A Hybrid Fitted Mesh Numerical Scheme for Solving Singularly Perturbed Reaction–Diffusion Equation With Mixed Shifts

This study develops a robust hybrid numerical scheme for a class of time-dependent, singularly perturbed differential-difference equations characterized by a small positive parameter multiplying the highest order derivative term and mixed spatial shifts (delay and advance) in the reaction terms. Such equations arise in various scientific computing and engineering model problems, whose solutions often exhibit abruptly changing twin boundary layers which are a challenging behavior to find analytical solutions. However, standard numerical approaches are not adequate to treat these problems as they do not consider the behaviors of the boundary layers. To address these difficulties, we propose a fitted mesh scheme that integrates the Crank–Nicolson method for temporal discretization with a hybrid spatial scheme, combining midpoint upwind and cubic spline finite difference methods on piecewise-uniform Shishkin meshes. The mesh is specifically constructed to align with the shift parameters, ensuring high-order accuracy within the layer regions. Theoretical analysis confirms that the proposed scheme is parameter-uniformly convergent. Numerical experiments validate the theoretical findings, demonstrating second-order accuracy in time and almost second-order accuracy in space subject to a logarithmic factor.