Geometric integration using discrete gradients

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 an ODE and one or more first integrals (i.e. constants of the motion) and/or Lyapunov functions, it is shown that the ODE can be rewritten as a 'linear-gradient system'. Discrete gradients are used to construct discrete approximations to the ODE which preserve the first integrals and Lyapunov functions exactly. The method applies to all Hamiltonian, Poisson and gradient systems, and also to many dissipative systems (those with a known first integral or Lyapunov function).

A local discontinuous Galerkin method for the Burgers–Poisson equation

“…It is believed that numerical methods preserving more invariants are advantageous: besides the high accuracy of numerical solutions, an invariant preserving scheme can preserve good stability properties after long-time numerical integration. Much more effort has been devoted in this topic for different integrable PDEs recently [7,18,30,31].…”

Conservative fourth-order time integration of non-linear dynamic systems

“…A second-order secant approximation was introduced in the form of a finite derivative or gradient into computational mechanics by Gonzalez [7,8] and McLachlan et al [9]. The finite derivative is a special form of a secant correction, used to obtain a special property over a finite interval, see e.g.…”

“…An important extension of the range of conservative time integration methods was attained by the introduction of the so-called finite derivative introduced by Gonzalez [7,8], and further developed under the name of the 'discrete gradient' by McLachlan et al [9]. The basic idea is to replace the original expression of the internal force by an augmented form including a correction in terms of the increment of the quantity to be conserved.…”

Numerical Implementation of Gradient Algorithms

“…This approach has been rather fruitful in providing numerical methods for Hamiltonian systems, whereas few methods have been proposed that preserve the Lyapunov function of gradient systems. To the best of our knowledge, the main proposals in this regard comprise the discrete gradient methods [12] and the projection methods [5]. On one hand, projection methods are too involved and, although they are formally explicit, they require solving a nonlinear equation at every step.…”