Arieh Iserles

Lie-group methods

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.

Efficient quadrature of highly oscillatory integrals using derivatives

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 is 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 (Filon 1928 Proc. R. Soc. Edinb. 49, 38–47). 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, perhaps counterintuitively, that their performance drastically improves as frequency grows. The paper is accompanied by numerical results that demonstrate the potential of this set of ideas.

On the generalized pantograph functional-differential equation

The generalized pantograph equation y ′( t ) = Ay ( t ) + By ( qt ) + Cy ′( qt ), y (0) = y 0 , 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 ∈ ℂ d × 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 lim t ⋅→∞ 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) = Y 0 .

On polynomials orthogonal with respect to certain Sobolev inner products

Abstract We are concerned with polynomials { p n ( λ ) } that are orthogonal with respect to the Sobolev inner product 〈 f , g 〉 λ = ∝ fg dϑ + λ ∝ f ′ g ′ dψ , where λ is a non-negative constant. We show that if the Borel measures dϑ and dψ obey a specific condition then the P n ( λ ) s can be expanded in the polynomials orthogonal with respect to dϑ in such a manner that, subject to correct normalization, the expansion coefficients, except for the last, are independent of n and are themselves orthogonal polynomials in λ. We explore several examples and demonstrate how our theory can be used for efficient evaluation of Sobolev-Fourier Coefficients.

Geometric integration: numerical solution of differential equations on manifolds

Since their introduction by Sir Isaac Newton, diffierential equations have played a decisive role in the mathematical study of natural phenomena. An important and widely acknowledged lesson of the last three centuries is that critical information about the qualitative nature of solutions of diffierential equations can be determined by studying their geometry. Perhaps the most important example of this approach was the formulation of the laws of mechanics by Alexander Rowan Hamilton, which allowed deep geometric tools to be used in understanding the dynamics of complex systems such as rigid bodies and the Solar System. Conserved quantities of a Hamiltonian system, such as energy, linear and angular momentum, could be understood in terms of the symmetries of the underlying Hamiltonian function, its ergodic properties determined from the underlying symplectic nature of the formulation and constraints on the system could be incorporated in a natural manner. The Hamiltonian geometric formulation of many other problems in science modelled by ordinary and partial diffierential equations, such as ocean dynamics, nonlinear optics and elastic deformations, continues to play a vital role in our qualitative understanding of these systems. An equally important geometric approach to the study of diffierential equations is the application of symmetry–based methods pioneered by Sophus Lie. Exploiting underlying symmetries of a partial or ordinary difierential equation, it can be often greatly simplified and sometimes solved altogether in closed form. Such methods, which lie at the heart of the construction of self–similar solutions of diffierential equations and the symmetry reduction of complex systems, have become increasingly popular with the development of symbolic algebra packages. It is no coincidence that the most important equations of mathematical physics are precisely those for which geometric and symmetry–based methods are most effiective. Arguably, these equations are really a shorthand for the deep underlying symmetries in nature that they encapsulate.

ON THE IMPLEMENTATION OF THE METHOD OF MAGNUS SERIES FOR LINEAR DIFFERENTIAL EQUATIONS

The method of Magnus series has recently been analysed by Iserles and Nørsett. It approximates the solution of linear differential equations y′ = a(t)y in the form y(t) = eσ(t)y0, solving a nonlinear differential equation for σ by means of an expansion in iterated integrals of commutators. An appealing feature of the method is that, whenever the exact solution evolves in a Lie group, so does the numerical solution.The subject matter of the present paper is practical implementation of the method of Magnus series. We commence by briefly reviewing the method and highlighting its connection with graph theory. This is followed by the derivation of error estimates, a task greatly assisted by the graph-theoretical connection. These error estimates have been incorporated into a variable-step fourth-order code. The concluding section of the paper is devoted to a number of computer experiments that highlight the promise of the proposed approach even in the absence of a Lie-group structure.

On the Global Error of Discretization Methods for Highly-Oscillatory Ordinary Differential Equations

AbstractCommencing from a global-error formula, originally due to Henrici, we investigate the accumulation of global error in the numerical solution of linear highly-oscillating systems of the form y″+ g(t)y = 0, where g(t)

Quadrature methods for multivariate highly oscillatory integrals using derivatives

g(t)\mathop \to \limits^{t \to \infty } \infty