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

@article{Celledoni2014AnIT,
  title={An introduction to Lie group integrators - basics, new developments and applications},
  author={Elena Celledoni and H{\aa}kon Marthinsen and Brynjulf Owren},
  journal={J. Comput. Phys.},
  year={2014},
  volume={257},
  pages={1040-1061}
}
We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion of discrete gradient methods is generalised to Lie groups. 

Figures, Tables, and Topics from this paper

Convergence of Lie group integrators
TLDR
It is proved for the first time that the Lie–Butcher theory of Lie group integrators leads to global error estimates from local bounds on Riemannian homogeneous spaces.
Symplectic Lie group methods
In this article, a unified approach to obtain symplectic integrators on T*G from Lie group integrators on a Lie group G is presented. The approach is worked out in detail for symplectic integrators
Lie Group Integrators for Cotangent Bundles of Lie Groups and Their Application to Systems of Dipolar Soft Spheres
The objective of this thesis is to study numerical integrators and their application to solving ordinary di erential equations arising from mechanical systems. Many mechanical problems are naturally
Lie Group Integrators
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
An introduction to Lie group integrators
The significance of the geometry of differential equations was well understood already in the nineteenth century, and in the last few decades such aspects have played an increasing role in numerical
1 6 A ug 2 01 8 Convergence of Lie group integrators
We relate two notions of local error for integration schemes on Riemannian homogeneous spaces, and show how to derive global error estimates from such local bounds. In doing so, we prove for the
Lie Group integrators for mechanical systems
TLDR
This work considers Lie group integrators for problems on cotangent bundles of Lie groups where a number of different formulations are possible and considers the practical aspects of the implementation of these methods, such as adaptive time stepping.
Preserving first integrals with symmetric Lie group methods
The discrete gradient approach is generalized to yield integral preserving methods for differential equations in Lie groups.
Dynamics of the N-fold Pendulum in the framework of Lie Group Integrators
TLDR
This work briefly introduces this class of Lie group integrators, considering some of the practical aspects of their implementation, and presents some mathematical background that allows them to apply to some families of Lagrangian mechanical systems.
A review of some geometric integrators
TLDR
The Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus, is presented, which leads to numerical schemes which preserve exactly the energy of the system.
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 79 REFERENCES
Lie-group methods
Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the
Commutator-free Lie group methods
TLDR
A new format of Lie group methods is proposed which does not involve commutators and which uses a much lower number of exponentials than those proposed by Crouch and Grossman, and which may be particularly useful when applied to problems on homogeneous manifolds with large isotropy groups, or when used for stiff problems.
Numerical integration of differential equations on homogeneous manifolds
TLDR
It is argued that homogeneous spaces are the natural structures for the study and the analysis of intrinsic integration schemes for differential equations evolving on manifolds and abstract Runge-Kutta methods.
Cost Efficient Lie Group Integrators in the RKMK Class
In this work a systematic procedure is implemented in order to minimise the computational cost of the Runge—Kutta—Munthe-Kaas (RKMK) class of Lie-group solvers. The process consists of the
A note on the numerical integration of the KdV equation via isospectral deformations
The main purpose of this paper is to test the performance of Lie group integrators as applied to a semi-discrete version of the KdV equation.
High order Runge-Kutta methods on manifolds
We present a family of Runge-Kutta type integration schemes of arbitrarily high order for differential equations evolving on manifolds. We prove that any classical Runge-Kutta method can be turned
Partitioned Runge–Kutta Methods in Lie-Group Setting
We introduce partitioned Runge–Kutta (PRK) methods as geometric integrators in the Runge–Kutta–Munthe-Kaas (RKMK) method hierarchy. This is done by first noticing that tangent and cotangent bundles
A Low Complexity Lie Group Method on the Stiefel Manifold
A low complexity Lie group method for numerical integration of ordinary differential equations on the orthogonal Stiefel manifold is presented. Based on the quotient space representation of the
On the Implementation of Lie Group Methods on the Stiefel Manifold
TLDR
This paper shows how Lie group methods can be implemented in a computationally competitive way, by exploiting that analytic functions of n×n matrices of rank 2p can be computed with O(np2) complexity.
Hamilton-Pontryagin Integrators on Lie Groups
In this thesis structure-preserving time integrators for mechanical systems whose configuration space is a Lie group are derived from a Hamilton-Pontryagin (HP) variational principle. In addition to
...
1
2
3
4
5
...