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Introduction

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September 1970 - present

## Publications

Publications (109)

High-order variants of the classical Størmer-Cowell methods are still a popular class of methods for computations in celestial mechanics. In this work we investigate the absolute stability of Størmer-Cowell methods close to zero, and present a characterization of the stability of methods of all orders. In particular, we show that many methods are n...

The concern of this paper is in expanding and computing initial-value problems of the form y′=f(y)+hω(t), where the function hω oscillates rapidly for ω≫1. Asymptotic expansions for such equations are well understood in the case of modulated Fourier oscillators and they can be used as an organizing principle for very accurate and affordable numeric...

We consider the use of eigenfunctions of polyharmonic operators, equipped with homogeneous Neumann boundary conditions, to
approximate nonperiodic functions in compact intervals. Such expansions feature a number of advantages in comparison with
classical Fourier series, including uniform convergence and more rapid decay of expansion coefficients. H...

We are concerned with the computation of spectra of highly oscillatory Fredholm problems, in particular with the Fox-Li operator ∫ -1 1 f(x)e iω(x-y) 2 dx=λf(y),-1≤y≤1, where ω≫1. Our main tool is the finite section method: an eigenfunction is expanded in an orthonormal basis of the underlying space, resulting in an algebraic eigenvalue problem. We...

We address in this paper the approximation of functions in an equilateral triangle by a linear combination of Laplace–Neumann
eigenfunctions. The Laplace–Neumann basis exhibits a number of advantages. The approximations converge fairly fast and their
speed of convergence can be much improved by using techniques familiar in Fourier analysis and spec...

Modified Fourier expansion is a powerful means for the approximation of nonperiodic smooth functions in a univariate or multivariate setting. In the current paper we consider further enhancement of this approach by two techniques familiar from conventional Fourier analysis: the polynomial subtraction and the hyperbolic cross. We demonstrate that, j...

The Fox–Li operator is a convolution operator over a finite interval with a special highly oscillatory kernel. It plays an important role in laser engineering. However, the mathematical analysis of its spec-trum is still rather incomplete. In this expository paper we survey part of the state of the art, and our emphasis is on showing how standard W...

Let be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K0(x, y)eiω|x–y|. We study the spectral problem for large ω, showing that the spectrum consists of infinitely many discrete (complex) eigenvalues and give a precise description of the
way in which they converge to the origin. In addition, we investigate the...

In this paper, we expand upon the theme of modified Fourier expansions and extend the theory to a multivariate setting and to expansions in eigenfunctions of the Laplace-Neumann operator. We pay detailed attention to expansions in a d-dimensional cube and to an effective derivation of expansion coefficients there by means of quadratures of highly o...

In this paper, we consider a modification of the classical Fourier expansion, whereby in [- 1, 1] the sin πnx functions are replaced by, n ≥ 1. This has a number of important advantages in the approximation of analytic, nonperiodic functions. In particular, expansion coefficients decay like (n -2), rather than like (n-1). We explore theoretical fea...

Given a monotone measure α(x), a positive function ω(x,μ), μεΩ and a sequence μ1,μ2 ,... εΩ, we consider monic polynomials that satisfy the bi-orthogonality conditions
òpm ( x )w( x,mk )da( x ) = 0, 1 \leqslant k \leqslant m, pm Î pm [ x ].\int {p_m \left( x \right)\omega \left( {x,\mu _k } \right)d\alpha \left( x \right) = 0,} 1 \leqslant k \leq...

Let Rn/m(z, γ)=Pn(z; γ)/(1-γz)m be a restricted rational approximation to exp(z), zεℂ, of order n for all real γ. In this paper we discuss how γ can be used
to obtain fitting at a real non-positive point z1. It is shown that there are exactly min(n+1, m) different positive values of γ with this property.

So far ODE-solvers have been implemented mostly on sequential computers. This has lead to development of methods that are
very difficult to parallelisize. In this paper we discuss how to develop Runge-Kutta methods that lead to parallel implementation
on computers with a small number of CPU's. Both explicit and implicit methods are discussed. Some...

We consider two types of highly oscillatory bivariate integrals with a nondegenerate stationary point. In each case we produce an asymptotic expansion and two kinds of quadrature algorithms: an asymptotic method and a Filon-type method. Our results emphasize the crucial role played by the behaviour at the stationary point and by the geometry of the...

Abstract While there exist eective methods for univariate highly oscillatory quadra- ture, this is not the case in a multivariate setting. In this paper we embark on a project, extending univariate theory to more variables. Inter alia, we demon- strate that, subject to a nonresonance condition, an integral over a simplex can be expanded,asymptotica...

