An advanced numerical approach for solving stiff initial value problems using a self-starting two-stage composite block scheme

Document Type : Research Paper

Authors

1 1. Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia.\\ 2. Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA Cawangan Sarawak, Kampus Samarahan 2, 94300 Kota Samarahan, Sarawak, Malaysia.

2 Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia.

Abstract

In this study, a two-point composite block method based on the backward differentiation formula (CBBDF) is introduced to solve stiff ordinary differential equations. The CBBDF method incorporates an additional intermediate point among the interpolating points, developed in two stages: the first stage employs the Euler's method as a fundamental building block, while the second stage utilizes CBBDF of order three. A key distinction of the method with the classical block method is the introduction of an independent parameter $\gamma$, which eliminates the need for an external startup calculation, while maintaining the accuracy and stability of numerical solutions. The theoretical analysis verifies that the proposed method is convergent and $A$-stable. It fulfills the essential properties of consistency and zero-stability, and it lies within the $A$-stability region. To demonstrate the effectiveness of the proposed approach, several stiff initial value problems of linear and non-linear are solved. For validation, the results are compared with existing literature. While approximating the solution at multiple points simultaneously, the composite block method offers the ability to use larger step sizes for solution approximation. The CBBDF method shows promising results, achieving a reliable degree of accuracy as indicated by its maximum error and average error measurements.

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Main Subjects

References

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Computational Methods for Differential Equations
Volume 14, Issue 2
April 2026
Pages 734-753

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History

  • Receive Date: 08 January 2025
  • Revise Date: 26 March 2025
  • Accept Date: 23 April 2025

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