We derive and analyse well-posed boundary conditions for the linear shallow water wave equation. The analysis is based on the energy method and it identifies the number, location and form of the boundary conditions so that the initial boundary value problem is well-posed. A finite volume method is developed based on the summation-by-parts framework with the boundary conditions implemented weakly using penalties. Stability is proven by deriving a discrete energy estimate analogous to the continuous estimate. The continuous and discrete analysis covers all flow regimes. Numerical experiments are presented verifying the analysis.

翻译：我们推导并分析了线性浅水波方程的适定边界条件。该分析基于能量方法，通过确定边界条件的数量、位置及形式，使得初边值问题具有适定性。基于和式分部求和框架，发展了一种有限体积方法，采用惩罚项对边界条件进行弱施加。通过推导与连续估计相对应的离散能量估计，证明了数值方法的稳定性。连续与离散分析覆盖了所有流动状态。数值实验验证了分析结果。