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VLSI Design
Volume 6 (1998), Issue 1-4, Pages 313-319
http://dx.doi.org/10.1155/1998/38298

Numerically Absorbing Boundary Conditions for Quantum Evolution Equations

1Fachbereich Mathematik, TU-Berlin, Berlin D-10623, Germany
2Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Copyright © 1998 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Citations to this Article [98 citations]

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  • M Ehrhardt, and A Arnold, “Discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics,” Journal Of Computational Physics, vol. 145, no. 2, pp. 611–638, 1998. View at Publisher · View at Google Scholar
  • M Ehrhardt, “Discrete transparent boundary conditions for general Schrodinger-type equations,” Vlsi Design, vol. 9, no. 4, pp. 325–338, 1999. View at Publisher · View at Google Scholar
  • David Yevick, Tilmann Friese, and Frank Schmidt, “A Comparison of Transparent Boundary Conditions for the Fresnel Equation,” Journal of Computational Physics, vol. 168, no. 2, pp. 433–444, 2001. View at Publisher · View at Google Scholar
  • A Arnold, “Mathematical concepts of open quantum boundary conditions,” Transport Theory And Statistical Physics, vol. 30, no. 4-6, pp. 561–584, 2001. View at Publisher · View at Google Scholar
  • Naoufel Ben Abdallah, Olivier Pinaud, Christian Ringhofer, and Carl L. Gardner, “A comparison of resonant tunneling based on Schrödinger's equation and quantum hydrodynamics,” VLSI Design, vol. 15, no. 4, pp. 695–700, 2002. View at Publisher · View at Google Scholar
  • Naoufel Ben Abdallah, and Olivier Pinaud, “A mathematical model for the transient evolution of a resonant tunneling diode,” Comptes Rendus Mathematique, vol. 334, no. 4, pp. 283–288, 2002. View at Publisher · View at Google Scholar
  • Christian Lubich, and Achim Schädle, “Fast Convolution for Nonreflecting Boundary Conditions,” SIAM Journal on Scientific Computing, vol. 24, no. 1, pp. 161–182, 2002. View at Publisher · View at Google Scholar
  • N. Ben Abdallah, P. Degond, and I. M. Gamba, “Coupling one-dimensional time-dependent classical and quantum transport models,” Journal of Mathematical Physics, vol. 43, no. 1, pp. 1, 2002. View at Publisher · View at Google Scholar
  • Olivier Pinaud, “Transient simulations of a resonant tunneling diode,” Journal of Applied Physics, vol. 92, no. 4, pp. 1987, 2002. View at Publisher · View at Google Scholar
  • M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Physical Review E, vol. 67, no. 3, 2003. View at Publisher · View at Google Scholar
  • A. Arnold, M. Ehrhardt, and I. Sofronov, “A fast method to implement non-local discrete transparent boundary conditions for the Schrödinger equation,” Pamm, vol. 2, no. 1, pp. 424–425, 2003. View at Publisher · View at Google Scholar
  • Matthias Ehrhardt, and Ronald E. Mickens, “Solutions to the discrete Airy equation: application to parabolic equation calculations,” Journal of Computational and Applied Mathematics, vol. 172, no. 1, pp. 183–206, 2004. View at Publisher · View at Google Scholar
  • Xavier Antoine, Christophe Besse, and Vincent Mouysset, “Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions,” Mathematics of Computation, vol. 73, no. 248, pp. 1779–1799, 2004. View at Publisher · View at Google Scholar
  • Jérémie Szeftel, “ Design of Absorbing Boundary Conditions for Schrödinger Equations in $\mathbbR$ d ,” SIAM Journal on Numerical Analysis, vol. 42, no. 4, pp. 1527–1551, 2004. View at Publisher · View at Google Scholar
  • Curt A. Moyer, “Numerov extension of transparent boundary conditions for the Schrödinger equation in one dimension,” American Journal of Physics, vol. 72, no. 3, pp. 351, 2004. View at Publisher · View at Google Scholar
  • Naoufel Ben Abdallah, Florian MÉHats, and Olivier Pinaud, “On An Open Transient Schrödinger–Poisson System,” Mathematical Models and Methods in Applied Sciences, vol. 15, no. 05, pp. 667–688, 2005. View at Publisher · View at Google Scholar
