Contractivity of continuous Runge-Kutta methods for delay differential equations
ABSTRACT In this paper the author investigates the stability of numerical methods for general delay differential equations of the type where α(t) ≤ t and y(t) is a vector complex-valued function. Contractivity conditions are found for Runge-Kutta methods as applied to linear and nonlinear scalar equations. As for systems, a general condition is found for the contractivity of the solution of (1) in any vector norm, and a numerical method is proposed which preserves the contractivity in the maximum norm.
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ABSTRACT: The present paper pursues the study of contractivity properties of Runge-Kutta methods for ODEs with respect to forcing terms which has been started in the papers by Torelli (1991) and Bellen and Zennaro (1992). Whereas these two papers addressed only the case of implicit methods for the discussion of optimal stability properties, here also explicit methods are considered and the regions of stability are introduced and investigated. Some of the problems opened in Bellen and Zennaro (1992) are partially settled and many examples of methods up to order p = 4 are discussed.Applied Numerical Mathematics. 01/1993;
- Siam Journal on Numerical Analysis - SIAM J NUMER ANAL. 01/1994; 31(2).