36 research outputs found
An introduction to Lie group integrators -- basics, new developments and applications
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
Lie Group integrators for mechanical systems
Since they were introduced in the 1990s, Lie group integrators have become a
method of choice in many application areas. These include multibody dynamics,
shape analysis, data science, image registration and biophysical simulations.
Two important classes of intrinsic Lie group integrators are the
Runge--Kutta--Munthe--Kaas methods and the commutator free Lie group
integrators.
We give a short introduction to these classes of methods. The Hamiltonian
framework is attractive for many mechanical problems, and in particular we
shall consider Lie group integrators for problems on cotangent bundles of Lie
groups where a number of different formulations are possible. There is a
natural symplectic structure on such manifolds and through variational
principles one may derive symplectic Lie group integrators. We also consider
the practical aspects of the implementation of Lie group integrators, such as
adaptive time stepping. The theory is illustrated by applying the methods to
two nontrivial applications in mechanics. One is the N-fold spherical pendulum
where we introduce the restriction of the adjoint action of the group
to , the tangent bundle of the two-dimensional sphere. Finally, we show
how Lie group integrators can be applied to model the controlled path of a
payload being transported by two rotors. This problem is modeled on
and put in a format where Lie group integrators can be applied.Comment: 35 page
Geometric integration of non-autonomous Hamiltonian problems
Symplectic integration of autonomous Hamiltonian systems is a well-known
field of study in geometric numerical integration, but for non-autonomous
systems the situation is less clear, since symplectic structure requires an
even number of dimensions. We show that one possible extension of symplectic
methods in the autonomous setting to the non-autonomous setting is obtained by
using canonical transformations. Many existing methods fit into this framework.
We also perform experiments which indicate that for exponential integrators,
the canonical and symmetric properties are important for good long time
behaviour. In particular, the theoretical and numerical results support the
well documented fact from the literature that exponential integrators for
non-autonomous linear problems have superior accuracy compared to general ODE
schemes.Comment: 20 pages, 3 figure
B-stability of numerical integrators on Riemannian manifolds
We propose a generalization of nonlinear stability of numerical one-step
integrators to Riemannian manifolds in the spirit of Butcher's notion of
B-stability. Taking inspiration from Simpson-Porco and Bullo, we introduce
non-expansive systems on such manifolds and define B-stability of integrators.
In this first exposition, we provide concrete results for a geodesic version of
the Implicit Euler (GIE) scheme. We prove that the GIE method is B-stable on
Riemannian manifolds with non-positive sectional curvature. We show through
numerical examples that the GIE method is expansive when applied to a certain
non-expansive vector field on the 2-sphere, and that the GIE method does not
necessarily possess a unique solution for large enough step sizes. Finally, we
derive a new improved global error estimate for general Lie group integrators
On the Lie enveloping algebra of a post-Lie algebra
We consider pairs of Lie algebras and , defined over a common
vector space, where the Lie brackets of and are related via a
post-Lie algebra structure. The latter can be extended to the Lie enveloping
algebra . This permits us to define another associative product on
, which gives rise to a Hopf algebra isomorphism between and
a new Hopf algebra assembled from with the new product.
For the free post-Lie algebra these constructions provide a refined
understanding of a fundamental Hopf algebra appearing in the theory of
numerical integration methods for differential equations on manifolds. In the
pre-Lie setting, the algebraic point of view developed here also provides a
concise way to develop Butcher's order theory for Runge--Kutta methods.Comment: 25 page
Integrators on homogeneous spaces: Isotropy choice and connections
We consider numerical integrators of ODEs on homogeneous spaces (spheres,
affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a
built-in symmetry. A numerical integrator respects this symmetry if it is
equivariant. One obtains homogeneous space integrators by combining a Lie group
integrator with an isotropy choice. We show that equivariant isotropy choices
combined with equivariant Lie group integrators produce equivariant homogeneous
space integrators. Moreover, we show that the RKMK, Crouch--Grossman or
commutator-free methods are equivariant. To show this, we give a novel
description of Lie group integrators in terms of stage trees and motion maps,
which unifies the known Lie group integrators. We then proceed to study the
equivariant isotropy maps of order zero, which we call connections, and show
that they can be identified with reductive structures and invariant principal
connections. We give concrete formulas for connections in standard homogeneous
spaces of interest, such as Stiefel, Grassmannian, isospectral, and polar
decomposition manifolds. Finally, we show that the space of matrices of fixed
rank possesses no connection