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.