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RUSSIAN FEDERAL NUCLEAR CENTER 
ALLRUSSIAN RESEARCH INSTITUTE OF EXPERIMENTAL PHYSICS 

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Issue N^{o} 1, 2024  NUMERICAL SOLUTION OF THE RADIATIVE HEAT TRANSFER EQUATION IN 2D PROBLEMS WITH PHASE TRANSITIONS ON ADAPTIVE MESHES
A. M. Mustafin, N. N. Veselova, S. N. Lebedev VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 1. P. 417.
The paper discusses the development of a numerical technique for solving a 2D radiative heat transfer equation with account for phase transitions on adaptive difference meshes with a ROMB scheme, including the construction of the adaptive mesh and the efficiency testing of the developed technique. The problems of heat transfer during phase transitions, referred to as Stephan´s type problems, are solved by a oncethrough method based on the use of a difference analog of the δfunction. A ROMB scheme on an adaptively embedded mesh is obtained and an algorithm for constructing an adaptive difference mesh is described. Provided are numerical solutions for two problems: heat wave propagation in a heterogeneous region and a thermal beam impact on ice. The first clearly shows advantages of mesh adaptation near the front of the heat wave in 2D space. Both demonstrate gains from the use of mesh adaptation in combination with oncethrough calculation before applying traditional approaches to Stephan´s type problems. Key words: Stephan´s problem, ROMB difference scheme, adaptive difference mesh, 2D radiative heat transfer equation, phase transition, heat wave.
 STOCHASTIC METHOD OF NUMERICAL SIMULATION OF RADIATION BELTS OF THE EARTH A. N. Zalialov, N. V. Ivanov, A. E. Shirokov VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 1. P. 1829.
The paper describe a stochastic method of numerical simulation of the dynamics of relativistic electrons in radiation belts of the Earth that uses approximation of the leading center that, opposite other approaches, allows describing relaxation of relatively quick perturbations in the radiation belts of the Earth (for example, during geomagnetic storm) in the regions of low orbits. A stationary offcentered dipole is used as a physical–mathematical model of the geomagnetic field. The solution of the kinetic equation in the Fokker  Plank approximation in this method is found using a MonteCarlo method basing on the simulation of the motion of relativistic electrons along the geomagnetic line with the account for their interaction with the atmosphere. A method of splitting by physical processes into three stages is used to construct the trajectory. The motion of the electron under the effect of the magnetic field with the account for the deceleration force is computed at the first stage, electron scattering is simulated at the second stage, and its radial diffusion is simulated at the third stage. The results of the numerical solution of the test problems is compared with the results of the analytical and difference solutions. Key words: relativistic electrons, radiation belt of the Earth, MonteCarlo method, radial diffusion.
 SPECIFIC FEATURES OF DETONATION PROPAGATION IN SMALL CROSSSECTION CHANNELS FOR THE PETNBASED COMPOSITION V. B. Titova, N. A. Volodina, M. O. Shirshova, E. V. Khaldeev, Y. M. Sustaeva VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 1. P. 3036.
The works to study the processes of detonation wave propagation in the channels equipped with PETNbased (pentraerythritoltetranitrate) composition are currently underway to upgrade the systems of detonation initiation in experimental setups with explosives. This paper studies specific features of the detonation wave propagation in the smallcrosssection channels using model experiments with numerical simulation in the ideal and kinematic approximations. It shows the necessity to account for detonation kinetics in such computations. As a result of the computational analysis, we obtained a characteristic picture of the detonation propagation when the detonation wave follows the shadow zone, and the size of the shadow zone region is estimated after the detonation wave passes the turn of the channel. Key words: detonation kinetics, PETN, channels interconnections, numerical simulation.
 ACCOUNT FOR KINETIC EFFECTS IN APPROXIMATION OF RADIATING HEAT TRANSFER TO COMPUTE RADIATION TRANSFER PROBLEMS D. A. Koshutin, A. A. Shestakov VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 1. P. 3741.
