2024/07/10 by Samardhi Yadav, Yadav, Samardhi, Vishal Vaibhav +1
Engineering · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Computational Physics (physics.comp-ph) #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #FOS: Physical sciences #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2408.10208
openalex publication_date 2024/07/10 · openalex created_date 2024/09/21 · openalex updated_date 2026/07/28
In this article, we discuss the efficient ways of implementing the transparent boundary condition (TBC) and its various approximations for the free Schrödinger equation on a hyperrectangular computational domain in \fieldRd with periodic boundary conditions along the (d-1) unbounded directions. In particular, we consider Padé 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.