New formulation of the transparent boundary conditions for the parabolic equation

2005 ◽  
Vol 31 (5) ◽  
pp. 400-402
Author(s):  
M. Yu. Trofimov
Keyword(s):  
Parabolic Equation ◽  
Boundary Conditions ◽  
Transparent Boundary Conditions ◽  
New Formulation

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

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.

Transparent boundary conditions for discretized non-Hermitian eigenvalue problems

2021 ◽  
pp. 2150138
Author(s):  
Jonathan Heinz ◽  
Miroslav Kolesik
Keyword(s):  
Quantum Mechanics ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Complex Scaling ◽  
Transparent Boundary Conditions ◽  
Drastic Reduction ◽  
Optical Cavities ◽  
Energy Dependent ◽  
Lossy Waveguides

A method is presented for transparent, energy-dependent boundary conditions for open, non-Hermitian systems, and is illustrated on an example of Stark resonances in a single-particle quantum system. The approach provides an alternative to external complex scaling, and is applicable when asymptotic solutions can be characterized at large distances from the origin. Its main benefit consists in a drastic reduction of the dimesnionality of the underlying eigenvalue problem. Besides application to quantum mechanics, the method can be used in other contexts such as in systems involving unstable optical cavities and lossy waveguides.

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