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A time dependent singularly perturbed differential-difference equation is considered. The problem involves time delay and general small space shift terms. Taylor series approximation is used to expand the space shift term. A robust numerical scheme based on the backward Euler scheme for the time and classical upwind scheme for space is proposed. The convergence analysis is carried out. It is observed that the proposed scheme converges almost first order up to a logarithm term and optimal first order in space on the Shishkin and Bakhvalov–Shishkin mesh, respectively. Numerical results confirm the efficiency of the proposed scheme, which are in agreement with the theoretical bounds.

Purpose The purpose of this study is to provide a robust numerical method for a two parameter singularly perturbed delay parabolic initial boundary value problem (IBVP). Design/methodology/approach To solve the problem, the authors have used a hybrid scheme combining the midpoint scheme, the upwind scheme and the second-order central difference scheme for the spatial derivatives. The backward Euler scheme on a uniform mesh is used to approximate the time derivative. Here, the authors have used Shishkin type meshes for spatial discretization. Findings It is observed that the proposed method converges uniformly with almost second-order spatial accuracy with respect to the discrete maximum norm. Originality/value This paper deals with the numerical study of a two parameter singularly perturbed delay parabolic IBVP. To solve the problem, the authors have used a hybrid scheme combining the midpoint scheme, the upwind scheme and the second-order central difference scheme for the spatial derivatives. The backward Euler scheme on a uniform mesh is used to approximate the time derivative. The convergence analysis is carried out. It is observed that the proposed method converges uniformly with almost second-order spatial accuracy with respect to the discrete maximum norm. Numerical experiments illustrate the efficiency of the proposed scheme.

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