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We construct necessary and sufficient conditions for the existence of solution of seminonlinear matrix boundary value problem for a parametric excitation system of ordinary differential equations. The convergent iteration algorithms for the construction of the solutions of the semi-nonlinear matrix boundary value problem for a parametric excitation system differential equations in the critical case have been found. Using the convergent iteration algorithms we expand solution of seminonlinear periodical boundary value problem for a parametric excitation Riccati type equation in the neighborhood of the generating solution. Estimates for the value of residual of the solutions of the seminonlinear periodical boundary value problem for a parametric excitation Riccati type equation are found.

Verification results of supersonic turbulent jets computational characteristics are presented. Numerical simulation of axisymmetric nozzle operating is realized using FlowVision CFD. Open test cases for CFD are used. The test cases include Seiner tests with exit Mach number of 2.0 both fully-expanded and under-expanded $(P/P_0 = 1.47)$. Fully-expanded nozzle investigated with wide range of flow temperature (300…3000 K). The considered studies include simulation downstream from the nozzle exit diameter. Next numerical investigation is presented at an exit Mach number of 2.02 and a free-stream Mach number of 2.2. Geometric model of convergent- divergent nozzle rebuilt from original Putnam experiment. This study is set with nozzle pressure ratio of 8.12 and total temperature of 317 K.

The paper provides a comparison of obtained FlowVision results with experimental data and another current CFD studies. A comparison of the calculated characteristics and experimental data indicates a good agreement. The best coincidence with Seiner's experimental velocity distribution (about 7 % at far field for the first case) obtained using two-equation $k–\varepsilon$ standard turbulence model with Wilcox compressibility correction. Predicted Mach number distribution at $Y/D = 1$ for Putnam nozzle presents accuracy of 3 %.

General guidelines for simulation of supersonic turbulent jets in the FlowVision software are formulated in the given paper. Grid convergence determined the optimal cell rate. In order to calculate the design regime, it is recommended to build a grid, containing not less than 40 cells from the axis of symmetry to the nozzle wall. In order to calculate an off-design regime, it is necessary to resolve the shock waves. For this purpose, not less than 80 cells is required in the radial direction. Investigation of the influence of turbulence model on the flow characteristics has shown that the version of the SST $k–\omega$ turbulence model implemented in the FlowVision software essentially underpredicts the axial velocity. The standard $k–\varepsilon$ model without compressibility correction also underpredicts the axial velocity. These calculations agree well with calculations in other CFD codes using the standard $k–\varepsilon$ model. The in-home $k–\varepsilon$ turbulence model KEFV with compressibility correction a little bit overpredicts the axial velocity. Since, the best results are obtained using the standard $k–\varepsilon$ model combined with the Wilcox compressibility correction, this model is recommended for the problems discussed.

The developed methodology can be regarded as a basis for numerical investigations of more complex nozzle flows.

The paper has methodical character; it is devoted to three classic partial differential equations (Laplace, Diffusion and Wave) solution using simple numerical methods in terms of Cellular Automata. Special attention was payed to the matter conservation law and the offensive effect of excessive hexagonal symmetry.

It has been shown that in contrary to finite-difference approach, in spite of terminological equivalence of CA local transition function to the pattern of computing double layer explicit method, CA approach contains the replacement of matrix technique by iterative ones (for instance, sweep method for three diagonal matrixes). This suggests that discretization of boundary conditions for CA-cells needs more rigid conditions.

The correct local transition function (LTF) of the boundary cells, which is valid at least for the boundaries of the rectangular and circular shapes have been firstly proposed and empirically given for the hexagonal grid and the conservative boundary conditions. The idea of LTF separation into «internal», «boundary» and «postfix» have been proposed. By the example of this problem the value of the Courant-Levy constant was re-evaluated as the CA convergence speed ratio to the solution, which is given at a fixed time, and to the rate of the solution change over time.

In present paper we reexamine the properties of CABARET numerical scheme formulated for a weakly compressible fluid flow basing the results of free shear layer modeling. Kelvin–Helmholtz instability and successive generation of two-dimensional turbulence provide a wide field for a scheme analysis including temporal evolution of the integral energy and enstrophy curves, the vorticity patterns and energy spectra, as well as the dispersion relation for the instability increment. The most part of calculations is performed for Reynolds number $\text{Re} = 4 \times 10^5$ for square grids sequentially refined in the range of $128^2-2048^2$ nodes. An attention is paid to the problem of underresolved layers generating a spurious vortex during the vorticity layers roll-up. This phenomenon takes place only on a coarse grid with $128^2$ nodes, while the fully regularized evolution pattern of vorticity appears only when approaching $1024^2$-node grid. We also discuss the vorticity resolution properties of grids used with respect to dimensional estimates for the eddies at the borders of the inertial interval, showing that the available range of grids appears to be sufficient for a good resolution of small–scale vorticity patches. Nevertheless, we claim for the convergence achieved for the domains occupied by large-scale structures.

