# Linearly implicit local and global energy-preserving methods for PDEs with a cubic Hamiltonian

## TL;DR

This paper introduces linearly implicit numerical methods that conserve energy for multi-symplectic PDEs with cubic invariants, demonstrating improved stability and efficiency over fully implicit schemes through tests on Korteweg-de Vries and Zakharov-Kuznetsov equations.

## Contribution

The paper develops novel linearly implicit energy-preserving methods for multi-symplectic PDEs with cubic invariants, enhancing computational efficiency and stability.

## Key findings

- Methods successfully conserve discrete energy laws.
- Numerical tests show good stability and efficiency.
- Methods outperform fully implicit schemes in speed.

## Abstract

We present linearly implicit methods that preserve discrete approximations to local and global energy conservation laws for multi-symplectic PDEs with cubic invariants. The methods are tested on the one-dimensional Korteweg-de Vries equation and the two-dimensional Zakharov-Kuznetsov equation; the numerical simulations confirm the conservative properties of the methods, and demonstrate their good stability properties and superior running speed when compared to fully implicit schemes.