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

Iranian Journal of Numerical Analysis and Optimization

مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 22 شهریور 1404

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2025.94194.1673

نویسندگان

Gholamreza Hojjati^{* 1}؛ Somayyeh Fazeli²؛ Afsaneh Moradi³

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

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

³Institute of Analysis and Numerics, Otto von Guericke University Magdeburg, Universit\"atsplatz 2, 39106 Magdeburg, Germany.

چکیده

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. This restriction impedes high-order methods for stiff systems prevalent in scientific applications. Leveraging techniques like variable stepsizes, implicit-explicit schemes, or generalized stability definitions, researchers have developed efficient methods to overcome this obstacle through diverse approaches. In this paper, we survey these strategies and analyze their impact on enhancing the stability and accuracy of resulting methods. A comprehensive understanding of these advances assists researchers in designing more effective algorithms for stiff problems.

کلیدواژه‌ها

Initial value problem؛ Stiff system؛ Linear multistep methods؛ convergence؛ Stability