On structure-preserving discontinuous Galerkin methods for Hamiltonian partial differential equations: Energy conservation and multi-symplecticity

TL;DR

This paper develops discontinuous Galerkin methods for Hamiltonian PDEs that preserve multi-symplectic structure and energy, demonstrating their effectiveness on various equations and highlighting flux choices for improved accuracy.

Contribution

The paper introduces DG schemes that simultaneously conserve energy and multi-symplecticity for Hamiltonian PDEs, with analysis of flux choices affecting accuracy.

Findings

DG methods preserve multi-symplectic structure and energy.

Certain flux choices improve accuracy and long-term stability.

Numerical experiments confirm theoretical properties.

Abstract

In this paper, we present and study discontinuous Galerkin (DG) methods for one-dimensional multi-symplectic Hamiltonian partial differential equations. We particularly focus on semi-discrete schemes with spatial discretization only, and show that the proposed DG methods can simultaneously preserve the multi-symplectic structure and energy conservation with a general class of numerical fluxes, which includes the well-known central and alternating fluxes. Applications to the wave equation, the Benjamin-Bona-Mahony equation, the Camassa-Holm equation, the Korteweg-de Vries equation and the nonlinear Schr\"odinger equation are discussed. Some numerical results are provided to demonstrate the accuracy and long time behavior of the proposed methods. Numerically, we observe that certain choices of numerical fluxes in the discussed class may help achieve better accuracy compared with the commonly used ones including the central fluxes.