Showing papers by "G. R. W. Quispel published in 1999"
15 Apr 1999-Philosophical transactions - Royal Society. Mathematical, physical and engineering sciences
TL;DR: The discrete analogue of the gradient of a function is discussed and how discrete gradients can be used in the numerical integration of ordinary differential equations (ODEs) is shown.
Abstract: This paper discusses the discrete analogue of the gradient of a function and shows how discrete gradients can be used in the numerical integration of ordinary differential equations (ODEs). Given a...
492 citations
TL;DR: In this paper, the existence of pre-symplectic or pre-implectic structures for ODEs and OΔEs was shown to imply a complex implectic structure.
Abstract: We present a number of alternative sufficient conditions for the existence of pre-symplectic or pre-implectic (Poisson) structures, for both ordinary differential (ODE) and ordinary difference (OΔE) equations. Four alternative sets of conditions are obtained for ODEs and OΔEs in n dependent variables: (1) A vector field in involution with the ODE and an integral (or two symmetries for an OΔE) imply a pre-implectic structure; (2) volume preservation and n −2 symmetries imply a pre-symplectic structure; (3) volume preservation and n −2 integrals imply a pre-implectic structure; (4) complex implectic structure implies infinitely many real implectic structures. In all but the first case the methods can give a set of distinct structures.
45 citations