High-accuracy cubic spline alternating group explicit methods for 1D quasi-linear parabolic equations†

This research work is dedicated to Late Prof. D.J. Evans

Abstract

In this article, we study the application of the alternating group explicit (AGE) and Newton-AGE iterative methods to a two-level implicit cubic spline formula of O(k ²+kh ²+h ⁴) for the solution of 1D quasi-linear parabolic equation u _xx =φ (x, t, u, u _x , u _t ), 0<x<1, t>0 subject to appropriate initial and natural boundary conditions prescribed, where k>0 and h>0 are mesh sizes in t- and x-directions, respectively. The proposed cubic spline methods require 3-spatial grid points and are applicable to problems in both rectangular and polar coordinates. The convergence analysis at advanced time level is briefly discussed. The proposed methods are then compared with the corresponding successive over relaxation (SOR) and Newton-SOR iterative methods both in terms of accuracy and performance.

This research work is dedicated to Late Prof. D.J. Evans

Keywords:

• quasi-linear parabolic equation
• cubic spline method
• AGE method
• Newton-AGE method
• singular equation

2000 AMS Subject Classification :

• 65M06

Acknowledgements

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

Notes

This research work is dedicated to Late Prof. D.J. Evans

Additional information

Notes on contributors

M. K. Jain

Present address: 4076, C/4, Vasant Kunj, New Delhi-110 070, India