Local error analysis of Evans–Sanugi, nonlinear one-step methods based on θ-means

Abstract

A new class of weighted, homogeneous, differentiable means is introduced, referred to as θ-means, which may be extended to the whole plane. These means are applied to Evans–Sanugi nonlinear one-step methods for initial value problems of scalar differential equations. Their general local truncation error is obtained, showing the first order of these methods for θ≠1/2 and second order for θ=1/2. Numerical results for scalar DETEST problems using Arithmetic, Harmonic, Contraharmonic, Quadratic, Geometric, Heronian, Centroidal and Logarithmic θ-means are presented. Both the local error of the methods and the global error of the numerical results have the same functional dependence with the parameters of the numerical method. The results show that a comparison of different methods for scalar problems may be based on the numerical evaluation of local truncation error.

Keywords:

• local error
• truncation error terms
• nonlinear one-step methods
• classical means

2000 AMS Subject Classification::

• 65L06
• 65L99
• 34L30
• 26E60

Acknowledgements

The research reported in this paper was supported by Project FIS2005-03191 from the Ministerio de Educación y Ciencia, Spain.