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RUSSIAN FEDERAL NUCLEAR CENTER 
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Issue N^{o} 2, 2024  A ROMB SCHEME FOR SOLVING THE RADIATIVE HEAT TRANSFER EQUATION ON ADAPTIVE SQUARE MESHES
N. N. Veselova, S. N. Lebedev VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 2. P. 317.
The paper describes a difference scheme for solving 2D radiative heat transfer equations on adaptive square meshes. The scheme is based on the approach used in the ROMB method. The 2D difference equations are solved by splitting to reduce the 2D problem to a number of simple 1D problems. Comparative calculations of model problems on adaptive fractional meshes are provided. Key words: mathematical modeling, numerical methods for solving equations, adaptive mesh, ROMB scheme, countermarching method, traveling heat wave.
 THREESTEP ALGORITHM FOR CALCULATING A LOCALLY COMONOTONIC CUBIC C^{1}CLASS SPLINE WITH APPROXIMATE PROJECTION S. V. Mzhachikh, Yu. N. Lapshina VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 2. P. 1827.
Ideas are presented to improve the computational efficiency of the previously proposed threestage method for calculating a locally comonotonic cubic spline of C^{1} class. Basing on numerical experiments, it was concluded that, of the three considered splines of the C^{1}class, the interpolant constructed using the threestage method deviates less than others from the classic cubic spline of the C^{2} class both in the C norm and in the L_{2} norm. However, the threestep spline calculation method may seem too expensive, especially when compared to the popular FritschCarlson method. The problem of accelerating a threestage algorithm appeared to be easily solved. To do this, the method of accurately projecting a point onto the boundary of the comonotony set normal to the ellipse line that requires an accurate solution of the fourth degree equation is replaced by a noniterative approximate projection method, which is considered in several versions. Testing has demonstrated that the results of accurate and approximate calculations practically coincide. Key words: cubic spline, monotone interpolation, local comonotonicity.
 A TEST PROBLEM WITH ANALYTICAL SOLUTION OF RADIATION AND ENERGY TRANSFER EQUATIONS IN TWODIMENSIONAL AXISYMMETRIC CASE A. A. Shestakov, A. N. Startsev VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 2. P. 2834.
A large number of works are devoted to the problem of designing tests to verify numerical methods for solving the radiation transport equation. Testing is one of the key tasks when developing any new algorithm or program. Universal verification tasks that immediately test all the capabilities of the program are very rare with complex algorithms. When testing programs, it is advisable to select tasks that have analytical solutions along with verification ones. Although some progress has been made in constructing analytical solutions for the radiation transport equation, these solutions are not always sufficient for different classes of transport problems. It is particularly difficult to obtain analytical solutions in curved coordinate systems. In this work, a test problem was designed in a twodimensional axisymmetric case basing on the expansion of the transport operator resolvent in von Neumann series, and numerical calculations were carried out at different absorption coefficients. A feature of the test problem is that it allows comparing the results of numerical calculations with an anisotropic distribution of intensity along the flight directions of photons with an accurate solution. This helps to estimate the accuracy of quadrature formulas used in the approximation of the radiation field in the phase space of the flight directions of photons. Key words: test problem, system of equations of thermal radiation transfer.
 MODELING THE DYNAMICS OF A HIGHCURRENT RELATIVISTIC ELECTRON BEAM IN A DRIFT CHAMBER WITH AN INHOMOGENEOUS MAGNETIC FIELD G. N. Kolesov, A. E. Dubinov VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 2. P. 3544.
The results of mathematical modeling of dynamics of a highcurrent relativistic electron beam in a vacuum cylindrical drift chamber, on which an external inhomogeneous magnetostatic field is superimposed, are presented. The work uses the KARAT code, Maxwell´s selfconsistent solving equations and relativistic equations of electron motion on a set computation mesh using the particleincell method. Instantaneous phase portraits and a time densitogram of electrons of a highcurrent relativistic electron beam are calculated. The evolution of the energy of electrons of a highcurrent relativistic electron beam has been investigated. The occurrence of a chain of virtual cathodes in the area of magnetic field increase was detected. Key words: highcurrent relativistic electron beam, mathematical simulation, KARAT code, particleincell method, inhomogeneous magnetic field, a virtual cathode.
