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

Abstract

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.

Keywords:

• singular perturbed parabolic problem
• cubic spline
• piecewise-uniform Shishkin mesh
• Crank–Nicolson scheme
• extended-trapezoidal scheme

2000 AMS Subject Classification :

• 65M06
• 65M12

CCS Category :

• G1.8

Acknowledgements

This work is supported by the Department of Science and Technology, Government of India under research grant SR/S4/MS:318/06.