(untitled) (Journal of Computational Mathematics, 2026)
This paper deals with the numerical solutions of two-dimensional (2D) semi-linear reaction-diffusion equations (SLRDEs) with piecewise continuous argument (PCA) in reaction term. A high-order compact difference method called I-type basic scheme is developed for solving the equations and it is proved under the suitable conditions that this method has the computational accuracy O(τ^2+h^4_x+h^4_y), where τ, h_x and h_y are the calculation stepsizes of the method in t-, x- and y-direction, respectively. With the above method and Newton linearized technique, a II-type basic scheme is also suggested. Based on the both basic schemes, the corresponding I- and II-type alternating direction implicit (ADI) schemes are derived. Finally, with a series of numerical experiments, the computational accuracy and efficiency of the four numerical schemes are further illustrated.