# Line Integral solution of Hamiltonian PDEs

## TL;DR

This paper introduces energy-conserving line integral methods within the Hamiltonian Boundary Value Methods class to numerically solve Hamiltonian PDEs, demonstrating their effectiveness on key equations like wave, Schrödinger, and KdV.

## Contribution

It presents a novel approach using line integral methods for Hamiltonian PDEs, emphasizing energy conservation and broad applicability to important equations.

## Key findings

- Effective energy conservation in numerical solutions
- Successful application to wave, Schrödinger, and KdV equations
- Potential for improved long-term stability in simulations

## Abstract

In this paper, we report about recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs), by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schr\"odinger equation, and the Korteweg-de Vries equation, to illustrate the main features of this novel approach.