Stability analyses of block hybrid methods for numerical solutions of differential equations

Articles in Press

Document Type : Research Article

Authors

Oludare Adedire

U. W. Sirisena

Department of Mathematics, Faculty of Natural Sciences, University of Jos, Plateau State, Nigeria.

10.22067/ijnao.2026.97215.1817

Abstract

This research presents a comparative analysis of stability polynomials and stability functions for investigating the stability of derived block hybrid methods. The methods—an explicit (BHEM) and two implicit (BHIM) formulations—were obtained through continuous formulation using multistep collocation and matrix inversion techniques. A novel theoretical result establishes that for consistent block methods, stability polynomials and stability functions must yield congruent stability conclusions, with any discrepancy indicating potential derivation errors. It is demonstrated that the use of stability polynomials offers straightforward determination of the angle $\alpha$ for methods that are $A(\alpha)$-stable. However, for investigating stronger stability properties such as $L$-stability and $L(\alpha)$-stability, the stability function is preferable due to its direct rational form, which allows immediate asymptotic analysis. Detailed experiments on block methods and block hybrid methods reveal the respective advantages and limitations of each stability analysis tool. The analysis conclusively answers the research question: the absence of poles in the left half-plane alone does not guarantee $A$-stability; both analyticity in the left half-plane and boundedness on the imaginary axis are required conditions. This research successfully extends stability analysis from the one-step domain to multi-step block formulations.

Keywords

Function

Polynomial

Region

Stability

Interval

Subjects

• Numerical methods for differential equations and its applications

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