High-Order Symplectic Partitioned Lie Group Methods

Bogfjellmo, Geir; Marthinsen, Håkon

doi:10.1007/s10208-015-9257-9

Cited by 21 publications

(39 citation statements)

References 21 publications

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“…As expected, we obtain near conservation of the Hamiltonian (owing to the symplectic quality) and exact conservation (up to round-off errors) of the Casimir functions (owing to the isospectral quality). 3 5.1. The generalized rigid body.…”

Section: Numerical Examplesmentioning

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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Section: Numerical Examplesmentioning

confidence: 99%

“…We apply the isospectral midpoint method with h = 0.1. In Figure 9 we plot the eigenvalues and the components variation for a randomly generated self-adjoint initial matrix W 0 of dimension 3 × 3 and N = diag (1,2,3). Figure 9 displays the exponential convergence to a similar diagonal matrix.…”

Section: 4mentioning

confidence: 99%

Lie–Poisson Methods for Isospectral Flows

Modin

Viviani

2019

Found Comput Math

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“…By replacing the discrete Lagrangian and action sum by other more advanced approximations, one can obtain various different variants of symplectic integrators on Lie groups, see e.g. [5,18].…”

Section: Variational Integrators On Lie Groupsmentioning

confidence: 99%

Lie Group Integrators

Owren

2018

Discrete Mechanics, Geometric Integration and Lie–Butcher Series

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In this survey we discuss a wide variety of aspects related to Lie group integrators. These numerical integration schemes for differential equations on manifolds have been studied in a general and systematic manner since the 1990s and the activity has since then branched out in several different subareas, focussing both on theoretical and practical issues. From two alternative setups, using either frames or Lie group actions on a manifold, we here introduce the most important classes of schemes used to integrate nonlinear ordinary differential equations on Lie groups and manifolds. We describe a number of different applications where there is a natural action by a Lie group on a manifold such that our integrators can be implemented. An issue which is not well understood is the role of isotropy and how it affects the behaviour of the numerical methods. The order theory of numerical Lie group integrators has become an advanced subtopic in its own right, and here we give a brief introduction on a somewhat elementary level. Finally, we shall discuss Lie group integrators having the property that they preserve a symplectic structure or a first integral.

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“…On the other hand, if we discontinuously reset the generator according to Eq. (20), we obtain the components seen in the bottom panel. The integrator does not need to evolve through the discontinuity, and everything in between is smooth and slowly varying, so integration of this quantity is much more efficient.…”

Section: B Precessing and Nutating Binarymentioning

confidence: 99%

“…Bottasso and Borri [15] developed an algorithm that appears to be essentially the Crouch-Grossman technique specialized to the rotation group, while simultaneously treating displacements. Numerous groups have devised other specialized low-order numerical algorithms for such integrations [16][17][18][19][20], as well as several adapted to closely related problems in Lagrangian and Hamiltonian dynamics [21][22][23]. Candy and Lasenby [24] outlined integrations of rotations using quaternions, and simultaneous treatment of rotations and translations using their natural generalizations in the conformal representation of threegeometry.…”

Section: Introductionmentioning

confidence: 99%

The Integration of Angular Velocity

Boyle

2017

Adv. Appl. Clifford Algebras

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A common problem in physics and engineering is determination of the orientation of an object given its angular velocity. When the direction of the angular velocity changes in time, this is a nontrivial problem involving coupled differential equations. Several possible approaches are examined, along with various improvements over previous efforts. These are then evaluated numerically by comparison to a complicated but analytically known rotation that is motivated by the important astrophysical problem of precessing black-hole binaries. It is shown that a straightforward solution directly using quaternions is most efficient and accurate, and that the norm of the quaternion is irrelevant. Integration of the generator of the rotation can also be made roughly as efficient as integration of the rotation. Both methods will typically be twice as efficient naive vector-or matrix-based methods. Implementation by means of standard general-purpose numerical integrators is stable and efficient, so that such problems can be readily solved as part of a larger system of differential equations. Possible generalization to integration in other Lie groups is also discussed.

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High-Order Symplectic Partitioned Lie Group Methods

Cited by 21 publications

References 21 publications

Lie–Poisson Methods for Isospectral Flows

Lie–Poisson Methods for Isospectral Flows

Lie Group Integrators

The Integration of Angular Velocity

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