Almost second-order uniformly convergent numerical method for singularly perturbed convection–diffusion–reaction equations with delay

Abstract

This paper deals with the numerical treatment of time-dependent singularly perturbed convection–diffusion–reaction equations with delay. In the considered equations, the highest order derivative term is multiplied by a perturbation parameter, ε taking arbitrary value in the interval $(0,1]$. For small ε, the solution of the equations exhibits a boundary layer on the right side of the spatial domain. The considered equations contain a small delay on the convection and reaction terms of the spatial variable. The terms involving the delay are approximated using the Taylor series approximation. The resulting singularly perturbed parabolic convection–diffusion–reaction equations are treated using the Crank Nicolson method in time derivative discretization and the mid-point upwind finite difference method on piecewise uniform Shishkin mesh for the space variable derivative discretization. The stability and the uniform convergence of the scheme are investigated well. The proposed scheme gives almost first-order convergence in the spatial direction. The Richardson extrapolation technique is applied to accelerate the rate of convergence of the scheme in the spatial direction to order almost two. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter and delay parameter.

Keywords:

• Mid-point upwind
• Shishkin mesh
• singularly perturbed
• uniformly convergent

2010 Mathematics Subject Classifications:

• 65M06
• 65M12
• 65M15

Acknowledgments

The authors acknowledge the referees for their constructive comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

No funding organization for this research work.