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RUSSIAN FEDERAL NUCLEAR CENTER 
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Issue N^{o} 2, 2021  NONLIENEAR CONSISTENT METHOD (NCMETHOD) TO ACCELERATE CONVERGENCE OF ITERATIONS FOR TRANSPORT EQUATION
R. M. Shagaliev, A. A. Busalov VANT. Ser.: Mat. Mod. Fiz. Proc. 2021. No 2. P. 310.
The paper is devoted to constructing a new convergence accelerating method, namely, the nonlinear consistent method (NCmethod) for onedimensional computations. The NCmethod is derived as applied to the grid approximation of the transport equation using a difference scheme that provides positive grid solutions on structured grids. The simplest case of a 1D transport equation in Cartesian coordinates is considered. The method admits generalization to a multidimensional case and can be also used to solve the transport equation in curvilinear coordinates. This is a twostage method. The first stage is finding an approximate solution to the transport equation with the simple iteration method using the sweep algorithm at all points of the phase space. The second stage is constructing a system of grid equations relative to the scalar flow function that associates its values at a given point of the space grid with the values in neighboring intervals. The way of constructing the second stage equations ensures the consistency of the NCmethod of accelerating the convergence of simple iterations. Results of numerical studies are given and they demonstrate a high efficiency of the NCmethod and its unconditional convergence. Key words: the transport equation, iterative methods for solving grid equations, acceleration methods, the NCmethod.
 INCREASING THE ALGEBRAIC ORDER OF ACCURACY FOR ES_{n}–QUADRATURE M. P. Pepelyaev, E. A. Irinichev VANT. Ser.: Mat. Mod. Fiz. Proc. 2021. No 2. P. 1123.
When solving the particle transport problem in kinetic approximation using difference schemes one needs to develop quadrature formulas in angular variables on a sphere. The ES_{n}quadrature with equal weights is one of the commonly used for this purpose. The equality of weights reduces an error of the quadrature formula. However, the ES_{n}quadrature has a relatively low algebraic order of accuracy. The authors developed the way of how to improve the accuracy of the ES_{n}quadrature in angular variables to solve the particle transport equation. The quadrature still has equal weights, while the directional cosines of the azimuthal angle are corrected so that even moment conditions are satisfied. The system of equations for the calculation of the particle flight direction is linearized with Newton’s method and iteratively solved using Gaussian method at each iteration. The resultant ES_{n}quadrature has a higher order of accuracy, as compared to the ES_{n}quadrature. This is proved by results of numerical studies: the calculation of integrals of the given function over the sphere surface, the solution of a model problem with an anisotropic source, and the K. Kobayashi symmetrical test. Key words: angular quadrature, algebraic order of accuracy, equal weights, the discrete ordinate method, even moment conditions, threedimensional transport equation, Cartesian coordinate system.
 ABOUT ONE TYPE OF TENSOR ARTIFICIAL VISCOSITY FOR SIMULATION OF 3D GAS DYNAMIC FLOWS A. O. Naumov VANT. Ser.: Mat. Mod. Fiz. Proc. 2021. No 2. P. 2443.
The paper presents the description of the tensorform artificial viscosity for computations with the shockcapturing method, it is applied in the LEGAK3D code developed and used at RFNCVNIIEF for the simulation of complex gas dynamic flows with severely strained contact boundaries. The motion and energy equation approximation owing to the viscosity effect is based on the approximation of continuous divergence operators and the arbitrary tensor gradient represented by vector projections on the computational grid lines. The viscous tensor includes a scalar factor which is based on the Kuropatenkotype viscosity and contains a function that eliminates the effect of viscosity under the conditions of shockfree compression, or rotation of a medium as a rigid body. To demonstrate advantages of the proposed viscosity, simulation results are presented for the three problems: the 3D version of Zaltsman´s problem, the Noh spherical test, and the problem of gas compressing by a heavy shell. Key words: gas dynamics, shock waves, the tensorform artificial viscosity, numerical simulation.
 ALGORITHMS USED IN TIM CODE TO CONTROL VELOCITY OF DETONATION FRONT PROPARATION S. S. Sokolov,   V. N. Motlokhov 
VANT. Ser.: Mat. Mod. Fiz. Proc.. 2021. No 2. P. 4455.
