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The purpose of the paper is a research of a 2nd order finite difference scheme based on the Z-scheme. This research is the numerical solution of several two-dimensional differential equations simulated the incompressible medium convection.

One of real tasks for similar equations solution is the numerical simulating of strongly non-stationary midscale processes in the Earth ionosphere. Because convection processes in ionospheric plasma are controlled by magnetic field, the plasma incompressibility condition is supposed across the magnetic field. For the same reason, there can be rather high velocities of heat and mass convection along the magnetic field.

Ionospheric simulation relevant task is the research of plasma instability of various scales which started in polar and equatorial regions first of all. At the same time the mid-scale irregularities having characteristic sizes 1–50 km create conditions for development of the small-scale instabilities. The last lead to the F-spread phenomenon which significantly influences the accuracy of positioning satellite systems work and also other space and ground-based radio-electronic systems.

The difference schemes used for simultaneous simulating of such multi-scale processes must to have high resolution. Besides, these difference schemes must to be high resolution on the one hand and monotonic on the other hand. The fact that instabilities strengthen errors of difference schemes, especially they strengthen errors of dispersion type is the reason of such contradictory requirements. The similar swing of errors usually results to nonphysical results at the numerical solution.

At the numerical solution of three-dimensional mathematical models of ionospheric plasma are used the following scheme of splitting on physical processes: the first step of splitting carries out convection along, the second step of splitting carries out convection across. The 2nd order finite difference scheme investigated in the paper solves approximately convection across equations. This scheme is constructed by a monotonized nonlinear procedure on base of the Z-scheme which is one of 2nd order schemes. At this monotonized procedure a nonlinear correction with so-called “oblique differences” is used. “Oblique differences” contain the grid nodes relating to different layers of time.

The researches were conducted for two cases. In the simulating field components of the convection vector had: 1) the constant sign; 2) the variable sign. Dissipative and dispersive characteristics of the scheme for different types of the limiting functions are in number received.

The results of the numerical experiments allow to draw the following conclusions.

1. For the discontinuous initial profile the best properties were shown by the SuperBee limiter.

2. For the continuous initial profile with the big spatial steps the SuperBee limiter is better, and at the small steps the Koren limiter is better.

3. For the smooth initial profile the best results were shown by the Koren limiter.

4. The smooth F limiter showed the results similar to Koren limiter.

5. Limiters of different type leave dispersive errors, at the same time dependences of dispersive errors on the scheme parameters have big variability and depend on the scheme parameters difficulty.

6. The monotony of the considered differential scheme is in number confirmed in all calculations. The property of variation non-increase for all specified functions limiters is in number confirmed for the onedimensional equation.

7. The constructed differential scheme at the steps on time which are not exceeding the Courant's step is monotonous and shows good exactness characteristics for different types solutions. At excess of the Courant's step the scheme remains steady, but becomes unsuitable for instability problems as monotony conditions not satisfied in this case.

An essential class of problems describes physical processes occurring in non-convex domains containing a corner greater than 180 degrees on the boundary. The solution in a neighborhood of a corner is singular and its finding using classical approaches entails a loss of accuracy. In the paper, we consider stationary, linearized by Picard’s iterations, Navier – Stokes equations governing the flow of a incompressible viscous fluid in the convection form in $L$-shaped domain. An $R_\nu$-generalized solution of the problem in special sets of weighted spaces is defined. A special finite element method to find an approximate $R_\nu$-generalized solution is constructed. Firstly, functions of the finite element spaces satisfy the law of conservation of mass in the strong sense, i.e. at the grid nodes. For this purpose, Scott – Vogelius element pair is used. The fulfillment of the condition of mass conservation leads to the finding more accurate, from a physical point of view, solution. Secondly, basis functions of the finite element spaces are supplemented by weight functions. The degree of the weight function, as well as the parameter $\nu$ in the definition of an $R_\nu$-generalized solution, and a radius of a neighborhood of the singularity point are free parameters of the method. A specially selected combination of them leads to an increase almost twice in the order of convergence rate of an approximate solution to the exact one in relation to the classical approaches. The convergence rate reaches the first order by the grid step in the norms of Sobolev weight spaces. Thus, numerically shown that the convergence rate does not depend on the corner value.

