Pentadiagonal alternating-direction-implicit finite-difference time-domain method for two-dimensional Schrödinger equation |
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Authors: | Wei Choon Tay Eng Leong Tan |
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Affiliation: | School of EEE, Nanyang Technological University, Singapore 639798, Singapore |
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Abstract: | In this paper, we have proposed a pentadiagonal alternating-direction-implicit (Penta-ADI) finite-difference time-domain (FDTD) method for the two-dimensional Schrödinger equation. Through the separation of complex wave function into real and imaginary parts, a pentadiagonal system of equations for the ADI method is obtained, which results in our Penta-ADI method. The Penta-ADI method is further simplified into pentadiagonal fundamental ADI (Penta-FADI) method, which has matrix-operator-free right-hand-sides (RHS), leading to the simplest and most concise update equations. As the Penta-FADI method involves five stencils in the left-hand-sides (LHS) of the pentadiagonal update equations, special treatments that are required for the implementation of the Dirichlet’s boundary conditions will be discussed. Using the Penta-FADI method, a significantly higher efficiency gain can be achieved over the conventional Tri-ADI method, which involves a tridiagonal system of equations. |
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Keywords: | Schrö dinger equation Alternating-direction-implicit (ADI) Finite-difference time-domain (FDTD) methods Pentadiagonal Tridiagonal |
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