  • Nicolae Carjan, Margarit Rizea, and Dan Strottman, “Improved boundary conditions for the decay of low lying metastable proton states in a time-dependent approach,” Computer Physics Communications, vol. 173, no. 1-2, pp. 41–60, 2005. View at Publisher · View at Google Scholar
  • A. Zisowsky, A. Arnold, M. Ehrhardt, and Th. Koprucki, “Discrete transparent boundary conditions for transientkp-Schrödinger equations with application to quantum heterostructures,” Zamm, vol. 85, no. 11, pp. 793–805, 2005. View at Publisher · View at Google Scholar
  • Houde Han, Jicheng Jin, and Xiaonan Wu, “A finite-difference method for the one-dimensional time-dependent schrödinger equation on unbounded domain,” Computers & Mathematics with Applications, vol. 50, no. 8-9, pp. 1345–1362, 2005. View at Publisher · View at Google Scholar
  • F. Delgado, J. G. Muga, D. G. Austing, and G. Garci´a-Caldero´n, “Resonant tunneling transients and decay for a one-dimensional double barrier potential,” Journal of Applied Physics, vol. 97, no. 1, pp. 013705, 2005. View at Publisher · View at Google Scholar
  • C.A. Moyer, “Numerical solution of the stationary state Schrodinger equation using transparent boundary conditions,” Computing in Science & Engineering, vol. 8, no. 4, pp. 32–40, 2006. View at Publisher · View at Google Scholar
  • Oleg Tolstikhin, “Siegert-state expansion for nonstationary systems: Coupled equations in the one-channel case,” Physical Review A, vol. 73, no. 6, 2006. View at Publisher · View at Google Scholar
  • Zhi-zhong Sun, and Xiaonan Wu, “The stability and convergence of a difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions,” Journal of Computational Physics, vol. 214, no. 1, pp. 209–223, 2006. View at Publisher · View at Google Scholar
  • Matthias Ehrhardt, and Andrea Zisowsky, “Fast calculation of energy and mass preserving solutions of Schrödinger–Poisson systems on unbounded domains,” Journal of Computational and Applied Mathematics, vol. 187, no. 1, pp. 1–28, 2006. View at Publisher · View at Google Scholar
  • Jiten C. Kalita, Puneet Chhabra, and Sudhanshu Kumar, “A semi-discrete higher order compact scheme for the unsteady two-dimensional Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 197, no. 1, pp. 141–149, 2006. View at Publisher · View at Google Scholar
  • Virginie Bonnaillie-Noël, Francis Nier, and Yassine Patel, “Computing the steady states for an asymptotic model of quantum transport in resonant heterostructures,” Journal of Computational Physics, vol. 219, no. 2, pp. 644–670, 2006. View at Publisher · View at Google Scholar
  • Zhi-zhong Sun, “The stability and convergence of an explicit difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions,” Journal of Computational Physics, vol. 219, no. 2, pp. 879–898, 2006. View at Publisher · View at Google Scholar
  • Xavier Antoine, Christophe Besse, and Stephane Descombes, “Artificial boundary conditions for one-dimensional cubic nonlinear Schrödinger equations,” SIAM Journal on Numerical Analysis, vol. 43, no. 6, pp. 2272–2293, 2006. View at Publisher · View at Google Scholar
  • Houde Han, Zhenli Xu, and Xiaonan Wu, “Adaptive absorbing boundary conditions for Schrödinger-type equations: Application to nonlinear and multi-dimensional problems,” Journal of Computational Physics, vol. 225, no. 2, pp. 1577–1589, 2007. View at Publisher · View at Google Scholar
  • Mehdi Dehghan, and Ali Shokri, “A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions,” Computers & Mathematics with Applications, vol. 54, no. 1, pp. 136–146, 2007. View at Publisher · View at Google Scholar
  • A. A. Zlotnik, “On the stability of the σ-scheme with transparent boundary conditions for parabolic equations,” Computational Mathematics and Mathematical Physics, vol. 47, no. 4, pp. 644–663, 2007. View at Publisher · View at Google Scholar
  • Matthias Ehrhardt, and Andrea Zisowsky, “Discrete non-local boundary conditions for split-step Padé approximations of the one-way Helmholtz equation,” Journal of Computational and Applied Mathematics, vol. 200, no. 2, pp. 471–490, 2007. View at Publisher · View at Google Scholar