One of the most difficult problems of radiation gas dynamics is solution of a multidimensional equation of radiative heat transfer, the difficulty of which mostly comes from the largedimensional character of the phase space. Because of this, they implement different approximations that simplify the radiation transfer problem. These approximations comprise, for example, approximation of radiative heat transfer. Here, complex enough problems of selecting an effective iteration method and construction of a monotonous scheme of upgraded accuracy for the kinetic equation are eliminated. When we implement approximation of radiative heat transfer, different correction factors are used. The need in accurate correction factors in approximation of the radiative heat transfer stimulated development of new mathematical models. In 2016, M. Yu. Kozmanov, D. A. Koshutin and A. A. Shestakov suggested replacing a system of 2D transfer equations with the equations of quasithermal conductivity, where the solutions are obtained by parabolic equations. This is a generalization of the quasitransfer method offered by M. Yu. Kozmanov and N. G. Karlykhanov. To solve a 2D system of parabolic equations of quasiheat transfer, they used the method splitting by directions and the difference scheme of the second order of approximation. In the approximation of quasiheat transfer, a parabolic system is solved, for which you can construct a monotonous difference scheme of the second order of approximation. The paper describes algorithms proposed in 2016, but not previously published. The ideas of quasithermal conductivity were further developed in the works by N. G. Karlykhanov, where averaging of multigroup equation of transfer by the spectrum of photons was additionally applied. Key words: radiative heat transfer, account for kinetic effects.
 VERIFICATION OF THE EQUATIONS OF STATE FIR THE EXPLOSION PRODUCTS OF THE PLASTICIZED HMX USING THE EXPERIMENTAL RESULTS ON LINERS ACCELERATION V. B. Titova, N. A. Volodina, M. O. Shirshova, M. N. Kityukhina, E. N. Bogdanov, A. A. Stanovov VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 1. P. 4247.
The work describes the experimental results where continuous registration was carried out for the motion of liners made of aluminum and the tungstennickeliron alloy thrown with the explosion products of the plasticized HMX. The computational analysis is provided for the experimental results on the research on the propelling properties of the plasticized HMX in order to verify the equations of state of its explosion products. Verification of the equations of state of the explosion products was carried out for three forms of the equations of state relying of the experimental data. Involvement of modern experimental research methods allowed for more highquality verification of the equations of state of the explosion products. Key words: explosion products, detonation, equation of state, kinetics, numerical simulation.
 50 YEARS OF FLECK PROBLEMS A. A. Shestakov VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 1. P. 4856.
In 1971, specialists on radiation transfer from Livermore National Lab (LLNA) of the USA suggested problems that with time became classical tests for radiation transfer. In all problems, a flat layer is heated with the radiation flux that corresponds to the temperature of 1 keV. The basic advantages of these problems are: – the simplicity of the setup of the boundary and initial conditions that allows computing these problems in different approximations; – simple analytical formulas for spectral runs; – a wide range of implementation of spectral coefficient of absorption; – simulation of disruptors of the spectral coefficient разрывов of absorption in space and energy spectrum; – the simplest equation of state of the material. The paper is devoted to the fiftyyear anniversary of Fleck tests in the method of numerical solution of heat radiation transfer. The paper provides the setups and numerical computations of two modifications of these problems. Key words: heat radiation transfer, Fleck problems.
 NUMERICAL SIMULATION ON THE BASIS OF LINEARIZED SYSTEM OF GASDYNAMIC EQUATIONS OF RISING SWIRL FLOWS WITH THE ACCOUNT FOR CORIOLIS FORCES A. A. Bugaenko, I. Yu. Krutova VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 1. P. 5763.
A system of gasdynamic equations is linearized with the account for Coriolis forces using an accurate solution (uniform rest). Origination and development of rising swirl flows are numerically simulated using the gas blow up the pipe. The results of numerical computations of the gasdynamic parameters for average by intensity cyclones are described. Key words: a system of gasdynamic equations, Coriolis force, linearization, numerical simulation, gas blow.
 ALGORITHMS AND SPECIFIC FEATURES OF PROGRAM REALIZATION OF SEMIEMPIRICAL MODEL OF WIDERANGE "ROSAMFI" EQUATION OF STATE IN "URSOF" PACKAGE A. S. Danilov, D. G. Gordeev, O. N. Shumilina, I. N. Arapov VANT. Ser.: Mat. Mod. Fiz. Proc. 2024. No 1. P. 6477.
The paper briefly describes algorithms that realize the computation of thermodynamic functions by semiempirical model of widerange ROSAMFI equations of state and specific features of their program realization in the library of programs of UFSOF package. The ROSAMFI model is the development of the earlier created ROSAMI model and it additionally accounts for the changes in the thermodynamic functions at a solid  a liquid phase transition (melting). Possibilities of the model in the description of thermodynamic functions in the region of melting are shown using the example of the copper equation of state. Program realization of the ROSAMFI model is done in Fortran 90 language using vectorized structures in multiarchitectural reference format. Using specially developed proprocessor, equivalent transformations of this format onto programs are done that account for the specific features of notvectorized and vectorized computations and the computations using the CUDA architecture. Key words: URSOF program library, algorithms of the ROSAMFI model, algorithms to compute thermodynamic functions at phase transitions, melting, evaporation, multiversion reference file, parallelization efficiency.
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