  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.
