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arXiv · 2408.10208

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

Samardhi Yadav·Vishal Vaibhav
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Abstract

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.

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Samardhi Yadav, Vishal Vaibhav. 2024-07-10. Nonreflecting Boundary Condition for the free Schr\"{o}dinger equation for hyperrectangular computational domains. https://arxiv.org/abs/2408.10208

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