Unconditionally stable difference methods for delay partial differential equations

This paper is concerned with the numerical solution of parabolic partial differential equations with time-delay. We focus in particular on the delay dependent stability analysis of difference methods that use a non-constrained mesh, i.e., the time step-size is not required to be a submultiple of the delay. We prove that the fully discrete system unconditionally preserves the delay dependent asymptotic stability of the linear test problem under consideration, when the following discretization is used: a variant of the classical second-order central differences to approximate the diffusion operator, a linear interpolation to approximate the delay argument, and, finally, the trapezoidal rule or the second-order backward differentiation formula to discretize the time derivative. We end the paper with some numerical experiments that confirm the theoretical results.