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Strong contractivity properties of numerical methods for ordinary and delay differential equations. (English) Zbl 0749.65042
Some new stability and contractivity concepts are introduced for ordinary differential equations for linear and nonlinear test problems. The linear problem is the standard constant coefficient test problem with the addition of a forcing term. The motivation for these test problems is to study stability properties of methods for delay differential equations.
Characterizations of these concepts are provided for the case of Runge- Kutta methods and some relationships are established among various concepts of stability for ordinary and delay differential equations. Runge-Kutta methods with upto two stages are studied in detail and the Lobatto III-C method is shown to be the only two-stage formula of order 2 satisfying certain stability properties.

MSC:
65L05 Numerical methods for initial value problems
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
34K05 General theory of functional-differential equations
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