On overcoming Dahlquist’s second barrier for$A$-stable linear multistep methods

Document Type : Research Article

Authors

• G. Hojjati ¹
• S. Fazeli ²
• A. Moradi ³

¹ Faculty of Mathematics, Statistics and Computer Science, University of Tabriz, Tabriz, Iran.

² Marand Technical College, University of Tabriz, Tabriz, Iran.

³ Institute of Analysis and Numerics, Otto von Guericke University Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany.

10.22067/ijnao.2025.94194.1673

Abstract

Dahlquist’s second barrier limits the order of $A$-stable linear multistep methods to at most two, posing significant challenges for achieving higher accuracy in the numerical solution of stiff ordinary differential equations. Leveraging various successful techniques, many efforts have been made to develop efficient methods that overcome this fundamental obstacle through different approaches. In this paper, we survey these techniques and analyze their impact on enhancing the stability and accuracy of the resulting methods. A comprehensive understanding of these advances can assist researchers in designing more effective algorithms for stiff problems.

Keywords

• Initial value problem
• Stiff system
• Linear multistep methods
• Dahlquist’s second barrier
• convergence
• Stability

Main Subjects

• Numerical methods for differential equations and its applications

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