Stability of θ-methods for delay integro-differential equations

Stability of θ-methods for delay integro-differential equations (DIDEs) is studied on the basis of the linear equationdudt=λu(t)+μu(t−τ)+κ∫t−τtu(σ) dσ,where λ,μ,κ are complex numbers and τ is a constant delay. It is shown that every A-stable θ-method possesses a similar stability property to P-stability, i.e., the method preserves the delay-independent stability of the exact solution under the condition that κ is real and τ/h is an integer, where h is a step-size. It is also shown that the method does not possess the same property if τ/h is not an integer. As a result, no θ-method can possess a similar stability property to GP-stability with respect to DIDEs.