Exponential integrators preserving first integrals or Lyapunov functions for conservative or dissipative systems

TL;DR

This paper introduces a new exponential integrator that preserves first integrals or Lyapunov functions for conservative or dissipative systems, combining exponential integrators with discrete gradients for improved structure preservation.

Contribution

The paper proposes a novel structure-preserving exponential scheme for systems with skew-symmetric or negative semidefinite matrices, integrating ideas from exponential integrators and discrete gradients.

Findings

The new scheme effectively preserves first integrals or Lyapunov functions.

Numerical results show the scheme outperforms existing structure-preserving methods.

The scheme is applicable to a broad class of conservative and dissipative systems.

Abstract

In this paper, combining the ideas of exponential integrators and discrete gradients, we propose and analyze a new structure-preserving exponential scheme for the conservative or dissipative system $\dot{y} = Q(M y + \nabla U (y))$, where $Q$ is a $d\times d$ skew-symmetric or negative semidefinite real matrix, $M$ is a $d\times d$ symmetric real matrix, and $U : \mathbb{R}^d\rightarrow\mathbb{R}$ is a differentiable function. We present two properties of the new scheme. The paper is accompanied by numerical results that demonstrate the remarkable superiority of our new scheme in comparison with other structure-preserving schemes in the scientific literature.