Energy-preserving multi-symplectic Runge-Kutta methods for Hamiltonian wave equations

TL;DR

This paper introduces a new class of multi-symplectic Runge-Kutta methods for Hamiltonian wave equations that can preserve geometric structures and approximate energy conservation simultaneously under certain conditions.

Contribution

It proposes a parametric multi-symplectic Runge-Kutta method capable of weak energy preservation, with theoretical proof of the energy-preserving parameter's existence.

Findings

The method effectively preserves geometric structures.

Numerical experiments confirm energy preservation.

Theoretical analysis supports the method's validity.

Abstract

It is well-known that a numerical method which is at the same time geometric structure-preserving and physical property-preserving cannot exist in general for Hamiltonian partial differential equations. In this paper, we present a novel class of parametric multi-symplectic Runge-Kutta methods for Hamiltonian wave equations, which can also conserve energy simultaneously in a weaker sense with a suitable parameter. The existence of such a parameter, which enforces the energy-preserving property, is proved under certain assumptions on the fixed step sizes and the fixed initial condition. We compare the proposed method with the classical multi-symplectic Runge-Kutta method in numerical experiments, which shows the remarkable energy-preserving property of the proposed method and illustrate the validity of theoretical results.