Numerical methods that preserve a Lyapunov function for Ordinary Differential Equations

TL;DR

This paper introduces a discrete gradient numerical method that preserves Lyapunov functions in ordinary differential equations, ensuring energy decay similar to the original system, and demonstrates its advantages over standard methods through numerical experiments.

Contribution

The paper develops a discrete gradient method for ODEs that guarantees Lyapunov function preservation, paving the way for higher-order schemes and improving energy decay in simulations.

Findings

Discrete gradient methods preserve Lyapunov functions where standard methods fail.

The proposed method outperforms conventional schemes in computational efficiency.

Numerical experiments validate the energy decay and stability advantages of the new method.

Abstract

The paper studies numerical methods that preserve a Lyapunov function of a dynamical system, i.e. numerical approximations whose energy decreases, just like in the original differential equation. With this aim, a discrete gradient method is implemented for numerical integration of a system of ordinary differential equations. In principle, this procedure yields first order methods, but the analysis paves the way to the design of higher-order methods. As a case in point, the proposed method is applied to the Duffing equation without external forcing, considering that in this case, preserving the Lyapunov function is more important than accuracy of particular trajectories. Results are validated by means of numerical experiments, where the discrete gradient method is compared to standard Runge-Kutta methods. As predicted by the theory, discrete gradient methods preserve the Lyapunov function, whereas conventional methods fail to do so, since either periodic solutions appear or the energy does not decrease. Besides, the discrete gradient method outperforms conventional schemes when these do preserve the Lyapunov function, in terms of computational cost, thus the proposed method is promising.