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12terms 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: pdubovsk@stevens.edu
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: denis@esipov.org
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
model, it is assumed that the fluid occupies the entire flow region, including the space occupied by particles.
The influence of the particles on the fluid flow is realized by adding a special force acting from the immersed
boundary. The direct forcing scheme is used [1] to satisfy the no-slip condition on the immersed boundary and
evaluate the total force acting on every particle.
The CFD part of the numerical model consists of the SIMPLE-like algorithm on the staggered mesh to solve
the Navier � Stokes equations. A moving Lagrangian mesh with an almost uniform distribution of nodes repre-
sents the boundaries of particles. In general, the nodes of these two meshes don�t coincide, and a discrete del-
ta function is used to interpolate the variables. The discrete element method (DEM) is used to track, rotate
and collide particles by integrating all motion and rotation equations. The main feature of the DEM is integra-
tion with a very small time step. Empirical formulae [2] form the basis for the interaction of particles through a
thin layer of fluid.
References
1. Uhlmann M. An immersed boundary method with direct forcing for the simulation of particulate flows //
J. Comput. Phys. 2005. V. 209, iss. 2. P. 448-476.
2. Legendre D., Zenit R., Daniel C., Guiraud P. A note on the modelling of the bouncing of spherical drops or solid
spheres on a wall in viscous fluid // Chem. Eng. Sci. 2006. V. 61, iss. 11. P. 3543-3549.
Levinson type algorithms for solving scattering problems for the Manakov model of nonlinear Schroedinger
equations
L. L. Frumin
Institute of Automation and Electrometry SB RAS
Email: llfrumin@gmail.com
DOI 10.24412/cl-35065-2021-1-00-14
We have presented a numerical approach for solving the inverse and direct spectral scattering problems
for the focusing Manakov system. We have found an algebraic group of 4-block matrices with ordinary matri-
ces in diagonal blocks and with off-diagonal blocks consisting of special vector-like matrices that help general-
ize the scalar problem's efficient Levinson type numerical algorithms [1, 2] to the vector case of the Manakov
system. The inverse scattering problem solution represents the inversion of block matrices of the discretized
system of Gelfand � Levitan � Marchenko integral equations. Like the Zakharov � Shabad system's scalar case,
the Toeplitz symmetry of the matrix of the discretized GLM equations system drastically speeds up numerical
computations as in the Levinson algorithm. The reversal of steps of the inverse scattering problem algorithm
solves the direct scattering problem. Numerical tests performed by comparing calculations with the known
exact analytical solution, the Manakov vector soliton, have confirmed the proposed algorithms' efficiency and
stability, sufficient for applications. The application of the algorithms illustrated by the numerical simulation of
the polarized
This work was (partially) supported by the Ministry of Science and Higher Education of Russian Federation (project
����-�21-121012190005-2).
References
1. Belai O. V., Frumin L. L., Podivilov E. V., and Shapiro D. A., Efficient numerical method of the fiber Bragg grating
synthesis, J. Opt. Soc. Am. B 24 (2007), P. 1451�1457.
2. Frumin L. L., Belai O. V., Podivilov E. V., and Shapiro D. A., Efficient numerical method for solving the direct
Zakharov � Shabat scattering problem, J. Opt. Soc. Am. B 32 (2015), P. 290�296.