The paper studies a multidimensional convection-diffusion equation with variable coefficients and a nonclassical boundary condition. Two cases are considered: in the first case, the first boundary condition contains the integral of the unknown function with respect to the integration variable $x_\alpha^{}$, and in the second case, the integral of the unknown function with respect to the integration variable $\tau$, denoting the memory effect. Similar problems arise when studying the transport of impurities along the riverbed. For an approximate solution of the problem posed, a locally one-dimensional difference scheme by A.A. Samarskii with order of approximation $O(h^2+\tau)$. In view of the fact that the equation contains the first derivative of the unknown function with respect to the spatial variable $x_\alpha^{}$, the wellknown method proposed by A.A. Samarskii in constructing a monotonic scheme of the second order of accuracy in $h_\alpha^{}$ for a general parabolic type equation containing one-sided derivatives taking into account the sign of $r_\alpha^{}(x,t)$. To increase the boundary conditions of the third kind to the second order of accuracy in $h_\alpha^{}$, we used the equation, on the assumption that it is also valid at the boundaries. The study of the uniqueness and stability of the solution was carried out using the method of energy inequalities. A priori estimates are obtained for the solution of the difference problem in the $L_2^{}$-norm, which implies the uniqueness of the solution, the continuous and uniform dependence of the solution of the difference problem on the input data, and the convergence of the solution of the locally onedimensional difference scheme to the solution of the original differential problem in the $L_2^{}$-norm with speed equal to the order of approximation of the difference scheme. For a two-dimensional problem, a numerical solution algorithm is constructed.

A 2D numerical model of the immersion of a cold oceanic plate into the thickness of the Earth’s upper mantle has been developed, where the stage of the initial immersion of the plate is preceded by the establishment of a regime of thermogravitational convection of the mantle substance. The model approximation of the mantle is a two-dimensional image of an incompressible Newtonian quasi-liquid in a Cartesian coordinate system, where, due to the high viscosity of the medium, the equations of mantle convection are accepted in the Stokes approximation. It is assumed that seawater that has leaked here enters the first horizons of the mantle together with the plate. With depth, the increase in pressure and temperature leads to certain losses of its light fractions and fluids, losses of water and gases of water-containing minerals of the plate, restructuring of their crystal lattice and, as a consequence, phase transformations. These losses cause an increase in the plate density and an uneven distribution of stresses along the plate (the initial sections of the plate are denser), which subsequently, together with the effect of mantle currents on the plate, causes its fragmentation. The state of mantle convection is considered when the plate and its individual fragments have descended to the bottom of the upper mantle. Computational schemes for solving the model equations have been developed. Mantle convection calculations are performed in terms of the Stokes approximation for vorticity and the stream function, and SPH is used to calculate the state and subsidence of the plate. A number of computational experiments have been performed. It is shown that fragmentation of the plate occurs due to the effect of mantle convection on the plate and the development of inhomogeneous stress fields along the plate. Following the equations of the model, the time of the final stage of subduction is estimated, i.e. the time of the entire oceanic plate reaching the bottom of the upper mantle. In geodynamics, this process is determined by the collision of plates that immediately follows subduction and is usually considered as the final stage of the Wilson cycle (i. e., the cycle of development of folded belts).

This paper presents a non-iterative method for solving an inverse problem for a parabolictype equation with an unknown time-dependent coefficient at the first spatial derivative. The overdetermination condition is specified as a definite integral of the unknown function with a weighting factor over the spatial domain or its subdomain. The study is motivated by the need to identify dynamic parameters in applied problems, particularly in modeling transport processes in biological fluids, where the flow velocity may vary over time. In contrast to conventional iterative methods that require substantial computational effort and careful selection of regularization parameters, an original approach based on solution decomposition is proposed. At each time layer, the solution is represented as a linear combination of solutions to two auxiliary systems with the same matrix and different right-hand sides, followed by the determination of the unknown coefficient from a discrete analogue of the overdetermination condition. This approach eliminates the need for an iterative procedure. In the presence of inexact overdetermination data, the highest reconstruction accuracy is achieved using a quasi-solution. Numerical experiments on test problems demonstrate high accuracy in reconstructing the unknown functions under small perturbations of the overdetermination condition. The results indicate strong potential for applications in medical diagnostics and other fields requiring rapid processing of experimental data.

Pattern formation described by the scalar Fisher–Kolmogorov–Petrovsky–Piscounov equation with nonlocal competition loses and convection linear on coordinates is considered numerically. Initial function localized around a point is shown to transform in a function localized around a ring with symmetrically sited local maxima. The ring radius and number of maxima depend on convection.

A mathematical model for the numerical study of thermal destruction of the "Chelyabinsk" meteorite when entering the Earth’s atmosphere is presented in the article. The study was conducted in the framework of an integrated approach, including the calculation of the meteorite trajectory associated with the physical processes connected with the meteorite motion. Together with the trajectory the flow field and radiation-convective heat
transfer were determined as well as warming and destruction of the meteorite under the influence of the calculated heat load. An integrated approach allows to determine the trajectories of space objects more precisely, predict the area of their fall and destruction.