The paper describes three algorithms used to control the velocity of the HE detonation front propagation, which were developed for unstructured polygonal and polyhedral grids. The first of them is the exact control algorithm, when the detonation time for all HEcontaining cells of the grid is determined once at the beginning of computations. The second is the stepbystep control algorithm that allows specifying in the computation process the time of the detonation wave arrival at each cell using the times of its arrival at neighboring cells. The both algorithms are costeffective, however, they have certain restrictions for a wide range of applied problems. The third algorithm represents by itself an improved version of the stepbystep control algorithm. In this algorithm the accuracy of calculating the detonation time for each HEcontaining cell is improved, because the direction of the moving detonation wave front is taken into account. In contrast to the basic stepbystep control algorithm, where the cell detonation time is corrected in the process of successively considering each detonated neighboring cell, in the third algorithm the detonation time for a given cell is corrected by considering neighbor cells from the first layer of cells surrounding the given cell. The third one is the generalpurpose algorithm that can be applied for calculations with the HE detonation control in regions of complex geometries, however, this is a time consuming algorithm in comparison with the first two algorithms. To demonstrate the applicability of all algorithms, the paper presents numerical results for several methodological problems on the simulation of a detonation wave propagating in HE using the TIM and TIM2D codes for solving the continuum mechanics problem on unstructured polygonal and polyhedral grids with an arbitrary number of links at nodes. Key words: the TIM code, a high explosive, steadystate detonation, detonation wave front, detonation wave propagation velocity, unstructured grids.
 ABOUT ONE LOCALLY COMONOTONE CUBIC C^{1}CLASS SPLINE S. V. Mzhachikh, Yu. N. Lapshina VANT. Ser.: Mat. Mod. Fiz. Proc. 2021. No 2. P. 5669.
In some studies it is required, from the standpoint of physics, that the data interpolating curve is monotone in each data monotonicity interval. The use of the classical cubic C^{2}class spline is not always possible for such problems. However, this problem is solvable and there are different ways to solve it. The paper presents a cubic C^{1}class spline for solving the monotone interpolation problem. This spline coincides with the classical cubic C^{2}class spline in the monotone behavior sections of the functional sequence under the condition that the classical spline is monotone in these sections. The only difference is observed near local extrema. Numerical results of the accuracy examination are presented, with the new interpolant being compared to other splines of some popular algorithms. Key words: cubic spline, monotone interpolation, local comonotonicity, the FritschCarlson method.
 A PROGRAM MODULE FOR GENERATION OF CLOSED SURFACE TRIANGULAR GRIDS IN "LOGOS" SOFTWARE PACKAGE V. A. Nikitin, A. V. Shurygin, I. G. Novikov, A. V. Egorov, S. S. Sokolov, A. I. Panov VANT. Ser.: Mat. Mod. Fiz. Proc. 2021. No 2. P. 7079.
RFNCVNIIEF developed the "Logos" engineeringanalysis software package for the integrated simulation of a broad class of problems on a supercomputer with massive parallelism. A prepostprocessor with a set of generators, including a program module for the generation of a closed surface triangular grid is used when preparing initial data for codes. The main purpose of this grid generator is to eliminate defects in the surface grids to be imported to the prepostprocessor, from external grid formats as well. Possible defects include some peculiarities of surfaces such as superpositions, overlapping of triangular cells, "holes", "hanging" triangles and vertices, small dihedral angles between neighboring triangles, and degenerated triangles. The generator operation stages are described: 1) preprocessing of an original surface, 2) generation of a closed surface near the original one; 3) mapping of the closed surface to the original surface, 4) postprocessing of the resultant surface. The general description of the approaches used in each stage is given, as well as their effect on the final result is shown. Key words: surface grid generator, closed surface triangular grid, vacuum packaging method, the "Logos" software package.
 ADAPTATION CRITERION IMPLEMENTATIONS IN OPTIMAL GRID CONSTRUCTING ALGORITHM O. V. Ushakova VANT. Ser.: Mat. Mod. Fiz. Proc. 2021. No 2. P. 8095.
For the algorithm of constructing optimal grids that meet two criteria of optimality – closeness to uniform grids and orthogonal grids, the paper offers numerical implementations of the third one that is the criterion of adaptation to the specified function. The optimal grid constructing algorithm is used in the numerical simulation of multicomponent media to construct threedimensional structured grids for complex geometries: solids of revolution, deformed solids of revolution, as well as volumes bounded by surfaces of revolution with parallel axes of revolution. Key words: adaptive grids, optimal grids, structured grids.
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