On the θ-method for delay differential equations with infinite lag

This paper discusses the θ-method for the numerical solution of delay differential equations with infinite lag. Our analysis is based on the test equation y′(t)=ay(λt) + by(t), where a,b ∈ C, and λ∈(0, 1). In order to solve the storage problem involved in studying the long time behaviour of the solution we use a grid with increased stepsizes. Monotonicity, uniform boundedness, asymptotic stability, algebraic decay and global discretisation error of the numerical solution are investigated.