Stability and convergence of compact finite difference method for parabolic problems with delay

The compact finite difference method becomes more acceptable to approximate the diffusion operator than the central finite difference method since it gives a better convergence result in spatial direction without increasing the computational cost. In this paper, we apply the compact finite difference method and the linear θ-method to numerically solve a class of parabolic problems with delay. Stability of the fully discrete numerical scheme is investigated by using the spectral radius condition. When θ∈[0,12), a sufficient and necessary condition is presented to show that the fully discrete numerical scheme is stable. When θ∈[12,1], the fully discrete numerical method is proved to be unconditionally asymptotically stable. Moreover, convergence of the fully discrete scheme is studied. Finally, several numerical examples are presented to illustrate our theoretical results.