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 Schrödinger 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.

在本文中，我们提出并研究了一维多辛哈密顿偏微分方程的间断Galerkin（DG）方法。我们特别关注仅具有空间离散化的半离散方案，并表明所提出的DG方法可以同时利用通用的一类数值通量（包括众所周知的中心通量和交变通量）来保留多辛结构和能量守恒。讨论了在波动方程，本杰明－波纳－莫哈尼方程，卡马萨－霍尔姆方程，科特维格－德弗里斯方程和非线性薛定ding方程中的应用。提供了一些数值结果，以证明所提出方法的准确性和长时间行为。在数值上