Compact θ-method for the generalized delay diffusion equation

The generalized diffusion equation with a delay has inherent complex nature because its analytical solutions are hard to obtain. Therefore, one has to seek numerical methods, especially the high-order accurate ones, for their approximate solutions. In this paper, we have established the results of the numerical asymptotic stability of the compact θ-method for the generalized delay diffusion equation. It shows that the compact θ-method is asymptotically stable if and only if (k+r)Δth2<10−cos(h)12(1+cos(h))(1−2θ) for θ∈[0,12) and is unconditionally asymptotically stable for θ∈[12,1], respectively. The convergent results in the maximum norm are studied according to the consistency analysis and Lax theorem. In the end, a series of numerical tests on stability and convergence are carried out to support our theoretical results.