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.

本文涉及具有时滞的瞬态奇异摄动对流-扩散-反应方程的数值处理。在所考虑的方程中，最高阶导数项乘以一个扰动参数，ε取区间内的任意值$(0,\mathrm{1个}]$. 对于小ε，方程的解在空间域的右侧显示出边界层。所考虑的方程包含空间变量的对流和反应项的小延迟。涉及延迟的项使用泰勒级数近似来近似。得到的奇摄动抛物线对流-扩散-反应方程在时间导数离散化中使用 Crank Nicolson 方法处理，在空间变量导数离散化中使用分段均匀 Shishkin 网格的中点逆风有限差分法处理。对该方案的稳定性和一致收敛性进行了很好的研究。所提出的方案在空间方向上给出了几乎一阶收敛。应用Richardson外推技术使格式在空间方向的收敛速度几乎达到二阶。为了验证该方案的适用性，针对不同的扰动参数和延迟参数值给出并求解了数值示例。