 ACCOUNTING FOR SPONTANEOUS MAGNETIC FIELDS IN 3D "FOCUS" PROGRAM I. V. Glazyrin, A. V. Ershova, N. A. Mikhailov VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 2. P. 4559.
A mathematical description of certain physical effect, which is referred to in the literature as spontaneous magnetic fields, is presented. Spontaneous magnetic fields can occur in an environment, initially without magnetic charge, provided that pressure and density gradients are noncollinear. In various experiments on the laser radiation interaction with matter in the forming plasma, significant magnetic fields were detected that affect the development of instabilities. A threedimensional Focus program is being developed to study the described processes. The Focus program uses the finite volume approach. Within the framework of the program, the solution of the equations of ideal magnetic gas dynamics has been implemented. Approximation of flows in the centers of cell faces is carried out according to the HLL scheme. To calculate the flows on the faces of the cells, the onedimensional Riemann problem is solved. The time solution is carried out according to the twostage RungeKutta scheme. Magnetic charge cleaning is performed using an artificial scalar potential. The program has been tested to solve onedimensional, twodimensional and threedimensional problems of ideal magnetic gas dynamics. Comparisons with solutions obtained under the FLASH program showed a good correspondence of the results. Using magnetic charge cleaning gives a more accurate result. Test problems with an analytical solution are proposed to verify the numerical implementation of accounting for spontaneous magnetic fields. The second order of approximation for calculating the gradient of smooth functions according to the Gauss´s theorem is obtained. The convergence of the numerical solution of the problem to the analytical one on thickening meshes is shown. Key words: magnetic gas dynamics, spontaneous magnetic field, magnetic charge cleaning, artificial scalar potential.
 MATHEMATICAL MODELING OF IMPURITY TRANSPORT IN SURFACE WATERS USING THE GERA COMPUTATION CODE K. A. Novikov VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 2. P. 6070.
A mathematical model of impurity transport in surface waters is considered. It is based on the diffusion wave approximation for shallow water equations and convectiondiffusion equations taking into account sorption in bottom sediments to simulate surface runoff and impurity transport in surface waters, respectively. The implementation of the model is based on a finitevolume flow sampling scheme and the use of the INMOST distributed computing platform. Three numerical experiments are described: the first two are used for verification, and the third is aimed at checking the effectiveness of the parallel implementation of the model. Verification tests demonstrate good agreement of the simulation solution with the analytical solution and experimental data on impurity concentration and surface water dynamics. The last test computes the acceleration of the runtime when running the model on 2, 4, 8, 16 and 32 cores. Good (up to 12 times) scalability is shown within the limits set by the used number of computation cells (138 849) and the method of solving systems of linear equations with a preconditioner based on incomplete triangular decomposition. Key words: surface runoff, transport in surface water, verification, parallel computations.
 FORMATION OF A LAYER OF PYRAMIDS FOR THE TRANSITION FROM HEXAHEDRAL TO TETRAHEDRAL MESHES E. Yu. Arapova, A. V. Tikhonov VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 2. P. 7183.
Algorithms of automatic generator forming transition layer of pyramids in a combined mesh that consists of hexahedrons and tetrahedrons are described. The generator is used when generating mesh models to solve strength problems in the LOGOS software package. At the initial stage, it is planned to obtain/generate the initial surface quadrilateral mesh with further division of each quadrilateral with a diagonal into two triangles. The next step is to generate a tetrahedral mesh. Then, performing transformations over tetrahedrons incident to the diagonal of the quadrilateral, a pyramid is built with a base in this quadrilateral. The process continues until each element of the quadrilateral mesh becomes the base of the pyramidal element. At the final stage, an algorithm is used that improves the quality of tetrahedrons that have common faces with the constructed pyramids. Key words: pyramid, transformation of tetrahedrons, quality of mesh elements, LOGOS software package.
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