Linear gradient structures and discrete gradient methods for conservative/dissipative differential-algebraic equations

TL;DR

This paper develops a unified framework for discrete gradient methods applied to differential-algebraic equations, introducing linear gradient structures and constructing new methods for index-1 DAEs to ensure conservation and dissipation laws.

Contribution

It introduces a novel linear gradient structure for DAEs and constructs the first discrete gradient method for index-1 DAEs, advancing the theoretical foundation.

Findings

Established a linear gradient structure for DAEs

Demonstrated that simple discrete gradients do not guarantee conservation/dissipation

Constructed a new discrete gradient method for index-1 DAEs

Abstract

In this paper, we consider the use of discrete gradients for differential-algebraic equations (DAEs) with a conservation/dissipation law. As one of the most popular numerical methods for conservative/dissipative ordinary differential equations, the framework of discrete gradient methods has been intensively developed over recent decades. Although discrete gradients have been applied to several specific conservative/dissipative DAEs, no unified framework for DAEs has yet been constructed. In this paper, we move toward the establishment of such a framework, and introduce concepts including an appropriate linear gradient structure for DAEs. Then, we reveal that the simple use of discrete gradients does not imply the discrete conservation/dissipation laws. Fortunately, however, we can successfully construct a new discrete gradient method for the case of index-1 DAEs. We believe this first attempt provides an indispensable basis for constructing a unified framework of discrete gradient methods for DAEs.