# Linearly implicit structure-preserving schemes for Hamiltonian systems

## TL;DR

This paper investigates and compares linearly implicit, structure-preserving numerical schemes for Hamiltonian systems, demonstrating their effectiveness on specific PDEs like Korteweg-de Vries and Camassa-Holm equations.

## Contribution

It introduces and analyzes linearly implicit schemes that preserve modified energy in Hamiltonian systems, with applications to important PDEs.

## Key findings

- Schemes effectively preserve modified energy.
- Numerical results demonstrate stability and accuracy.
- Applicable to integrable PDEs like KdV and Camassa-Holm.

## Abstract

Kahan's method and a two-step generalization of the discrete gradient method are both linearly implicit methods that can preserve a modified energy for Hamiltonian systems with a cubic Hamiltonian. These methods are here investigated and compared. The schemes are applied to the Korteweg-de Vries equation and the Camassa-Holm equation, and the numerical results are presented and analysed.