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

In this paper, the use of discrete gradients is considered 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 the discrete gradient method has been intensively developed over recent decades. Although discrete gradients have been applied to several specific DAEs, no unified framework has yet been constructed. In this paper, the author moves toward the establishment of such a framework, and introduces concepts including an appropriate linear gradient structure for DAEs. Then, it is revealed that the simple use of discrete gradients does not imply the discrete conservation/dissipation laws. Fortunately, however, for the case of index-1 DAEs, an appropriate reformulation and a new discrete gradient enable us to successfully construct a novel scheme, which satisfies both of the discrete conservation/dissipation law and the constraint. This first attempt may provide an indispensable basis for constructing a unified framework of discrete gradient methods for DAEs.