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Liegroup methods
 ACTA NUMERICA
, 2000
"... Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having ..."
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Cited by 110 (19 self)
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Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup 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 Liegroup structure, highlighting theory, algorithmic issues and a number of applications.
Improved high order integrators based on the Magnus expansion
 BIT
, 1999
"... We build high order efficient numerical integration methods for solving the linear differential equation X = A(t)X based on Magnus expansion. These methods ..."
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Cited by 23 (3 self)
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We build high order efficient numerical integration methods for solving the linear differential equation X = A(t)X based on Magnus expansion. These methods
The Magnus expansion and some of its applications
, 2008
"... Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an ..."
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Cited by 9 (0 self)
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Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as TimeDependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial resummation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related nonperturbative
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 M ..."
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Cited by 6 (1 self)
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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.
Convergence of the Magnus series
 Found. Comput. Math
"... The Magnus series is an infinite series which arises in the study of linear ordinary differential equations. If the series converges, then the matrix exponential of the sum equals the fundamental solution of the differential equation. The question considered in this paper is: When does the series co ..."
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Cited by 5 (0 self)
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The Magnus series is an infinite series which arises in the study of linear ordinary differential equations. If the series converges, then the matrix exponential of the sum equals the fundamental solution of the differential equation. The question considered in this paper is: When does the series converge? The main result establishes a sufficient condition for convergence, which improves on several earlier results. 1
A Magnus expansion for the equation . . .
, 2000
"... The subject matter of this paper is the representation of the solution of the linear differential equation Y 0 = AY \Gamma Y B, Y (0) = Y0 , in the form Y (t) = e\Omega\Gamma t) Y0 and the representation of the function\Omega as a generalisation of the classical Magnus expansion. An immediate a ..."
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Cited by 5 (1 self)
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The subject matter of this paper is the representation of the solution of the linear differential equation Y 0 = AY \Gamma Y B, Y (0) = Y0 , in the form Y (t) = e\Omega\Gamma t) Y0 and the representation of the function\Omega as a generalisation of the classical Magnus expansion. An immediate application is a new recursive algorithm for the derivation of the BakerCampbellHausdorff formula and its symmetric generalisation. 1 Introduction This paper is concerned with the solution of the linear ordinary differential system Y 0 = AY \Gamma Y B; t 0; Y (0) = Y 0 ; (1.1) where both A and B are Lipschitz functions that map [0; 1) into Mm , the set of m \Theta m matrices, and Y 0 2 Mm . The equation (1.1) features in numerous applications and the approximation of its solution is of interest. Moreover, solutions of this equation often display interesting geometry. For example, B = A results in the isospectral flow Y 0 = AY \Gamma Y A; t 0; Y (0) = Y 0 ; (1.2) whose invariant...
Evaluating the Evans function: Order reduction in numerical methods
 MATH. COMP
, 2008
"... We consider the numerical evaluation of the Evans function, a Wronskianlike determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The ..."
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Cited by 3 (2 self)
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We consider the numerical evaluation of the Evans function, a Wronskianlike determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The problem becomes stiff as the spectral parameter grows. Consequently, the Gauss–Legendre method has previously been used for such problems; however more recently, methods based on the Magnus expansion have been proposed. Here we extensively examine the stiff regime for a general scalar Schrödinger operator. We show that although the fourthorder Magnus method suffers from order reduction, a fortunate cancellation when computing the Evans matching function means that fourthorder convergence in the end result is preserved. The Gauss–Legendre method does not suffer from order reduction, but it does not experience the cancellation either, and thus it has the same order of convergence in the end result. Finally we discuss the relative merits of both methods as spectral tools.
RCMS: Right Correction Magnus Series approach for oscillatory ODEs
, 2005
"... We consider RCMS, a method for integrating differential equations of the form y ′ =[�A + A1(t)]y with highly oscillatory solution. It is shown analytically and numerically that RCMS can accurately integrate problems using stepsizes determined only by the characteristic scales of A1(t), typically muc ..."
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We consider RCMS, a method for integrating differential equations of the form y ′ =[�A + A1(t)]y with highly oscillatory solution. It is shown analytically and numerically that RCMS can accurately integrate problems using stepsizes determined only by the characteristic scales of A1(t), typically much larger than the solution “wavelength”. In fact, for a given t grid the error decays with, or is independent of, increasing solution oscillation. RCMS consists of two basic steps, a transformation which we call the right correction and solution of the right correction equation using a Magnus series. With suitable methods of approximating the highly oscillatory integrals appearing therein, RCMS has high order of accuracy with little computational work. Moreover, RCMS respects evolution on a Lie group. We illustrate with application to the 1D Schrödinger equation and to Frenet–Serret equations. The concept of right correction integral series schemes is suggested and right correction Neumann schemes are discussed. Asymptotic analysis for a large class of ODEs is included which gives certain numerical integrators converging to exact asymptotic behaviour. © 2005 Elsevier B.V. All rights reserved.
CASE FOR SUPPORT: EFFICIENT EVANS FUNCTION CALCULATIONS VIA NEUMANN AND MAGNUS EXPANSIONS
"... Summary, significance and objectives. A practical problem when integrating systems of linear ordinary differential equations, is that if we wish to sample the solution for a different value of an inherent parameter, then we must reintegrate. This is particularly inefficient when we want to accurate ..."
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Summary, significance and objectives. A practical problem when integrating systems of linear ordinary differential equations, is that if we wish to sample the solution for a different value of an inherent parameter, then we must reintegrate. This is particularly inefficient when we want to accurately sample the solution over a continuous widespread set of parameter values. Though continuity methods resolve this issue locally, they still involve some degree of reintegration. A particular application we have in mind is that of evaluating the Evans function for different values of the spectral parameter, which involves repeated integration of the spectral equations. In this project we propose to extensively study a set of new efficient numerical methods based on Neumann and Magnus expansions, that completely avoid the need for reintegration. These methods were recently proposed by Aparicio, Malham & Oliver (2002); they extend and generalize work by Moan (1998). The basic idea is that we expand either the Neumann or Magnus series solution for such systems as a power series in the parameter(s) in question. The coefficients of the series can be precomputed to any required accuracy. Then we evaluate the series for any of the parameter values we wish to sample. This proposal is intended to crystallize and then implement these ideas. The realization of these methods