General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs

TL;DR

This paper introduces a flexible framework for high-order local energy-preserving algorithms for Hamiltonian PDEs, combining Runge-Kutta and spectral spatial discretizations, with demonstrated long-term stability.

Contribution

It presents a novel general approach to construct arbitrarily high-order local energy-preserving integrators for Hamiltonian PDEs using continuous Runge-Kutta and spectral methods.

Findings

Effective long-term energy preservation demonstrated in numerical experiments.

Superior performance compared to classical multi-symplectic schemes.

Applicable to nonlinear Schrödinger equations with external fields.

Abstract

In this paper we propose and investigate a general approach to constructing local energy-preserving algorithms which can be of arbitrarily high order in time for solving Hamiltonian PDEs. This approach is based on the temporal discretization using continuous Runge-Kutta-type methods, and the spatial discretization using pseudospectral methods or Gauss--Legendre collocation methods. The local energy conservation law of our new schemes is analyzed in detail. The effectiveness of the novel local energy-preserving integrators is demonstrated by coupled nonlinear Schr\"odinger equations and 2D nonlinear Schr\"odinger equations with external fields. Our new schemes are compared with some classical multi-symplectic and symplectic schemes in numerical experiments. The numerical results show the remarkable \emph{long-term} behaviour of our new schemes.