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

This paper discusses the theta-method for the numerical solution of delay differential equations with infinite lag. Our analysis is based on the test equation y'(t)=ay(lambda t)+by(t), where a,b is an element of C, and lambda is an element of(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.