The application of a conservative numerical method of fluxes is examined for studying the vortex structures in the massive, fast-turned compact astrophysical objects, which are in self-gravity conditions. The simulation is accomplished for the objects with different mass and rotational speed. The pictures of the vortex structure of objects are visualized. In the calculations the gas-dynamic model is used, in which gas is accepted perfected and nonviscous. Numerical procedure is based on the finite-difference approximation of the conservation laws of the additive characteristics of medium for the finite volume. The “upwind” approximations of the densities of distribution of mass, components of momentum and total energy are applied. For the simulation of the objects, which possess fast-spin motion, the control of conservation for the component of moment of momentun is carried out during calculation. Evolutionary calculation is carried out on the basis of the parallel algorithms, realized on the computer complex of cluster architecture. Algorithms are based on the standardized system of message transfer Message Passing Interface (MPI). The blocking procedures of exchange and non-blocking procedures of exchange with control of the completion of operation are used. The parallelization on the space in two or three directions is carried out depending on the size of integration area and parameters of computational grid. For each subarea the parallelization based on the physical factors is carried out also: the calculations of gas dynamics part and gravitational forces are realized on the different processors, that allows to raise the efficiency of algorithms. The real possibility of the direct calculation of gravitational forces by means of the summation of interaction between all finite volumes in the integration area is shown. For the finite volume methods this approach seems to more consecutive than the solution of Poisson’s equation for the gravitational potential. Numerical calculations were carried out on the computer complex of cluster architecture with the peak productivity 523 TFlops. In the calculations up to thousand processors was used.

In this paper, a time-dependent natural convective heat transfer in a closed square cavity filled with non- Newtonian fluid was considered in the presence of an isothermal energy source located on the lower wall of the region under consideration. The vertical boundaries were kept at constant low temperature, while the horizontal walls were completely insulated. The behavior of a non-Newtonian fluid was described by the Ostwald de Ville power law. The process under study was described by transient partial differential equations using dimensionless non-primitive variables “stream function – vorticity – temperature”. This method allows excluding the pressure field from the number of unknown parameters, while the non-dimensionalization allows generalizing the obtained results to a variety of physical formulations. The considered mathematical model with the corresponding boundary conditions was solved on the basis of the finite difference method. The algebraic equation for the stream function was solved by the method of successive lower relaxation. Discrete analogs of the vorticity equation and energy equation were solved by the Thomas algorithm. The developed numerical algorithm was tested in detail on a class of model problems and good agreement with other authors was achieved. Also during the study, the mesh sensitivity analysis was performed that allows choosing the optimal mesh.

As a result of numerical simulation of unsteady natural convection of a non-Newtonian power-law fluid in a closed square cavity with a local isothermal energy source, the influence of governing parameters was analyzed including the impact of the Rayleigh number in the range 104–106, power-law index $n = 0.6–1.4$, and also the position of the heating element on the flow structure and heat transfer performance inside the cavity. The analysis was carried out on the basis of the obtained distributions of streamlines and isotherms in the cavity, as well as on the basis of the dependences of the average Nusselt number. As a result, it was established that pseudoplastic fluids $(n < 1)$ intensify heat removal from the heater surface. The increase in the Rayleigh number and the central location of the heating element also correspond to the effective cooling of the heat source.

The article presents the results of a numerical study of the flow structure in a two-dimensional flat diffuser. A feature of diffusers is that they have a complex anisotropic turbulent flow, which occurs due to recirculation flows. The turbulent RANS models, which are based on the Boussinesq hypothesis, are not able to describe the flow in diffusers with sufficient accuracy. Because the Boussinesq hypothesis is based on isotropic turbulence. Therefore, to calculate anisotropic turbulent flows, models are used that do not use this hypothesis. One of such directions in turbulence modeling is the methods of Reynolds stresses. These methods are complex and require rather large computational resources. In this work, a relatively recently developed two-fluid turbulence model was used to study the flow in a flat diffuser. This model is developed on the basis of a two-fluid approach to the problem of turbulence. In contrast to the Reynolds approach, the two-fluid approach allows one to obtain a closed system of turbulence equations using the dynamics of two fluids. Consequently, if empirical equations are used in RANS models for closure, then in the two-fluid model the equations used are exact equations of dynamics. One of the main advantages of the two-fluid model is that it is capable of describing complex anisotropic turbulent flows. In this work, the obtained numerical results for the profiles of the longitudinal velocity, turbulent stresses in various sections of the channel, as well as the friction coefficient are compared with the known experimental data. To demonstrate the advantages of the used turbulence model, the numerical results of the Reynolds stress method EARSM are also presented. For the numerical implementation of the systems of equations of the two-fluid model, a non-stationary system of equations was used, the solution of which asymptotically approached the stationary solution. For this purpose, a finite-difference scheme was used, where the viscosity terms were approximated by the central difference implicitly, and for the convective terms, an explicit scheme against the flow of the second order of accuracy was used. The results are obtained for the Reynolds number Re = 20 000. It is shown that the two-fluid model, despite the use of a uniform computational grid without thickening near the walls, is capable of giving a more accurate solution than the rather complex Reynolds stress method with a high resolution of computational grids.