Linear gradient structures and discrete gradient methods for conservative/dissipative differential-algebraic equations
For researchers in numerical methods for DAEs, this work provides a foundational step toward a unified framework, though it is limited to index-1 DAEs.
The paper tackles the lack of a unified framework for discrete gradient methods for differential-algebraic equations (DAEs) with conservation/dissipation laws. It introduces a linear gradient structure for DAEs and constructs a new discrete gradient method for index-1 DAEs that preserves discrete conservation/dissipation laws.
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.