  • Mehdi Dehghan, and Davoud Mirzaei, “The meshless local Petrov–Galerkin (MLPG) method for the generalized two-dimensional non-linear Schrödinger equation,” Engineering Analysis with Boundary Elements, vol. 32, no. 9, pp. 747–756, 2008. View at Publisher · View at Google Scholar
  • Andrea Zisowsky, and Matthias Ehrhardt, “Discrete artificial boundary conditions for nonlinear Schrödinger equations,” Mathematical and Computer Modelling, vol. 47, no. 11-12, pp. 1264–1283, 2008. View at Publisher · View at Google Scholar
  • Matthias Ehrhardt, “Discrete transparent boundary conditions for Schrödinger-type equations for non-compactly supported initial data,” Applied Numerical Mathematics, vol. 58, no. 5, pp. 660–673, 2008. View at Publisher · View at Google Scholar
  • Anton Arnold, and Maike Schulte, “Discrete Transparent Boundary Conditions For The Schrodinger Equation - A Compact Higher Order Scheme,” Kinetic And Related Models, vol. 1, no. 1, pp. 101–125, 2008. View at Publisher · View at Google Scholar
  • Anton Arnold, and Maike Schulte, “Transparent boundary conditions for quantum-waveguide simulations,” Mathematics and Computers in Simulation, vol. 79, no. 4, pp. 898–905, 2008. View at Publisher · View at Google Scholar
  • Mehdi Dehghan, and Davoud Mirzaei, “Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method,” International Journal for Numerical Methods in Engineering, vol. 76, no. 4, pp. 501–520, 2008. View at Publisher · View at Google Scholar
  • Naoufel Ben Abdallah, and Jihene Kefi-Ferhane, “Mathematical analysis of the two-band Schrödinger model,” Mathematical Methods in the Applied Sciences, vol. 31, no. 10, pp. 1131–1151, 2008. View at Publisher · View at Google Scholar
  • Houde Han, and Zhiwen Zhang, “An analysis of the finite-difference method for one-dimensional Klein–Gordon equation on unbounded domain,” Applied Numerical Mathematics, vol. 59, no. 7, pp. 1568–1583, 2009. View at Publisher · View at Google Scholar
  • M. Heinen, and H.-J. Kull, “Radiation boundary conditions for the numerical solution of the three-dimensional time-dependent Schrödinger equation with a localized interaction,” Physical Review E, vol. 79, no. 5, 2009. View at Publisher · View at Google Scholar
  • Akbar Mohebbi, and Mehdi Dehghan, “The use of compact boundary value method for the solution of two-dimensional Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 124–134, 2009. View at Publisher · View at Google Scholar
  • Bernard Ducomet, Alexander Zlotnik, and Ilya Zlotnik, “On A Family Of Finite-Difference Schemes With Approximate Transparent Boundary Conditions For A Generalized 1D Schrodinger Equation,” Kinetic And Related Models, vol. 2, no. 1, pp. 151–179, 2009. View at Publisher · View at Google Scholar
  • Xavier Antoine, Christophe Besse, and Pauline Klein, “Absorbing boundary conditions for the one-dimensional Schrödinger equation with an exterior repulsive potential,” Journal of Computational Physics, vol. 228, no. 2, pp. 312–335, 2009. View at Publisher · View at Google Scholar
  • Marie Doumic, Frédéric Duboc, François Golse, and Rémi Sentis, “Simulation of laser beam propagation with a paraxial model in a tilted frame,” Journal of Computational Physics, vol. 228, no. 3, pp. 861–880, 2009. View at Publisher · View at Google Scholar
  • Virginie Bonnaillie-Noë, Ali Faraj, and Francis Nier, “Simulation of resonant tunneling heterostructures: Numerical comparison of a complete Schrödinger-Poisson system and a reduced nonlinear model,” Journal of Computational Electronics, vol. 8, no. 1, pp. 11–18, 2009. View at Publisher · View at Google Scholar
  • A. A. Zlotnik, and A. V. Lapukhina, “Stability of a Numerov type finite–difference scheme with approximate transparent boundary conditions for the nonstationary Schrödinger equation on the half-axis,” Journal of Mathematical Sciences, vol. 169, no. 1, pp. 84–97, 2010. View at Publisher · View at Google Scholar
  • Jicheng Jin, and Xiaonan Wu, “Convergence of a finite element scheme for the two-dimensional time-dependent Schrödinger equation in a long strip,” Journal of Computational and Applied Mathematics, vol. 234, no. 3, pp. 777–793, 2010. View at Publisher · View at Google Scholar
