In this paper, we develop three conservative discontinuous Galerkin (DG) schemes for the one-dimensional nonlinear dispersive Serre equations, including two conserved schemes for the equations in conservative form and a Hamiltonian conserved scheme for the equations in non-conservative form. One of the schemes owns the well-balanced property via constructing a high order approximation to the source term for the Serre equations with a non-flat bottom topography. By virtue of the Hamiltonian structure of the Serre equations, we introduce an Hamiltonian invariant and then develop a DG scheme which can preserve the discrete version of such an invariant. Numerical experiments in different cases are performed to verify the accuracy and capability of these DG schemes for solving the Serre equations.

在本文中，我们为一维非线性色散Serre方程开发了三个保守的不连续Galerkin（DG）方案，包括两个守恒形式的守恒方案和一个非保守形式的哈密顿守恒方案。其中一种方案通过为非平坦底部地形的Serre方程构建与源项的高阶近似，从而拥有良好的平衡特性。借助于Serre方程的哈密顿结构，我们引入了哈密顿不变量，然后开发了可以保留这种不变量的离散形式的DG方案。在不同情况下进行了数值实验，以验证这些DG方案求解Serre方程的准确性和能力。