An Optimized Perfectly Matched Layer for the Schrödinger Equation

An Optimized Perfectly Matched Layer for the Schrödinger Equation

Year:    2011

Communications in Computational Physics, Vol. 9 (2011), Iss. 1 : pp. 147–179

Abstract

We derive a perfectly matched layer (PML) for the Schrödinger equation using a modal ansatz. We derive approximate error formulas for the modeling error from the outer boundary of the PML and the error from the discretization in the layer and show how to choose layer parameters so that these errors are matched and optimal performance of the PML is obtained. Numerical computations in 1D and 2D demonstrate that the optimized PML works efficiently at a prescribed accuracy for the zero potential case, with a layer of width less than a third of the de Broglie wavelength corresponding to the dominating frequency.

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.010909.010410a

Communications in Computational Physics, Vol. 9 (2011), Iss. 1 : pp. 147–179

Published online:    2011-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    33

Keywords:   

  1. Dispersion relations of the modes for open nonhomogeneous waveguides terminated by perfectly matched layers

    Zhu, Jianxin | Shen, Zheqi | Chen, Zengsi

    Journal of the Optical Society of America B, Vol. 29 (2012), Iss. 9 P.2524

    https://doi.org/10.1364/JOSAB.29.002524 [Citations: 4]
  2. On the non-equivalence of perfectly matched layers and exterior complex scaling

    Scrinzi, A. | Stimming, H.P. | Mauser, N.J.

    Journal of Computational Physics, Vol. 269 (2014), Iss. P.98

    https://doi.org/10.1016/j.jcp.2014.03.007 [Citations: 9]
  3. Time Dependent Phase Space Filters

    Introduction

    Soffer, Avy | Stucchio, Chris | Tran, Minh-Binh

    2023

    https://doi.org/10.1007/978-981-19-6818-1_1 [Citations: 0]
  4. CEM methods in R.F and microwave engineering in the context of parameters that influence the outcome of modeling

    Mohammed, Mohammed Ismail | Gebremicaheal, Mahder Girmay | Yohannes, Gebremichael

    The European Physical Journal Plus, Vol. 135 (2020), Iss. 10

    https://doi.org/10.1140/epjp/s13360-020-00854-2 [Citations: 0]
  5. Drift of Spectrally Stable Shifted States on Star Graphs

    Kairzhan, Adilbek | Pelinovsky, Dmitry E. | Goodman, Roy H.

    SIAM Journal on Applied Dynamical Systems, Vol. 18 (2019), Iss. 4 P.1723

    https://doi.org/10.1137/19M1246146 [Citations: 15]
  6. Accelerated convergence for Schrödinger equations with non-smooth potentials

    Kieri, Emil

    BIT Numerical Mathematics, Vol. 54 (2014), Iss. 3 P.729

    https://doi.org/10.1007/s10543-013-0465-x [Citations: 1]
  7. Review and Recent Developments on the Perfectly Matched Layer (PML) Method for the Numerical Modeling and Simulation of Elastic Wave Propagation in Unbounded Domains

    Pled, Florent | Desceliers, Christophe

    Archives of Computational Methods in Engineering, Vol. 29 (2022), Iss. 1 P.471

    https://doi.org/10.1007/s11831-021-09581-y [Citations: 36]
  8. Adaptive Absorbing Boundary Layer for the Nonlinear Schrödinger Equation

    Stimming, Hans Peter | Wen, Xin | Mauser, Norbert J.

    Computational Methods in Applied Mathematics, Vol. 24 (2024), Iss. 3 P.797

    https://doi.org/10.1515/cmam-2023-0096 [Citations: 1]
  9. Finite‐Difference Time‐Domain Simulation of Strong‐Field Ionization: A Perfectly Matched Layer Approach

    Kamban, Høgni C. | Christensen, Sigurd S. | Søndergaard, Thomas | Pedersen, Thomas G.

    physica status solidi (b), Vol. 257 (2020), Iss. 5

    https://doi.org/10.1002/pssb.201900467 [Citations: 3]
  10. Quantum Simulation for Quantum Dynamics with Artificial Boundary Conditions

    Jin, Shi | Li, Xiantao | Liu, Nana | Yu, Yue

    SIAM Journal on Scientific Computing, Vol. 46 (2024), Iss. 4 P.B403

    https://doi.org/10.1137/23M1563451 [Citations: 0]
  11. Application of the Reflectionless Discrete Perfectly Matched Layer for Acoustic Wave Simulation

    Gao, Yingjie | Zhu, Meng-Hua

    Frontiers in Earth Science, Vol. 10 (2022), Iss.

    https://doi.org/10.3389/feart.2022.883160 [Citations: 0]
  12. The influence of perforation on electrostatic and damping forces in thick SOI MEMS structures

    Ya'akobovitz, Assaf | Krylov, Slava

    Journal of Micromechanics and Microengineering, Vol. 22 (2012), Iss. 11 P.115006

    https://doi.org/10.1088/0960-1317/22/11/115006 [Citations: 6]
  13. Theoretical and quantitative evaluation of hybrid PML-ABCs for seismic wave simulation

    Wang, Yuhang | Zhang, Wei

    Earthquake Science, Vol. 35 (2022), Iss. 2 P.105

    https://doi.org/10.1016/j.eqs.2022.05.002 [Citations: 0]
  14. Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations

    Kieri, Emil | Kreiss, Gunilla | Runborg, Olof

    Advances in Applied Mathematics and Mechanics, Vol. 7 (2015), Iss. 6 P.687

    https://doi.org/10.4208/aamm.2013.m411 [Citations: 4]
  15. Nested dissection solver for transport in 3D nano-electronic devices

    Zhao, Y. | Hetmaniuk, U. | Patil, S. R. | Qi, J. | Anantram, M. P.

