A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations

X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle

Commun. Comput. Phys. Vol. 4, Number 4, (2008), 729-796. (open-access article)


In this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present and compare several approaches to implement the classical transparent boundary condition into finite difference and finite element discretizations. We present in detail the approaches of the authors and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.

Supplementary Material

[1] Software
There is a Matlab GUI written for numerical experiments; currently it covers the following implementations of TBCs/ABCs: (Antoine - Besse, Arnold - Ehrhardt, Baskakov-Popov, Di Menza, Fast convolution, Fevens-Jiang, Kuska, Pade, PML, PML-FEM, Pole Condition, Shibata, Szeftel)
Click here for a Screenshot of the Software (Version 3) (jpg-format)

Minisymposium Artificial Boundary Conditions for linear and nonlinear Schrödinger equations

6th International Congress on Industrial and Applied Mathematics, July 16-20, Zürich, Switzerland.

[1] M. Ehrhardt,
A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations
[2] X. Antoine,
Construction of stable discretization schemes of the transparent boundary
[3] A. Arnold,
Open Boundary Conditions for Quantum Wave guide
[4] F. Schmidt,
Construction of transparent boundary conditions by the pole-condition method
[5] C. Besse,
Construction of 2D artificial boundary conditions for the linear Schrödinger equation via fractional pseudo-differential operators
[6] C. Zheng,
Exact absorbing boundary conditions for the Schrödinger equation with periodic potentials at infinity
[6a] M. Ehrhardt and C. Zheng,
Exact artificial boundary conditions for problems with periodic structures J. Comput. Phys. Vol. 227, Issue 14, (2008), 6877-6894.
[6b] M. Ehrhardt, H. Han and C. Zheng,
Numerical simulation of waves in periodic structures, Commun. Comput. Phys. Vol. 5, Number 5, (2009), 849-872.
[7] J. Szeftel,
Absorbing boundary conditions for one-dimensional nonlinear Schrödinger equations
[8] S. Descombes,
Construction and numerical approximation of artificial boundary conditions for a nonlinear Schrödinger equation

mailbox ehrhardt@math.uni-wuppertal.de