On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms

Benettin, Giancarlo; Giorgilli, Antonio

doi:10.1007/bf02188219

“…We now show that when the numerical approximation scheme generates a symplectic semigroup, the associated modi"ed equations share the same property. This result is already well known (see Mackay (1992), Sanz-Serna (1992), Reich (1993Reich ( , 1996, Benettin & Giorgilli (1994), Calvo, Murua & Sanz-Serna (1994) and Hairer (1994) for example), and we simply provide a new proof using the unifying technique of this article.…”

Section: Symplecticity Of the Modi"ed Semigroupsupporting

confidence: 57%

“…For further results on the usefulness and applicability of the modi"ed equation approach, especially in the context of ordinary differential equations, see Grif"ths & Sanz-Serna (1986), Beyn (1991), Reich (1993Reich ( , 1996, Calvo, Murua & Sanz-Serna (1994), Hairer (1994), Reddien (1995), Fiedler & Scheurle (1996) and Sanz-Serna & Murua (1997). A major application has been the derivation of exponentially small error estimates, starting with the work of Neishtadt (1984) and continued in, for example, Benettin & Giorgilli (1994), Hairer & Lubich (1997), and Reich (1996).…”

Section: Introductionmentioning

confidence: 99%

Qualitative properties of modified equations

González¹,

Higham

²

,

Stuart

³

1999

IMA Journal of Numerical Analysis

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Suppose that a consistent one-step numerical method of order r is applied to a smooth system of ordinary differential equations. Given any integer m 1, the method may be shown to be of order r + m as an approximation to a certain modi"ed equation. If the method and the system have a particular qualitative property then it is important to determine whether the modi"ed equations inherit this property. In this article, a technique is introduced for proving that the modi"ed equations inherit qualitative properties from the method and the underlying system. The technique uses a straightforward contradiction argument applicable to arbitrary one-step methods and does not rely on the detailed structure of associated power series expansions. Hence the conclusions apply, but are not restricted, to the case of Runge-Kutta methods. The new approach uni"es and extends results of this type that have been derived by other means: results are presented for integral preservation, reversibility, inheritance of "xed points, Hamiltonian problems and volume preservation. The technique also applies when the system has an integral that the method preserves not exactly, but to order greater than r . Finally, a negative result is obtained by considering a gradient system and gradient numerical method possessing a global property that is not shared by the associated modi"ed equations.

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“…Hence, in particular, the bounds are valid only for "nite time intervals and they are not uniform in the index N . However, by optimizing over the index, Neishtadt (1984), Benettin & Giorgilli (1994), Hairer & Lubich (1997) and Reich (1996) have shown that the difference between the numerical approximation and a modi"ed equation remains exponentially small over arbitrarily long time intervals as ∆t → 0 for a variety of problems of interest.…”

Section: Remarksmentioning

confidence: 99%

“…For further results on the usefulness and applicability of the modi"ed equation approach, especially in the context of ordinary differential equations, see Grif"ths & Sanz-Serna (1986), Beyn (1991), Reich (1993), Calvo, Murua & Sanz-Serna (1994, Hairer (1994), Reddien (1995), Fiedler & Scheurle (1996) and Sanz-Serna & Murua (1997). A major application has been the derivation of exponentially small error estimates, starting with the work of Neishtadt (1984) and continued in, for example, Benettin & Giorgilli (1994), Hairer & Lubich (1997), and Reich (1996).…”

Section: Introductionmentioning

confidence: 99%

“…For early references in the numerical analysis literature see Mackay (1992), Sanz-Serna (1992) and and for some early applications see Auerbach & Friedman (1991) and Yoshida (1993). More recently, modi"ed equations for problems with special structure have been explored in greater detail using several different approaches, as in Hairer (1994), Calvo, Murua & Sanz-Serna (1994), Benettin & Giorgilli (1994), Reich (1993Reich ( , 1996, Hairer & Stoffer (1997) and Sanz-Serna & Murua (1997).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the Qualitative Properties of Modified Equations

González

¹

,

Stuart²

1997

Foundations of Computational Mathematics

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Suppose that a consistent one-step numerical method of order r is applied to a smooth system of ordinary differential equations. Given any integer m 1, the method may be shown to be of order r + m as an approximation to a certain modi"ed equation. If the method and the system have a particular qualitative property then it is important to determine whether the modi"ed equations inherit this property. In this article, a technique is introduced for proving that the modi"ed equations inherit qualitative properties from the method and the underlying system. The technique uses a straightforward contradiction argument applicable to arbitrary one-step methods and does not rely on the detailed structure of associated power series expansions. Hence the conclusions apply, but are not restricted, to the case of Runge-Kutta methods. The new approach uni"es and extends results of this type that have been derived by other means: results are presented for integral preservation, reversibility, inheritance of "xed points, Hamiltonian problems and volume preservation. The technique also applies when the system has an integral that the method preserves not exactly, but to order greater than r . Finally, a negative result is obtained by considering a gradient system and gradient numerical method possessing a global property that is not shared by the associated modi"ed equations.

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