Compact θ-method for the generalized delay diffusion equation.

Abstract 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 ) Δ t h 2 10 − cos ( h ) 12 ( 1 + cos ( h ) ) ( 1 − 2 θ ) for θ ∈ [ 0 , 1 2 ) and is unconditionally asymptotically stable for θ ∈ [ 1 2 , 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.