On the Abramov approach for the approximation of whispering gallery modes in prolate spheroids
•‘Whispering gallery’ mode.•Separation of variables.•Multi-parameter spectral problems.•Prüfer angle.•High accuracy finite differences. In this paper, we present the Abramov approach for the numerical simulation of the whispering gallery modes in prolate spheroids. The main idea of this approach is...
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| Published in | Applied mathematics and computation Vol. 409; p. 125599 |
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| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
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| Abstract | •‘Whispering gallery’ mode.•Separation of variables.•Multi-parameter spectral problems.•Prüfer angle.•High accuracy finite differences.
In this paper, we present the Abramov approach for the numerical simulation of the whispering gallery modes in prolate spheroids. The main idea of this approach is the Newton–Raphson technique combined with the quasi-time marching. In the first step, a solution of a simpler problem, as an initial guess for the Newton–Raphson iterations, is provided. Then, step-by-step, this simpler problem is converted into the original problem, while the quasi-time parameter τ runs from τ=0 to τ=1. While following the involved imaginary path two numerical approaches are realized, the first is based on the Prüfer angle technique, the second on high order finite difference schemes. |
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| AbstractList | •‘Whispering gallery’ mode.•Separation of variables.•Multi-parameter spectral problems.•Prüfer angle.•High accuracy finite differences.
In this paper, we present the Abramov approach for the numerical simulation of the whispering gallery modes in prolate spheroids. The main idea of this approach is the Newton–Raphson technique combined with the quasi-time marching. In the first step, a solution of a simpler problem, as an initial guess for the Newton–Raphson iterations, is provided. Then, step-by-step, this simpler problem is converted into the original problem, while the quasi-time parameter τ runs from τ=0 to τ=1. While following the involved imaginary path two numerical approaches are realized, the first is based on the Prüfer angle technique, the second on high order finite difference schemes. |
| ArticleNumber | 125599 |
| Author | Weinmüller, Ewa B. Amodio, Pierluigi Levitina, Tatiana Settanni, Giuseppina Arnold, Anton |
| Author_xml | – sequence: 1 givenname: Pierluigi surname: Amodio fullname: Amodio, Pierluigi email: pierluigi.amodio@uniba.it organization: Dipartimento di Matematica, Università di Bari, Italy – sequence: 2 givenname: Anton surname: Arnold fullname: Arnold, Anton email: anton.arnold@tuwien.ac.at organization: Institute for Analysis and Scientific Computing, Vienna University of Technology, Austria – sequence: 3 givenname: Tatiana surname: Levitina fullname: Levitina, Tatiana email: levitina@mps.mpg.de organization: Max-Planck-Institut für Sonnensystemforschung, Göttingen, Germany – sequence: 4 givenname: Giuseppina surname: Settanni fullname: Settanni, Giuseppina email: giuseppina.settanni@uniba.it organization: Dipartimento di Matematica, Università di Bari, Italy – sequence: 5 givenname: Ewa B. surname: Weinmüller fullname: Weinmüller, Ewa B. email: e.weinmueller@tuwien.ac.at organization: Institute for Analysis and Scientific Computing, Vienna University of Technology, Austria |
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| Cites_doi | 10.1016/0041-5553(89)90158-4 10.1109/JSTQE.2005.862952 10.1016/j.optcom.2019.125036 10.1016/0022-247X(78)90160-9 10.1088/1751-8113/41/1/015209 10.1063/1.3637866 10.1007/s12190-013-0657-1 10.1006/jmaa.2001.7695 10.1063/1.4912449 10.1016/j.optlaseng.2019.105968 10.1016/j.cpc.2013.12.012 10.1016/j.cam.2012.04.011 10.1016/j.cnsns.2014.05.032 10.1016/0030-4018(94)90603-3 10.1109/JSTQE.2005.862943 |
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| Keywords | ‘Whispering gallery’ mode Prüfer angle 65L15 Multi-parameter spectral problems Separation of variables High accuracy finite differences 34B09 65L10 |
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