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41
Commutatorfree Lie group methods
 FGCS
, 2002
"... RKMK methods and CrouchGrossman methods are two classes of Lie group methods. The former is using flows and commutators of a Lie algebra... ..."
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RKMK methods and CrouchGrossman methods are two classes of Lie group methods. The former is using flows and commutators of a Lie algebra...
Lie Group Methods for Rigid Body Dynamics and Time Integration on Manifolds
 Computer Methods in Applied Mechanics and Engineering
, 1999
"... 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 l ..."
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Cited by 10 (2 self)
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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...
Adjoint and selfadjoint Liegroup methods
, 1999
"... In this paper we discuss Liegroup methods and their dependence on centering coordinate charts. The definition of adjoint of a numerical method is thus subordinate to the method itself and the choice of the coordinate map. We study Liegroup numerical methods and their adjoint, and define selfadjoin ..."
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Cited by 9 (2 self)
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In this paper we discuss Liegroup methods and their dependence on centering coordinate charts. The definition of adjoint of a numerical method is thus subordinate to the method itself and the choice of the coordinate map. We study Liegroup numerical methods and their adjoint, and define selfadjoint numerical methods. The latter are defined in terms of classical selfadjoint RungeKutta schemes and symmetric coordinates, based on a geodesic midpoint or on a flow midpoint. As a result, the proposed selfadjoint Liegroup numerical schemes obey timesymmetry both for linear and nonlinear problems, a property that is illustrated with three numerical experiments. 1 Introduction In a series or recent papers [16, 11, 23, 19, 3] a number of Liegroup methods have been introduced. Liegroup methods are numerical methods that solve the Liegroup differential equation y 0 = fl(y)y; y(0) = y 0 ; t 2 [0; T ]; (1.1) whereby y(t) 2 G, a Liegroup, and fl : G ! g is a sufficiently smooth function...
Bseries and order conditions for exponential integrators
"... Abstract. We introduce a general format of numerical ODEsolvers which include many of the recently proposed exponential integrators. We derive a general order theory for these schemes in terms of Bseries and bicolored rooted trees. To ease the construction of specific schemes we generalize an idea ..."
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Abstract. We introduce a general format of numerical ODEsolvers which include many of the recently proposed exponential integrators. We derive a general order theory for these schemes in terms of Bseries and bicolored rooted trees. To ease the construction of specific schemes we generalize an idea of Zennaro [Math. Comp., 46 (1986), pp. 119–133] and define natural continuous extensions in the context of exponential integrators. This leads to a relatively easy derivation of some of the most popular recently proposed schemes. The general format of schemes considered here makes use of coefficient functions which will usually be selected from some finite dimensional function spaces. We will derive lower bounds for the dimension of these spaces in terms of the order of the resulting schemes. Finally, we illustrate the presented ideas by giving examples of new exponential integrators of orders 4 and 5.
DiffMan  an object oriented MATLAB toolbox for solving differential equations on manifolds
, 1999
"... We describe an object oriented MATLAB toolbox for solving differential equations on manifolds. The software reflects recent development within the area of geometric integration. Through the use of elements from differential geometry, in particular Lie groups and homogeneous spaces, coordinate free f ..."
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Cited by 8 (3 self)
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We describe an object oriented MATLAB toolbox for solving differential equations on manifolds. The software reflects recent development within the area of geometric integration. Through the use of elements from differential geometry, in particular Lie groups and homogeneous spaces, coordinate free formulations of numerical integrators are developed. The strict mathematical definitions and results are well suited for implementation in an object oriented language, and, due to its simplicity, the authors have chosen MATLAB as the working environment. The basic ideas of DiffMan are presented, along with particular examples that illustrate the working of and the theory behind the software package.
On the Hopf Algebraic Structure of Lie Group Integrators
, 2006
"... Abstract A commutative but not cocommutative graded Hopf algebra HN, based on ordered rooted trees, is studied. This Hopf algebra generalizes the Hopf algebraic structure of unordered rooted trees HC, developed by Butcher in his study of Runge–Kutta methods and later rediscovered by Connes and Mosco ..."