  • Zhen F. Tian, and P.X. Yu, “High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation,” Computer Physics Communications, vol. 181, no. 5, pp. 861–868, 2010. View at Publisher · View at Google Scholar
  • Chunxiong Zheng, “Fast evaluation of exact transparent boundary condition for one-dimensional cubic nonlinear Schrödinger equation,” Frontiers of Mathematics in China, vol. 5, no. 3, pp. 589–606, 2010. View at Publisher · View at Google Scholar
  • Ansgar Jüngel, and Jan-Frederik Mennemann, “Time-dependent simulations of quantum waveguides using a time-splitting spectral method,” Mathematics and Computers in Simulation, vol. 81, no. 4, pp. 883–898, 2010. View at Publisher · View at Google Scholar
  • Jiwei Zhang, Zhizhong Sun, Desheng Wang, and Xiaonan Wu, “Analysis of high-order absorbing boundary conditions for the Schrödinger equation,” Communications in Computational Physics, vol. 10, no. 3, pp. 742–766, 2011. View at Publisher · View at Google Scholar
  • Simen Kvaal, “Multiconfigurational time-dependent Hartree method to describe particle loss due to absorbing boundary conditions,” Physical Review A, vol. 84, no. 2, 2011. View at Publisher · View at Google Scholar
  • I. A. Zlotnik, “Family of finite-difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a semi-infinite strip,” Computational Mathematics and Mathematical Physics, vol. 51, no. 3, pp. 355–376, 2011. View at Publisher · View at Google Scholar
  • Zhen Gao, and Shusen Xie, “Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations,” Applied Numerical Mathematics, vol. 61, no. 4, pp. 593–614, 2011. View at Publisher · View at Google Scholar
  • Tao Li, Guo-Dong Wang, and Zi-Wu Jiang, “A numerical method for two-dimensional schrödinger equation using MPS,” Communications in Computer and Information Science, vol. 243, no. 1, pp. 44–51, 2011. View at Publisher · View at Google Scholar
  • M. Amiri, and F. Ebrahimi, “Full counting statistics of one and two-electron systems in the presence of external gaussian white noise,” Journal of Applied Physics, vol. 110, no. 6, pp. 063712, 2011. View at Publisher · View at Google Scholar
  • Ilya Zlotnik, and Alexander Zlotnik, “Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation,” Kinetic and Related Models, vol. 5, no. 3, pp. 639–667, 2012. View at Publisher · View at Google Scholar
  • Eric Blayo, and Véronique Martin, “Building generalized open boundary conditions for fluid dynamics problems,” International Journal for Numerical Methods in Fluids, vol. 71, no. 4, pp. 506–521, 2012. View at Publisher · View at Google Scholar
  • Xinghui Cai, XinLi Sun, Zhn Li, Guoxun Ji, and Jiangren Lu, “The Element-Free Galerkin Method for Two-dimensional Schrödinger Equation,” Procedia Engineering, vol. 31, pp. 1108–1114, 2012. View at Publisher · View at Google Scholar
  • Julien Coatléven, “Transparent Boundary Conditions for Evolution Equations in Infinite Periodic Strips,” SIAM Journal on Scientific Computing, vol. 34, no. 3, pp. A1563–A1583, 2012. View at Publisher · View at Google Scholar
  • Julien Coatleven, “Transparent Boundary Conditions For Evolution Equations In Infinite Periodic Strips,” Siam Journal on Scientific Computing, vol. 34, no. 3, pp. A1563–A1583, 2012. View at Publisher · View at Google Scholar
  • Hang Xie, Feng Jiang, Heng Tian, Xiao Zheng, Yanho Kwok, Shuguang Chen, ChiYung Yam, YiJing Yan, and Guanhua Chen, “Time-dependent quantum transport: An efficient method based on Liouville-von-Neumann equation for single-electron density matrix,” The Journal of Chemical Physics, vol. 137, no. 4, pp. 044113, 2012. View at Publisher · View at Google Scholar
  • Xavier Antoine, Weizhu Bao, and Christophe Besse, “Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations,” Computer Physics Communications, 2013. View at Publisher · View at Google Scholar
  • S. Abbasbandy, H. Roohani Ghehsareh, and I. Hashim, “A meshfree method for the solution of two-dimensional cubic nonlinear Schrödinger equation,” Engineering Analysis with Boundary Elements, vol. 37, no. 6, pp. 885–898, 2013. View at Publisher · View at Google Scholar
