A new algorithm based on spline in tension approximation for 1D quasi-linear parabolic equations on a variable mesh

Abstract

In this paper, we propose a new two-level implicit method of order two in time and four in space directions, based on spline in tension approximation for the numerical solution of one space dimensional quasi-linear parabolic partial differential equation on a uniform mesh. We have discussed the derivation of the proposed method in detail and have also discussed the stability analysis for a model problem. We have extended the method to non-uniform mesh. Numerical results are given to illustrate the usefulness of the proposed methods.

Keywords:

• quasi-linear parabolic equations
• spline in tension
• non-polynomial spline
• stability analysis
• variable mesh
• Burgers’ equation

2010 AMS Subject Classifications:

• 65M06
• 65M12

Acknowledgements

The authors thank the reviewers for their valuable suggestions, which substantially improved the standard of the paper.

Disclosure Statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by ‘The Council of Scientific and Industrial Research’ under research grant no. 09/045(0836)2009-EMR-I.