Numerical Integration of Highly Oscillatory Functions with and without Stationary Points

Author:

Lovetskiy Konstantin P.1ORCID,Sevastianov Leonid A.12ORCID,Hnatič Michal234ORCID,Kulyabov Dmitry S.25ORCID

Affiliation:

1. Department of Computational Mathematics and Artificial Intelligence, RUDN University, 6 Miklukho-Maklaya St., Moscow 117198, Russia

2. Joint Institute for Nuclear Research, 6 Joliot-Curie St., Dubna 141980, Russia

3. Faculty of Science, Šafárik University, 040 01 Košice, Slovakia

4. Institute of Experimental Physics, Slovak Academy of Sciences, 040 14 Košice, Slovakia

5. Department of Probability Theory and Cyber Security, RUDN University, 6 Miklukho-Maklaya St., Moscow 117198, Russia

Abstract

This paper proposes an original approach to calculating integrals of rapidly oscillating functions, based on Levin’s algorithm, which reduces the search for an anti-derivative function to solve an ODE with a complex coefficient. The direct solution of the differential equation is based on the method of integrating factors. The reduction in the original integration problem to a two-stage method for solving ODEs made it possible to overcome the instability that arises in the standard (in the form of solving a system of linear algebraic equations) approach to the solution. And due to the active use of Chebyshev interpolation when using the collocation method on Gauss–Lobatto grids, it is possible to achieve high speed and stability when taking into account a large number of collocation points. The presented spectral method of integrating factors is both flexible and reliable and allows for avoiding the ambiguities that arise when applying the classical method of collocation for the ODE solution (Levin) in the physical space. The new method can serve as a basis for solving ordinary differential equations of the first and second orders when creating high-efficiency software, which is demonstrated by solving several model problems.

Funder

RUDN University Scientific Projects Grant System

Ministry of Education, Science, Research and Sport of the Slovak Republic

RUDN University Strategic Academic Leadership Program

Russian Science Foundation

Publisher

MDPI AG

Reference42 articles.

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3. Comparison of numerical integration methods for calculating diffraction of a plane electromagnetic wave by a rectangular aperture;Mokeev;Comput. Opt.,2021

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5. Sveshnikov, A.G., and Mogilevskiy, I.Y. (2012). Selected Mathematical Problems of the Theory of Diffraction, Faculty of Physics, Moscow State University. (In Russian).

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