The last few years have witnessed substantive developments in the computation of highly oscillatory integrals in one or more dimensions. The availability of new asymptotic expansions and a Stokes-type theorem allow for a comprehensive analysis of a number of old (although enhanced) and new quadrature techniques: the asymptotic, Filon-type and Levin...

The main theme of this paper is the construction of efficient, reliable and affordable error bounds for two families of quadrature methods for highly oscillatory integrals. We demonstrate, using asymptotic expansions, that the error can be bounded very precisely indeed at the cost of few extra derivative evaluations. Moreover, in place of derivativ...

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...

Commencing with a brief survey of Lie-group theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Lie-group structure: Runge–Kutta–Munthe-Kaas schemes, Fer and Magnus expansions. This is followed by derivation of the computational cost of Fer and Magnus expansions, whose conclusio...

In this paper we consider an application of Sobolev-orthogonal functions and radial basis function to the numerical solution of partial differential equations. We develop the fundamentals of a spectral method, present examples via reaction--diffusion partial differential equations and discuss briefly some links with theory of wavelets.

In this paper we consider an application of Sobolev-orthogonal functions and radial basis function to the numerical solution of partial diierential equations. We develop the funda-mentals of a spectral method, present examples via reaction{diiusion partial diierential equations and discuss brieey some links with theory of wavelets.

Commencing with a brief survey of Lie-group theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Lie-group structure: Runge--Kutta--Munthe-Kaas schemes, Fer and Magnus expansions. This is followed by complexity analysis of Fer and Magnus expansions, whose conclusion is that for o...

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 geom...

The subject of this paper is the investigation of the Magnus expansion of a solution of the linear differential equation y 0 = a(t)y, y(0) 2 G, where G is a Lie group and a : R + ! g, g being the Lie algebra of G. We commence with a brief survey of recent work in this area. Next, building on earlier work of Iserles and Nørsett, we prove that an app...

The method of Magnus series has recently been analysed by Iserles and Nrsett. It approximates the solution of linear differential equations y = a(t)y in the form y(t) = e
(t)
y
0, solving a nonlinear differential equation for by means of an expansion in iterated integrals of commutators. An appealing feature of the method is that, whenever the exac...

The subject matter of this paper is the solution of the linear differential equation y′ = a(t)y, y(0) = y0, where y0 ∈ 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 indexe by binary trees. This relationship between the in...

The subject matter of this paper is the solution of the linear differential equation y = a(t)y, y(0) = y0 , where y0 2 G, a( Delta ) : R ! g and g is a Lie algebra of the Lie group G. By building upon an earlier work of Wilhelm Magnus (Magnus 1954), we represent the solution as an infinite series whose terms are indexed by binary trees. This relati...

Runge-Kutta research in Trondheim began in 1970 when Syvert P. Nørsett was appointed to the NTH. Although the group has worked on various aspects of Runge-Kutta methods, we have elected, in this paper, to focus on DIRK methods, linear stability, order stars, parallel methods and continuous explicit methods.

The solution of Y 0 = A(t)Y; Y (0) = Y 0 , where Y is in a matrical Lie group G and A(t) is in the Lie algebra of G, is discussed in this article. The basis of our approach are generalized iterated commutators. In that manner we can approximate the solution to an arbitrarily high order. The exact solution is given as an infinite series of commutato...

The authors have presented in [1] a technique to generate transformations T of the set n of nth degree polynomials to itself such that if if p ϵ n has all its zeros in (c, d) then T{p} has all its zeros in (a, b), where (a, b) and (c, d) are given real intervals. The technique rests upon the derivation of an explicit form of biorthogonal polynomial...

We survey briefly recent developments in geometric integration and numerical methods on manifolds. The underlying philosophy is that numerical methods should, whenever practicable, recover correctly known qualitative behaviour of the underlying differential system. Classical nume rical methods retain invariants very poorly indeed, and this justifie...

Let (x, ) and (x,) be two functions,x[a, b] and {"Abs1">Let (x, ) and (x,) be two functions,x[a, b] and {
jj
}}
j=1j=1
and { and {
jj
}}
j=1j=1
be two sequences where be two sequences where
ii
jj
and 1206K002G167/xxlarge956.gif" alt="mgr" align="MIDDLE" border="0">
ii
jj
whenij. We define the vector spacesU whenij. We define the vector space...

