  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.
