layer-resolving mesh method for convection-diffusion delay problems with boundary turning points

This paper introduces a numerical scheme designed to solve time-delay singularly perturbed parabolic convection-diffusion problems with turning points. a small parameter induces boundary layers, making standard methods ineffective. to tackle these challenges, we developed a layer-resolving numerical scheme using the crank-nicolson method (time) and a central finite difference method on a shishkin mesh (space). the stability and parameter-uniform convergence analysis show that the error decreases quadratically. numerical results demonstrate higher accuracy than existing approaches.