Nonreflecting Boundary Condition for the free Schr\"odinger equation for hyperrectangular computational domains
<|reference_start|>Nonreflecting Boundary Condition for the free Schr\"odinger equation for hyperrectangular computational domains: 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.<|reference_end|>
@article{yadav2024nonreflecting,
    title={Nonreflecting Boundary Condition for the free Schr\"{o}dinger equation
  for hyperrectangular computational domains},
    author={Samardhi Yadav, Vishal Vaibhav},
    journal={arXiv preprint arXiv:2408.10208},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.10208},
    primaryClass={math.NA cs.NA physics.comp-ph}
}