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International audienceMany 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 minimisation 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

Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This new edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems.

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The subject matter of this paper is the solution of the linear differential equation y = a(t)y, y(0) = y 0 , where y 0 ∈ G, a(•) : R + → g and g is a Lie algebra of the Lie group G. By building upon an earlier work of Wilhelm Magnus, we represent the solution as an infinite series whose terms are indexed by binary trees. This relationship between the infinite series and binary trees leads both to a convergence proof and to a constructive computational algorithm. This numerical method requires the evaluation of a large number of multivariate integrals, but this can be accomplished in a tractable manner by using quadrature schemes in a novel manner and by exploiting the structure of the Lie algebra.

In this paper we explore quadrature methods for highly oscillatory integrals. Generalizing the method of stationary phase, we expand such integrals into asymptotic series in inverse powers of the frequency. The outcome are two families of methods, one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of Filon. Both kinds of methods approximate the integral as a linear combination of function values and derivatives, with coefficients that may depend on frequency. We determine asymptotic properties of these methods, proving that, perhaps counterintuitively, their performance drastically improves as frequency grows. The paper is accompanied by numerical results that demonstrate the potential of this set of ideas.

The generalized pantograph equation y′(t) = Ay(t) + By(qt) + Cy′(qt), y(0) = y0, where q ∈ (0, 1), has numerous applications, as well as being a useful paradigm for more general functional-differential equations with monotone delay. Although many special cases have been already investigated extensively, a general theory for this equation is lacking–its development and exposition is the purpose of the present paper. After deducing conditions on A, B, C ∈ ℂd×d that are equivalent to well-posedness, we investigate the expansion of y in Dirichlet series. This provides a very fruitful form for the investigation of asymptotic behaviour, and we duly derive conditions for limt⋅→∞y(t) = 0. The behaviour on the stability boundary possesses no comprehensive explanation, but we are able to prove that, along an important portion of that boundary, y is almost periodic and, provided that q is rational, it is almost rotationally symmetric. The paper also addresses itself to a detailed analysis of the scalar equation y′(t) = by(qt), y(0) = 1, to high-order pantograph equations, to a phenomenon, similar to resonance, that occurs for specific configurations of eigenvalues of A, and to the equation Y′(t) = AY(t) + Y(qt) B, Y(0) = Y0.

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