  • Mehdi Dehghan, and Faezeh Emami-Naeini, “The Sinc-collocation and Sinc-Galerkin methods for solving the two-dimensional Schrödinger equation with nonhomogeneous boundary conditions,” Applied Mathematical Modelling, 2013. View at Publisher · View at Google Scholar
  • Jan-Frederik Mennemann, Ansgar Jüngel, and Hans Kosina, “Transient Schrödinger–Poisson simulations of a high-frequency resonant tunneling diode oscillator,” Journal of Computational Physics, vol. 239, pp. 187–205, 2013. View at Publisher · View at Google Scholar
  • A.H. Bhrawy, M.A. Abdelkawy, and Anjan Biswas, “Optical solitons in (1+1) and (2+1) dimensions,” Optik - International Journal for Light and Electron Optics, 2013. View at Publisher · View at Google Scholar
  • A. A. Zlotnik, and I. A. Zlotnik, “Finite element method with discrete transparent boundary conditions for the one-dimensional nonstationary Schrödinger equation,” Doklady Mathematics, vol. 86, no. 3, pp. 750–755, 2013. View at Publisher · View at Google Scholar
  • Mohammad Najafi, and Somayeh Arbabi, “Exact Solutions of Five Complex Nonlinear Schrödinger Equations by Semi-Inverse Variational Principle,” Communications in Theoretical Physics, vol. 62, no. 3, pp. 301–307, 2014. View at Publisher · View at Google Scholar
  • L.W. Zhang, Y.J. Deng, K.M. Liew, and Y.M. Cheng, “The improved complex variable element-free Galerkin method for two-dimensional Schrödinger equation,” Computers & Mathematics with Applications, 2014. View at Publisher · View at Google Scholar
  • V. F. Elesin, “Transient processes in two-barrier nanostructures,” Journal of Experimental and Theoretical Physics, vol. 118, no. 6, pp. 951–958, 2014. View at Publisher · View at Google Scholar
  • Jan-Frederik Mennemann, and Ansgar Jüngel, “Perfectly Matched Layers versus discrete transparent boundary conditions in quantum device simulations,” Journal of Computational Physics, 2014. View at Publisher · View at Google Scholar
  • Farhad Fakhar-Izadi, and Mehdi Dehghan, “A spectral element method using the modal basis and its application in solving second-order nonlinear partial differential equations,” Mathematical Methods in the Applied Sciences, 2014. View at Publisher · View at Google Scholar
  • L.W. Zhang, and K.M. Liew, “An element-free based solution for nonlinear Schrödinger equations using the ICVMLS-Ritz method,” Applied Mathematics and Computation, vol. 249, pp. 333–345, 2014. View at Publisher · View at Google Scholar
  • Jan-Frederik Mennemann, and Ansgar Juengel, “Perfectly Matched Layers versus discrete transparent boundary conditions in quantum device simulations,” Journal of Computational Physics, vol. 275, pp. 1–24, 2014. View at Publisher · View at Google Scholar
  • Behzad Ghanbari, “An Analytical Study for (2+1)-Dimensional Schrodinger Equation,” Scientific World Journal, 2014. View at Publisher · View at Google Scholar
  • V. V. Kapaev, “Nonlinear theory of the narrow-band generation and detection of terahertz radiation in resonant tunneling heterostructures,” Journal of Experimental and Theoretical Physics, vol. 121, no. 2, pp. 303–320, 2015. View at Publisher · View at Google Scholar
  • Ratikanta Behera, and Mani Mehra, “A Dynamic Adaptive Wavelet Method for Solution of the Schrodinger Equation,” Journal of Multiscale Modelling, pp. 1450001, 2015. View at Publisher · View at Google Scholar
  • Alexander Zlotnik, “The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis,” Kinetic and Related Models, vol. 8, no. 3, pp. 587–613, 2015. View at Publisher · View at Google Scholar
  • Alexander Zlotnik, and Ilya Zlotnik, “The High Order Method with Discrete TBCs for Solving the Cauchy Problem for the 1D Schrodinger Equation,” Computational Methods In Applied Mathematics, vol. 15, no. 2, pp. 233–245, 2015. View at Publisher · View at Google Scholar
  • Olivier Pinaud, “Absorbing layers for the Dirac equation,” Journal of Computational Physics, 2015. View at Publisher · View at Google Scholar
