Faster Convergence of Extrapolation Technique for System of Singularly Perturbed Time‐Delay Parabolic PDEs

ABSTRACT In this article, we investigate extrapolation based higher‐order numerical approximation to a system of coupled singularly perturbed convection–diffusion time‐delay parabolic partial differential equations (PDEs) with multiple perturbation parameters. In practice, such systems often model multi‐species population dynamics in complex ecological systems, where spatial diffusion captures heterogeneous population distributions, and time delays arise from biological processes such as maturation, gestation, or digestion. The solutions of these problems exhibit overlapping boundary layers, when the diffusion coefficients are sufficiently small, posing significant challenge to solve numerically. To address this, the governing system of PDEs is discretized by an implicit‐Euler method (IE) in the temporal direction and the upwind scheme in the spatial direction on an appropriate generalized S‐mesh, denoted by S()‐mesh. The main goal of this paper is two‐fold. First, to derive a priori bounds of the analytical solution and its derivatives, and pursue the convergence analysis of the proposed numerical method on S()‐mesh. Second, to analyze the Richardson extrapolation technique to accelerate the rate of convergence of the method both in the spatial and the temporal directions, yielding faster higher‐order rate of convergence on S()‐mesh than the usual Shishkin mesh for . Due to the presence of time‐delay term, error analysis is carried out by partitioning the domain into different time segments. Finally, the theoretical estimates before and after the extrapolation are corroborated in practice with numerical results.