We employ B-series to analyze the order of Runge-Kutta (RK) methods that use simple iteration, modified Newton iteration or full Newton iteration to compute their internal stage values. Our assumptions about the initial guess for the internal stage values cover a wide range of practical schemes. Moreover, the analytical techniques developed in this...

We discuss preconditioning and overlapping of waveform relaxation methods for sparse linear differential systems. It is demonstrated that these techniques significantly improve the speed of convergence of the waveform relaxation iterations resulting from application of various modes of block Gauss-Jacobi and block Gauss-Seidel methods to differenti...

Given a parametrised weight function $\omega(x,\mu)$ such that the quotients
of its consecutive moments are M\"obius maps, it is possible to express the
underlying biorthogonal polynomials in a closed form \cite{IN2}. In the present
paper we address ourselves to two related issues. Firstly, we demonstrate that,
subject to additional assumptions, ev...

. We examine the potential for parallelism in Runge-Kutta (RK) methods based on formulas in standard one-step form. Both negative and positive results are presented. Many of the negative results are based on a theorem that bounds the order of a RK formula in terms of the minimum polynomial for its coefficient matrix. The positive results are largel...

We consider the numerical solution of initial value problems for both ordinary differential equations and differential-algebraic equations by Runge-Kutta (RK) formulas. We assume that the internal stage values of the RK formula are computed by some iterative scheme for solving nonlinear equations, such as Newton's method. Using Butcher series and r...

Orthogonal polynomials feature frequently in the analysis of numerical methods for ODEs, but the usefulness of orthogonality is limited. In this paper the authors survey a range of problems whose analysis calls for the broader concept of biorthogonality. They commence by demonstrating how biorthogonality arises naturally in the investigation of two...

We examine the numerical solution of the delay differential equation y ' (t)=αy(t)+βy(1 2t)+γy ' (1 2t),t≥0,y(0)=1, where α, β and γ are complex numbers, using a Runge-Kutta approach. Sufficient conditions for the asymptotic stability of the numerical solution, i.e. lim n→∞ y n =0, are found and particular attention is given to the simple case of c...

This chapter is devoted to the study of multistep and general multivalue methods. After retracing their historical development (Adams, Nyström, Milne, BDF) we study in the subsequent sections the order, stability and convergence properties of these methods. Convergence is most elegantly set in the framework of onestep methods in higher dimensions....

This first chapter contains the classical theory of differential equations, which we judge useful and important for a profound understanding of numerical processes and phenomena. It will also be the occasion of presenting interesting examples of differential equations and their properties.

Numerical methods for ordinary differential equations fall naturally into two classes: those which use one starting value at each step (“one-step methods”) and those which are based on several values of the solution (“multistep methods” or “multi-value methods”). The present chapter is devoted to the study of one-step methods, while multistep metho...

We are concerned with polynomials {pn(λ)} that are orthogonal with respect to the Sobolev inner product 〈 f, g 〉λ = ∝ fgdϑ + 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 Pn(λ)'s can be expanded in the polynomials orthogonal with respect to dϑ in such a manner that,...

We survey certain transformations of the set 1r, [x of ri-th degree polynomials into themselves. These transformations share the property that polynomials with all their zeros in a certain real interval are mapped to polynomials with all their zeros in another real interval. Rich sources of such ̈zero-mapping̈ transformations can be found in the La...

The purpose of this paper is to create a theoretical framework for parallelization of Runge-Kutta methods. We investigate
the inherent potential for parallelism by considering digraphs of Runge-Kutta matrices. By highlighting the important role
of the underlying sparsity pattern, this approach narrows the field down to certain types of methods. The...

Transformations that map polynomials with zeros in a certain interval into polynomials with zeros in another interval are considered here. By using the theory of bi-orthogonal polynomials, a general technique for the construction of such transformations is developed. Finally, a list of 16 different transformations formed by using the authors’ techn...

In this paper we survey order fitting and A-acceptability of rational approximants to the exponential. Our exposition is centered around two theorems of great generality: the first imposes a bound on the order of an A-acceptable scheme with given fitting and the second presents an order bound when an A-acceptable approximant possesses a given numbe...

It is proved that biorthogonal polynomials obey two different kinds of Christoffel-Darboux-type formulae, one linking polynomials with a different parameter and one combining polynomials with different degrees. This is used to produce a mixed recurrence relation, which is valid for all biorthogonal polynomials. This recurrence relation establishes...

