Remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrödinger equation on the half-axis

AbstractWe consider the generalized time-dependent Schrödinger equation on the half-axis and a broad family of finite-difference schemes with the discrete transparent boundary conditions (TBCs) to solve it. We first rewrite the discrete TBCs in a simplified form explicit in space step

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Splitting in Potential Finite-Difference Schemes with Discrete Transparent Boundary Conditions for the Time-Dependent Schrödinger Equation

Lecture Notes in Computational Science and Engineering - Numerical Mathematics and Advanced Applications - ENUMATH 2013 ◽

10.1007/978-3-319-10705-9_20 ◽

2014 ◽

pp. 203-211 ◽

Cited By ~ 1

Author(s):

Alexander Zlotnik ◽

Bernard Ducomet ◽

Ilya Zlotnik ◽

Alla Romanova

Keyword(s):

Finite Difference ◽

Schrödinger Equation ◽

Boundary Conditions ◽

Schrodinger Equation ◽

Difference Schemes ◽

Time Dependent ◽

Finite Difference Schemes ◽

Transparent Boundary Conditions

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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 ◽

10.1134/s0965542511030122 ◽

2011 ◽

Vol 51 (3) ◽

pp. 355-376 ◽

Cited By ~ 8

Author(s):

I. A. Zlotnik

Keyword(s):

Finite Difference ◽

Schrödinger Equation ◽

Boundary Conditions ◽

Schrodinger Equation ◽

Difference Schemes ◽

Finite Difference Schemes ◽

Infinite Strip ◽

Transparent Boundary Conditions ◽

Nonstationary Schrödinger Equation

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Stability of a symmetric finite-difference scheme with approximate transparent boundary conditions for the time-dependent Schrödinger equation

Doklady Mathematics ◽

10.1134/s1064562407020214 ◽

2007 ◽

Vol 75 (2) ◽

pp. 260-265

Author(s):

A. A. Zlotnik ◽

B. Ducomet

Keyword(s):

Finite Difference ◽

Schrödinger Equation ◽

Boundary Conditions ◽

Difference Scheme ◽

Finite Difference Scheme ◽

Schrodinger Equation ◽

Time Dependent ◽

Transparent Boundary Conditions

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Transparent boundary conditions for higher-order finite-difference schemes of the Schrödinger equation in (1+1)D

Computer Physics Communications ◽

10.1016/j.cpc.2019.107048 ◽

2020 ◽

Vol 250 ◽

pp. 107048 ◽

Cited By ~ 1

Author(s):

Mathias Schwendt ◽

Walter Pötz

Keyword(s):

Finite Difference ◽

Schrödinger Equation ◽

Boundary Conditions ◽

Schrodinger Equation ◽

Higher Order ◽

Difference Schemes ◽

Finite Difference Schemes ◽

Transparent Boundary Conditions

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Family of finite-difference schemes with transparent boundary conditions for the nonstationary Schrödinger equation in a semi-infinite strip

Doklady Mathematics ◽

10.1134/s1064562411010066 ◽

2011 ◽

Vol 83 (1) ◽

pp. 12-18 ◽

Cited By ~ 4

Author(s):

A. A. Zlotnik ◽

I. A. Zlotnik

Keyword(s):

Finite Difference ◽

Schrödinger Equation ◽

Boundary Conditions ◽

Schrodinger Equation ◽

Difference Schemes ◽

Finite Difference Schemes ◽

Infinite Strip ◽

Transparent Boundary Conditions ◽

Nonstationary Schrödinger Equation

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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 ◽

10.1007/s10958-010-0040-9 ◽

2010 ◽

Vol 169 (1) ◽

pp. 84-97 ◽

Cited By ~ 5

Author(s):

A. A. Zlotnik ◽

A. V. Lapukhina

Keyword(s):

Finite Difference ◽

Schrödinger Equation ◽

Boundary Conditions ◽

Difference Scheme ◽

Finite Difference Scheme ◽

Schrodinger Equation ◽

Transparent Boundary Conditions ◽

Nonstationary Schrödinger Equation ◽

Half Axis

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On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation

Kinetic and Related Models ◽

10.3934/krm.2009.2.151 ◽

2009 ◽

Vol 2 (1) ◽

pp. 151-179 ◽

Cited By ~ 15

Author(s):

Bernard Ducomet ◽

◽

Alexander Zlotnik ◽

Ilya Zlotnik ◽

◽

...

Keyword(s):

Finite Difference ◽

Schrödinger Equation ◽

Boundary Conditions ◽

Schrodinger Equation ◽

Difference Schemes ◽

Finite Difference Schemes ◽

Transparent Boundary Conditions

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A Family of Finite-Difference Schemes with Discrete Transparent Boundary Conditions for a Parabolic Equation on the Half-Axis

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2013-0006 ◽

2013 ◽

Vol 13 (2) ◽

pp. 119-138

Author(s):

Alexander Zlotnik ◽

Natalya Koltsova

Keyword(s):

Parabolic Equation ◽

Finite Difference ◽

Boundary Conditions ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Difference Schemes ◽

Finite Difference Schemes ◽

Transparent Boundary Conditions ◽

Half Axis ◽

Two Parameters

Abstract. An initial-boundary value problem for the 1D self-adjoint parabolic equation on the half-axis is solved. We study a broad family of two-level finite-difference schemes with two parameters related to averages both in time and space. Stability in two norms is proved by the energy method. Also discrete transparent boundary conditions are rigorously derived for schemes by applying the method of reproducing functions. Results of numerical experiments are included as well.

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