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The stability analysis of the \(\theta\)-methods for delay differential equations. (English) Zbl 0857.65081
This paper analyses the behaviour (in terms of monotonicity, uniform boundedness, asymptotic stability, algebraic decay and global discretization errors) of the \(\theta\)-method when applied to a linear delay differential equation with infinite lag. In particular, this paper resolves some of the gaps in earlier work of M. D. Buhmann and A. Iserles [IMA J. Numer. Anal. 12, No. 3, 339-363 (1992; Zbl 0759.65056)] in terms of their wrong prediction of the long term behaviour of numerical methods due to insufficient computer memory. This is done by constructing a stepsize grid in which stepsizes increase geometrically after an initial stage. The ensuring recurrence relation is now of fixed order but has variable coefficients.

65L20 Stability and convergence of numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34K05 General theory of functional-differential equations
65L70 Error bounds for numerical methods for ordinary differential equations