Since the works of Barwell (1975)on the P-stability, the stability analysis in presence of delays was mainly meant in the delay-independent sense. Results on both asymptotic stability and contractivity with respect to initial functions were obtained for the class of test equations satisfying the wanted stability property "for all delays". In this sense, carrying out the stability analysis for the classes of test equation as well as for the class of Runge-Kutta methods is much easier and the theory has been satisfactorily developed by Zennaro, i 'nt Hout, Torelli, Koto, Mitsui et al. On the contrary, determining the stability region for the classes of test equation in terms of the given "fixed delay", and finding stable numerical methods is much harder. Recent results are due to Guglielmi (1998) and Guglielmi-Hairer (1999) for linear scalar equations with real coefficients and by Maset (1999) for the complex case. Despite a satisfactory description of the asymptotic stability region is not available yet for systems, Maset was able to prove the unexpected and disappointing result that in this case no Runge-Kutta method can be asymptotically stable.

MINISIMPOSIUM SESSION: 3. Numerical methods for delay equations (2 sessions)