Research Article Open Access

A Non-Linear Absolutely-Stable Explicit Numerical Integration Algorithm for Stiff Initial-Value Problems

Essam R. El-Zahar1
  • 1 Department of Mathematics, College of Sciences and Humanities, Salman Bin Abdulaziz University, P.O. Box 83, Alkharj, 11942, Saudi Arabia

Abstract

The time-step in integration process has two restrictions. The first one is the time step restriction due to accuracy requirement τac and the second one is the time-step restriction due to stability requirement τst. The most of explicit methods have small stability regions and consequently small τst. It obliges us to solve stiff problems with small step size τst << τac. The implicit methods work well with stiff problems but these methods require more work per step than the explicit methods. In this study, a non-linear absolutly stable explicit one step numerical integration algorithm is proposed for solving non linear stiff initial-value problems in ordinary differential equations. The algorithm is based on deriving a non-linear relation between the dependent variable and its derivatives from the well known Taylor expansion. The accuracy of the method depends on some unknown parameter inserted in Taylor expansion and determined from the error analysis. The accuracy and stability properties of the method are investigated and shown to yield at least third-order and A-stable. The results obtained in the numerical experiments show the efficiency of the present method in solving stiff initial value problems.

American Journal of Applied Sciences
Volume 10 No. 11, 2013, 1363-1370

DOI: https://doi.org/10.3844/ajassp.2013.1363.1370

Submitted On: 7 June 2012 Published On: 11 September 2013

How to Cite: El-Zahar, E. R. (2013). A Non-Linear Absolutely-Stable Explicit Numerical Integration Algorithm for Stiff Initial-Value Problems. American Journal of Applied Sciences, 10(11), 1363-1370. https://doi.org/10.3844/ajassp.2013.1363.1370

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Keywords

  • Initial-Value Problems
  • Non-Linear Methods
  • Explicit Methods
  • A-Stability
  • Stiff Problems