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In this paper the convergence of a waveform relaxation method applied to an initial value problem for the Volterra functional-differential system is discussed. It is shown that the method is convergent under the assumption that the splitting function satisfies only the one side Lipschitz condition with respect to some arguments and the Lipschitz condition with respect to the others. The conditions given in the paper also guarantee the existence and uniqueness of the solution to the initial problem discussed in the paper. The convergence of the perturbed continuous time waveform relaxation method is also discussed. ᮊ

In this paper the problem of delay-dependent error estimates for waveform relaxation methods applied to systems of delay-differential equations is discussed. Under suitable conditions imposed on the so-called splitting function it is shown how the error estimates depend on the character of delays and how much these estimates are better than the known error estimates for relaxation methods. We attempt to derive the error estimates as sharp as possible under the assumed conditions. Our approach takes into account the specific properties of the considered equations. It is also proved that under some assumptions the exact solution can be obtained after a finite number of steps of the iterative process; i.e., we prove that the waveform relaxation methods have the same property as the classical method of steps for solving delay-differential equations with nonvanishing delays. From the given estimates the number of these steps can be found.

Various issues related to the development of a new code for nonstiff differential equations are discussed. This code is based on two-step Runge-Kutta methods of order five and stage order five. Numerical experiments are presented which demonstrate that the new code is competitive with the Matlab ode45 program for all tolerances.