Numerical treatment of oscillatory functional differential equations

In this paper we are concerned with oscillatory   functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear ϑϑ-methods will also be oscillatory. We conclude with some general theory.