Parameter robust higher‐order finite difference method for convection‐diffusion problem with time delay

Abstract

This paper deals with the study of a higher‐order numerical approximation for a class of singularly perturbed convection‐diffusion problems with time delay. The method combines a higher‐Order Difference with an Identity Expansion (HODIE) scheme over a piece‐wise uniform mesh in the spatial direction and the backward Euler method on a uniform mesh for discretization in the temporal direction. A priori bounds for the continuous solution and its derivatives are derived by splitting the solution into regular and singular components. These bounds are useful in the error analysis of the proposed scheme. The present scheme converges ε$$ \varepsilon $$ ‐uniformly with the order of convergence one in time and almost second‐order in space direction. Further, to increase the rate of convergence in the time variable, we implemented the Richardson extrapolation technique. Thus, finally, the resultant scheme with Richardson extrapolation in time becomes almost second‐order ε$$ \varepsilon $$ ‐uniformly convergent in both the space and time variable. The detailed stability and convergence analysis have been done using the derived a priori estimates. We consider three test problems to validate the predicted theory and show that numerical results are in good agreement with our theoretical findings.