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An alternating-direction hybrid implicit-explicit finite-difference time-domain method for the Schrodinger equation

Pieter Decleer (UGent) , Arne Van Londersele (UGent) , Hendrik Rogier (UGent) and Dries Vande Ginste (UGent)
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Abstract
This paper proposes a novel hybrid FDTD method for solving the time-dependent Schrodinger equation, which is fundamental for modeling materials and designing nanoscale devices. The wave function is propagated on nonuniform grids by applying explicit updates in part of the grid and implicit updates elsewhere. The latter are based on the Alternating-Direction Implicit (ADI) scheme while the former are constructed with a central difference for the time derivative. A rigorous stability analysis proves that spatial steps can be selectively removed from the stability criterion thus combining the unconditional stability of the ADI scheme with fast explicit calculations. The scheme excels in its flexibility by efficiently discretizing and balancing explicit with implicit updates, as such expediting the computations. Moreover, it retains the linear complexity of explicit schemes with respect to space and time, making it especially scalable to numerically large problems. Several numerical experiments, including a laterally tunnel coupled quantum wire and a nanowire double-barrier resonant-tunneling diode, show the validity of the scheme by demonstrating its high accuracy and decreased CPU time compared to traditional methods. (C) 2021 Elsevier B.V. All rights reserved.
Keywords
FDTD METHOD, BOUNDARY-CONDITIONS, SCHEME, STABILITY, TRANSMISSION, SIMULATIONS, ALGORITHM, TRANSPORT, Finite-difference time-domain (FDTD), Schrodinger equation, Stability, Nonuniform, Alternating-direction hybrid, implicit-explicit (ADHIE)

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Citation

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MLA
Decleer, Pieter, et al. “An Alternating-Direction Hybrid Implicit-Explicit Finite-Difference Time-Domain Method for the Schrodinger Equation.” JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, vol. 403, Elsevier, 2022, pp. 1–19, doi:10.1016/j.cam.2021.113881.
APA
Decleer, P., Van Londersele, A., Rogier, H., & Vande Ginste, D. (2022). An alternating-direction hybrid implicit-explicit finite-difference time-domain method for the Schrodinger equation. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 403, 1–19. https://doi.org/10.1016/j.cam.2021.113881
Chicago author-date
Decleer, Pieter, Arne Van Londersele, Hendrik Rogier, and Dries Vande Ginste. 2022. “An Alternating-Direction Hybrid Implicit-Explicit Finite-Difference Time-Domain Method for the Schrodinger Equation.” JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 403: 1–19. https://doi.org/10.1016/j.cam.2021.113881.
Chicago author-date (all authors)
Decleer, Pieter, Arne Van Londersele, Hendrik Rogier, and Dries Vande Ginste. 2022. “An Alternating-Direction Hybrid Implicit-Explicit Finite-Difference Time-Domain Method for the Schrodinger Equation.” JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 403: 1–19. doi:10.1016/j.cam.2021.113881.
Vancouver
1.
Decleer P, Van Londersele A, Rogier H, Vande Ginste D. An alternating-direction hybrid implicit-explicit finite-difference time-domain method for the Schrodinger equation. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS. 2022;403:1–19.
IEEE
[1]
P. Decleer, A. Van Londersele, H. Rogier, and D. Vande Ginste, “An alternating-direction hybrid implicit-explicit finite-difference time-domain method for the Schrodinger equation,” JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, vol. 403, pp. 1–19, 2022.
@article{8727679,
  abstract     = {{This paper proposes a novel hybrid FDTD method for solving the time-dependent Schrodinger equation, which is fundamental for modeling materials and designing nanoscale devices. The wave function is propagated on nonuniform grids by applying explicit updates in part of the grid and implicit updates elsewhere. The latter are based on the Alternating-Direction Implicit (ADI) scheme while the former are constructed with a central difference for the time derivative. A rigorous stability analysis proves that spatial steps can be selectively removed from the stability criterion thus combining the unconditional stability of the ADI scheme with fast explicit calculations. The scheme excels in its flexibility by efficiently discretizing and balancing explicit with implicit updates, as such expediting the computations. Moreover, it retains the linear complexity of explicit schemes with respect to space and time, making it especially scalable to numerically large problems. Several numerical experiments, including a laterally tunnel coupled quantum wire and a nanowire double-barrier resonant-tunneling diode, show the validity of the scheme by demonstrating its high accuracy and decreased CPU time compared to traditional methods. (C) 2021 Elsevier B.V. All rights reserved.}},
  articleno    = {{113881}},
  author       = {{Decleer, Pieter and Van Londersele, Arne and Rogier, Hendrik and Vande Ginste, Dries}},
  issn         = {{0377-0427}},
  journal      = {{JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS}},
  keywords     = {{FDTD METHOD,BOUNDARY-CONDITIONS,SCHEME,STABILITY,TRANSMISSION,SIMULATIONS,ALGORITHM,TRANSPORT,Finite-difference time-domain (FDTD),Schrodinger equation,Stability,Nonuniform,Alternating-direction hybrid,implicit-explicit (ADHIE)}},
  language     = {{eng}},
  pages        = {{113881:1--113881:19}},
  publisher    = {{Elsevier}},
  title        = {{An alternating-direction hybrid implicit-explicit finite-difference time-domain method for the Schrodinger equation}},
  url          = {{http://dx.doi.org/10.1016/j.cam.2021.113881}},
  volume       = {{403}},
  year         = {{2022}},
}

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