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

TL;DR

This paper develops and analyzes efficient transparent boundary conditions for the free Schrödinger equation on hyperrectangular domains, employing rational approximations and spectral methods, with numerical tests demonstrating stability and convergence.

Contribution

It introduces a rational approximation-based TBC and a high-frequency approximation, combined with spectral discretization and time-stepping methods, for improved simulation of the Schrödinger equation.

Findings

The proposed methods are stable under tested conditions.

Numerical results show good convergence behavior.

The approach effectively simulates wave propagation with minimal boundary reflections.

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.