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2638 Section 1
The research was partly supported by the Ministry of Science and Higher Education of the Russian Federation within
the framework of the state assignment (project Nos. 121030500137-5 and ����-�19-119051590004-5).
References
1. Belyaev V. A. Solving a Poisson equation with singularities by the least-squares collocation method // Numer. Anal.
Appl. 2020. V. 13, N. 3, P. 207-218.
Numerical simulation of a stabilizing Poiseuille-type polymer fluid flow in the channel with elliptical
cross-section
A. M. Blokhin1,2, B. V. Semisalov1
1
Novosibirsk State University
2
Sobolev Institute of Mathematics SB RAS
Email: blokhin@math.nsc.ru, vibis@ngs.ru
DOI 10.24412/cl-35065-2021-1-00-04
Stabilization of the Poiseuille-type flows of an incompressible viscoelastic polymer fluid is studied using
non-linear rheological relations from [1]. Channels of elliptical and circular cross-sections are considered. In [2]
it was shown that the corresponding stationary formulation admits three different solutions. The process of
stabilization of the flow after the jump of pressure gradient in the channel was simulated using the algorithm
from [3]. The stabilized flow shows which of the three solutions of the stationary problem is implemented in
practice. Simulations in a wide range of values of the physical parameters enable us to discover the effect of
"switching" the limiting solution of the non-stationary problem from one solution of the stationary equations
to another. The scenario of this switch is discussed in detail.
The research has been done under the financial support of the Russian Science Foundation (project No. 20-11-
20036)
References
1. Altukhov Yu. A., Gusev A. S., Pyshnograi G. V. Introduction to the Mesoscopic Theory of Flowing Polymer Systems.
Barnaul: Altai State Pedagogical Academy Press, 2012 [in Russian].
2. Blokhin, A. M., Semisalov, B. V. Simulation of the Stationary Nonisothermal MHD Flows of Polymeric Fluids in
Channels with Interior Heating Elements // J. Appl. Ind. Math. 2020. V. 14. P. 222�241.
3. Semisalov B. V. Fast Nonlocal Algorithm for Solving Neumann�Dirichlet Boundary Value Problems with an Error
Control // Vychisl. Metody. Programmirovanie. 2016. V. 17, N. 4. P. 500�522 [in Russian].
The discontinuous shapeless particle method for quasi-linear transport
S. V. Bogomolov1, A. E. Kuvshinnikov2
1
Lomonosov Moscow State University
2
Keldysh Institute of Applied Mathematics RAS
Email: kuvsh90@yandex.ru
DOI 10.24412/cl-35065-2021-1-00-06
Simulation of gas dynamic problems deals with the appearance of discontinuities, more precisely, strong
gradients. The quality of computational methods is assessed primarily by their ability to convey this behavior
of a solution as adequately as possible. In our opinion, the discontinuous particle method [1-3] allows one to
cope with these difficulties better than alternative, traditionally more commonly used difference and finite
element methods. This is achieved because the particle method is based on the Lagrange approach, and this,
in turn, provides automatic mesh generation.
Methods of computational algebra and solving mathematical physics equations
A new variant of the discontinuous particle method is presented. We use a new particle rearrangement
criterion without analyzing particle overlaps. It is assumed that the nonlinear elastic transport preserves not
only the mass of the particles, but also the mass located between the centers of these particles. This requirement
leads to the fact that the change in the distance between the particles in the process of their movement
and the conservation of mass in the space between them leads to a change in the density of one of the particles.
The new version applies to solving the one-dimensional and two-dimensional quasi-linear transport equation
problems. The main feature of the new variant is minimal smearing of discontinuities.
References
1. Bogomolov S.V., Esikova N.B., Kuvshinnikov A.E. Micro-macro Fokker�Planck�Kolmogorov models for a gas of rigid
spheres // Math. Models Comput. Simul. 2016. V. 8, N. 5. P. 533-547.
2. Bogomolov S.V., Kuvshinnikov A.E. Discontinuous particle method on gas dynamic examples // Math. Models
Comput. Simul. 2019. V. 11, N. 5. P. 768-777.
3. Bogomolov S.V., Filippova M.A., Kuvshinnikov A.E. A discontinuous particle method for the inviscid Burgers�
equation // J. Phys.: Conf. Ser. 2021. V. 1715. 012066.
On the optimal approximation of functions in the boundary layer
I. V. Boikov, V. A. Ryazantsev
Penza State University
Email: i.v.boykov@gmail.com
DOI 10.24412/cl-35065-2021-1-00-07
In this study we consider the problem of approximation of functions belonging to the class of functions
with high gradients in boundary layer. For such functions we build an algorithm of approximation both in one-
dimensional and multidimensional cases. The idea of the algorithm is based on the results for optimal approximations
of specific functional classes [1-3]. These classes include functions with modules of derivatives having
power-type singularity that is a function of distance from the point to the boundary of the domain.
In order to develop the proposed algorithm we introduce the specific functional class and use the apparatus
of continuous local splines providing approximation of functions from the mentioned class that is optimal
with respect to accuracy. Solving model problems demonstrate the efficiency of the proposed method.
References
1. Boikov I. V., Ryazantsev V. A. On the optimal approximation of geophysical field // Siberian J. Num. Math. 2021.
V. 24, N. 1. P. 17-34.
2. Boykov I. V., Ryazantsev V. A. On a difference method of potential fields� extension // University proceedings.
Volga region. Physical and mathematical sciences. 2014. N. 2 (30). P. 20-33.
3. Boikov I.V. Approximation of some classes of functions by local splines // Computational Mathematics and
Mathematical Physics. 1998. V. 38, N. 1. P. 21�29.
On one iterative method for solving the amplitude-phase problem
I. V. Boikov, Ya. V. Zelina, D. I. Vasyunin
Penza State University
Email: zelinayana@gmail.com
DOI 10.24412/cl-35065-2021-1-00-08
Methods for solving amplitude and phase problems for one and two-dimensional discrete signals are proposed.
Methods are based on using nonlinear singular integral equations. In the one-dimensional case ampli
