Stability of θ-methods for delay integro-differential equations

Stability of theta-methods for delay integro-differential equations (DIDEs) is studied on the basis of the linear equation du/dt = lambdau(t) + muu(t - tau) + kappa integral(l-tau)(t) u(sigma)dsigma, where, lambda, mu, kappa are complex numbers and tau is a constant delay. It is shown that every A-stable theta-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 kappa is real and tau/h is an integer, where h is a step-size. It is also shown that the method does not possess the same property if tau/h is not an integer. As a result, no theta-method can possess a similar stability property to GP-stability with respect to DIDEs. (C) 2003 Elsevier B.V. All rights reserved.