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High Order Scheme for Schrödinger Equation with Discontinuous Potential I: Immersed Interface Method

Published online by Cambridge University Press:  28 May 2015

Hao Wu*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing, 10084, China
*
*Corresponding author.Email address:[email protected]
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Abstract

The immersed interface method is modified to compute Schrödinger equation with discontinuous potential. By building the jump conditions of the solution into the finite difference approximation near the interface, this method can give at least second order convergence rate for the numerical solution on uniform cartesian grids. The accuracy of this algorithm is tested via several numerical examples.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Ben Abdallah, N., On a multidimensional Schrödinger-Poisson scattering model for semiconductors, Joural of Mathematical Physics, 41(2000), no. 7, 42414261.CrossRefGoogle Scholar
[2]Ben Abdallah, N., Degond, P. and Markowich, P.A., On a one-dimensional Schrödinger-Poisson scattering model, Zeitschrift für Angewandte Mathematik und Physik., 48(1997), 135155.CrossRefGoogle Scholar
[3]Ben Abdallah, N., Negulescu, C., Mouis, M. and Polizzi, E., Simulation Schemes in 2D Nanoscale MOSFETs: A WKB Based Method, Journal of Computational Electronics, 3(2004), 397400.Google Scholar
[4]Ben Abdallah, N. and Pinaud, O., Multiscale simulation of transport in an open quantum system: Resonances and WKB interpolation, Journal of Computational Physics, 213(2006), no. 1, 288310.CrossRefGoogle Scholar
[5]Ben Abdallah, N. and Wu, H., A generalized stationary algorithm for resonant tunneling: multi-mode approximation and high dimension, Communications in Computational Physics, 10(2011), no. 4, 882898.CrossRefGoogle Scholar
[6]Adams, L. and Li, Z.L., The immersed interface/multigrid methods for interface problems, SIAM Journal on Scientific Computing, 24(2002), no. 2, 463479.CrossRefGoogle Scholar
[7]Bao, W.Z., Jin, S. and P.Markowich, A., On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, Journal of Computational Physics, 175(2002), 487524.CrossRefGoogle Scholar
[8]Bao, W.Z., Jin, S. and Markowich, P.A., Numerical studies of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regime, SIAM Journal on Scientific Computing, 25(2003), no. 1, 2764.CrossRefGoogle Scholar
[9]Datta, S., Electronic Transport in Mesoscopic Systems, Cambridge University Press, 1995.CrossRefGoogle Scholar
[10]Datta, S., Quantum Transport: Atom to Transistor, Cambridge University Press, 2005.CrossRefGoogle Scholar
[11]Harrison, P., Quantum wells, wires and dots: theoretical and computational physics of semiconductor nanostructures, Jonh Wiley & Sons, New York, 2000.Google Scholar
[12]Huang, H.X. and Li, Z.L., Convergence analysis of the immersed interface method, IMA Journal of Numerical Analysis, 19(1999), 583608.CrossRefGoogle Scholar
[13]Jahnke, T. and Lubich, C., Error bounds for exponential operator splittings, BIT, 40(2000), no. 4, 735744.CrossRefGoogle Scholar
[14]Jin, S., Markowich, P. and Sparber, C., Mathematical and computational methods for semiclas-sical Schrödinger equations, Acta Numerica, to appear in 2011.CrossRefGoogle Scholar
[15]Lai, M.C. and Li, Z.L., The immersed interface method for the Navier-Stokes equations with singular forces, Journal of Computational Physics, 171(2001), 822842.CrossRefGoogle Scholar
[16]Lee, L. and Leveque, R.J., An immersed interface method for incompressible Navier-Stokes equations, SIAM Journal on Scientific Computing, 25(2003), no. 3, 832856.CrossRefGoogle Scholar
[17]Lent, C. and Kirkner, D., The quantum transmitting boundary method, Journal of Applied Physics, 67(1990), 63536359.CrossRefGoogle Scholar
[18]Leveque, R.J. and Li, Z.L., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM Journal on Numerical Analysis, 31(1994), no. 4, 10191044.CrossRefGoogle Scholar
[19]Leveque, R.J. and Z.Li, L., Immersed interface method for Stokes flow with elastic boundaries or surface tension, SIAM Journal on Scientific Computing, 18(1997), 709735.CrossRefGoogle Scholar
