Higher-order time accurate numerical methods for singularly perturbed parabolic partial differential equations

This article presents two numerical methods for singularly perturbed time-dependent reaction-diffusion initial-boundary-value problems. The spatial derivative is replaced by a hybrid scheme, which is a combination of the cubic spline and the classical central difference scheme in both the methods. In the first method, the time derivative is replaced by the Crank-Nicolson scheme, whereas in the second method the time derivative is replaced by the extended-trapezoidal scheme. These schemes are applied on the layer resolving piecewise-uniform Shishkin mesh. Some numerical examples are carried out to show the accuracy and efficiency of these methods.