  • Mohammad Najafi, and Somayeh Arbabi, “Traveling wave solutions for nonlinear Schrödinger equations,” Optik - International Journal for Light and Electron Optics, vol. 126, no. 23, pp. 3992–3997, 2015. View at Publisher · View at Google Scholar
  • Shengliang Zhang, and Siyan Chen, “A meshless symplectic method for two-dimensional Schrödinger equation with radial basis functions,” Computers & Mathematics with Applications, 2016. View at Publisher · View at Google Scholar
  • Ahmad Golbabai, and Ahmad Nikpour, “Computing a numerical solution of two dimensional non-linear Schrödinger equation on complexly shaped domains by RBF based differential quadrature method,” Journal of Computational Physics, 2016. View at Publisher · View at Google Scholar
  • Lei Bian, Gang Pang, Shaoqiang Tang, and Anton Arnold, “ALmost EXact boundary conditions for transient Schrödinger-Poisson system,” Journal of Computational Physics, 2016. View at Publisher · View at Google Scholar
  • Rong-Jun Cheng, and Yu-Min Cheng, “Solving unsteady Schrödinger equation using the improved element-free Galerkin method,” Chinese Physics B, vol. 25, no. 2, pp. 020203, 2016. View at Publisher · View at Google Scholar
  • Somayeh Arbabi, and Mohammad Najafi, “Exact solitary wave solutions of the complex nonlinear Schrödinger equations,” Optik - International Journal for Light and Electron Optics, 2016. View at Publisher · View at Google Scholar
  • Alexander Zlotnik, and Ilya Zlotnik, “Remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrodinger equation on the half-axis,” Russian Journal Of Numerical Analysis And Mathematical Modelling, vol. 31, no. 1, pp. 51–64, 2016. View at Publisher · View at Google Scholar
  • Lei Bian, Songsong Ji, Gang Pang, and Shaoqiang Tang, “Accurate boundary treatment for transient Schrödinger equation under polar coordinates,” Computers & Mathematics with Applications, 2016. View at Publisher · View at Google Scholar
  • Christophe Besse, Pascal Noble, and David Sanchez, “Discrete transparent boundary conditions for the mixed KDV-BBM equation,” Journal of Computational Physics, 2017. View at Publisher · View at Google Scholar
  • Mostafa Abounouh, Hassan Al Moatassime, and Abderrazak Chrifi, “Existence of global attractor for one-dimensional weakly damped nonlinear Schrödinger equation with Dirac interaction and artificial boundary condition in half-line,” Advances in Difference Equations, vol. 2017, no. 1, 2017. View at Publisher · View at Google Scholar
  • Dongdong He, and Kejia Pan, “An unconditionally stable linearized CCD–ADI method for generalized nonlinear Schrödinger equations with variable coefficients in two and three dimensions,” Computers & Mathematics with Applications, 2017. View at Publisher · View at Google Scholar
  • Andrej Bugajev, Raimondas Čiegis, Rima Kriauzienė, Teresė Leonavičienė, and Julius Žilinskas, “On the Accuracy of Some Absorbing Boundary Conditions for the Schrödinger Equation,” Mathematical Modelling and Analysis, pp. 1–16, 2017. View at Publisher · View at Google Scholar
  • Elyas Shivanian, and Ahmad Jafarabadi, “An efficient numerical technique for solution of two-dimensional cubic nonlinear Schrödinger equation with error analysis,” Engineering Analysis with Boundary Elements, vol. 83, pp. 74–86, 2017. View at Publisher · View at Google Scholar
  • Dan Li, Jiwei Zhang, and Zhimin Zhang, “The Numerical Computation of the Time Fractional Schrödinger Equation on an Unbounded Domain,” Computational Methods in Applied Mathematics, no. 0, 2017. View at Publisher · View at Google Scholar
  • Mostafa Abounouh, Hassan Al Moatassime, and Abderrazak Chrifi, “Artificial boundary condition for one-dimensional nonlinear Schrödinger problem with Dirac interaction: existence and uniqueness results,” Boundary Value Problems, vol. 2018, no. 1, 2018. View at Publisher · View at Google Scholar
  • Christophe Besse, Benoît Mésognon-Gireau, and Pascal Noble, “Artificial boundary conditions for the linearized Benjamin–Bona–Mahony equation,” Numerische Mathematik, 2018. View at Publisher · View at Google Scholar