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14tude 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: [email protected], [email protected], [email protected]
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: [email protected]
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
terms are made proportional to the weak local residuals originally introduced in [2]. The performance of the
proposed schemes will be demonstrated on a number of challenging benchmarks.
References
1. Jiang Y., Shu C.-W., Zhang, M. An alternative formulation of finite difference weighted ENO schemes with Lax-
Wendroff time discretization for conservation laws // SIAM J. Sci. Comput.. 2013. V. 35 (2). P. A1137-A1160.
2. Karni S., Kurganov A. Local error analysis for approximate solutions of hyperbolic conservation laws // Adv.
Comput. Math. V. 22. P. 79-99.
3. Kurganov A., Liu Y. New adaptive artificial viscosity method for hyperbolic systems of conservation laws // J.
Comput. Phys. 2012. V. 231. P. 8114-8132.
Dual approach as empirical reliability for fractional differential equations
P. B. Dubovski, J. Slepoi
Stevens Institute of Technology
Email: [email protected]
DOI 10.24412/cl-35065-2021-1-00-12
We consider fractional differential equations with derivatives in the Gerasimov-Caputo sense. Computa-
tional methods for these equations exhibit essential instability. Even a minor modification of the coefficients
or other entry data may switch good results to divergent. The goal of this talk is to suggest a reliable dual ap-
proach which fixes this inconsistency. We suggest to use of two parallel computational methods based on the
transformation of fractional derivatives through (1) integration by parts and (2) substitution method. The by-
parts method is known whereas the substitution method is novel. We prove the stability theorem for the sub-
stitution method and introduce a proper discretization scheme that fits the grid points for both methods. The
solution is treated as reliable (robust) only if both schemes produce the same results. In order to demonstrate
the proposed dual approach, we apply it to linear, quasilinear and semilinear equations and obtain very good
precision. The provided examples and counterexamples support the necessity to use the dual approach be-
cause either method, used separately, may produce incorrect results. The order of the exactness is close to the
exactness of the approximations of fractional derivatives.
References
1. Gerasimov A. N. A generalization of the deformation laws and its application to the problems of internal friction //
Appl. Math. and Mechanics. 1948. V. 12. P. 251-260.
2. Zhukovskaya N. V., Kilbas A. A. Solving homogeneous fractional differential equations of Euler type // Differential
Equations. 2011. V. 47, N. 12. P. 1714-1725.
3. Albadarneh R., M. Zerqat M., Batiha I. Numerical Solutions for linear and non-linear fractional differential
equations // International J. of Pure and Applied Mathematics. 2016. V. 106, N. 3. P. 859-871.
4. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and applications of Fractional Differential equations. Amsterdam:
Elsevier Science 2006.
The CFD-DEM numerical model for simulation of a fluid flow with large particles
D. V. Esipov
Kutateladze Institute of Thermophysics SB RAS
Email: [email protected]
DOI 10.24412/cl-35065-2021-1-00-13
The mathematical model for considering flows consists of the Navier � Stokes equations for a viscous in-
compressible fluid flow and a set of motion and rotation equations for every particle. The no-slip boundary
condition posed at the surface of the particles connects these two different sets of equations. In the numerical
