On the convergence of the iterative methods

Abstract

This article is focused on the discussion and clarification of some issues related with the convergence of the usual iterative methods to solve heat transfer and fluid flow problems using a segregated approach. These are not new issues, but no such discussion and clarification are found in the literature. It includes clarification of the necessary and sufficient conditions for convergence of the iterative methods to solve the linear discretization equations’ systems, assessment of those conditions from the discretization equations’ coefficients, discussion of the convergence criterion usually referred to as the Scarborough criterion, and clarification why the linear equations’ systems can be solved even when the required condition for convergence is not met. It also highlights the similarity between stability and convergence, both related with successive multiplications of numbers lower than 1 by themselves. This article has also the objective of put together those clarifications and discussions and make them available for students, instructors, programmers and CFD users, in a simultaneously accurate and simple way.

Keywords:

• Conditions for convergence
• convergence
• convergence criteria
• iterative methods
• numerical solution
• segregated approach

Declaration of interest

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in the article entitled “On the convergence of the iterative methods” submitted for publication in the Numerical Heat Transfer, Part B: Fundamentals.

Additional information

Funding

This work was supported by the projects UIDB/00481/2020 and UIDP/00481/2020 – FCT – Fundação para a Ciência e a Tecnologia; and CENTRO-01-0145-FEDER-022083 – Centro Portugal Regional Operational Programme (Centro2020), under the PORTUGAL 2020 Partnership Agreement, through the European Regional Development Fund.