Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-30T20:15:54.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Implementation of fictitious absorbing layers with deceleration effects for one-dimensional Schrödinger equations

Published online by Cambridge University Press:  01 October 2020

Hitoshi Kanai Open the ORCID record for Hitoshi Kanai [Opens in a new window]  and
Tomo Tatsuno
Hitoshi Kanai*
Affiliation:
Department of Computer and Network Engineering, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo182-8585, Japan
Tomo Tatsuno
Affiliation:
Department of Computer and Network Engineering, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo182-8585, Japan
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Absorbing boundary conditions or layers are used in simulations to reduce or eliminate wave reflections from the boundary; one of the most widely used absorbing layers is Berenger's perfectly matched layer (PML). In this paper, PML is extended to a compound absorbing layer which has multiple effects of damping and deceleration, and is applied to linear and nonlinear Schrödinger equations. The deceleration extends the time to damp out the modes with higher phase velocities, leading to remarkably reduced total reflection for dispersive waves. By invoking the two effects independently, the flexibility and performance are enhanced. Since this method is based on the WKB formalism, it requires an absorbing layer of a moderate size.

Keywords

plasma simulationplasma waves
Type
Research Article
Information
Journal of Plasma Physics , Volume 86 , Issue 5 , October 2020 , 905860507
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Antoine, X., Arnold, A., Besse, C. & Schädle, M. E. A. 2008 A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Commun. Comput. Phys. 4, 729.Google Scholar
Antoine, X. & Besse, C. 2003 Unconditionally stable discretization schemes of non-reflecting boundary condition for the one-dimensional Schrödinger equation. J. Comput. Phys. 188, 157.CrossRefGoogle Scholar
Baskakov, V. A. & Popov, A. V. 1991 Implementation of transparent boundaries for numerical solution of the Schrödinger equation. Wave Motion 14, 123.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers I. Springer ScienceCrossRefGoogle Scholar
Berenger, J.-P. 1994 A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185.CrossRefGoogle Scholar
Birn, J., Drake, J. F., Shay, M. A., Rogers, B. N., Denton, R. E., Hesse, M., Kuznetsova, M., Ma, Z. W., Bhattacharjee, A., Otto, A., et al. 2001 Geospace environmental modeling (GEM) magnetic reconnection challenge. J. Geophys. Res. 106, 3715.CrossRefGoogle Scholar
Chew, W. C. & Weedon, W. H. 1994 A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates. Micro. Opt. Tech. Lett. 7, 599.CrossRefGoogle Scholar
Colonius, T. 2004 Modeling artificial boundary conditions for compressible flow. Annu. Rev. Fluid Mech. 36, 315.CrossRefGoogle Scholar
Dalrymple, R. & Martin, P. 1992 Perfect boundary conditions for parabolic water-wave models. Proc. R. Soc. Lond. A 437, 41.Google Scholar
Daughton, W., Scudder, J. & Karimabadi, H. 2006 Fully kinetic simulations of undriven magnetic reconnection with open boundary conditions. Phys. Plasmas 13, 072101.CrossRefGoogle Scholar
Dedner, A., Kröner, D., Sofronov, I. L. & Wesenberg, M. 2001 Transparent boundary conditions for MHD simulations in stratified atmospheres. J. Comput. Phys. 171, 448.CrossRefGoogle Scholar
Ehrhardt, M. & Zheng, C. 2008 Exact artificial boundary conditions for problems with periodic structures. J. Comput. Phys. 227, 6877.CrossRefGoogle Scholar
Fromang, S. & Nelson, R. P. 2006 Global MHD simulations of stratified and turbulent protoplanetary discs I. Model properties. Astron. Astrophys. 457, 343.CrossRefGoogle Scholar
Hanasoge, S. M., Komatitsch, D. & Gizon, L. 2010 An absorbing boundary formulation for the stratified, linearized, ideal MHD equations based on an unsplit, convolutional perfectly matched layer. Astron. Astrophys. 522, A87.CrossRefGoogle Scholar
Hasegawa, A. & Uberoi, C. 1982 The Alfvén wave. DOE critical review series. US DOE Crit. Rev. Ser. DOE/TIC-11197 (1982).Google Scholar
Hellums, J. & Frensley, W. 1994 Non-Markovian open-system boundary conditions for the time-dependent Schrödinger equation. Phys. Rev. B 49, 2904.CrossRefGoogle ScholarPubMed