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Abstract A commutative but not cocommutative graded Hopf algebra HN, based on ordered rooted trees, is studied. This Hopf algebra generalizes the Hopf algebraic structure of unordered rooted trees HC, developed by Butcher in his study of Runge–Kutta methods and later rediscovered by Connes and Moscovici in the context of noncommutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that HN is naturally obtained from a universal object in a category of noncommutative derivations, and in particular, it forms a foundation for the study of numerical integrators based on noncommutative Lie group actions on a manifold. Recursive and nonrecursive definitions of the coproduct and the antipode are derived. It is also shown that the dual of HN is a Hopf algebra of Grossman and Larson. HN contains two wellknown Hopf algebras as special cases: The Hopf algebra HC of Butcher–Connes–Kreimer is identified as a proper subalgebra of HN using the image of a tree symmetrization operator. The Hopf algebra HF of the Free Associative Algebra is obtained from HN by a quotient construction.
Interpolation in Lie groups and homogeneous spaces
 SIAM J. NUMER. ANAL
, 1998
"... We consider interpolation in Lie groups and homogeneous spaces. Based on points on the manifold together with tangent vectors at (some of) these points, we construct Hermite interpolation polynomials. If the points and tangent vectors are produced in the process of integrating an ordinary differenti ..."
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Cited by 6 (3 self)
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We consider interpolation in Lie groups and homogeneous spaces. Based on points on the manifold together with tangent vectors at (some of) these points, we construct Hermite interpolation polynomials. If the points and tangent vectors are produced in the process of integrating an ordinary differential equation on a Lie group or a homogeneous space, we use the truncated inverse of the differential of the exponential mapping and the truncated BakerCampbellHausdorff formula to relatively cheaply construct an interpolation polynomial. Much effort has lately been put into research on geometric integration, i.e. the process of integrating a differential equation in such away that the configuration space is respected by the numerical solution. Some of these methods may be viewed as generalizations of classical methods, and we investigate the construction of intrinsic dense output devices as generalizations of the continuous RungeKutta methods.
On the Method of Neumann Series for Highly Oscillatory Equations
 BIT
, 2004
"... The main purpose of this paper is to describe and analyse techniques for the numerical solution of highily oscillatory ordinary di#erential equations by exploying a Neumann expansion. Once the variables in the di#erential system are changed with respect to a rapidly rotating frame of reference, the ..."
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Cited by 6 (1 self)
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The main purpose of this paper is to describe and analyse techniques for the numerical solution of highily oscillatory ordinary di#erential equations by exploying a Neumann expansion. Once the variables in the di#erential system are changed with respect to a rapidly rotating frame of reference, the Neumann method becomes very e#ective indeed. However, this e#ectiveness rests upon suitable quadrature of highly oscillatory multivariate integrals, and we devote part of this paper to describe how to accomplish this to high accuracy with a modest computational e#ort. 1
Complexity theory for Liegroup solvers
, 1999
"... Commencing with a brief survey of Liegroup theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Liegroup structure: RungeKuttaMuntheKaas schemes, Fer and Magnus expansions. This is followed by complexity analysis of Fer and ..."
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Commencing with a brief survey of Liegroup theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Liegroup structure: RungeKuttaMuntheKaas schemes, Fer and Magnus expansions. This is followed by complexity analysis of Fer and Magnus expansions, whose conclusion is that for order four, six and eight an appropriately discretized Magnus method is always cheaper than a Fer method of the same order. Each Liegroup method of the kind surveyed in this paper requires the computation of a matrix exponential. Classical methods, e.g. Krylovsubspace and rational approximants, may fail to map elements in a Lie algebra to a Lie group. Therefore we survey a number of approximants based on the splitting approach and demonstrate that their cost is compatible (and often superior) to classical methods. 1 Introduction A central goal of classical numerical analysis is to design, implement and analyse computational algorithms that ...