Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation

This paper addresses the problem of the construction of stable approximation schemes for the one-dimensional linear Schrodinger equation set in an unbounded domain. After a study of the initial boundary-value problem in a bounded domain with a transparent boundary condition, some unconditionally stable discretization schemes are developed for this kind of problem. The main difficulty is linked to the involvement of a fractional integral operator defining the transparent operator. The proposed semi-discretization of this operator yields with a very different point of view the one proposed by Yevick, Friese and Schmidt [J. Comput. Phys. 168 (2001) 433]. Two possible choices of transparent boundary conditions based on the Dirichlet-Neumann (DN) and Neumann-Dirichlet (ND) operators are presented. To preserve the stability of the fully discrete scheme, conform Galerkin finite element methods are employed for the spatial discretization. Finally, some numerical tests are performed to study the respective accuracy of the different schemes.

[1]  Frensley,et al.  Non-Markovian open-system boundary conditions for the time-dependent Schrödinger equation. , 1994, Physical review. B, Condensed matter.

[2]  F. Collino.  Perfectly Matched Absorbing Layers for the Paraxial Equations , 1997 .

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

[4]  M. Levy.  Parabolic Equation Methods for Electromagnetic Wave Propagation , 2000 .

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

[6]  Thomas Fevens,et al.  ABSORBING BOUNDARY CONDITIONS FOR THE SCHR ¨ ODINGER EQUATION , 1999 .

[7]  Frank Schmidt,et al.  Discrete transparent boundary conditions for the numerical solution of Fresnel's equation , 1995 .

[8]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[9]  Matthias Ehrhardt,et al.  Discrete transparent boundary conditions for the Schrödinger equation , 2001 .

[10]  Patrick Joly,et al.  A generalized image principle for the wave equation with absorbing boundary conditions and applications to fourth order schemes , 1990 .

[11]  Laurent Di Menza.  Transparent and absorbing boundary conditions for the schrödinger equation in a bounded domain , 1997 .

[12]  Matthias Ehrhardt.  Discrete Transparent Boundary Conditions for General Schrödinger-type Equations , 1999, VLSI Design.

[13]  Christophe Besse,et al.  CONSTRUCTION, STRUCTURE AND ASYMPTOTIC APPROXIMATIONS OF A MICRODIFFERENTIAL TRANSPARENT BOUNDARY CONDITION FOR THE LINEAR SCHRÖDINGER EQUATION , 2001 .

[14]  Frank Schmidt,et al.  Discrete transparent boundary conditions for Schrödinger-type equations , 1997 .

[15]  Shibata.  Absorbing boundary conditions for the finite-difference time-domain calculation of the one-dimensional Schrödinger equation. , 1991, Physical review. B, Condensed matter.

[16]  Matthias Ehrhardt,et al.  Discrete Transparent Boundary Conditions for Wide Angle Parabolic Equations in Underwater Acoustics , 1998 .

[17]  Frank Schmidt,et al.  A comparison of transparent boundary conditions for the Fresnel equation , 2001 .

[18]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

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

[20]  Patrick Joly,et al.  A principle of images for absorbing boundary conditions , 1994 .