The generated turbulence evolution is consistent with theoretical concepts imposing the emergence of large vortices, which collect all the kinetic energy of motion, and solitary small-scale eddies. The latter resemble the coherent structures surviving in the filamentation process and almost noninteracting with other scales. The dissipative characteristics of numerical method employed are discussed in terms of kinetic energy dissipation rate calculated directly and basing theoretical laws for incompressible (via enstrophy curves) and compressible (with respect to the strain rate tensor and dilatation) fluid models. The asymptotic behavior of the kinetic energy and enstrophy cascades comply with two-dimensional turbulence laws $E(k) \propto k^{−3}, \omega^2(k) \propto k^{−1}$. Considering the instability increment as a function of dimensionless wave number shows a good agreement with other papers, however, commonly used method of instability growth rate calculation is not always accurate, so some modification is proposed. Thus, the implemented CABARET scheme possessing remarkably small numerical dissipation and good vorticity resolution is quite competitive approach compared to other high-order accuracy methods

In the article we have obtained some estimates of the rate of convergence for the recently proposed by Yu. E.Nesterov method of minimization of a convex Lipschitz-continuous function of two variables on a square with a fixed side. The idea of the method is to divide the square into smaller parts and gradually remove them so that in the remaining sufficiently small part. The method consists in solving auxiliary problems of one-dimensional minimization along the separating segments and does not imply the calculation of the exact value of the gradient of the objective functional. The main result of the paper is proved in the class of smooth convex functions having a Lipschitz-continuous gradient. Moreover, it is noted that the property of Lipschitzcontinuity for gradient is sufficient to require not on the whole square, but only on some segments. It is shown that the method can work in the presence of errors in solving auxiliary one-dimensional problems, as well as in calculating the direction of gradients. Also we describe the situation when it is possible to neglect or reduce the time spent on solving auxiliary one-dimensional problems. For some examples, experiments have demonstrated that the method can work effectively on some classes of non-smooth functions. In this case, an example of a simple non-smooth function is constructed, for which, if the subgradient is chosen incorrectly, even if the auxiliary one-dimensional problem is exactly solved, the convergence property of the method may not hold. Experiments have shown that the method under consideration can achieve the desired accuracy of solving the problem in less time than the other methods (gradient descent and ellipsoid method) considered. Partially, it is noted that with an increase in the accuracy of the desired solution, the operating time for the Yu. E. Nesterov’s method can grow slower than the time of the ellipsoid method.

We present the iterative algorithm that solves numerically both Urysohn type Fredholm and Volterra nonlinear one-dimensional nonsingular integral equations of the second kind to a specified, modest user-defined accuracy. The algorithm is based on descending recursive sequence of quadratures. Convergence of numerical scheme is guaranteed by fixed-point theorems. Picard’s method of integrating successive approximations is of great importance for the existence theory of integral equations but surprisingly very little appears on numerical algorithms for its direct implementation in the literature. We show that successive approximations method can be readily employed in numerical solution of integral equations. By that the quadrature algorithm is thoroughly designed. It is based on the explicit form of fifth-order embedded Runge–Kutta rule with adaptive step-size self-control. Since local error estimates may be cheaply obtained, continuous monitoring of the quadrature makes it possible to create very accurate automatic numerical schemes and to reduce considerably the main drawback of Picard iterations namely the extremely large amount of computations with increasing recursion depth. Our algorithm is organized so that as compared to most approaches the nonlinearity of integral equations does not induce any additional computational difficulties, it is very simple to apply and to make a program realization. Our algorithm exhibits some features of universality. First, it should be stressed that the method is as easy to apply to nonlinear as to linear equations of both Fredholm and Volterra kind. Second, the algorithm is equipped by stopping rules by which the calculations may to considerable extent be controlled automatically. A compact C++-code of described algorithm is presented. Our program realization is self-consistent: it demands no preliminary calculations, no external libraries and no additional memory is needed. Numerical examples are provided to show applicability, efficiency, robustness and accuracy of our approach.

We consider a finite-dimensional optimization problem, the formulation of which in addition to the required variables contains parameters. The solution to this problem is a dependence of optimal values of variables on parameters. In general, these dependencies are not functions because they can have ambiguous meanings and in the functional case be nondifferentiable. In addition, their domain of definition may be narrower than the domains of definition of functions in the condition of the original problem. All these properties make it difficult to solve both the original parametric problem and other tasks, the statement of which includes these dependencies. To overcome these difficulties, usually methods such as non-differentiable optimization are used.

This article proposes an alternative approach that makes it possible to obtain solutions to parametric problems in a form devoid of the specified properties. It is shown that such representations can be explored using standard algorithms, based on the Taylor formula. This form is a function smoothly approximating the solution of the original problem for any parameter values, specified in its statement. In this case, the value of the approximation error is controlled by a special parameter. Construction of proposed approximations is performed using special functions that establish feedback (within optimality conditions for the original problem) between variables and Lagrange multipliers. This method is described for linear problems with subsequent generalization to the nonlinear case.

From a computational point of view the construction of the approximation consists in finding the saddle point of the modified Lagrange function of the original problem. Moreover, this modification is performed in a special way using feedback functions. It is shown that the necessary conditions for the existence of such a saddle point are similar to the conditions of the Karush – Kuhn – Tucker theorem, but do not contain constraints such as inequalities and conditions of complementary slackness. Necessary conditions for the existence of a saddle point determine this approximation implicitly. Therefore, to calculate its differential characteristics, the implicit function theorem is used. The same theorem is used to reduce the approximation error to an acceptable level.