[20]Li, Z.L., A note on immersed interface methods for three dimensional elliptic equations, Computers & Mathematics with Applications, 31(1996), no. 3, 917.CrossRefGoogle Scholar
[21]Li, Z.L., A fast iterative algorithm for elliptic interface problems, SIAM Journal on Numerical Analysis, 35(1998), no. 1, 230254.CrossRefGoogle Scholar
[22]Li, Z.L., The immersed interface method using a finite element formulation, Applied Numerical Mathematics, 27(1998), 253267.CrossRefGoogle Scholar
[23]Li, Z.L., An overview of the immersed interface method and its applications, Taiwanese Journal of Mathematics, 7(2003), no. 1, 149.CrossRefGoogle Scholar
[24]Li, Z.L. and Ito, K., Maximum principle preserving schemes for interface problems with discontinuous coefficients, SIAM Journal on Scientific Computing, 23(2001), no. 1, 339361.CrossRefGoogle Scholar
[25]Li, Z.L. and Mayo, A., ADI methods for heat equations with discontinuties along an arbitrary interface, Proceedings of Symposia in Applied Mathematics, 48(1993), 311315.CrossRefGoogle Scholar
[26]Lu, T. and Cai, W., A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schrödinger-Poisson equations with discontinuous potentials, Journal of Computational and Applied Mathematics, 220(2008), 588614.CrossRefGoogle Scholar
[27]Lu, T., Cai, W. and Zhang, P.W., Conservative local discontinuous Galerkin methods for time dependent Schrödinger equation, International Journal of numerical analysis and Modeling, 2(2005), no. 1, 7584.Google Scholar
[28]Markowich, P.A., Pietra, P. and Pohl, C., Numerical approximation of quadratic obaservables of Schrödinger-type equations in the semiclassical limit, Numerische Mathematik, 81(1999), no. 4, 595630.CrossRefGoogle Scholar
[29]Markowich, P.A., Pietra, P., Pohl, C. and Stimming, H.P., A wigner-measure analysis of the Dufort-Frankel scheme for the Schrödinger equation, SIAM Journal on Numerical Analysis, 40(2002), no. 4, 12811310.CrossRefGoogle Scholar
[30]Markowich, P.A., Ringhofer, C. and Schmeiser, C., Semiconductor Equations, Springer Verlag Wien, 1990.CrossRefGoogle Scholar
[31]Mizuta, H. and Tanou, T., The Physics and Applications of Resonant Tunnelling Diodes, Cambridge University Press, 1995.CrossRefGoogle Scholar
[32]Peskin, C.S., The immersed boundary method, Acta Numerica, 11(2002), 479517.CrossRefGoogle Scholar
[33]Piraux, J. and Lombard, B., A new interface method for hyperbolic problems with discontinuous coefficients. One-dimensional acoustic example, Journal of Computational Physics, 168(2001), 227248.CrossRefGoogle Scholar
[34]Polizzi, E. and Ben Abdallah, N., Subband decomposition approach for the simulation of quantum electron transport in nanostructures, Journal of Computational Physics 202(2005), no. 1, 150180.CrossRefGoogle Scholar
[35]Tornberg, A.K. and Engquist, B., Numerical approximations of singular source terms in differential equations, Journal of Computational Physics, 200(2004), no. 2, 462488.CrossRefGoogle Scholar
[36]Wang, W. and Shu, C.W., The WKB local discontinuous Galerkin method for the simulation of Schrödinger equation in a resonant tunneling diode, Journal of Scientific Computing, 40(2009), no. 1-3, 360374.CrossRefGoogle Scholar
[37]Weisbuch, C. and Vinter, B., Quantum Semiconductor Structures: Fundamentals and Applications, Academic Press, 1991.CrossRefGoogle Scholar
[38]Wu, L.X., Dufort-Frankel-type methods for linear and nonlinear Schrödinger equations, SIAM Journal on Numerical Analysis, 33(1996), no. 4, 15261533.CrossRefGoogle Scholar
[39]Wu, H., High order scheme for Schrödinger equation with discontinuous potential II: novel methods for dynamic problems in the semiclassical regime, in preparation.Google Scholar
[40]Xu, Y. and Shu, C.W., Local discontinuous Galerkin methods for nonlinear Schrödinger equations, Journal of Computational Physics, 205(2005), 7297.CrossRefGoogle Scholar
[41]Zhang, C.M. and Leveque, R.J., The immersed interface method for acoustic wave equations with discontinuous coefficients, Wave Motion, 25(1997), 237263.CrossRefGoogle Scholar