Vol 30, No 4 (2022)


Constitutive tensor in the geometrized Maxwell theory

Korolkova A.V.


It is generally accepted that the main obstacle to the application of Riemannian geometrization of Maxwell’s equations is an insufficient number of parameters defining a geometrized medium. In the classical description of the equations of electrodynamics in the medium, a constitutive tensor with 20 components is used. With Riemannian geometrization, the constitutive tensor is constructed from a Riemannian metric tensor having 10 components. It is assumed that this discrepancy prevents the application of Riemannian geometrization of Maxwell’s equations. It is necessary to study the scope of applicability of the Riemannian geometrization of Maxwell’s equations. To determine whether the lack of components is really critical for the application of Riemannian geometrization. To determine the applicability of Riemannian geometrization, the most common variants of electromagnetic media are considered. The structure of the dielectric and magnetic permittivity is written out for them, the number of significant components for these tensors is determined. Practically all the considered types of electromagnetic media require less than ten parameters to describe the constitutive tensor. In the Riemannian geometrization of Maxwell’s equations, the requirement of a single impedance of the medium is critical. It is possible to circumvent this limitation by moving from the complete Maxwell’s equations to the approximation of geometric optics. The Riemannian geometrization of Maxwell’s equations is applicable to a wide variety of media types, but only for approximating geometric optics.

Discrete and Continuous Models and Applied Computational Science. 2022;30(4):305-317
pages 305-317 views

Implementation of hyperbolic complex numbers in Julia language

Korolkova A.V., Gevorkyan M.N., Kulyabov D.S.


Hyperbolic complex numbers are used in the description of hyperbolic spaces. One of the well-known examples of such spaces is the Minkowski space, which plays a leading role in the problems of the special theory of relativity and electrodynamics. However, such numbers are not very common in different programming languages. Of interest is the implementation of hyperbolic complex in scientific programming languages, in particular, in the Julia language. The Julia language is based on the concept of multiple dispatch. This concept is an extension of the concept of polymorphism for object-oriented programming languages. To implement hyperbolic complex numbers, the multiple dispatching approach of the Julia language was used. The result is a library that implements hyperbolic numbers. Based on the results of the study, we can conclude that the concept of multiple dispatching in scientific programming languages is convenient and natural.

Discrete and Continuous Models and Applied Computational Science. 2022;30(4):318-329
pages 318-329 views

On a dispersion curve of a waveguide filled with inhomogeneous substance

Kroytor O.K., Malykh M.D.


The paper discusses the relationship between the modes traveling along the axis of the waveguide and the standing modes of a cylindrical resonator, and shows how this relationship can be explored using the Sage computer algebra system. In this paper, we study this connection and, on its basis, describe a new method for constructing the dispersion curve of a waveguide with an optically inhomogeneous filling. The aim of our work was to find out what computer algebra systems can give when calculating the points of the waveguide dispersion curve. Our method for constructing the dispersion curve of a waveguide with optically inhomogeneous filling differs from those proposed earlier in that it reduces this problem to calculating the eigenvalues of a self-adjoint matrix, i.e., a well-studied problem. The use of a selfadjoint matrix eliminates the occurrence of artifacts associated with the appearance of a small imaginary addition to the eigenvalues. We have composed a program in the Sage computer algebra system that implements this method for a rectangular waveguide with rectangular inserts and tested it on SLE modes. The obtained results showed that the program successfully copes with the calculation of the points of the dispersion curve corresponding to the hybrid modes of the waveguide, and the points found fit the analytical curve with graphical accuracy even when with a small number of basis elements taken into account.

Discrete and Continuous Models and Applied Computational Science. 2022;30(4):330-341
pages 330-341 views

On a modification of the Hamming method for summing discrete Fourier series and its application to solve the problem of correction of thermographic images

Laneev E.B., Baaj O.


