Analysis of a third-order absorbing boundary condition for the Schrödinger equation discretized in space.
type: Item
instance of: scholarly article
title: Analysis of a third-order absorbing boundary condition for the Schrödinger equation discretized in space.
zbMATH Open document ID: 1067.65097
DOI: 10.1016/S0893-9659(04)90030-4
published in: Applied Mathematics Letters
publication date: 2005-02-25T00:00:00Z
review text: A Schrödinger type equation of the form \[ \partial_t u = -{i \over c} ( \partial^2_x u +V u) \] for \(x\in \mathbb R,\;t>0\) and with \(c>0, V \in \mathbb R\) is considered especially from the numerical point of view. For obtaining the reliable numerical solution by space discretization a finite spatial subdomain must be used with appropriate boundary conditions. One of the wide spread techniques is to use local boundary conditions (ABC) obtained by approximation the transparent, or reflection free, boundary conditions (TBC). ABC boundary conditions are obtained by interpolatory techniques and usually are denoted as \(ABC(j_1,j_2)\) when \(j_1+j_2+1\) interpolatory nodes are used. It was proved in previous papers that the matrices associated with \(ABC(1,0)\) as well \(ABC(1,1)\) have all eigenvalues with negative real part and the semidiscrete problem is weak ill-posed when the spatial step size \(h\) goes to zero. A semidiscrete alternative to these ABC was developed denoted by \(SABC(j_1,j_2)\). Similar properties as \(ABC(1,1)\) for \(SABC(1,1)\), that means weakly ill-posed, are proved in this paper. The essential difference is in the possibility to use bigger values of the spatial discretiazation in order to obtain a certain absorption, what is important because of the weak instability of both problems. Stability of the matrix for the resulting semidiscrete problem obtained for \(SABC(1,1)\) is proved and a bound for the possible growth of the norm of the solution is obtained.
reviewed by: Angela Handlovičová
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 2138569
zbMATH Keywords: semidiscretization
zbMATH Keywords: absorbing boundary conditions
zbMATH Keywords: well posedness
zbMATH Keywords: Schrödinger-type equation
zbMATH Keywords: stability
author: Nuria Reguera
MaRDI profile type: Publication
cites work: Weak Ill-Posedness of Spatial Discretizations of Absorbing Boundary Conditions for Schrödinger-Type Equations
cites work: Absorbing Boundary Conditions for the Schrödinger Equation
cites work: Stability of a class of matrices with applications to absorbing boundary conditions for Schrödinger-type equations.
cites work: Discrete Absorbing Boundary Conditions for Schrödinger-Type Equations. Construction and Error Analysis
cites work: Absorbing Boundary Conditions for the Discretization Schemes of the One-Dimensional Wave Equation
cites work: Discrete absorbing boundary conditions for Schrödinger-type equations. Practical implementation
cites work: Q4288951
cites work: Transparent and absorbing boundary conditions for the schrödinger equation in a bounded domain
full work available at URL: s0893-9659(04)90030-4
OpenAlex ID: W2086928224
description: scientific article; zbMATH DE number 2138569