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6A 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 require-
ment 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 parti-
cles. The new version applies to solving the one-dimensional and two-dimensional quasi-linear transport equa-
tion 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 approx-
imations 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 appa-
ratus 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 pro-
posed. Methods are based on using nonlinear singular integral equations. In the one-dimensional case ampli-
tude and phase problems are modeled by corresponding nonlinear singular integral equations. In the two-
dimensional case amplitude and phase problems are modeled by corresponding nonlinear bisingular integral
equations. Several approaches are presented for modeling two-dimensional problems:
1) reduction of amplitude and phase problems to systems of nonlinear singular integral equations;
2) using methods of the theory of functions of the complex variable, problems are reduced to nonlinear
bisingular integral equations.
To solve the constructed nonlinear singular equations, methods of collocation and mechanical quadrature
are used. These methods lead to systems of nonlinear algebraic equations, which are solved by the continuous
method for solution of nonlinear operator equations. The choice of this method is due to the fact that it is sta-
ble against perturbations of coefficients in the right-hand side of the system of equations. In addition, the
method is realizable even in cases where the Frechet and Gateaux derivatives degenerate at a finite number of
steps in the iterative process.
Some model examples have shown effectiveness of proposed methods and numerical algorithms.
Numerical simulation of the dynamics of a heated turbulent mixing zone in a linear stratified medium
G. G. Chernykh1,2, A. V. Fomina3, N. P. Moshkin2,4
1Federal Research Center for Information and Computational Technologies
2Novosibirsk State University
3Novokuznetsk Institute (branch) of the Kemerovo State University, Novokuznetsk
4Lavrentyev Institute of Hydrodynamics SB RAS
Email: chernykh@ict.nsc.ru, fav@rdtc.ru, nikolay.moshkin@gmail.com
DOI 10.24412/cl-35065-2021-1-00-10
Evolution of localized regions of turbulized fluid (turbulent spots) has a decisive effect on the formation of
fine microstructure of hydrophysical fields in the ocean [1].
Based on an algebraic model of Reynolds stresses and fluxes, a numerical model of the dynamics of a flat
localized region of turbulent perturbations of non-zero buoyancy in a linearly stratified medium was con-
structed. Presence of non-zero buoyancy leads to increase in the geometrical dimensions of the turbulent spot
and generation of internal waves of greater amplitude in comparison with a spot of non-zero buoyancy. The
work is a continuation and development of research [2].
References
1. Monin A. S., Yaglom A. M. Statistical fluid mechanics. V. 1. Mechanics of turbulence. Dover Books on Physics. 2007,
784 p.
2. Chernykh G. G., Fomina A. V., Moshkin N. P. Numerical Simulation of Dynamics of Weakly Heated Turbulent Mixing
Zone in Linearly Stratified Medium // J. of Engineering Thermophysics. 2020. Vol.29. Iss. 4. P. 674-685.
A-WENO schemes based on adaptive artificial viscosity
S. Chu, A. Kurganov
Southern University of Science and Technology, Shenzhen, China
Email: alexander@sustech.edu.cn
DOI 10.24412/cl-35065-2021-1-00-11
We will introduce new finite-difference A-WENO schemes for hyperbolic systems of conservation laws.
The proposed schemes are fifth-order accurate in space and stable. Unlike the original A-WENO schemes [1],
the stabilization is achieved using the adaptive artificial viscosity (AAV) approach introduced in [3]: the AAV