    Journal of Computational Electronics, Vol. 15 (2016), Iss. 2 P.708

    https://doi.org/10.1007/s10825-015-0778-x [Citations: 2]
  16. Perfectly matched layer for computing the dynamics of nonlinear Schrödinger equations by pseudospectral methods. Application to rotating Bose-Einstein condensates

    Antoine, Xavier | Geuzaine, Christophe | Tang, Qinglin

    Communications in Nonlinear Science and Numerical Simulation, Vol. 90 (2020), Iss. P.105406

    https://doi.org/10.1016/j.cnsns.2020.105406 [Citations: 19]
  17. A simple pseudospectral method for the computation of the time-dependent Dirac equation with Perfectly Matched Layers

    Antoine, Xavier | Lorin, Emmanuel

    Journal of Computational Physics, Vol. 395 (2019), Iss. P.583

    https://doi.org/10.1016/j.jcp.2019.06.020 [Citations: 13]
  18. Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations

    Antoine, Xavier | Bao, Weizhu | Besse, Christophe

    Computer Physics Communications, Vol. 184 (2013), Iss. 12 P.2621

    https://doi.org/10.1016/j.cpc.2013.07.012 [Citations: 273]
  19. LPSE: A 3-D wave-based model of cross-beam energy transfer in laser-irradiated plasmas

    Myatt, Jason F. | Shaw, John G. | Follett, Russell K. | Edgell, Dana H. | Froula, Dustin H. | Palastro, John P. | Goncharov, Valeri N.

    Journal of Computational Physics, Vol. 399 (2019), Iss. P.108916

    https://doi.org/10.1016/j.jcp.2019.108916 [Citations: 22]
  20. Optimizing perfectly matched layers in discrete contexts

    Modave, A. | Delhez, E. | Geuzaine, C.

    International Journal for Numerical Methods in Engineering, Vol. 99 (2014), Iss. 6 P.410

    https://doi.org/10.1002/nme.4690 [Citations: 36]
  21. Numerical comparison of high-order absorbing boundary conditions and perfectly matched layers for a dispersive one-dimensional medium

    Lancioni, Giovanni

    Computer Methods in Applied Mechanics and Engineering, Vol. 209-212 (2012), Iss. P.74

    https://doi.org/10.1016/j.cma.2011.10.015 [Citations: 21]
  22. Multi-hump potentials for efficient wave absorption in the numerical solution of the time-dependent Schrödinger equation

    Silaev, A A | Romanov, A A | Vvedenskii, N V

    Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 51 (2018), Iss. 6 P.065005

    https://doi.org/10.1088/1361-6455/aaa69c [Citations: 17]
  23. Quantum Anti-Reflection for Electron Transport

    Shin, Gwangjin | Park, Q-Han

    2024 Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR), (2024), P.1

    https://doi.org/10.1109/CLEO-PR60912.2024.10676452 [Citations: 0]
  24. Second-order imaginary differential operator for effective absorption in the numerical solution of the time-dependent Schrödinger equation

    Silaev, A. A. | Romanov, A. A. | Silaeva, M. V. | Vvedenskii, N. V.

    Physical Review A, Vol. 108 (2023), Iss. 1

    https://doi.org/10.1103/PhysRevA.108.013118 [Citations: 3]
  25. Nonuniform and Higher-order FDTD Methods for the Schrödinger Equation

    Decleer, Pieter | Van Londersele, Arne | Rogier, Hendrik | Vande Ginste, Dries

    Journal of Computational and Applied Mathematics, Vol. 381 (2021), Iss. P.113023

    https://doi.org/10.1016/j.cam.2020.113023 [Citations: 11]
  26. A reflectionless discrete perfectly matched layer

    Chern, Albert

    Journal of Computational Physics, Vol. 381 (2019), Iss. P.91

    https://doi.org/10.1016/j.jcp.2018.12.026 [Citations: 26]
  27. Numerical solution of the general coupled nonlinear Schrödinger equations on unbounded domains

    Li, Hongwei | Guo, Yue

    Physical Review E, Vol. 96 (2017), Iss. 6

    https://doi.org/10.1103/PhysRevE.96.063305 [Citations: 8]
  28. Eliminating Artificial Boundary Conditions in Time-Dependent Density Functional Theory Using Fourier Contour Deformation

    Kaye, Jason | Barnett, Alex | Greengard, Leslie | De Giovannini, Umberto | Rubio, Angel

    Journal of Chemical Theory and Computation, Vol. 19 (2023), Iss. 5 P.1409

    https://doi.org/10.1021/acs.jctc.2c01013 [Citations: 3]