symmetric and
volume-preserving, but no longer symplectic. We study its
long-time energy conservation and give theoretical
arguments, supported by numerical experiments, which
show the possibility of a drift in the energy (linear or like a random walk).
With respect to energy conservation, this article provides empirical
and theoretical data concerning the importance of using a symplectic
integrator.

LA - eng

KW - Symmetric and symplectic integrators; geometric
numerical integration; modified differential equation; energy conservation;
Hénon-Heiles problem; N-body problem in molecular dynamics.; symmetric and symplectic integrators; geometric numerical integration

UR - http://eudml.org/doc/250589

ER -

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