Nonreflecting Boundary Condition for the free Schr\"{o}dinger equation for hyperrectangular computational domains

• Main Authors: Yadav, Samardhi, Vaibhav, Vishal
• Format: Journal Article
• Language: English
• Published: 10.07.2024
• Subjects: Computer Science - Numerical Analysis; Mathematics - Numerical Analysis; Physics - Computational Physics
• DOI: 10.48550/arxiv.2408.10208

In this article, we discuss the efficient ways of implementing the transparent boundary condition (TBC) and its various approximations for the free Schr\"{o}dinger equation on a hyperrectangular computational domain in $\field{R}^d$ with periodic boundary conditions along the $(d-1)$ unbounded directions. In particular, we consider Pad\'e approximant based rational approximation of the exact TBC and a spatially local form of the exact TBC obtained under its high-frequency approximation. For the spatial discretization, we use a Legendre-Galerkin spectral method with a boundary-adapted basis to ensure the bandedness of the resulting linear system. Temporal discretization is then addressed with two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). Finally, several numerical tests are presented to demonstrate the effectiveness of the methods where we study the stability and convergence behaviour empirically.