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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.

##### MSC:
 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