Analysis of a numerical technique for a singularly perturbed parabolic convection–diffusion interface problem with time delay

Abstract

In this article, we are interested in the numerical analysis of a singularly perturbed parabolic differential equation with time delay. The source term of the considered problem has discontinuities in the spatial variable along the interface ${\mathrm{\Gamma }}_d:=\{(d,t):t\in (0,\mathcal{T}]\}$, 0<d<1, and the diffusion coefficient is a small positive perturbation parameter. The problem's solution exhibits interior and boundary layers as the perturbation parameter approaches zero, which makes the problem challenging to establish the uniform convergence of any applied classical numerical techniques with respect to the perturbation parameter. The solution to the considered problem is decomposed into regular and singular components. Some appropriate a priori bounds on derivatives of these components have been given. The domain is discretized using Shishkin mesh in the spatial direction and uniform mesh in the time direction. On the mesh points which are not on the interface, the problem is discretized using a central-difference upwind scheme. Along the interface, the problem is discretized using an especial central-difference upwind scheme that uses the average value of the source term. The ε-uniform convergence analysis of the scheme is given using the decomposition of the solution. Some numerical experiments are conducted to corroborate the efficiency of the method.

Keywords:

• Singularly perturbed
• delay differential equations
• parabolic problems
• convection–diffusion
• interior and boundary layers
• ε-uniform convergence

Mathematics Subject Classifications:

• 65M06
• 65M12

Acknowledgments

The authors gratefully acknowledge the valuable comments and suggestions of the anonymous reviewers. The authors also acknowledge the IIT Delhi HPC facility for computational resources.

Data availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research work is supported by the Science and Engineering Research Board (SERB) under Project No. MTR/2019/000614.