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Numerical Analysis and Applications

2022 year, number 1

On one method for modeling an nonhomogeneous Poisson point process

T.A. Averina1,2
1Institute of Computational Mathematics and Mathematical Geophysics of Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
Keywords: nonhomogeneous Poisson point process, stochastic differential equations, Monte Carlo methods

Abstract >>
In the statistical solution to problems of analysis, synthesis and filtration for systems of the diffusion-discontinuous type, it is required to simulate an inhomogeneous Poisson point process. To simulate the latter, an algorithm is sometimes used based on the property of the ordinariness of the process. In this paper, a modification of this algorithm is constructed using an efficient method for modeling random variables. The statistical adequacy of the method developed was checked by solving test problems.

Solving two-dimensional problems of gas dynamics using an implicit scheme for the discontinuous Galerkin method on unstructured triangular grids

R.V. Zhalnin1, V.F. Masyagin1, V.F. Tishkin2
1Ogarev Mordovia State University, Saransk, Russia
2Keldysh Applied Mathematics Institute, Academy of Sciences of the USSR, Moscow, Russia
Keywords: gas dynamics equations, discontinuous Glerkin method, implicit scheme, NVIDIA AmgX

Abstract >>
An implicit scheme of the discontinuous Galerkin method for solving gas dynamics equations on unstructured triangular grids is constructed. The implicit scheme is based on the representation of a system of grid equations in the so-called «delta» form. To solve the resulting SLAE for each moment of time, solvers from the NVIDIA AmgX library are used. To verify the numerical algorithm, a series of calculations were performed for the flow over the NACA0012 symmetric airfoil profile at various angles of attack, and the problem of the flow over the RAE2822 airfoil profile was solved. The results of calculations are presented.

A method of the magnetotelluric field numerical modelling in a horizontally homogeneous medium: difference schemes, estimates of convergence

O.B. Zabinyakova1, S.N. Sklyar2
1Research Station of the Russian Academy of Sciences, Bishkek, Kyrgyzstan
2American University of Central Asia, Bishkek, Kyrgyzstan
Keywords: Tikhonov-Cagniard model, direct one-dimensional problem of magnetotelluric sounding, numerical solution, interpolation of an approximate solution, matrix exponent, local integral equation, convergence estimates

Abstract >>
This paper proposes a method for the numerical solution of the direct one-dimensional problem of magnetotelluric sounding. The construction of difference schemes is realized by the local integral equations method. A natural variant of the interpolation of an approximate solution is considered. The estimate of convergence of the approximate solution to the exact one and the estimate of the interpolation error are proved.

On matrices whose cosquares are diagonalizable and have real spectra

Kh.D. Ikramov
Lomonosov Moscow State University, Moscow,Russia
Keywords: congruence, cosquare, rational algorithm, diagonalizability via similarity transformation

Abstract >>
It is shown that the algorithm for verifying congruence of square roots of Hermitian matrices proposed earlier by the author can be extended to the considerably more broad class of matrices whose cosquares are diagonalizable and have real spectra.

On one approach to the qualitative analysis of nonlinear dynamical systems

V.D. Irtegov, T.N. Titorenko
Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, Irkutsk, Russia
Keywords: nonlinear dynamical systems, qualitative analysis, computer algebra, invariant sets, stability

Abstract >>
By an example of the investigation of the Euler equations on Lie algebras, we discuss an approach to the qualitative analysis of differential equations arising in a number of problems of mathematical physics, including rigid body dynamics. The approach proposed is based on a combination of methods of computer algebra and qualitative analysis of differential equations. We consider the applications of computer algebra in the problems of finding stationary invariant sets and studying their stability. For the equations under study, stationary invariant sets of various dimension have been found and their stability in the Lyapunov sense has been investigated.

On an approach to numerical solutions of the Dirichlet problem of an arbitrary dimension

B.V. Semisalov1,2
1Novosibirsk State University, Novosibirsk, Russia
2S.L. Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
Keywords: Dirichlet boundary value problem, decrease of computational costs, pseudospectral method, collocation method, relaxation method

Abstract >>
A method for the search for numerical solutions to the Dirichlet boundary value problems for nonlinear partial differential equations of the elliptic type and of an arbitrary dimension is proposed. It ensures low consumptions of memory and computer time for the problems with smooth solutions. The method is based on the modified interpolation polynomials with the Chebyshev nodes for approximation of the sought for function and on the new approach to constructing and solving the problems of linear algebra corresponding to the given differential equations. The analysis of spectra and condition numbers of matrices of the designed algorithm is made by applying the interval methods. The theorems on approximation and stability of the algorithm proposed for the linear case are proved. It is shown that the algorithm ensures an essential decrease in computational costs as compared to the classical collocation methods and to finite difference schemes.