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59
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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Cited by 18 (4 self)
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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...
On the Discretization of DoubleBracket Flows
 Found. Comput. Math
, 2001
"... This paper extends the method of Magnus series to Liealgebraic equations originating in doublebracket flows. We show that the solution of the isospectral flow Y = [[Y; N ]; Y ], Y (0) = Y0 2 Sym(n), can be represented in the form Y (t) = e \Omega\Gamma Y0e , where the Taylor expansion of\Om ..."
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Cited by 15 (5 self)
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This paper extends the method of Magnus series to Liealgebraic equations originating in doublebracket flows. We show that the solution of the isospectral flow Y = [[Y; N ]; Y ], Y (0) = Y0 2 Sym(n), can be represented in the form Y (t) = e \Omega\Gamma Y0e , where the Taylor expansion of\Omega can be constructed explicitly, termbyterm, identifying individual expansion terms with certain rooted trees with bicolour leaves. This approach is extended to other Liealgebraic equations that can be appropriately expressed in terms of a finite `alphabet'. 1
Efficient approximation of SturmLiouville problems using Liegroup methods
, 1998
"... We present a new approach to the numerical solution of SturmLiouville eigenvalue problems based on Magnus expansions. Our algorithms are closely related to Pruess' methods [Pre73], but provide for high order approximations at nearly the same cost as the secondorder Pruess method. By using New ..."
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Cited by 15 (0 self)
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We present a new approach to the numerical solution of SturmLiouville eigenvalue problems based on Magnus expansions. Our algorithms are closely related to Pruess' methods [Pre73], but provide for high order approximations at nearly the same cost as the secondorder Pruess method. By using Newton iteration to solve for the eigenvalues, we are able to present an efficient algorithm for computing a range of eigenvalues and eigenfunctions. Numerical experiments display promising results, and asymptotic corrections are made to improve further the accuracy of the schemes.
On the Hopf algebraic structure of Lie group integrators
 J. Found. Comput. Math., “Online First
, 2007
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Hopf algebras of formal diffeomorphisms and numerical integration on manifolds
, 2009
"... Bseries originated from the work of John Butcher in the 1960s as a tool to analyze numerical integration of differential equations, in particular Runge–Kutta methods. Connections to renormalization have been established in recent years. The algebraic structure of classical Runge–Kutta methods is de ..."
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Cited by 12 (4 self)
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Bseries originated from the work of John Butcher in the 1960s as a tool to analyze numerical integration of differential equations, in particular Runge–Kutta methods. Connections to renormalization have been established in recent years. The algebraic structure of classical Runge–Kutta methods is described by the Connes–Kreimer Hopf algebra. Lie–Butcher theory is a generalization of Bseries aimed at studying Liegroup integrators for differential equations evolving on manifolds. Liegroup integrators are based on general Lie group actions on a manifold, and classical Runge–Kutta integrators appear in this setting as the special case of R n acting upon itself by translations. Lie–Butcher theory combines classical Bseries on R n with Lieseries on manifolds. The underlying Hopf algebra HN combines the Connes–Kreimer Hopf algebra with the shuffle Hopf algebra of free Lie algebras. Aimed at a general mathematical audience, we give an introduction to Hopf algebraic structures and their relationship to structures appearing in numerical analysis. In particular we explore the close connection between Lie series, timedependent Lie series and Lie–Butcher series for diffeomorphisms on manifolds. The role of the Euler and Dynkin idempotents in numerical analysis is discussed. A noncommutative version of a Faà di Bruno bialgebra is introduced, and the relation to noncommutative Bell polynomials is explored.
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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Cited by 11 (4 self)
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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.
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 10 (3 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
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 10 (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...
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.