scholarly journals Numerically Absorbing Boundary Conditions for Quantum Evolution Equations

VLSI Design ◽  
1998 ◽  
Vol 6 (1-4) ◽  
pp. 313-319 ◽  
Author(s):  
Anton Arnold
Keyword(s):  
Boundary Conditions ◽  
Ad Hoc ◽  
Evolution Equations ◽  
Absorbing Boundary Conditions ◽  
Quantum Evolution ◽  
Transparent Boundary Conditions ◽  
Absorbing Boundary ◽  
Quantum Devices ◽  
Non Local ◽  
The Given

Transparent boundary conditions for the transient Schrödinger equation on a domain Ω can be derived explicitly under the assumption that the given potential V is constant outside of this domain. In 1D these boundary conditions are non-local in time (of memory type). For the Crank-Nicolson finite difference scheme, discrete transparent boundary conditions are derived, and the resulting scheme is proved to be unconditionally stable. A numerical example illustrates the superiority of discrete transparent boundary conditions over existing ad-hoc discretizations of the differential transparent boundary conditions. As an application of these boundary conditions to the modeling of quantum devices, a transient 1D scattering model for mixed quantum states is presented.

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

2017 ◽  
Vol 22 (3) ◽  
pp. 408-423 ◽  
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.

scholarly journals Discrete Transparent Boundary Conditions for General Schrödinger-type Equations

VLSI Design ◽  
1999 ◽  
Vol 9 (4) ◽  
pp. 325-338 ◽  
Author(s):  
Matthias Ehrhardt
Keyword(s):  
Boundary Conditions ◽  
Underwater Acoustics ◽  
Wide Angle ◽  
Transparent Boundary Conditions ◽  
Space Problem ◽  
Whole Space ◽  
Non Local ◽  
Memory Type ◽  
The Given ◽  
Stable Scheme

Transparent boundary conditions (TBCs) for general Schrödinger-type equations on a bounded domain can be derived explicitly under the assumption that the given potential V is constant on the exterior of that domain. In 1D these boundary conditions are non-local in time (of memory type).Existing discretizations of these TBCs have accuracy problems and render the overall Crank–Nicolson finite difference method only conditionally stable. In this paper a novel discrete TBC is derived directly from the discrete whole-space problem that yields an unconditionally stable scheme. Numerical examples illustrate the superiority of the discrete TBC over other existing consistent discretizations of the differential TBCs.As an application of these boundary conditions to wave propagation problems in underwater acoustics results for the so-called standard and wide angle “parabolic” equation (SPE, WAPE) models are presented.

Toward perfectly absorbing boundary conditions for Euler equations

AIAA Journal ◽  
1999 ◽  
Vol 37 ◽  
pp. 912-918
Author(s):  
M. E. Hayder ◽  
Fang Q. Hu ◽  
M. Y. Hussaini
Keyword(s):  
Boundary Conditions ◽  
Euler Equations ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary

scholarly journals Dirichlet absorbing boundary conditions for classical and peridynamic diffusion-type models

2020 ◽  
Vol 66 (4) ◽  
pp. 773-793 ◽  
Author(s):  
Arman Shojaei ◽  
Alexander Hermann ◽  
Pablo Seleson ◽  
Christian J. Cyron
Keyword(s):  
Boundary Conditions ◽  
Materials Science ◽  
Unbounded Domains ◽  
Diffusion Models ◽  
Absorbing Boundary Conditions ◽  
The Finite Element Method ◽  
Absorbing Boundary ◽  
Diffusion Type ◽  
2D And 3D ◽  
Accuracy And Stability

Abstract Diffusion-type problems in (nearly) unbounded domains play important roles in various fields of fluid dynamics, biology, and materials science. The aim of this paper is to construct accurate absorbing boundary conditions (ABCs) suitable for classical (local) as well as nonlocal peridynamic (PD) diffusion models. The main focus of the present study is on the PD diffusion formulation. The majority of the PD diffusion models proposed so far are applied to bounded domains only. In this study, we propose an effective way to handle unbounded domains both with PD and classical diffusion models. For the former, we employ a meshfree discretization, whereas for the latter the finite element method (FEM) is employed. The proposed ABCs are time-dependent and Dirichlet-type, making the approach easy to implement in the available models. The performance of the approach, in terms of accuracy and stability, is illustrated by numerical examples in 1D, 2D, and 3D.

Absorbing boundary conditions for the TLM method

1992 ◽  
Vol 40 (11) ◽  
pp. 2095-2099 ◽  
Author(s):  
J.A. Morente ◽  
J.A. Porti ◽  
M. Khalladi
Keyword(s):  
Boundary Conditions ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary ◽  
Tlm Method
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