An introduction to Lie group integrators – basics, new developments and applications

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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“…Note that a combination of port Hamiltonian structure and DEC has been considered in [249]. Liegroup integrators can also be cited [250].…”

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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“…Only an extremely brief discussion of Lie-Butcher theory is provided to lay the foundation for the following computations. For a friendly introduction to the theory of Lie group integrators we refer the reader to [CMO14].…”

Section: Lie-butcher Theory and Lie Series Estimatesmentioning

confidence: 99%

Convergence of Lie group integrators

Curry

Schmeding

2019

Numer. Math.

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“…The chimney can be plotted by discretizing the circle of initial points and calculating threads on the chimney. We have implemented this using a Runge-Kutta method for Lie groups [29] which ensuresn remains normalized. The results are consistent with the preceding analysis.…”

Section: Examplesmentioning

confidence: 99%

Flag-based control of quantum purity forn=2systems

2016

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This paper investigates the fast Hamiltonian control of n = 2 density operators by continuously varying the flag (i.e. the eigenspaces) as one moves away from the completely mixed state. In general, the critical points and zeros of the purity derivative can only be solved analytically in the limit of minimal purity. We derive differential equations that maintain these features as the purity increases. In particular, there is a thread of points in the Bloch ball that locally maximizes the purity derivative, and a corresponding thread that minimizes it. Additionally, we show there is a closed surface of points inside of which the purity derivative is positive, and inside of which is negative. We argue that this approach may be useful in studying higher-dimensional systems.

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An introduction to Lie group integrators – basics, new developments and applications

Cited by 80 publications

References 70 publications

A review of some geometric integrators

A review of some geometric integrators

Convergence of Lie group integrators

Flag-based control of quantum purity forn=2systems

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