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742 Section 1
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 theimmersed 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 numericalmodel consists of the SIMPLE.like algorithm on the staggered mesh to solve the Navier�Stokes equations. A moving Lagrangian mesh with an almost uniform distributionof 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. Empiricalformulae [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 sphericaldrops or solid spheres on a wallin viscous fluid // Chem. Eng.Sci. 2006. V. 61, iss. 11.P. 3543.3549.
LevinsontypealgorithmsforsolvingscatteringproblemsfortheManakovmodelofnonlinearSchroedingerequations
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 theinversion 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. Numericaltests 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 numericalmethod 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.
