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Accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations on rectangular domains

Songsong Ji, Yibo Yang, Gang Pang, Xavier Antoine
2018 Computer Physics Communications  
The aim of this paper is to design some accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations in rectangular domains.  ...  The Laplace transform in time and discrete Fourier transform in space are applied to get the Green's functions of the semi-discretized equations in unbounded domains with single-source.  ...  Antoine thanks the support of the LIASFMA (funding from the University of Lorraine) and the French ANR grant Bond (ANR-13-BS01-0009-01).  ... 
doi:10.1016/j.cpc.2017.09.019 fatcat:thiwfy5bkfgq5k3b3kvuyfcmqa

Absorbing boundary conditions for the time-dependent Schrödinger-type equations in R^3 [article]

Xiaojie Wu, Xiaotao Li
2019 arXiv   pre-print
Absorbing boundary conditions are presented for three-dimensional time-dependent Schr\"odinger-type of equations as a means to reduce the cost of the quantum-mechanical calculations.  ...  This approach can be applied to domains with arbitrary geometry. The stability of the zeroth order and first order absorbing boundary conditions is proved.  ...  Namely, we first consider the semi-discrete approximation of (1) in the entire domain, and regard it as a full/exact model. This has been done for the acoustic wave equations in [53] .  ... 
arXiv:1908.02456v2 fatcat:4wanqzhh3ncxnlyz2usgsjaioq

Adaptive absorbing boundary conditions for Schrödinger-type equations: Application to nonlinear and multi-dimensional problems

Zhenli Xu, Houde Han, Xiaonan Wu
2007 Journal of Computational Physics  
We propose an adaptive approach in picking the wave-number parameter of absorbing boundary conditions for Schrödinger-type equations.  ...  Numerical examples of nonlinear Schrödinger equations in one- and two dimensions are presented to demonstrate the properties of the discussed absorbing boundary conditions.  ...  The first author thanks Prof. J. Deng for helpful discussion. The authors also thank the anonymous referees for many comments and suggestions in the earlier draft of this paper.  ... 
doi:10.1016/j.jcp.2007.02.004 fatcat:ezwiqk2jsvbgnovxucmxli3yty

Fractional calculus and non-reflecting boundary conditions in wave propagation

Xavier Antoine
2008 Journal Europeen des Systemes Automatises  
A special attention is dedicated to the construction and discretization of truncation boundary conditions to model the exterior infinite domain arising in such situations.  ...  The aim of this paper is to provide a comprehensive introduction to the modelling of wave phenomena by linear and nonlinear Schrödingertype equations.  ...  It is proved that the resulting schemes for the linear and nonlinear Schrödinger equations in an infinite medium are unconditionnally stable.  ... 
doi:10.3166/jesa.42.895-910 fatcat:yhqaishqovfhjdec6ydhvwyswi

A friendly review of absorbing boundary conditions and perfectly matched layers for classical and relativistic quantum waves equations

X. Antoine, E. Lorin, Q. Tang
2017 Molecular Physics  
The aim of this paper is to describe concisely the recent theoretical and numerical developments concerning absorbing boundary conditions and perfectly matched layers for solving classical and relativistic  ...  The equations considered in this paper are the Schrödinger, Klein-Gordon and Dirac equations.  ...  Acknowledgement The three authors warmly thanks Professor Olivier Pinaud (Colorado State University) for discussions about PMLs for the Dirac equation and for providing the simulations from Figure 8 .  ... 
doi:10.1080/00268976.2017.1290834 fatcat:mv5myuhirveyhmborgq4dhgkva

A pseudo-spectral splitting method for linear dispersive problems with transparent boundary conditions [article]

Lukas Einkemmer, Alexander Ostermann, Mirko Residori
2019 arXiv   pre-print
The goal of the present work is to solve a linear dispersive equation with variable coefficient advection on an unbounded domain.  ...  In this setting, transparent boundary conditions are vital to allow waves to leave (or even re-enter) the, necessarily finite, computational domain.  ...  A previous work which combines splitting scheme with absorbing boundary conditions for the Schrödinger equation can be found in [4] .  ... 
arXiv:1904.10751v2 fatcat:l5fhsiq4tjfdfbbsyhnzpz5g6u

A fast and robust computational method for the ionization cross sections of the driven Schroedinger equation using an O(N) multigrid-based scheme [article]

Siegfried Cools, Wim Vanroose
2015 arXiv   pre-print
in a fast and scalable solution method for the complex-valued Schroedinger break-up problem for any energy regime.  ...  This paper improves the convergence and robustness of a multigrid-based solver for the cross sections of the driven Schroedinger equation.  ...  Acknowledgments This research is funded by the Research Council of the University of Antwerp. The authors additionally thank Bram Reps for fruitful discussions on the subject.  ... 
arXiv:1412.1953v4 fatcat:hmoreznqcjdibiim5qt3wbodei

High-Order Local Absorbing Boundary Conditions for Heat Equation in Unbounded Domains