Horiuchi, R., Pei, W. & Sato, T. 2001 Collisionless driven reconnection in an open system. Earth Planets Space 53, 439.CrossRefGoogle Scholar
Hu, F. Q. 1996 On absorbing boundary conditions for linearized Euler equations by a perfectly matched layer. J. Comput. Phys. 129, 201.CrossRefGoogle Scholar
Hu, F. Q., Li, X. & Lin, D. 2008 Absorbing boundary conditions for nonlinear Euler and Navier–Stokes equations based on the perfectly matched layer technique. J. Comput. Phys. 227, 4398.CrossRefGoogle Scholar
Huang, Y.-M. & Bhattacharjee, A. 2010 Scaling laws of resistive magnetohydrodynamic reconnection in the high-Lundquist-number, plasmoid-unstable regime. Phys. Plasmas 17, 062104.CrossRefGoogle Scholar
Kako, T. & Ohi, Y. 2005 Perfectly matched layer for numerical method in unbounded region. RIMS Kokyuroku 1441, 187.Google Scholar
Kingsep, A. S., Chukbar, K. V. & Yan'kov, V. V. 1990 Electron magnetohydrodynamics. Rev. Plasma Phys. 16, 243.Google Scholar
LeVeque, R. J. 1992 Numerical Methods for Conservation Laws. 2nd edn. Birkhäuser.CrossRefGoogle Scholar
Lin, D., Li, X. & Hu, F. 2009 PML absorbing boundary condition for nonlinear Euler equations in primitive variables. In 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition.CrossRefGoogle Scholar
Meier, E. T., Glasser, A. H. & Shumlak, U. 2012 Modeling open boundaries in dissipative MHD simulation. J. Comput. Phys. 231, 2963.CrossRefGoogle Scholar
Mur, G. 1981 Absorbing boundary conditions for the finite-difference approximation of time-domain electromagnetic-field equations. IEEE Trans. Electromagn. Compat. 23, 377.CrossRefGoogle Scholar
Nataf, F. 2006 A new approach to perfectly matched layers for the linearized Euler system. J. Comput. Phys. 214, 757.CrossRefGoogle Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 2007 Numerical Recipes in C. Cambridge University Press.Google Scholar
Ruprecht, D., Achädle, A. & Schmidt, F. 2013 Transparent boundary conditions based on the pole condition for time-dependent, two-dimensional problems. Numer. Meth. Partial Differ. Eqn. 29, 1367.CrossRefGoogle Scholar
Ruprecht, D., Schädle, A., Schmidt, F. & Zschiedrich, L. 2008 Transparent boundary conditions for time-dependent problems. SIAM J. Sci. Comput. 30, 2358.CrossRefGoogle Scholar
Sato, T. & Hayashi, T. 1979 Externally driven magnetic reconnection and a powerful magnetic energy converter. Phys. Fluids 22, 1189.Google Scholar
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Quataert, E. & Tatsuno, T. 2009 Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. Astrophys. J. Suppl. 182, 310.CrossRefGoogle Scholar
Scholer, M. 1989 Undriven magnetic reconnection in an isolated current sheet. J. Geophys. Res. 94, 8805.CrossRefGoogle Scholar
Shibata, K. & Uchida, Y. 1985 A magnetodynamic mechanism for the formation of astrophysical jets. I – dynamical effects of the relaxation of nonlinear magnetic twists. Publ. Astron. Soc. Jpn. 37, 31.Google Scholar
Sonoda, D., Ohi, Y. & Tatsuno, T. 2017 Implementation of fictitious absorbing layer for two-dimensional vorticity equations based on the perfectly matched layer. Trans. Japan Soc. Indust. Appl. Maths 27, 84 [in Japanese].Google Scholar
Stone, J. M. & Norman, M. L. 1992 ZEUS-2D: a radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. II. The magnetohydrodynamic algorithms and tests. Astrophys. J. 80, 791.CrossRefGoogle Scholar
Strang, G. 1968 On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506.CrossRefGoogle Scholar
Tourrette, T. & Halpern, L. 2001 Absorbing Boundaries and Layers, Domain Decomposition Methods. Nova Science.Google Scholar
Tretler, R., Tatsuno, T. & Hosokawa, K. 2020 Plasma sheet thinning due to loss of near-Earth magnetotail plasma. arXiv:2006.04289.Google Scholar
Tsynkov, S. V. 1998 Numerical solution of problems on unbounded domains. A review. Appl. Numer. Maths 27, 465.CrossRefGoogle Scholar
Yan, M., Lee, M. C. & Priest, E. R. 1992 Fast magnetic reconnection with small shock angles. J. Geophys. Res. 97, 8277.CrossRefGoogle Scholar
Zheng, C. 2006 Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equation. J. Comput. Phys. 215, 552.CrossRefGoogle Scholar
Zheng, C. 2007 A perfectly matched layer approach to nonlinear Schrödinger wave equations. J. Comput. Phys. 227, 537.CrossRefGoogle Scholar
Zheng, C. 2008 An exact absorbing boundary condition for the Schrödinger equation with sinusoidal potentials at infinity. Commun. Comput. Phys. 3, 641.Google Scholar