Geometric integration using discrete gradients

This paper discusses the discrete analogue of the gradient of a function an d shows how discrete gradients can be used in the numerical integration of ordinary differential equations (ODEs). Given an ODE and one or more first integrals (i.e. constants of the motion) and/or Lyapunov functions, it is s hown that the ODE can be rewritten as a 'linear-gradient system'. Discrete gradients are used to construct discrete approximations to the ODE which pr eserve the first integrals and Lyapunov functions exactly. The method appli es to all Hamiltonian, Poisson and gradient systems, and also to many dissi pative systems (those with a known first integral or Lyapunov function).