On the Abramov approach for the approximation of whispering gallery modes in prolate spheroids

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 runs from 0 to 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.