Stability by order stars for non-linear theta-methods based on means

Abstract

The linear stability analysis of non-linear one-step methods based on means is studied by means of the concept of stability regions and order stars. Concretely, non-linear θ-methods based on harmonic, contraharmonic, quadratic, geometric, Heronian, centroidal and logarithmic means are considered. Their stability diagrams and order stars show their A-stability for θ≥1/2, and L-stability in some cases. Order stars in the Riemann surface are a requirement for non-linear one-step methods. The advantages and disadvantages of this technique are presented.

Keywords:

• stability regions
• order stars
• non-linear multistep methods
• non-linear one-step methods
• classical means

2000 AMS Subject Classification :

• 65L20
• 65L12
• 65L05
• 65M12
• 30F99

Acknowledgements

This author would like to thank an anonymous reviewer of a previous version of this paper and Dr. Juan I. Ramos for the helpful suggestions which have improved the paper. The research reported in this paper was supported by project FIS2005-03191 from the Ministerio de Educación y Ciencia, Spain.