Invariance of the Discrete Gradient Schemes for Hamiltonian Equations under Changes of Riemannian Structures

Abstract

Abstract. The discrete gradient method is a method to derive energy-conservative or-dissipative numerical schemes for Hamiltonian equations or gradient flows. This method discretizes the equation by using a discrete gradient, which is a discrete analogue of a gradient. Because a gradient and hence a discrete gradient depend on the inner product, the resultant scheme apparently depends on the underlying Riemannian structure of the space. However, when the method is applied to Hamiltonian systems it often turns out that the scheme is actually independent of the inner product. In this paper, we investigate this invariance of the discrete gradient method.