Abstract—In this paper, a higher-order numerical method for time-dependent singularly perturbed problems is constructed on the Shishkin mesh. The method consists of Crank-Nicolson method for the time discretization and a hybrid difference scheme that combines the midpoint upwind difference scheme on the coarse mesh and the central difference scheme on the fine mesh for the spatial discretization. We prove that the method is uniformly convergent with respect to the singular perturbation parameter, having order near two in space and order two in time. Finally, numerical results support the convergence behavior.

Index Terms—Time-dependent problems, Crank-Nicolson method, Hybrid finite difference method, Shishkin mesh, uniform higher-order error estimate.

The authors are with North China University of Technology, Beijing, China (email: zhengq@ncut.edu.cn).

Cite: Quan Zheng, Ke Jin, "A Higher-Order Numerical Method on the Shishkin Mesh for Time-Dependent Problems with Boundary Layers," International Journal of Applied Physics and Mathematics vol. 10, no. 1, pp. 1-7, 2020.

Copyright © 2020 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).