One-step collocation methods are known to be a subclass of implicit Runge-Kutta methods. Further, one-leg methods are special multistep one-point collocation methods. In this paper we extend both of these collocation ideas to multistep collocation methods with $k$ previous meshpoints and $m$ collocation points. By construction, the order is at leas...

An automatic technique for solving discontinuous initial-value problems is developed and justified. The technique is based on the use of local interpolants such as those that have been developed for use with Runge-Kutta formula pairs. Numerical examples are presented to illustrate the significant improvement in efficiency and reliability that resul...

Let j (x, m) be a distribution in x Î R for every fj, in a real parameter set W. Subject to additional technical conditions, we study rath degree monic polynomials pm that satisfy the biorthogonality conditions (formula presented) for a distinct sequence m1, m2,&.Î W. Necessary and sufficient conditions for existence and uniqueness are established,...

Let $\varphi(x, \mu)$ be a distribution in $x \in R$ for every $\mu$ in a real parameter set $\Omega$. Subject to additional technical conditions, we study $m$th degree monic polynomials $p_m$ that satisfy the biorthogonality conditions $$\int^\infty_{-\infty} p_m(x) d\varphi(x, \mu_l) = 0, l = 1, 2,\ldots,m, m \geq 1,$$ for a distinct sequence $\m...

We study order and zero stability of two step methods of Obrechkoff type forn ordinary differential equations. A relation between order and properties of mth degree polynomials orthogonal to xßl, 1 < i ^ m, where -1 < \ix < \i2 < • • • < fim, is established. These polynomials are investigated, focusing on their explicit form, Rodrigues type formula...

We study order and zero-stability of two-step methods of Obrechkoff type for ordinary differential equations. A relation between order and properties of $m$th degree polynomials orthogonal to $x^{\mu_i}, 1 \leqslant i \leqslant m$, where $-1

Let D and E be two real intervals. We consider transformations that map polynomials with zeros in D into polynomials with zeros in E. A general technique for the derivation of such transformations is presented. It is based on identifying the transformation with a parametrised distribution φ(x, μ), xϵE, μϵD, and forming the bi-orthogonal polynomial...

LetR
n/m(z∶γ)=P
n(z∶γ)/(1−γz)m
be a rational approximation to exp (z),z ∈C, of ordern for all real positiveγ. In this paper we show there exists exactly one value ofγ in each of min(n+1,m) interpolation intervals such that the uniform error overR
− is at a local minimum.

A general procedure for the construction of interpolants for Runge-Kutta (RK) formulas is presented. As illustrations, this approach is used to develop interpolants for three explicit RK formulas, including those employed in the well-known subroutines RKF45 and DVERK. A typical result is that no extra function evaluations are required to obtain an...

This paper describes some problems that are encountered in the implementation of a class of Singly Diagonally Implicit Runge-Kutta (SDIRK) methods. The contribution to the local error from the local truncation error and the residual error from the algebraic systems involved are analysed. A section describes a special interpolation formula. This is...

Several methods for the numerical solution of stiff ordinary differential equations require approximation of an exponential of a matrix. In the present paper we present a technique for estimating the error incurred in replacing a matrix exponential by a rational approximation. This estimation is done by introducing another approximation, of superio...

In this paper we discuss the automatic switch between modified Newton iteration with the Jacobian not equal zero and fix-point iteration, i.e. the Jacobian equal zero. The switching strategy is based on the ratio between the norm of the displacement and the norm of the residual. Several examples are discussed both from non-stiff and stiff systems....

The stopping criterion in the iterative solution of the non-linear equations that arise in connection with the use of implicit
methods for the solution of stiff systems of ordinary differential equations is studied. The conclusion is that there must
be a consistent choice between the type of error estimator and the vector used for the test of conve...

Systems of ordinary differential equations (ODE) or ordinary differential/algebraic equations (DAE) are well-known mathematical models. The numerical solution of such systems are discussed. For (ODE) we mention some available codes and stress the need of type insensitive versions. Further the term stiffness is redefined, and ideas on handling disco...

The question of A-acceptability in regard to derivatives of Padé approximation to the exponential, is examined for a range of values of m and n. It is proven that are A-acceptable and that numerous other choices of m and n lead to non-A-acceptability. The results seem to indicate that the A-acceptability pattern of displays an intriguing generaliza...

Dahlquist proved that under the condition of zero-stability the order of lineark-step methods is bounded by 2[(k+2)/2]. In the present paper we provide a proof of this celebrated result by using the theory of order stars.

