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arXiv · 1805.04824

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

Shun Sato
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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.

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Shun Sato. 2018-05-13. Linear gradient structures and discrete gradient methods for conservative/dissipative differential-algebraic equations. https://arxiv.org/abs/1805.04824

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