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 recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.

Abstract In this paper, structure-preserving time-integrators for rigid body-type mechanical

In this paper, Smale’s α theory is generalized to the context of intrinsic Newton iteration on geodesically complete analytic Riemannian and Hermitian manifolds. Results are valid for analytic mappings from a manifold to a linear space of the same dimension, or for analytic vector fields on the manifold. The invariant γ is defined by means of high order covariant derivatives. Bounds on the size of the basin of quadratic convergence are given. If the ambient manifold has negative sectional curvature, those bounds depend on the curvature. A criterion of quadratic convergence for Newton iteration from the information available at a point is also given. 1 Introduction and main results. Numerical problems posed in manifolds arise in many natural contexts. Classical examples are given by the eigenvalue problem, the symmetric eigenvalue problem, invariant subspace computations, minimization problems with orthogonality constraints, optimization problems with equality constraints... etc. In the first

Given an ordinary differential equation on a homogeneous manifold, one can construct a “geometric integrator ” by determining a compatible ordinary differential equation on the associated Lie group, using a Lie group integration scheme to construct a discrete time approximation of the solution curves in the group, and then mapping the discrete trajectories onto the homogeneous manifold using the group action. If the points of the manifold have continuous isotropy, a vector field on the manifold determines a continuous family of vector fields on the group, typically with distinct discretizations. If sufficient isotropy is present, an appropriate choice of vector field can yield improved capture of key features of the original system. In particular, if the algebra of the group is “full”, then the order of accuracy of orbit capture (i.e. approximation of trajectories modulo time reparametrization) within a specified family of integration schemes can be increased by an appropriate choice of isotropy element. We illustrate the approach developed here with comparisons of several integration schemes for the reduced rigid body equations on the sphere. 1 Introduction. Geometric integration techniques have become increasingly popular in the modern approach to numerical

Recently there has been an increasing interest in time integrators for ordinary dierential equations which use Lie group actions as a primitive in the design of the methods. These methods are usually phrased in an abstract sense for arbitrary Lie groups and actions. We show here how the methods look when applied to the rigid body equations in particular and indicate how the methods work in general. An important part of the Lie group methods involves the computation of a coordinate map and its derivative. Various options are available, and they vary in cost, accuracy and ability to approximately conserve invariants. We discuss how the computation of these maps can be optimized for the rigid body case, and we provide numerical experiments which give an idea of the performance of Lie group methods compared to other known integration schemes. AMS Subject Classication: 65L05 Key Words: time integration, geometric integration, numerical integration of ordinary dierential equati...

In this paper new integration algorithms for linear differential equations up to eighth order are obtained. Starting from Magnus expansion, methods based on Cayley transformation and Fer expansion are also built. The structure of the exact solution is retained while the computational cost is reduced compared to similar methods. Their relative performance is tested on some illustrative examples.

Many discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is the calculation of commutators, a task that can be made significantly cheaper by the introduction of appropriate bases of function values and exploitation of redundancies inherent in a Lie-algebraic structure by means of graded spaces. In many Lie groups of practical interest a convenient alternative to the exponential map is a Cayley transformation and the subject of this paper is the investigation of graded algebras that occur in this context. To this end we introduce a new concept, a hierarchical algebra, a Lie algebra equipped with a countable number of m-nary multilinear operations which display alternating symmetry and a `hierarchy condition'. We present explicit formulae for the dimension of graded subspaces of free hirarchical algebras and an algorithm for the construction of their basis. The paper is concluded by reviewing a number of applications of our results to numerical methods in a Lie-algebraic setting. 1. Graded algebras and Lie-group methods 1.1. Geometric integration and general Lie-group solvers An increasing attention has been paid in recent years to discretization of differential equations that evolve on smooth manifolds. The main reason is that invariants and conservation laws of a differential system can be phrased by restricting the configuration space to a manifold. Discretization methods that respect manifold structure are an important example of geometric integrators, computational methods that preserve the underlying geometry and qualitative attributes of the differential system [1]. Perhaps the most ubiquitous (and arguably most important) type of a m...

In this paper we apply geometric integrators of the RKMK type to the problem of integrating Lie--Poisson systems numerically. By using the coadjoint action of the Lie group G on the dual Lie algebra g to advance the numerical flow, we devise methods of arbitrary order that automatically stay on the coadjoint orbits. First integrals known as Casimirs are retained to machine accuracy by the numerical algorithm. Within the proposed class of methods we find integrators that also conserve the energy. These schemes are implicit and of second order. Nonlinear iteration in the Lie algebra and linear error growth of the global error are discussed. Numerical experiments with the rigid body, the heavy top and a finite--dimensional truncation of the Euler equations for a 2D incompressible fluid are used to illustrate the properties of the algorithm. AMS Subject Classification: 65L06, 34A50, 34A26, 70H99 Key Words: Lie--Poisson systems, geometric integration, energy conserving, coadjoint actio...

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 are the natural domains for the differential equations to be solved. Next, we equip the (co)tangent bundle of a Lie group with a group structure and treat it as a Lie group. The structure of the differential equations on the (co)tangent-bundle Lie group is such that partitioned versions of the RKMK methods are naturally introduced. Numerical examples are included to illustrate the new methods. AMS Subject Classification: 65L06, 34C40 Key Words: Partitioned Runge--Kutta method, RKMK method, Magnus method, Cayley method, Crouch--Grossman method, geometric integration, tangent bundle of Lie group, semidirect product, differential equations on manifolds 1 Introduction The Runge--Kutta--Munthe-Kaas (RKMK) method was introduced in [17] to solve differential equations on homogeneous spaces, t...

There is growing recognition in the last few years that Lie groups and homogeneous spaces are often the right configuration space for the discretization of time-dependent differential equations. In this paper we review briefly recent advances in Lie-group calculations, concentrating mainly on approximation methods that advance a trivialised version of the differential equation in a Lie algebra in terms of either Magnus or Cayley expansions.