S. Nørsett - Academia.edu (original) (raw)

Papers by S. Nørsett

Journal of Pseudo-Differential Operators and Applications, 2015

Journal of Computational and Applied Mathematics, 1987

BIT Numerical Mathematics, 1996

ACM Transactions on Mathematical Software, 1986

A general procedure for the construction of interpolants for Runge-Kutta (RK) formulas is present... more 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 O ( h 5 ) 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 O ( h 6 ) local truncation error for this RK formula.

Applied Numerical Mathematics, 1994

SIAM Journal on Numerical Analysis, 1995

Applied Mathematics and Computation, 1988

On quadrature methods for highly oscillatory integrals and their implementations by

The method of Magnus series has recently been analysed by Iserles & Nørsett (1997). It approximat... more The method of Magnus series has recently been analysed by Iserles & Nørsett (1997). It approximates the solution of linear differential equations y 0 = a(t)y in the form y#t#=e ##t# , 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 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 structur...

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2013

The concern of this paper is in expanding and computing initial-value problems of the form y ′= f... more The concern of this paper is in expanding and computing initial-value problems of the form y ′= f ( y )+ h ω ( t ), where the function h ω 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 numerical solvers. However, there is no similar theory for more general oscillators, and there are sound reasons to believe that approximations of this kind are unsuitable in that setting. We follow in this paper an alternative route, demonstrating that, for a much more general family of oscillators, e.g. linear combinations of functions of the form e i ωg k ( t ) , it is possible to expand y ( t ) in a different manner. Each r th term in the expansion is for some ς >0 and it can be represented as an r -dimensional highly oscillatory integral. Because computation of multivariate highly oscillatory integrals is fairly well understoo...

So far ODE-solvers have been implemented mostly on sequential computers. This has lead to develop... more 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 initial experiments on a 2 processor CRAY X-MP and a 6 processor Alliant are presented.

Journal of Approximation Theory, 1985

ABSTRACT The question of A-acceptability in regard to derivatives of Padé approximation to the ex... more ABSTRACT 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 generalization of the Wanner-Hairer-Nørsett theorem on the A-acceptability of .

Algorithms for Approximation II, 1990

Numerical Mathematics and Advanced Applications, 2006

Contributions in Numerical Mathematics, 1993

ABSTRACT

Lecture Notes in Mathematics, 1985

ABSTRACT Given a monotone measure α(x), a positive function ω(x,μ), μεΩ and a sequence μ1,μ2 ,...... more ABSTRACT 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 \leqslant m, p_m \in \pi _m \left[ x \right]. Questions of existence, uniqueness, location of zeros and existence of Rodrigues-type formulae are investigated. Polynomials of this type arise in numerical analysis of two-step multistage methods for ordinary differential equations.