# Numerical solutions of Hamiltonian PDEs: a multi-symplectic integrator in light-cone coordinates

## TL;DR

This paper presents a simple, multi-symplectic numerical integrator for Hamiltonian PDEs that conserves stress-energy with high precision over long times, demonstrated on a nonlinear wave equation.

## Contribution

It introduces a novel, easy-to-implement multi-symplectic integrator that preserves key geometric properties of Hamiltonian PDEs, including stress-energy conservation.

## Key findings

- High-precision long-term conservation of stress-energy tensor
- Effective implementation for nonlinear wave equations in one dimension
- Extension framework for higher-dimensional problems

## Abstract

We introduce a novel numerical method to integrate partial differential equations representing the Hamiltonian dynamics of field theories. It is a multi-symplectic integrator that locally conserves the stress-energy tensor with an excellent precision over very long periods. Its major advantage is that it is extremely simple (it is basically a centered box scheme) while remaining locally well defined. We put it to the test in the case of the non-linear wave equation (with quartic potential) in one spatial dimension, and we explain how to implement it in higher dimensions. A formal geometric presentation of the multi-symplectic structure is also given as well as a technical trick allowing to solve the degeneracy problem that potentially accompanies the multi-symplectic structure.