Geometric Generalisations of shake and rattle

McLachlan, Robert I.; Modin, Klas; Verdier, Olivier; Wilkins, Matt

doi:10.1007/s10208-013-9163-y

Cited by 29 publications

(35 citation statements)

References 14 publications

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“…The identities (19) and (20) show that the right hand sides of equations (21), (22) have the same form. Therefore, it is enough to prove equation (21).…”

Section: 3mentioning

confidence: 99%

“…The same conclusion for the symplectic Runge-Kutta method will follow straightforwardly from this. We want to check equation (21) for any W n ∈ gl(n, C) and A as above. The right hand side is…”

Section: 3mentioning

confidence: 99%

“…• The Lie-Poisson system on g * g can be extended to a constrained canonical Hamiltonian system on T * G T G ⊂ T GL(n, C). One can then use the symplectic RATTLE method (or higher order versions of it) for the constrained system (see [13,21]).…”

Section: Introductionmentioning

confidence: 99%

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Lie–Poisson Methods for Isospectral Flows

Modin

Viviani

2019

Found Comput Math

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The theory of isospectral flows comprises a large class of continuous dynamical systems, particularly integrable systems and Lie-Poisson systems. Their discretization is a classical problem in numerical analysis. Preserving the spectra in the discrete flow requires the conservation of high order polynomials, which is hard to come by. Existing methods achieving this are complicated and usually fail to preserve the underlying Lie-Poisson structure.Here we present a class of numerical methods of arbitrary order for Hamiltonian and non-Hamiltonian isospectral flows, which preserve both the spectra and the Lie-Poisson structure. The methods are surprisingly simple, and avoid the use of constraints or exponential maps. Furthermore, due to preservation of the Lie-Poisson structure, they exhibit near conservation of the Hamiltonian function. As an illustration, we apply the methods to several classical isospectral flows.

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“…The identities (19) and (20) show that the right hand sides of equations (21), (22) have the same form. Therefore, it is enough to prove equation (21).…”

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

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Lie–Poisson Methods for Isospectral Flows

Modin

Viviani

2019

Found Comput Math

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“…Towards this goal, a fruitful approach is to regard the two-sphere S 2 as a coadjoint orbit of the Lie-Poisson manifold so(3) * corresponding to the Lie algebra so(3) of skew-symmetric matrices (for details on Lie-Poisson manifolds, see [7] and references therein). Then one can use variational discretizations, as those developed by Moser and Veselov [17] for some classical integrable systems, particularly the free rigid body (see also [15,13] for the extension to arbitrary Lie-Poisson manifolds and Hamiltonians). This discrete Moser-Veselov (DMV) algorithm is formulated as an SO(3)-symmetric symplectic map on the phase space T * SO (3).…”

Section: Introductionmentioning

confidence: 99%

Geometry of Discrete-Time Spin Systems

2016

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Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space (S 2 ) n . In this paper we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features, that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity.The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on T * R 2n for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with Kähler geometry, this provides another geometric proof of symplecticity.

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“…This method can thus be regarded as a particular case of the SHAKE algorithm for constrained mechanical systems [13, Sect. VII.1.4] [21], to Hamiltonian PDEs with constraints. The advantages of the proposed multi-symplectic method, which for the standard wave map equation reduces to (2), are summarized as follows:…”

Section: Introductionmentioning

confidence: 99%

MultiSymplectic Discretization of Wave Map Equations

Cohen

Verdier

2016

SIAM J. Sci. Comput.

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We present a new multi-symplectic formulation of constrained Hamiltonian partial differential equations, and we study the associated local conservation laws. A multi-symplectic discretisation based on this new formulation is exemplified by means of the Euler box scheme. When applied to the wave map equation, this numerical scheme is explicit, preserves the constraint and can be seen as a generalisation of the SHAKE algorithm for constrained mechanical systems. Furthermore, numerical experiments show excellent conservation properties of the numerical solutions. Keywords

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Geometric Generalisations of shake and rattle

Cited by 29 publications

References 14 publications

Lie–Poisson Methods for Isospectral Flows

Lie–Poisson Methods for Isospectral Flows

Geometry of Discrete-Time Spin Systems

MultiSymplectic Discretization of Wave Map Equations

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