Research Article
Open Access
A Stiffly Stable Adaptive Fifth-Order Block Time-Stepping Method for Stiff Oscillatory Differential Equations
1
Department of Statistics, Federal Polytechnic Kaura-Namoda, Zamfara State, Nigeria
2
Department of Mathematics, Umaru Musa Yar'adua University, Katsina, Katsina State, Nigeria
3
Department of Mathematics and Statistics, Al-Qalam University Katsina, Katsina State, Nigeria
4
Department of Mathematics, Isah Kaita College of Education Dutsinma, Katsina, Katsina State, Nigeria
* Corresponding Author:
Najamuddeen Bala,
[email protected]
Volume 2, Issue 3
2026
Received: 15 June 2026
Revised: 15 July 2026
Accepted: 26 August 2026
Published: 06 September 2026
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CC BY 4.0 License
Article Information
Volume/Issue
Volume 2, Issue 3, 2026
Pages
209-228
Abstract
This paper presents an adaptive block time-stepping algorithm designed for the numerical integration of first-order stiff oscillatory problems, which typically involve multiple time scales and stringent stability conditions. The proposed method integrates a block formulation enabling the simultaneous evaluation of several solution points with an adaptive step size mechanism to effectively regulate local truncation errors. A predictor-corrector strategy is employed, where an explicit predictor generates initial estimates, and an implicit block corrector enhances stability for stiff systems. The resulting nonlinear equations are resolved using Newton's iterative method. The scheme is proven to be consistent, zero-stable, and achieves fifth-order accuracy. Its stability characteristics are further examined through A-stability analysis using locus boundary techniques, validating its applicability to stiff problems. Additionally, an error control procedure is incorporated to dynamically adjust the step size, ensuring an optimal balance between computational efficiency and solution accuracy. Numerical experiments on benchmark stiff oscillatory systems indicate that the proposed method outperforms the fourth-order variable step size block backward differentiation formula (VSBBDF4), as well as MATLAB solvers ODE15s and ODE23s, in terms of stability, accuracy, and computational efficiency.
Graphical Abstract
Keywords
stiff ordinary differential equations
variable step size
block backward differentiation formula
A-stability
predictor-corrector method
Data Availability Statement
Data will be made available on request.
Funding
This work was supported without any funding.
Conflicts of Interest
The authors declare no conflicts of interest.
AI Use Statement
The authors declare that no generative AI was used in the preparation of this manuscript.
Ethical Approval and Consent to Participate
Not applicable.
References
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APA Style
Bala, N., Musa, H., Alhassan, B., & Lawal, A. (2026). A Stiffly Stable Adaptive Fifth-Order Block Time-Stepping Method for Stiff Oscillatory Differential Equations. Journal of Mathematics and Interdisciplinary Applications, 2(3), 209-228. https://doi.org/10.62762/JMIA.2026.142673
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TY - JOUR AU - Bala, Najamuddeen AU - Musa, Hamisu AU - Alhassan, Buhari AU - Lawal, Aliyu PY - 2026 DA - 2026/09/06 TI - A Stiffly Stable Adaptive Fifth-Order Block Time-Stepping Method for Stiff Oscillatory Differential Equations JO - Journal of Mathematics and Interdisciplinary Applications T2 - Journal of Mathematics and Interdisciplinary Applications JF - Journal of Mathematics and Interdisciplinary Applications VL - 2 IS - 3 SP - 209 EP - 228 DO - 10.62762/JMIA.2026.142673 UR - https://www.icck.org/article/abs/JMIA.2026.142673 KW - stiff ordinary differential equations KW - variable step size KW - block backward differentiation formula KW - A-stability KW - predictor-corrector method AB - This paper presents an adaptive block time-stepping algorithm designed for the numerical integration of first-order stiff oscillatory problems, which typically involve multiple time scales and stringent stability conditions. The proposed method integrates a block formulation enabling the simultaneous evaluation of several solution points with an adaptive step size mechanism to effectively regulate local truncation errors. A predictor-corrector strategy is employed, where an explicit predictor generates initial estimates, and an implicit block corrector enhances stability for stiff systems. The resulting nonlinear equations are resolved using Newton's iterative method. The scheme is proven to be consistent, zero-stable, and achieves fifth-order accuracy. Its stability characteristics are further examined through A-stability analysis using locus boundary techniques, validating its applicability to stiff problems. Additionally, an error control procedure is incorporated to dynamically adjust the step size, ensuring an optimal balance between computational efficiency and solution accuracy. Numerical experiments on benchmark stiff oscillatory systems indicate that the proposed method outperforms the fourth-order variable step size block backward differentiation formula (VSBBDF4), as well as MATLAB solvers ODE15s and ODE23s, in terms of stability, accuracy, and computational efficiency. SN - 3070-393X PB - Institute of Central Computation and Knowledge LA - English ER -
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@article{Bala2026A,
author = {Najamuddeen Bala and Hamisu Musa and Buhari Alhassan and Aliyu Lawal},
title = {A Stiffly Stable Adaptive Fifth-Order Block Time-Stepping Method for Stiff Oscillatory Differential Equations},
journal = {Journal of Mathematics and Interdisciplinary Applications},
year = {2026},
volume = {2},
number = {3},
pages = {209-228},
doi = {10.62762/JMIA.2026.142673},
url = {https://www.icck.org/article/abs/JMIA.2026.142673},
abstract = {This paper presents an adaptive block time-stepping algorithm designed for the numerical integration of first-order stiff oscillatory problems, which typically involve multiple time scales and stringent stability conditions. The proposed method integrates a block formulation enabling the simultaneous evaluation of several solution points with an adaptive step size mechanism to effectively regulate local truncation errors. A predictor-corrector strategy is employed, where an explicit predictor generates initial estimates, and an implicit block corrector enhances stability for stiff systems. The resulting nonlinear equations are resolved using Newton's iterative method. The scheme is proven to be consistent, zero-stable, and achieves fifth-order accuracy. Its stability characteristics are further examined through A-stability analysis using locus boundary techniques, validating its applicability to stiff problems. Additionally, an error control procedure is incorporated to dynamically adjust the step size, ensuring an optimal balance between computational efficiency and solution accuracy. Numerical experiments on benchmark stiff oscillatory systems indicate that the proposed method outperforms the fourth-order variable step size block backward differentiation formula (VSBBDF4), as well as MATLAB solvers ODE15s and ODE23s, in terms of stability, accuracy, and computational efficiency.},
keywords = {stiff ordinary differential equations, variable step size, block backward differentiation formula, A-stability, predictor-corrector method},
issn = {3070-393X},
publisher = {Institute of Central Computation and Knowledge}
}
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Copyright © 2026 by the Author(s). Published by Institute of Central Computation and Knowledge. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
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