X. N. Wu, J. W. Zhang
2010 Journal of Computational Mathematics  
This paper is devoted to the high-order local absorbing boundary conditions (ABCs) for heat equation.  ...  Enquist and Majda [6] proposed a whole family of local boundary conditions for wave equation, which not only resulted in stable difference approximation, but also minimized the unphysical reflections.  ...  We thank the referees for their careful reading of the paper and for suggestions that led to an improved version of the paper.  ... 
doi:10.4208/jcm.1004-m3195 fatcat:khpxvbq535cavj3gqqazvrsnze

Continuum and Discrete Initial-Boundary Value Problems and Einstein's Field Equations

Olivier Sarbach, Manuel Tiglio
2012 Living Reviews in Relativity  
The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial-boundary value problems arising from hyperbolic partial differential equations and to  ...  discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein's equations, and we discuss multi-domain high-order finite  ...  evolution equations on the unbounded domain R .  ... 
doi:10.12942/lrr-2012-9 pmid:28179838 pmcid:PMC5256023 fatcat:wa5g3wgzu5e4jpjyhyaxiffgw4

Semi-global approach for propagation of the time-dependent Schrödinger equation for time-dependent and nonlinear problems

Ido Schaefer, Hillel Tal-Ezer, Ronnie Kosloff
2017 Journal of Computational Physics  
A detailed exposition of highly efficient and accurate method for the propagation of the time-dependent Schr\"odinger equation is presented.  ...  In this paper, a thorough presentation of the basic principles of the propagator is given. We give also a special emphasis on the details of the numerical implementation of the method.  ...  Acknowledgements We want to thank Christiane Koch, Lutz Marder and Erik Torrontegui for their interest and help in this project.  ... 
doi:10.1016/j.jcp.2017.04.017 fatcat:ygjzmw5m7nhwdhm4g46kkdhvki

A Numerical Study of Quantum Decoherence

Riccardo Adami, Claudia Negulescu
2012 Communications in Computational Physics  
The time dependent two-particle Schrödinger equation is solved by means of a time-splitting method.  ...  The damping undergone by the non-diagonal terms of the heavy particle density matrix is estimated numerically as well as the error in the Joos-Zeh approximation formula.  ...  The authors would like to thank the Laboratoire LATP of the University of Provence and the University of Milano Bicocca for the support and the kind hospitality.  ... 
doi:10.4208/cicp.011010.010611a fatcat:v72jd7duyve7bp5tc2vx6ptmda

Numerical Stabilization of the Melt Front for Laser Beam Cutting [chapter]

Torsten Adolph, Willi Schönauer, Markus Niessen, Wolfgang Schulz
2010 Numerical Mathematics and Advanced Applications 2009  
The resulting linear systems are solved alternatively by the Multigrid method and Krylov subspace methods. We treat two types of boundary conditions: Dirichlet and natural boundary condition.  ...  Numerical experiments will be done for simple linear problems, the Burger's equation, and for the Euler equations with high Mach numbers. 24 This contribution deals with the modeling of soft biological  ...  In this paper we derive well-posed boundary and interface conditions as well as stability for the semi-discrete problem before implementing using high order finite difference methods.  ... 
doi:10.1007/978-3-642-11795-4_6 fatcat:nx4nvuxaxfbcdjknopny53ck5e

Runge-Kutta convolution quadrature and FEM-BEM coupling for the time-dependent linear Schrödinger equation

Jens Markus Melenk, Alexander Rieder
2017 Journal of Integral Equations and Applications  
We propose a numerical scheme to solve the time dependent linear Schrödinger equation.  ...  In this paper we analyze the resulting fully discrete scheme in terms of stability and convergence rate. Numerical experiments confirm the theoretical findings.  ...  Acknowledgments: Financial support by the Austrian Science Fund (FWF) through the doctoral school "Dissipation and Dispersion in Nonlinear PDEs" (project W1245, A.R.).  ... 
doi:10.1216/jie-2017-29-1-189 fatcat:xlrcwfugbjb3djvhv3ksjjxjee

VSP—a quantum-electronic simulation framework

Oskar Baumgartner, Zlatan Stanojevic, Klaus Schnass, Markus Karner, Hans Kosina
2013 Journal of Computational Electronics  
The Vienna Schrödinger-Poisson (VSP) simulation framework for quantum-electronic engineering applications is presented.  ...  The design principles of the software are explained. Software abstraction is divided into the data, modeling and algebraic level resulting in a flexible physical modeling tool.  ...  Acknowledgements This work has been supported by the Austrian Science Fund program F025 (IR-ON), and the Austrian Research Promotion Agency, project 838551 (NeGFQTS).  ... 
doi:10.1007/s10825-013-0535-y fatcat:cgaco3p53rgrphmmagsk53ufki

Global error control of the time-propagation for the Schrödinger equation with a time-dependent Hamiltonian

K. Kormann, S. Holmgren, H.O. Karlsson
2011 Journal of Computational Science  
We use a posteriori error estimation theory to derive a relation between local and global error in the propagation for the time-dependent Schrödinger equation.  ...  We provide results for models of several different small molecules including bounded and dissociative states, illustrating the efficiency and wide applicability of the new methods.  ...  Summary and Conclusions We have proposed and analyzed an h, p-adaptive propagator for the time-dependent Schrödinger equation that allows for a global error control.  ... 
doi:10.1016/j.jocs.2011.02.003 fatcat:bfb3jlpw4bhafd2dekuoynr5uy
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