On the equivalence between SOR-type methods for linear systems and discrete gradient methods for gradient systems

TL;DR

This paper establishes a novel connection between SOR-type iterative methods for linear systems and discrete gradient methods for gradient systems, providing new insights and interpretations for these numerical techniques.

Contribution

It introduces a new link between SOR methods and discrete gradient methods, leading to fresh derivations and interpretations of relaxation parameters.

Findings

SOR methods monotonically decrease a quadratic function

A new discrete gradient was derived

New interpretations for relaxation parameters

Abstract

The iterative nature of many discretisation methods for continuous dynamical systems has led to the study of the connections between iterative numerical methods in numerical linear algebra and continuous dynamical systems. Certain researchers have used the explicit Euler method to understand this connection, but this method has its limitation. In this study, we present a new connection between successive over-relaxation (SOR)-type methods and gradient systems; this connection is based on discrete gradient methods. The focus of the discussion is the equivalence between SOR-type methods and discrete gradient methods applied to gradient systems. The discussion leads to new interpretations for SOR-type methods. For example, we found a new way to derive these methods; these methods monotonically decrease a certain quadratic function and obtain a new interpretation of the relaxation parameter. We also obtained a new discrete gradient while studying the new connection.