ABSORBING BOUNDARY CONDITIONS FOR THE TWO-DIMENSIONAL SCHRÖDINGER EQUATION WITH AN EXTERIOR POTENTIAL PART I: CONSTRUCTION AND A PRIORI ESTIMATES

The aim of this paper is to construct some classes of absorbing boundary conditions for the two-dimensional Schrödinger equation with a time and space varying exterior potential and for general convex smooth boundaries. The construction is based on asymptotics of the inhomogeneous pseudodifferential operators defining the related Dirichlet-to-Neumann operator. Furthermore, a priori estimates are developed for the truncated problems with various increasing order boundary conditions. The effective numerical approximation will be treated in a second paper.

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Local absorbing boundary conditions for two‐dimensional nonlinear Schrödinger equation with wave operator on unbounded domain

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.7703 ◽

2021 ◽

Author(s):

Hongwei Li

Keyword(s):

Schrödinger Equation ◽

Boundary Conditions ◽

Nonlinear Schrödinger Equation ◽

Unbounded Domain ◽

Wave Operator ◽

Schrodinger Equation ◽

Nonlinear Schrodinger Equation ◽

Absorbing Boundary Conditions ◽

Two Dimensional ◽

Absorbing Boundary

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A Priori Estimates for Mixed Finite Element Approximations of Second-Order Hyperbolic Equations with Absorbing Boundary Conditions

SIAM Journal on Numerical Analysis ◽

10.1137/0733026 ◽

1996 ◽

Vol 33 (2) ◽

pp. 492-504 ◽

Cited By ~ 35

Author(s):

Lawrence C. Cowsar ◽

Todd F. Dupont ◽

Mary F. Wheeler

Keyword(s):

Finite Element ◽

Boundary Conditions ◽

A Priori Estimates ◽

Hyperbolic Equations ◽

Mixed Finite Element ◽

A Priori ◽

Absorbing Boundary Conditions ◽

Absorbing Boundary ◽

Finite Element Approximations ◽

Priori Estimates

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Absorbing boundary conditions for the two-dimensional Schrödinger equation with an exterior potential

Numerische Mathematik ◽

10.1007/s00211-013-0542-8 ◽

2013 ◽

Vol 125 (2) ◽

pp. 191-223 ◽

Cited By ~ 15

Author(s):

Xavier Antoine ◽

Christophe Besse ◽

Pauline Klein

Keyword(s):

Schrödinger Equation ◽

Boundary Conditions ◽

Schrodinger Equation ◽

Absorbing Boundary Conditions ◽

Two Dimensional ◽

Absorbing Boundary

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Remarks on global a priori estimates for the nonlinear Schrödinger equation

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-2010-10487-2 ◽

2010 ◽

Vol 138 (12) ◽

pp. 4359-4359

Author(s):

J. Colliander ◽

M. Grillakis ◽

N. Tzirakis

Keyword(s):

Schrödinger Equation ◽

Nonlinear Schrödinger Equation ◽

Schrodinger Equation ◽

A Priori Estimates ◽

A Priori ◽

Nonlinear Schrodinger Equation ◽

Nonlinear Schrödinger ◽

Priori Estimates

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Stability and Error Analysis for a Second-Order Fast Approximation of the One-dimensional Schrödinger Equation Under Absorbing Boundary Conditions

SIAM Journal on Scientific Computing ◽

10.1137/17m1162111 ◽

2018 ◽

Vol 40 (6) ◽

pp. A4083-A4104 ◽

Cited By ~ 2

Author(s):

Buyang Li ◽

Jiwei Zhang ◽

Chunxiong Zheng

Keyword(s):

Schrödinger Equation ◽

Error Analysis ◽

Boundary Conditions ◽

Schrodinger Equation ◽

Absorbing Boundary Conditions ◽

Second Order ◽

Absorbing Boundary ◽

One Dimensional ◽

The One ◽

Stability And Error Analysis

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Absorbing boundary conditions for the one-dimensional Schrödinger equation with an exterior repulsive potential

Journal of Computational Physics ◽

10.1016/j.jcp.2008.09.013 ◽

2009 ◽

Vol 228 (2) ◽

pp. 312-335 ◽

Cited By ~ 34

Author(s):

Xavier Antoine ◽

Christophe Besse ◽

Pauline Klein

Keyword(s):

Schrödinger Equation ◽

Boundary Conditions ◽

Schrodinger Equation ◽

Absorbing Boundary Conditions ◽

Repulsive Potential ◽

Absorbing Boundary ◽

One Dimensional ◽

The One

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ON THE ACCURACY OF SOME ABSORBING BOUNDARY CONDITIONS FOR THE SCHRODINGER EQUATION

Mathematical Modelling and Analysis ◽

10.3846/13926292.2017.1306725 ◽

2017 ◽

Vol 22 (3) ◽

pp. 408-423 ◽

Cited By ~ 2

Author(s):

Andrej Bugajev ◽

Raimondas Čiegis ◽

Rima Kriauzienė ◽

Teresė Leonavičienė ◽

Julius Žilinskas

Keyword(s):

Schrödinger Equation ◽

Boundary Conditions ◽

Schrodinger Equation ◽

Rational Functions ◽

Detailed Comparison ◽

Absorbing Boundary Conditions ◽

Benchmark Problems ◽

Transparent Boundary Conditions ◽

Absorbing Boundary ◽

L2 Norm

A detailed analysis of absorbing boundary conditions for the linear Schrodinger equation is presented in this paper. It is focused on absorbing boundary conditions that are obtained by using rational functions to approximate the exact transparent boundary conditions. Different strategies are investigated for the optimal selection of the coefficients of these rational functions, including the Pade approximation, the L2 norm approximations of the Fourier symbol, L2 minimization of a reflection coefficient, and two error minimization techniques for the chosen benchmark problems with known exact solutions. The results of computational experiments are given and a detailed comparison of the efficiency of these techniques is presented.

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