Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia2369710.22363/2658-4670-2020-28-1-62-76Research ArticleCalculation of the normal modes of closed waveguidesMalykhMikhail D.<p>Doctor of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics</p>malykh-md@rudn.ruDivakovDmitriy V.<p>Candidate of Physical and Mathematical Sciences, Assistant of Department of Applied Probability and Informatics</p>divakov-dv@rudn.ruEgorovAlexandre A.<p>Doctor of Physical and Mathematical Sciences, Chief Researcher of Department of Oscillations</p>yegorov@kapella.gpi.ruKuzivYaroslav Yu.<p>PhD student of Department of Applied Probability and Informatics</p>yaroslav.kuziw@yandex.ruPeoples’ Friendship University of Russia (RUDN University)A. M. Prokhorov General Physics Institute Russian Academy of Sciences15122020281627609052020Copyright © 2020, Malykh M.D., Divakov D.V., Egorov A.A., Kuziv Y.Y.2020<p>The aim of the work is the development of numerical methods for solving waveguiding problems of the theory of waveguides, as well as their implementation in the form of software packages focused on a wide range of practical problems from the classical issues of microwave transmission to the design of optical waveguides and sensors. At the same time, we strive for ease of implementation of the developed methods in computer algebra systems (Maple, Sage) or in software oriented to the finite element method (FreeFem++). The work uses the representation of electromagnetic fields in a waveguide using four potentials. These potentials do not reduce the number of sought functions, but even in the case when the dielectric permittivity and magnetic permeability are described by discontinuous functions, they turn out to be quite smooth functions. A simple check of the operability of programs by calculating the normal modes of a hollow waveguide is made. It is shown that the relative error in the calculation of the first 10 normal modes does not exceed 4%. These results indicate the efficiency of the method proposed in this article.</p>integrated opticsclosed waveguidecomputer simulationfinite element methodfour potential methodинтегральная оптиказакрытый волноводкомпьютерное моделированиеметод конечных элементовметод четырёх потенциалов[A. Taflove and S. C. Hagness, Computational electrodynamics: the finite difference time domain method, 2nd ed. London: Artech House, 2000.][A. A. Egorov and A. V. 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