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P. Amodio, A. Arnold, T. Levitina, E. Weinmüller:
"On the Abramov approach for the approximation of whispering gallery modes in prolate spheroids";
in: "ASC Report 31/2020", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2020, ISBN: 978-3-902627-13-1, 1 - 20.

In this paper, we present the Abramov approach for the numerical simulation of the whis-pering 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 Pr¨ufer angle technique, the second on high order finite difference schemes.

Separation of variables, `Whispering gallery´ mode, multi-parameter spectral problems, Prüfer angle, High accuracy finite differences

http://www.asc.tuwien.ac.at/preprint/2020/asc31x2020.pdf