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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.

A Higher-Order Numerical Method on the Shishkin Mesh for Time-Dependent Problems with Boundary Layers