Numerical Solutions for Turning Point Convection-Diffusion Problems with Time Delay on Modified Graded Meshes

Abstract

This paper investigates the numerical solutions of convection-diffusion problems with turning points and time delays, using modified graded meshes. Turning point problems, characterized by sharp gradients in solutions, possess significant challenges to numerical methods, particularly in terms of stability and convergence. To address these challenges, we use upwind difference scheme on a modified graded mesh to handle the steep gradients at turning points as well as stability and convergence more effectively. Numerical experiments are conducted to demonstrate the efficiency of the modified graded meshes. Results shows significant improvements and accuracy in the solution, especially near turning points. The paper concludes with a discussion on the implications of our findings.