A qualitative numerical solution of the equations of mathematical physics is intimately connected with ensuring a high accuracy of approximation of all differential operators included in these equations. The solution of this problem for the first and second derivatives in the equations of mathematical physics, which are used to describe a wide range of scientific and technical problems has been described in numerous literary publications. At the same time, mixed derivatives are not so often present in the equations of mathematical physics and, therefore, issues related to the quality of finite-difference approximation of these derivatives are not given enough attention in literary publications. One of the main reasons for the appearance of mixed derivatives in the equations of mathematical physics is the use of an affine transformation of the coordinate system, which provides the transition to domain of a substantially simpler form. The solution of this problem is the subject of the present paper. The problem is solved by the example of approximation of mixed derivatives on rectangular domain of definition of the required function with constant steps in each direction. A detailed derivation of the finite-difference relations used for the finite-difference approximation of mixed derivatives in all typical nodes of the function domain is given, which makes it possible to develop the proposed technique on domains of different types.
Горский В.В., Реш В.Г. Конечно-разностная аппроксимация смешанных производных в уравнениях математической физики. Математическое моделирование и численные методы, 2021, № 4, с. 58–79
Spherical aircraft elements are related to a number of their fragments influ-enced by the maximum heat fluxes. As a result calculating thermal and force loading of a hemisphere is of particular interest. This problem is one of the most acute within the extremely high Reynolds numbers that correspond to laminar-turbulent gas flow in a boundary layer over a hemisphere. At the same time, the fundamental monograph  on the aircraft convective heat exchange, which summarizes the results of many years of research on this problem, doesn’t pay proper attention to this issue. At the same time, the work  shows that using the approaches for a hemisphere outlined in the monograph  is associated with a number of significant errors within the range of extremely high Reynolds numbers. Calculating heat transfer and friction on the porous wall which correct solution has been missing by now is one of the least studied problems of laminar-turbulent heat exchange. The purpose of this article is to develop a highly accurate engineering method for calculating heat transfer and friction in a laminar-turbulent boundary layer on a hemisphere.
Горский В.В., Локтионова А.Г. Моделирование теплообмена и трения в тонком воздушном ламинарно-турбулентном пограничном слое над поверхностью полусферы. Математическое моделирование и численные методы, 2019, № 2, с. 51–67.
The study of aerothermal destruction of silicon carbide is constantly given increased at-tention. This is primarily due to its widespread use as a protection against oxidation of carbon materials. The rate of aerothermochemical destruction of materials largely de-pends on the specific heat flux and on the degree of recombination of oxygen atoms on the wall. In this paper, we consider a method for solving the complex problem of high-temperature non-equilibrium air stream flowing of a material coated with a silicon car-bide film under conditions of finite catalyticity of the surface of this film.
Горский В.В., Адаменко Р.А. Моделирование химического состава газа в неравновесном воздушном пограничном слое на стенке, обладающей конечной ка-талитической активностью. Математическое моделирование и численные методы, 2018, № 4, с. 93–106.
Currently, carbon-based materials are widely used as an ablative thermal protection for high-heat constructional elements in air and space en-gineering. In its turn, forecasting a change in shape of external surfaces of the specified elements with the course of time, which is determined by scorching of thermal protection, cannot be separated from the use of calcula-tion and theoretical procedures, which describe course of different phys-ico-chemical and mechanical processes associated with course of the considered event. Thereby, it is mandatory to approve such procedures through the re-sults of experi-mental studies, conducted in jets inside aerodynamic tunnels.
Among primary evidences of ablation of carbon-based materials is erosion (mass loss), which is usually observed in high-pressure gas flows. Meanwhile, it is required to carry out studies on large-scale models in the course of development tests, and that had de-termined the wide use of under-expanded jets of combustion gases of liquid-propellant rocket engines (LPE) for simulation of erosion of thermal protection.
Among the basic problems, encountering during solution of this kind of tasks, is calcu-lation of laminar-turbulent heat exchange in the conditions of gradient flow past blunt-ed point of the tested model by a diverging gas jet. This article is dedicated to the solu-tion of this problem and includes the modified version of semi-empirical model of ap-parent turbulent viscosity, approved by the results of experimental studies. This article shows that the use of this method makes it possible to specify significantly thermal con-di-tions of the model in comparison with the use of effective length method, which is used universally in practical work.
Горский В.В., Ковальский М.Г. Методика численного решения уравнений ламинарно-турбулентного пограничного слоя на осесимметричном затупленном теле в струе продуктов сгорания ЖРД. Математическое моделирование и численные методы, 2018, № 2, с. 96–108.
The article presents estimated accuracy of the engineering design procedure of the mass flow rate of gas through the laminar boundary layer on a hemisphere of . A similar engineering method of extra accuracy is proposed.
Gorskiy V., Sysenko V. Simulation of gas flow through the laminar boundary layer on the hemisphere surface in a supersonic air flow. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 88-94
Testing thermal protection in jets of propulsion system combustion products is one of the priority types of its experimental development. The informational content of such trials depends to a large extent on the mathematical modeling of the processes taking place in the experiments under consideration. They include flow past a model made of ablative thermal protection, convective heat transfer and friction in the laminar-turbulent boundary layer formed on the model surface, ablation of thermal protection and change in the shape of the model. The article describes the mathematical modeling of the first of the above processes, associated with solving the complex problem of constructing the field of gas dynamic functions in the underexpanded high-pressure jet of the liquid rocket engine combustion products in a flooded space and the divergent jet flow around the model surface. The results of comparison of analytical results with experimental data are presented.
V.V. Gorsky, M.G. Kovalsky Techniques for numerical simulation flow around an axisymmetric blunt body in an underexpanded jet of liquid rocket engine combustion products. Маthematical Modeling and Coтputational Methods, 2017, №2 (14), pp. 65-80