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

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.