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

Abstract The aim of this paper is to design some accurate artificial boundary conditions for the semi-discretized linear Schrodinger and heat equations in rectangular domains. The Laplace transform in time and discrete Fourier transform in space are applied to get Green’s functions of the semi-discretized equations in unbounded domains with single-source. An algorithm is given to compute these Green’s functions accurately through some recurrence relations. Furthermore, the finite-difference method is used to discretize the reduced problem with accurate boundary conditions. Numerical simulations are presented to illustrate the accuracy of our method in the case of the linear Schrodinger and heat equations. It is shown that the reflection at the corners is correctly eliminated.

[1]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[2]  A. V. Popov,et al.  Implementation of transparent boundaries for numerical solution of the Schrödinger equation , 1991 .

[3]  Hong Jiang,et al.  Absorbing Boundary Conditions for the Schrödinger Equation , 1999, SIAM J. Sci. Comput..

[4]  Christophe Besse,et al.  Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation , 2003 .

[5]  E. Schrödinger.  An Undulatory Theory of the Mechanics of Atoms and Molecules , 1926 .

[6]  Matthias Ehrhardt,et al.  Discrete transparent boundary conditions for the Schr ¨ odinger equation on circular domains , 2012 .

[7]  Patrick Po-Yen Lin,et al.  On the numerical solution of the heat equation in unbounded domains , 1993 .

[8]  Jiwei Zhang,et al.  Local artificial boundary conditions for Schrödinger and heat equations by using high-order azimuth derivatives on circular artificial boundary , 2014, Comput. Phys. Commun..

[9]  R. Higdon.  Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation , 1986 .

[10]  R. Hazeltine,et al.  The Framework Of Plasma Physics , 1998 .

[11]  Sophie Papst,et al.  Computational Methods For Fluid Dynamics , 2016 .

[12]  Ivar Giaever.  Charles Palmer Bean , 1997 .

[13]  Xiaonan Wu,et al.  Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions , 2004 .

[14]  Xavier Antoine,et al.  A friendly review of absorbing boundary conditions and perfectly matched layers for classical and relativistic quantum waves equations , 2017 .

[15]  Christophe Besse,et al.  Absorbing boundary conditions for the two-dimensional Schrödinger equation with an exterior potential , 2013, Numerische Mathematik.

[16]  Christophe Besse,et al.  A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations , 2008 .

[17]  Houde Han,et al.  EXACT ARTIFICIAL BOUNDARY CONDITIONS FOR SCHRÖDINGER EQUATION IN R2∗ , 2004 .

[18]  Christophe Besse,et al.  Absorbing Boundary Conditions for General Nonlinear Schrödinger Equations , 2011, SIAM J. Sci. Comput..

[19]  Christophe Besse,et al.  A Review of artificial boundary conditions for the Schrödinger equation , 2007 .

[20]  Houde Han,et al.  Exact artificial boundary conditions for the Schrödinger equation in $R ^2$ , 2004 .

[21]  Christophe Besse,et al.  Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions , 2004, Math. Comput..

[22]  Sakari Inawashiro,et al.  Lattice Green's Functions for the Rectangular and the Square Lattices at Arbitrary Points , 1971 .

[23]  Shaoqiang Tang,et al.  Time history kernel functions for square lattice , 2011 .

[24]  Christophe Besse,et al.  Communi-cations Computational methods for the dynamics of the nonlinear Schr̈odinger / Gross-Pitaevskii equations , 2013 .

[25]  J. Vukusic.  Optical Fiber Communications: Principles and Practice , 1986 .

[26]  R. Woolley.  Molecular quantum electrodynamics , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[27]  Christophe Besse,et al.  Absorbing boundary conditions for the one-dimensional Schrödinger equation with an exterior repulsive potential , 2009, J. Comput. Phys..

[28]  Matthias Ehrhardt,et al.  Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability , 2003 .

[29]  Christophe Besse,et al.  Absorbing Boundary Conditions for the Two-Dimensional Schrödinger Equation with an Exterior Potential. Part I: Construction and a priori Estimates , 2012 .

[30]  Jiwei Zhang,et al.  HIGH-ORDER LOCAL ABSORBING BOUNDARY CONDITIONS FOR HEAT EQUATION IN UNBOUNDED DOMAINS , 2010 .

[31]  Houde Han,et al.  Exact and approximating boundary conditions for the parabolic problems on unbounded domains , 2002 .

[32]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.