Extended One‐Step Methods: An Exponential Fitting Approach

Abstract

Exponentially fitted versions of the extended one‐step methods for first‐order ODEs are constructed following the six‐step flow chart introduced by Ixaru and Vanden Berghe in Exponential fitting, (Mathematics and Its Application 568, Kluwer Academic Publishers, 2004) and their properties are examined. It is shown how formulae for optimal frequencies can be constructed. Some numerical experiments illustrate the use of the developed algorithms. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Exponentially Fitted Error Correction Methods for Solving Initial Value Problems

“…It is well known that the A-stability of numerical methods is a key issue for stiff systems because one can choose the step size based only on accuracy without worrying about its stability constraint. Using higher-order derivatives, the idea of exponentially fitted methods (EFM), which was originally proposed by Liniger and Willoughby [9], has received considerable attention [1,5,6,8,13]. The basic idea of exponential fitting is to derive integration formulae containing free parameters and then to choose these parameters so that the given exponential function satisfies the integration formula exactly.…”

A Symplectic Trigonometrically Fitted Modified Partitioned Runge-Kutta Method for the Numerical Integration of Orbital Problems

“…The numerical integration of Hamiltonian systems by symplectic methods has been considered by many authors (Hairer [1], McLachlan [3], Ruth [6], Sanz-Serna [7], Yoshida [14]). On the other hand Hamiltonian systems have an oscillatory behaviour and have been solved in the literature with exponentially and trigonomatrically fitted methods (see Ixaru and Vanden Berghe [2],Vanden Berghe [12], Simos [8], [9], Psihoyios [5], [4],Van Daele M. [11] ). In this work we construct a symplecric modified partitioned Runge-Kutta method of second order with the trigonometrically fitted property.…”