# Geometric Exponential Integrators

## TL;DR

This paper introduces geometric exponential integrators for semilinear Poisson systems that preserve structure or energy, offering improved long-term stability and computational efficiency over traditional methods.

## Contribution

The paper constructs two new types of exponential integrators that preserve either Poisson structure or energy, enhancing stability and efficiency for Hamiltonian PDE discretizations.

## Key findings

- Geometric exponential integrators outperform non-geometric ones in long-term stability.
- They are more computationally efficient than symplectic and discrete gradient methods.
- Numerical experiments confirm their effectiveness for semilinear Poisson systems.

## Abstract

In this paper, we consider exponential integrators for semilinear Poisson systems. Two types of exponential integrators are constructed, one preserves the Poisson structure, and the other preserves energy. Numerical experiments for semilinear Possion systems obtained by semi-discretizing Hamiltonian PDEs are presented. These geometric exponential integrators exhibit better long time stability properties as compared to non-geometric integrators, and are computationally more efficient than traditional symplectic integrators and energy-preserving methods based on the discrete gradient method.