Numerical Integration of Lattice Systems with a Lyapunov Function

Abstract

In this contribution we implement and assess numerical methods for gradient systems, i.e. dynamical systems that possess a Lyapunov function, and consequently are stable. In particular, we claim that discrete gradient methods are well suited to so-called lattice systems, i.e. systems of ordinary differential equations that can reach high dimensionality. For these systems, reproducing the stable qualitative behaviour is more important than achieving an overly accurate quantitative approximation. The presented results show that discrete gradient methods outperform conventional Runge-Kutta methods, since these latter algorithms destroy the stability of the original system.