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Volume 11, Issue 1
Comparison of Solvers for 2D Schrödinger Problems

F. J. Gaspar, C. Rodrigo, R. Ciegis & A. Mirinavicius

Int. J. Numer. Anal. Mod., 11 (2014), pp. 131-147.

Published online: 2014-11

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  • Abstract

This paper deals with the numerical solution of both linear and non-linear Schrödinger problems, which mathematically model many physical processes in a wide range of applications of interest. In particular, a comparison of different solvers and different approaches for these problems is developed throughout this work. Two finite difference schemes are analyzed: the classical Crank-Nicolson approach, and a high-order compact scheme. Solvers based on geometric multigrid, Fast Fourier Transform and Alternating Direction Implicit methods are compared. Finally, the efficiency of the considered solvers is tested for a linear Schrödinger problem, proving that the computational experiments are in good agreement with the theoretical predictions. In order to test the robustness of the MG solver two additional Schrödinger problems with a nonconstant potential and nonlinear right-hand side are solved by the MG solver, since the efficiency of this solver depends on such data.

  • Keywords

finite difference method, Schrödinger problem, multigrid method, Alternating Direction Implicit method, Fast Fourier Transform method.

  • AMS Subject Headings

35R35, 49J40, 60G40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-11-131, author = {Gaspar , F. J.Rodrigo , C.Ciegis , R. and Mirinavicius , A.}, title = {Comparison of Solvers for 2D Schrödinger Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2014}, volume = {11}, number = {1}, pages = {131--147}, abstract = {

This paper deals with the numerical solution of both linear and non-linear Schrödinger problems, which mathematically model many physical processes in a wide range of applications of interest. In particular, a comparison of different solvers and different approaches for these problems is developed throughout this work. Two finite difference schemes are analyzed: the classical Crank-Nicolson approach, and a high-order compact scheme. Solvers based on geometric multigrid, Fast Fourier Transform and Alternating Direction Implicit methods are compared. Finally, the efficiency of the considered solvers is tested for a linear Schrödinger problem, proving that the computational experiments are in good agreement with the theoretical predictions. In order to test the robustness of the MG solver two additional Schrödinger problems with a nonconstant potential and nonlinear right-hand side are solved by the MG solver, since the efficiency of this solver depends on such data.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/518.html} }
TY - JOUR T1 - Comparison of Solvers for 2D Schrödinger Problems AU - Gaspar , F. J. AU - Rodrigo , C. AU - Ciegis , R. AU - Mirinavicius , A. JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 131 EP - 147 PY - 2014 DA - 2014/11 SN - 11 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/518.html KW - finite difference method, Schrödinger problem, multigrid method, Alternating Direction Implicit method, Fast Fourier Transform method. AB -

This paper deals with the numerical solution of both linear and non-linear Schrödinger problems, which mathematically model many physical processes in a wide range of applications of interest. In particular, a comparison of different solvers and different approaches for these problems is developed throughout this work. Two finite difference schemes are analyzed: the classical Crank-Nicolson approach, and a high-order compact scheme. Solvers based on geometric multigrid, Fast Fourier Transform and Alternating Direction Implicit methods are compared. Finally, the efficiency of the considered solvers is tested for a linear Schrödinger problem, proving that the computational experiments are in good agreement with the theoretical predictions. In order to test the robustness of the MG solver two additional Schrödinger problems with a nonconstant potential and nonlinear right-hand side are solved by the MG solver, since the efficiency of this solver depends on such data.

F. J. Gaspar, C. Rodrigo, R. Ciegis & A. Mirinavicius. (1970). Comparison of Solvers for 2D Schrödinger Problems. International Journal of Numerical Analysis and Modeling. 11 (1). 131-147. doi:
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