New approach of convergent numerical method for singularly perturbed delay parabolic convection-diffusion problems

• Published in: Research in mathematics (Philadelphia, Pa.) Vol. 10; no. 1
• Main Authors: Hassen, Zerihun Ibrahim, Duressa, Gemechis File
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis Ltd 31.12.2023
• Subjects: Approximation; Asymptotic series; Convection-diffusion equation; Crank-Nicholson method; Differential equations; Exact solutions; Finite difference method; Mathematical analysis; Mathematics; Methods; Numerical analysis; Numerical methods; Operators (mathematics); Parameters; Partial differential equations; Singular perturbation methods
• ISSN: 2768-4830; 2768-4830
• DOI: 10.1080/27684830.2023.2225267

In this paper, a parameter-uniform convergent numerical scheme is provided for solving singularly perturbed parabolic convection-diffusion differential equations with a large delay. A priori bounds on the exact solution and its derivatives derived by asymptotic analysis of the problem are provided.. The problem is discretized by the Crank-Nicolson method on uniform mesh in time direction and a fitted operator upwind finite difference method on uniform mesh in spatial direction. The fitting factor is derived from the zeroth-order asymptotic expansion of the exact solution and then introduced for the term containing the singular perturbation parameter. The convergence analysis is given for the proposed numerical method using the barrier function approach and the Peano kernel. We proved the scheme is uniformly convergent with first -order in space and second order in time, both independent of the perturbation parameter. Numerical experiments are presented to support the theoretical findings.