This paper addresses the numerical implementation of the transparent boundary condition (TBC) and its various approximations for the free Schr\"odinger equation on a rectangular computational domain. In particular, we consider the exact TBC and its spatially local approximation under high frequency assumption along with an appropriate corner condition. For the spatial discretization, we use a Legendre-Galerkin spectral method where Lobatto polynomials serve as the basis. Within variational formalism, we first arrive at the time-continuous dynamical system using spatially discrete form of the initial boundary-value problem incorporating the boundary conditions. This dynamical system is then discretized using various time-stepping methods, namely, the backward-differentiation formula of order 1 and 2 (i.e., BDF1 and BDF2, respectively) and the trapezoidal rule (TR) to obtain a fully discrete system. Next, we extend this approach to the novel Pad\'e based implementation of the TBC presented by Yadav and Vaibhav [arXiv:2403.07787(2024)]. Finally, several numerical tests are presented to demonstrate the effectiveness of the boundary maps (incorporating the corner conditions) where we study the stability and convergence behavior empirically.

翻译：本文研究了矩形计算域上自由薛定谔方程透明边界条件及其各类近似的数值实现。具体而言，我们考虑了精确透明边界条件及其在高频假设下的空间局部近似，并结合了适当的角点条件。在空间离散化方面，我们采用勒让德-伽辽金谱方法，其中以洛巴托多项式作为基函数。在变分形式体系内，我们首先结合边界条件，利用初边值问题的空间离散形式得到时间连续动力系统。随后采用多种时间步进方法对该动力系统进行离散化，即一阶和二阶后向差分公式以及梯形法则，从而获得全离散系统。接下来，我们将此方法推广到Yadav与Vaibhav提出的基于Padé近似的透明边界条件新实现方案。最后，通过若干数值实验验证了结合角点条件的边界映射的有效性，并实证研究了其稳定性与收敛性。