The objective of this paper is to construct and analyze a fitted operator finite difference method (FOFDM) for the family of time-dependent singularly perturbed parabolic convection-diffusion problems. The solution to the problems we consider exhibits an interior layer due to the presence of a turning point. We first establish sharp bounds on the solution and its derivatives. Then, we discretize the time variable using the classical Euler method. This results in a system of singularly perturbed interior layer two-point boundary value problems. We propose a FOFDM to solve the system above. Through a rigorous error analysis, we show that the scheme is uniformly convergent of order one with respect to both time and space variables. Moreover, we apply Richardson extrapolation to enhance the accuracy and the order of convergence of the proposed scheme. Numerical investigations are carried out to demonstrate the efficacy and robustness of the scheme. KEYWORDS error bounds, finite difference methods, interior layer, singularly perturbed problems, uniform convergence 1 Numer Methods Partial Differential Eq. 2019;35:2407-2422. wileyonlinelibrary.com/journal/num

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