Syvert P. Nørsett

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.

Order stars and stability theorems

AbstractThis paper clears up to the following three conjectures:1.The conjecture of Ehle [1] on theA-acceptability of Padé approximations toez, which is true;2.The conjecture of Nørsett [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, which is true, generalizing the well-known Theorem of Dahlquist. We further give necessary as well as sufficient conditions forA-stable (acceptable) rational approximations, bounds for the highest order of “restricted” Padé approximations and prove the non-existence ofA-acceptable restricted Padé approximations of order greater than 6.The method of proof, just looking at “order stars” and counting their “fingers”, is very natural and geometric and never uses very complicated formulas.

Interpolants for Runge-Kutta formulas

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 interpolant with <italic>O</italic>(<italic>h</italic><supscrpt>5</supscrpt>) local truncation error for the fifth-order RK formula used in RKF45; two extra function evaluations per step are required to obtain an interpolant with <italic>O</italic>(<italic>h</italic><supscrpt>6</supscrpt>) local truncation error for this RK formula.

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.

C-Polynomials for rational approximation to the exponential function

SummaryA 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.

The potential for parallelism in Runge-Kutta methods. Part 1: RK formulas in standard form

The authors 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 an RK formula in terms of the minimal polynomial associated with its coefficient matrix. The positive results are largely examples of prototypical formulas that offer a potential for effective “coarse-grain” parallelism on machines with a few processors.

Attainable order of rational approximations to the exponential function with only real poles

Rational approximations of the form Σi=0maiqi/Πi=1n (1+γiq) 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-stage semi-implicit Runge-Kutta method cannot exceedv+1.

Quadrature methods for multivariate highly oscillatory integrals using derivatives

While there exist effective methods for univariate highly oscillatory quadrature, 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 demonstrate that, in the absence of critical points and subject to a nonresonance condition, an integral over a simplex can be expanded asymptotically using only function values and derivatives at the vertices, a direct counterpart of the univariate case. This provides a convenient avenue towards the generalization of asymptotic and Filon-type methods, as formerly introduced by the authors in a single dimension, to simplices and, more generally, to polytopes. The nonresonance condition is bound to be violated once the boundary of the domain of integration is smooth: in effect, its violation is equivalent to the presence of stationary points in a single dimension. We further explore this issue and propose a technique that often can be used in this situation. Yet, much remains to be done to understand more comprehensively the influence of resonance on the asymptotics of highly oscillatory integrals.

Order conditions for Rosenbrock type methods

SummaryBased 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.