Solving the eikonal equation by the FSM method in Julia language

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Abstract

There are two main approaches to the numerical solution of the eikonal equation: reducing it to asystemofODES(methodofcharacteristics)andconstructingspecializedmethodsforthenumericalsolutionof this equation in the form of a partial differential equation. The latter approach includes the FSM (Fast sweeping method) method. It is reasonable to assume that a specialized method should have greater versatility. The purpose of this work is to evaluate the applicability of the FSM method for constructing beams and fronts. The implementation of the FSM method in the Eikonal library of the Julia programming language was used. The method was used for numerical simulation of spherical lenses by Maxwell, Luneburg and Eaton. These lenses were chosen because their optical properties have been well studied. A special case of flat lenses was chosen as the easiest to visualize and interpret the results. The results of the calculations are presented in the form of images of fronts and rays for each of the lenses. From the analysis of the obtained images, it is concluded that the FSM method is well suited for constructing electromagnetic wave fronts. An attempt to visualize ray trajectories based on the results of his work encounters a number of difficulties and in some cases gives an incorrect visual picture.

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1. Introduction In this article, we use the FSM (Fast Sweeping Method) method to solve the Eikonal equation using the example of three classical lenses: Luneburg, Maxwell and Eaton. These examples illustrate the limitations of the FSM method-it does a good job of calculating wave fronts, but is poorly applicable for calculating the trajectory of rays. To model lenses, we use the Julia language and the Eikonal library, which implements the FSM method. The simulation results are visualized using the Mackie.jl library. Schematic illustrations are created using a separate vector graphics language, Asymptote. 1.1. Structure of the article The paper consists of an introduction, a theoretical part, a description of the FSM (Fast sweeping method) [1-5], a description of implementation this method, a visualisation and discussion of the results. In the theoretical part, the eikonal equation in Cartesian coordinate system is given, and the spherical lenses used for numerical experiments are schematically described. In the next part, the numerical scheme of the FSM method with detailed formulas for the twodimensional case is described, and its advantages and disadvantages compared to the feature method are briefly analyzed. Below is a brief description of the Eikonal library [6] for the Julia language [7] and a description of the program we have written that implements a numerical experiment. In the final part, images of fronts and rays are presented, the results are analysed and conclusions are made about the advantages and disadvantages of the FSM method concerning the visualisation of the calculations. © Stepa C. A., Fedorov A. V., Gevorkyan M. N., Korolkova A. V., Kulyabov D. S., 2024 This work is licensed under a Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by-nc/4.0/legalcode 1.2. Designations and agreements For the purposes of this paper, we have followed standard notation, centred on the classic monograph [8]. All vector quantities are in bold, e.g., the position of a point
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About the authors

Christina A. Stepa

RUDN University

Email: 1042210111@pfur.ru
ORCID iD: 0000-0002-4092-4326
ResearcherId: GLS-1445-2022

PhD student of Probability Theory and Cyber Security

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Arseny V. Fedorov

RUDN University

Email: 1042210107@rudn.ru
ORCID iD: 0000-0002-3036-0117
Scopus Author ID: 57219092618
ResearcherId: AGY-9849-2022

PhD student of Probability Theory and Cyber Security

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Migran N. Gevorkyan

RUDN University

Email: gevorkyan-mn@rudn.ru
ORCID iD: 0000-0002-4834-4895
Scopus Author ID: 57190004380
ResearcherId: E-9214-2016

Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Probability Theory and Cyber Security

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Anna V. Korolkova

RUDN University

Email: korolkova-av@rudn.ru
ORCID iD: 0000-0001-7141-7610
Scopus Author ID: 36968057600
ResearcherId: I-3191-2013

Docent, Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Probability Theory and Cyber Security

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Dmitry S. Kulyabov

RUDN University; Joint Institute for Nuclear Research

Author for correspondence.
Email: kulyabov-ds@rudn.ru
ORCID iD: 0000-0002-0877-7063
Scopus Author ID: 35194130800
ResearcherId: I-3183-2013

Professor, Doctor of Sciences in Physics and Mathematics, Professor of the Department of Probability Theory and Cyber Security of Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University); Senior Researcher of Laboratory of Information Technologies, Joint Institute for Nuclear Research

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation; 6 Joliot-Curie St, Dubna, 141980, Russian Federation

References

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Copyright (c) 2024 Stepa C.A., Fedorov A.V., Gevorkyan M.N., Korolkova A.V., Kulyabov D.S.

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This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

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