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Stability of Runge–Kutta methods in the numerical solution of equation u ' (t)=au(t)+a 0 u([t]). (English) Zbl 1052.65070

This paper deals with the numerical solution of the delay differential equations with piecewise continuous arguments u ' (t)=f(t,u(t),u(α(t))). The stability of the Runge-Kutta methods is analyzed regarding stability regions. Conditions that the analytic stability region is contained in the numerical stability regions are obtained. Numerical results of the following problem

u 1 ' (t)=-20u 1 (t)-10·3u 1 ([t]),u 1 (0)=1u 2 ' (t)=10u 2 (t)-10·001u 2 ([t]),u 2 (0)=1

are also given.

65L20Stability and convergence of numerical methods for ODE
34K28Numerical approximation of solutions of functional-differential equations
65L05Initial value problems for ODE (numerical methods)
65L06Multistep, Runge-Kutta, and extrapolation methods