We derive well-posed boundary conditions for the linearized Serre equations in one spatial dimension by utilizing the energy method. An energy stable and conservative discontinuous Galerkin spectral element method with simple upwind numerical fluxes is proposed for solving the initial boundary value problem. We derive discrete energy estimates for the numerical approximation and prove a priori error estimates in the energy norm. Detailed numerical examples are provided to verify the theoretical analysis and show convergence of numerical errors.

翻译：本文通过能量方法推导出一维线性化Serre方程的适定边界条件。针对初边值问题求解，提出一种采用简单迎风数值通量的能量稳定且守恒的间断Galerkin谱元法。我们推导了数值逼近的离散能量估计，并证明了能量范数下的先验误差估计。通过详细的数值算例验证理论分析，并展示数值误差的收敛性。