The purpose of this report is to construct 3-stage SDIRK-methods (Singly Diagonally Implicit Runge-Kutta) to be used for the code SIMPLE. The local error control is performed by embedding techniques. Pairs with and without extrapolation are given.

Normally, the methods for solving ordinary initial value problems are viewed as discrete algorithms. However, a subclass of these schemes can be constructed by aiming at a global continuous approximation to the unknown solution. In this paper we approximate the solution by a spline of degree m and continuity k, 0 ≤ k ≤ m - 1. In each subinterval th...

Nice proofs of convergence and asymptotic expansions are known for one-step methods for ordinary differential equations. It is shown that these proofs can be generalized in a natural way to “extended” one-step methods for Volterra integral equations of the second kind. Furthermore, the convergence of “mixed” one-step methods is investigated. For bo...

Rational approximations to the exponential function are considered. Let R = P/Q, deg P = deg Q = n, R(z) = exp(z) + B(zln~,) and R(±iT) = exp(±/T) for a given posi- tive number T. We show that this approximation is /(-acceptable if and only if T belongs to one of intervals, whose endpoints are related to zeros of certain Bessel functions. The exist...

Rational approximations to the exponential function are considered. Let $R = P/Q, \deg P = \deg Q = n, R(z) = \exp(z) + \mathscr{O}(z^{2n - 1})$ and $R(\pm iT) = \exp(\pm iT)$ for a given positive number $T$. We show that this approximation is $A$-acceptable if and only if $T$ belongs to one of intervals, whose endpoints are related to zeros of cer...

The present paper develops the theory of general Runge-Kutta methods for Volterra integral equations of the second kind. The order conditions are derived by using the theory of /'-series, which for our problem reduces to the theory of K-series. These results are then applied to two special classes of Runge-Kutta methods introduced by Pouzet and by...

This paper deals with the question of the attainable order of convergence in the numerical solution of Volterra and Abel integral equations by collocation methods in certain piecewise polynomial spaces and which are based on suitable interpolatory quadrature for the resulting moment integrals. The use of a (nonlinear) variation of constants formula...

It is well known thatsome implicit Runge-Kutta methods are equivalent to collocation methods. This fact permits very short and natural proofs of order andA, B, AN, BN-stability properties for this subclass of methods (see [9] and [10]). The present paper answers the natural question, ifall RK methods can be considered as a somewhat perturbed colloc...

Based on the theory of Butcher series this paper developes the order conditions for Rosenbrock methods and its extensions to Runge-Kutta methods with exact Jacobian dependent coefficients. As an application a third order modified Rosenbrock method with local error estimate is constructed and tested on some examples.

Rational approximations to the exponential function with real poles only are studied with respect to stability at infinity and maximal order. Along each half line in the parameter space it is shown that these two properties occur in an alternating way (or do not occur at all). As an application of the general results the special approximations with...

This paper clears up to the following three conjectures:1.
The conjecture of Ehle [1] on theA-acceptability of Pad approximations toe
z
, which is true;
2.
The conjecture of Nrsett [5] on the zeros of the E-polynomial, which is false;
3.
The conjecture of Daniel and Moore [2] on the highest attainable order of certainA-stable multistep methods, wh...

In the rational Pade approximation to exp $\exp ( - q),q \in \mathbb{C}$, the parameters in the numerator and denominator are chosen to give maximum order. The zeros of the denominator of these approximations are distinct with at most one being real. When solving stiff systems, some methods have a close relation to rational approximations to $\exp...

Rational approximations of the form Σ
i=0m
a
i
q
i
/Π
i=1n
(1+γi
q) to exp(−q),qεC, are studied with respect to order and error constant. It is shown that the maximum obtainable order ism+1 and that the approximation of orderm+1 with least absolute value of the error constant has γ1=γ2=...=γn
. As an application it is shown that the order of av-sta...

Due to practical reasons one is interested inv-stage Runge-Kutta methods whose defining matrix has just one realv-fold eigenvalue. The purpose of this note is to show that methods of this type can be constructed by the method of collocation using the ratio between the zeros of certain Laguerre polynomials as collocation points.

A unique correspondence between (m, n) rational approximations to exp (q) of order at leastm and a polynomial of degreen, theC-polynomial, is obtained. This polynomial is then used to find an effective result regarding theA-acceptability of these approximations.

One-step methods of Hermite type with coefficients equal to the derivatives of Laguerre polynomials at certain points are considered. The methods areA-stable of order 1, 2, 3, 5 and for order higher than 5 they are nearlyA-stable. Used with special linear problems the matrix inversion turns out to be simple.