Features of the practical implementation feedback function method, including estimates of the rate of convergence to the exact solution are demonstrated for several specific classes of parametric optimization problems. Specifically, tasks searching for the global extremum of functions of many variables and the problem of multiple extremum (maximin-minimax) are considered. Optimization problems that arise when using multicriteria mathematical models are also considered. For each of these classes, there are demo examples.

The numerical solving of the system of high-temperature radiative gas dynamics (HTRGD) equations is a computationally laborious task, since the interaction of radiation with matter is nonlinear and non-local. The radiation absorption coefficients depend on temperature, and the temperature field is determined by both gas-dynamic processes and radiation transport. The method of splitting into physical processes is usually used to solve the HTRGD system, one of the blocks consists of a joint solving of the radiative transport equation and the energy balance equation of matter under known pressure and temperature fields. Usually difference schemes with orders of convergence no higher than the second are used to solve this block. Due to computer memory limitations it is necessary to use not too detailed grids to solve complex technical problems. This increases the requirements for the order of approximation of difference schemes. In this work, bicompact schemes of a high order of approximation for the algorithm for the joint solution of the radiative transport equation and the energy balance equation are implemented for the first time. The proposed method can be applied to solve a wide range of practical problems, as it has high accuracy and it is suitable for solving problems with coefficient discontinuities. The non-linearity of the problem and the use of an implicit scheme lead to an iterative process that may slowly converge. In this paper, we use a multiplicative HOLO algorithm named the quasi-diffusion method by V.Ya.Goldin. The key idea of HOLO algorithms is the joint solving of high order (HO) and low order (LO) equations. The high-order equation (HO) is the radiative transport equation solved in the energy multigroup approximation, the system of quasi-diffusion equations in the multigroup approximation (LO₁) is obtained by averaging HO equations over the angular variable. The next step is averaging over energy, resulting in an effective one-group system of quasi-diffusion equations (LO₂), which is solved jointly with the energy equation. The solutions obtained at each stage of the HOLO algorithm are closely related that ultimately leads to an acceleration of the convergence of the iterative process. Difference schemes constructed by the method of lines within one cell are proposed for each of the stages of the HOLO algorithm. The schemes have the fourth order of approximation in space and the third order of approximation in time. Schemes for the transport equation were developed by B.V. Rogov and his colleagues, the schemes for the LO₁ and LO₂ equations were developed by the authors. An analytical test is constructed to demonstrate the declared orders of convergence. Various options for setting boundary conditions are considered and their influence on the order of convergence in time and space is studied.

Conjugate gradient algorithms represent an important class of unconstrained optimization algorithms with strong local and global convergence properties and simple memory requirements. These algorithms have advantages that place them between the steep regression method and Newton’s algorithm because they require calculating the first derivatives only and do not require calculating and storing the second derivatives that Newton’s algorithm needs. They are also faster than the steep descent algorithm, meaning that they have overcome the slow convergence of this algorithm, and it does not need to calculate the Hessian matrix or any of its approximations, so it is widely used in optimization applications. This study proposes a novel method for image restoration by fusing the convex combination method with the hybrid (CG) method to create a hybrid three-term (CG) algorithm. Combining the features of both the Fletcher and Revees (FR) conjugate parameter and the hybrid Fletcher and Revees (FR), we get the search direction conjugate parameter. The search direction is the result of concatenating the gradient direction, the previous search direction, and the gradient from the previous iteration. We have shown that the new algorithm possesses the properties of global convergence and descent when using an inexact search line, relying on the standard Wolfe conditions, and using some assumptions. To guarantee the effectiveness of the suggested algorithm and processing image restoration problems. The numerical results of the new algorithm show high efficiency and accuracy in image restoration and speed of convergence when used in image restoration problems compared to Fletcher and Revees (FR) and three-term Fletcher and Revees (TTFR).

Calculations of radiation transport in the shockwave layer of a descent space vehicle cause essential difficulties due to complex multi-resonance dependence of the absorption macroscopic cross sections from the photon energy. The convergence of two approximate spectrum averaging methods to the results of exact pointwise spectrum calculations is investigated. The first one is the well known multigroup method, the second one is the Lebesgue averaging method belonging to methods of the reduction of calculation points by means of aggregation of spectral points which are characterized by equal absorption strength. It is shown that convergence of the Lebesgue averaging method is significantly faster than the multigroup approach as the number of groups is increased. The only 100–150 Lebesgue groups are required to achieve the accuracy of pointwise calculations even in the shock layer at upper atmosphere with sharp absorption lines. At the same time the number of calculations is reduced by more than four order. Series of calculations of the radiation distribution function in 2D shock layer around a sphere and a blunt cone were performed using the local flat layer approximation and the Lebesgue averaging method. It is shown that the shock wave radiation becomes more significant both in value of the energy flux incident on the body surface and in the rate of energy exchange with the gas-dynamic flow in the case of increasing of the vehicle’s size.