Integrators for Lie-Poisson dynamical systems

Channell, Paul J.; Scovel, James C.

doi:10.1016/0167-2789(91)90081-j

Cited by 61 publications

(52 citation statements)

References 9 publications

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“…If exp(tA i ) is not available, one can use ϕ(t) = r i=1 ϕ i (t) where ϕ i (t) approximates exp(tA i ) to first order. In the symplectic case, a suitable ϕ is generated by the generating function of the third kind q t p − tH(q, p); for the Lie-Poisson case, a suitable (Poisson) ϕ is constructed in [6]. If one is not worried about staying in the right group, ϕ(t) could be Euler's method 1 + tX.…”

Section: Counting the Determining Equationsmentioning

confidence: 99%

On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods

McLachlan¹

1995

SIAM J. Sci. Comput.

335

353

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Abstract. Differential equations of the formẋ = X = A + B are considered, where the vector fields A and B can be integrated exactly, enabling numerical integration of X by composition of the flows of A and B. Various symmetric compositions are investigated for order, complexity, and reversibility. Free Lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order. A new, more accurate way of applying the methods thus obtained to compositions of an arbitrary first-order integrator is described and tested. The determining equations are explored, and new methods up to 100 times more accurate (at constant work) than those previously known are given. Composition methods.Composition methods are particularly useful for numerically integrating differential equations when the equations have some special structure which it is advantageous to preserve. They tend to have larger local truncation errors than standard (Runge-Kutta, multistep) methods [4,5], but this defect can be more than compensated for by their superior conservation properties.Capital letters such as X will denote vector fields on some space with coordinates x, with flows exp(tX), i.e.,ẋ = X(x) ⇒ x(t) = exp(tX)(x(0)). The vector field X is given and is to be integrated numerically with fixed time step t. Composition methods apply when one can writein such a way that exp(tA), exp(tB) can both be calculated explicitly. Then the most elementary such method is the map (essentially the "Lie-Trotter" formulaThe advantage of composing exact solutions in this way is that many geometric properties of the true flow exp(tX) are preserved: group properties in particular. If X, A, and B are Hamiltonian vector fields then both exp(tX) and the map ϕ 1991 Mathematics Subject Classification. 65L05, 70-08, 58D07, 17B66.

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Section: Counting the Determining Equationsmentioning

confidence: 99%

On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods

McLachlan¹

1995

SIAM J. Sci. Comput.

335

353

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“…Other geometry-based integrators exist. We can cite, for instance, algorithms built to preserve a Lie-Poisson structure on a discrete manifold [246,247]. Other examples are schemes based on a port-Hamiltonian, or more generally, on a Dirac structure, which aim at correctly handling the interconnection between subsystems [248].…”

Section: Resultsmentioning

confidence: 99%

A review of some geometric integrators

Razafindralandy

Hamdouni

Chhay

2018

Adv. Model. and Simul. in Eng. Sci.

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Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler-Lagrange equations) and the conservation laws (Noether's theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

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“…Based on this, we provide an elementary construction of symplectic integrators for Lie-Poisson systems associated to subgroups of the general matrix group GL( n). This provides an alternative to the construction given by Ge and Marsden [ 5 ], Ge [4], and Channell and Scovel [2]. In contrast to our approach, their construction, although more general, is based on the generating function method and on a coordinatization of the group.…”

Section: Q' = V P H(qp) P' = -V Q H(qp)-v Q G(q)mentioning

confidence: 97%

“…The equations of motion on gl are given by the reduced Hamiltonian on gl and by the corresponding Lie-Poisson structure on gl which is defined in the following way [11,13]: Let F L ,Gl be two functions on gl = TmGL and let { , } denote the canonical Poisson bracket on TGL. Then the reduced Lie-Poisson bracket { , }_ on gl is given by [11] Special integrators for Hamiltonian systems on gl*, gl, respectively (and also for Hamiltonian systems on sub-algebras of gl*, gl, respectively), that conserve the Lie-Poisson structure have been derived by Ge and Marsden [5], Channell and Scovel [2], and Ge [4]. These schemes are based on the generating function method and require a coordinatization of the group under consideration.…”

Section: Dh(qp) = (V Q H(qp)dq) + (V P H(qp)mentioning

confidence: 99%

Momentum conserving symplectic integrators

Reich¹

1994

Physica D: Nonlinear Phenomena

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Sebastian Reich Momentum conserving symplectic integrators U n i v e r s i t ä t P o t s d a m AbstractIn this paper, we show that symplectic partitioned Runge-Kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of Hamiltonian differential equations by extended point transformation. We also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of Lie-Poisson systems on certain submanifolds of the general matrix group GL(n).

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Integrators for Lie-Poisson dynamical systems

Cited by 61 publications

References 9 publications

On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods

On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods

A review of some geometric integrators

Momentum conserving symplectic integrators

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