The paper considers mathematical methods of correction of thermographic images (thermograms) in the form of temperature distribution on the surface of the object under study, obtained using a thermal imager. The thermogram reproduces the image of the heat-generating structures located inside the object under study. This image is transmitted with distortions, since the sources are usually removed from its surface and the temperature distribution on the surface of the object transmits the image as blurred due to the processes of thermal conductivity and heat exchange. In this paper, the continuation of the temperature function as a harmonic function from the surface deep into the object under study in order to obtain a temperature distribution function near sources is considered as a correction principle. This distribution is considered as an adjusted thermogram. The continuation is carried out on the basis of solving the Cauchy problem for the Laplace equation - an ill-posed problem. The solution is constructed using the Tikhonov regularization method. The main part of the constructed approximate solution is presented as a Fourier series by the eigenfunctions of the Laplace operator. Discretization of the problem leads to discrete Fourier series. A modification of the Hamming method for summing Fourier series and calculating their coefficients is proposed.

Discrete and Continuous Models and Applied Computational Science. 2022;30(4):342-356
pages 342-356 views

Profile thickness synthesis of thin-film waveguide Luneburg lens

Lovetskiy K.P., Sevastianov A.L., Zorin A.V.


In the work the link between the focusing inhomogeneity of the effective refractive index of waveguide Luneburg lens and the irregularity of the waveguide layer thickness generating this inhomogeneity is demonstrated. For the dispersion relation of irregular thin-film waveguide in the model of adiabatic waveguide modes the problem of mathematical synthesis and computer-aided design of the waveguide layer thickness profile for the Luneburg thin-film generalized waveguide lens with a given focal length is being solved. The calculations are carried out in normalized (in a special way) coordinates to adapt the used relations to computer calculations. The obtained solution is compared with the same solution within the cross-section’s method. The performance of the algorithm implemented in Delphi, was demonstrated by plotting the dispersion curves and plotting a family of dispersion curves, demonstrating a critical convergence. As an additional result, the thickness profiles of additional (irregular in thickness) waveguide layer, forming a thin film generalized waveguide Luneburg lens were synthesized. This result generalizes Southwell’s results.

Discrete and Continuous Models and Applied Computational Science. 2022;30(4):357-363
pages 357-363 views

Conservative finite difference schemes for dynamical systems

Ying Y., Lu Z.


The article presents the implementation of one of the approaches to the integration of dynamical systems, which preserves algebraic integrals in the original fdm for Sage system. This approach, which goes back to the paper by del Buono and Mastroserio, makes it possible, based on any two explicit difference schemes, including any two explicit Runge-Kutta schemes, to construct a new numerical algorithm for integrating a dynamical system that preserves the given integral. This approach has been implemented and tested in the original fdm for Sage system. Details and implementation difficulties are discussed. For testing, two Runge-Kutta schemes were taken having the same order, but different Butcher tables, which does not complicate the method due to paralleling. Two examples are considered - a linear oscillator and a Jacobi oscillator with two quadratic integrals. The second example shows that the preservation of one integral of motion does not lead to the conservation of the other. Moreover, this method allows us to propose a practical application of the well-known ambiguity in the definition of Butcher tables.

Discrete and Continuous Models and Applied Computational Science. 2022;30(4):364-373
pages 364-373 views

Approximation of radial structure of unstable ion-sound modes in rotating magnetized plasma column by eikonal equation

Marusov N.A.


The problem of the correct asymptotic construction of the radial structure of linearly unstable ion-sound electrostatic eigenmodes is studied. The eigenvalue problem with boundary conditions of the first and second kind (electrodynamic and hydrodynamic types) for the oscillations that propagate in a uniform cylindrical column of magnetized plasma along an axial homogeneous magnetic field is formulated. A method for constructing a discrete spectrum of small-scale unstable oscillations of the system based on the basic principles of geometric optics is proposed. The main idea of the method is an explicit idea of the type of boundary conditions - the conductivity and absorbing properties of the wall bounding the plasma cylinder. A dispersion relation for unstable small-scale modes destabilized due to the effects of differential rotation is derived from the Eikonal equation. For the correct construction instability growth rates spectra an universal recipe for the selection of radial wave numbers of small-scale eigenmodes in accordance with any of the types of boundary conditions is proposed.

Discrete and Continuous Models and Applied Computational Science. 2022;30(4):374-378
pages 374-378 views

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