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Discrete transparent boundary conditions for transient kp ‐Schrödinger equations with application to quantum heterostructures
Author(s) -
Zisowsky A.,
Arnold A.,
Ehrhardt M.,
Koprucki Th.
Publication year - 2005
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200510231
Subject(s) - mathematics , boundary (topology) , boundary value problem , convolution (computer science) , mathematical analysis , crank–nicolson method , transformation (genetics) , transient (computer programming) , schrödinger equation , stability (learning theory) , scheme (mathematics) , computer science , biochemistry , chemistry , machine learning , artificial neural network , gene , operating system
This work is concerned with transparent boundary conditions (TBCs) for systems of Schrödinger‐type equations , namely the time‐dependent kp‐Schrödinger equations . These TBCs are constructed for the fully discrete scheme (Crank‐Nicolson, finite differences), in order to maintain unconditional stability of the scheme and to avoid numerical reflections. The discrete transparent boundary conditions (DTBCs) are discrete convolutions in time and are constructed using the ‐transformed solution of the exterior problem . We will analyse the numerical error of these convolution coefficients caused by the inverse ‐transformation. Since the DTBCs are non‐local in time and thus very costly to evaluate, we present approximate DTBCs of a sum‐of‐exponentials form that allow for a fast calculation of the boundary terms.