A new class of energy-preserving numerical integration methods

Quispel, G.; McLaren, David

doi:10.1088/1751-8113/41/4/045206

Cited by 390 publications

(310 citation statements)

References 19 publications

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“…In the special case when (1) is a canonical Hamiltonian system, the derivation of methods that preserve the energy has been the subject of an extensive research in the last years. Among them, we may mention the so called "averaged vector field" of Quispel and Mc Laren [13], generalized later by Hairer in [9]. In more details, [9] describes the limit formulae of a particular instance of Hamiltonian BVMs, as is shown in [5].…”

Section: Y Y Y(t) = F(y Y Y(t)) ≡ B(y Y Y(t)) ∇H(y Y Y(t)) T ∈ [0 Hmentioning

confidence: 99%

Recent Advances on Preserving Methods for Poisson Systems

Brugnano¹,

Calvo²,

Montijano³

et al. 2011

AIP Conference Proceedings

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Abstract. We present and analyze energy-conserving methods for the numerical integration of IVPs of Poisson type systems. They are also able to preserve some Casimirs. Their derivation and analysis is done following the ideas of Hamiltonian BVMs (HBVMs) (see [3] and references therein). The proposed methods turn out to be equivalent to those recently derived in [8], giving therefore an alternative point of view that provides additional insight on the methods. Sufficient conditions that ensure the existence of a unique solution of the implicit equations defining the formulae are given. A study of the implementation of the methods is provided.

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Section: Y Y Y(t) = F(y Y Y(t)) ≡ B(y Y Y(t)) ∇H(y Y Y(t)) T ∈ [0 Hmentioning

confidence: 99%

Recent Advances on Preserving Methods for Poisson Systems

Brugnano¹,

Calvo²,

Montijano³

et al. 2011

AIP Conference Proceedings

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“…The conditions to be energy-preserving i.e., the annihilator of T n H were rst given in [10], and a spanning set of T n H is given in [19]. The analogue for maps is studied in [5].…”

Section: Theorem 3 a Basis Of F Nmentioning

confidence: 99%

“…The rst to be discovered was the Average Vector Field (AVF) method rst written down in [14] and identi ed as energy-preserving and as a B-series method in [19]: for the di erential equatioṅ x = f (x), x ∈ R n , (1) the AVF method is the map x → x de ned by…”

mentioning

confidence: 99%

Energy-Preserving Integrators and the Structure of B-series

Celledoni¹,

McLachlan

Owren³

et al. 2010

Found Comput Math

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“…Such a relationship does exist in the case of the energy invariant of canonical Hamiltonian systems: see [3,5] on energy-preserving B-series. In this article we show that for any polynomial Hamiltonian function, there exists an RK method of any order that preserves it.The key is the average vector field (AVF) method first written down in [9] and identified as energy-preserving and as a B-series method in [10]: for the differential equatioṅthe AVF method is the map x → x defined by …”

mentioning

confidence: 99%

“…The key is the average vector field (AVF) method first written down in [9] and identified as energy-preserving and as a B-series method in [10]: for the differential equatioṅ…”

mentioning

confidence: 99%

Energy-preserving Runge-Kutta methods

Celledoni¹,

McLachlan²,

et al. 2009

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Abstract.We show that while Runge-Kutta methods cannot preserve polynomial invariants in general, they can preserve polynomials that are the energy invariant of canonical Hamiltonian systems.Mathematics Subject Classification. 65P10, 65L06. All Runge-Kutta (RK) methods preserve arbitrary linear invariants [12], and some (the symplectic) RK methods preserve arbitrary quadratic invariants [4]. However, no RK method can preserve arbitrary polynomial invariants of degree 3 or higher of arbitrary vector fields [1]; the linear systemẋ = x,ẏ = y,ż = −2z, with invariant xyz, provides an example. (Preservation would require R(h) 2 R(−2h) ≡ 1, where R is the stability function of the method 4 ; but this requires R(h) = e h , which is impossible [6].) This result does not rule out, however, the existence of RK methods that preserve particular (as opposed to arbitrary) invariants. Since the invariant does not appear in the RK method, this will require some special relationship between the invariant and the vector field. Such a relationship does exist in the case of the energy invariant of canonical Hamiltonian systems: see [3,5] on energy-preserving B-series. In this article we show that for any polynomial Hamiltonian function, there exists an RK method of any order that preserves it.The key is the average vector field (AVF) method first written down in [9] and identified as energy-preserving and as a B-series method in [10]: for the differential equatioṅthe AVF method is the map x → x defined by

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A new class of energy-preserving numerical integration methods

Abstract: The first ever energy-preserving B-series numerical integration method for (ordinary) differential equations is presented and applied to several Hamiltonian systems. Related novel Lie algebraic results are also discussed.

Cited by 390 publications

References 19 publications

Recent Advances on Preserving Methods for Poisson Systems

Recent Advances on Preserving Methods for Poisson Systems

Energy-Preserving Integrators and the Structure of B-series

Energy-preserving Runge-Kutta methods

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