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On the construction and properties of WENO schemes order five, seven, nine, eleven and thirteen. Part 1. Construction and stability
Computer Research and Modeling, 2016, v. 8, no. 5, pp. 721753Views (last year): 9. Citations: 1 (RSCI).Currently, different nonlinear numerical schemes of the spatial approximation are used in numerical simulation of boundary value problems for hyperbolic systems of partial differential equations (e. g. gas dynamics equations, MHD, deformable rigid body, etc.). This is due to the need to improve the order of accuracy and perform simulation of discontinuous solutions that are often occurring in such systems. The need for nonlinear schemes is followed from the barrier theorem of S. K. Godunov that states the impossibility of constructing a linear scheme for monotone approximation of such equations with approximation order two or greater. One of the most accurate nonlinear type schemes are ENO (essentially non oscillating) and their modifications, including WENO (weighted, essentially non oscillating) scemes. The last received the most widespread, since the same stencil width has a higher order of approximation than the ENO scheme. The benefit of ENO and WENO schemes is the ability to maintain a highorder approximation to the areas of nonmonotonic solutions. The main difficulty of the analysis of such schemes comes from the fact that they themselves are nonlinear and are used to approximate the nonlinear equations. In particular, the linear stability condition was obtained earlier only for WENO5 scheme (fifthorder approximation on smooth solutions) and it is a numerical one. In this paper we consider the problem of construction and stability for WENO5, WENO7, WENO9, WENO11, and WENO13 finite volume schemes for the Hopf equation. In the first part of this article we discuss WENO methods in general, and give the explicit expressions for the coefficients of the polynomial weights and linear combinations required to build these schemes. We prove a series of assertions that can make conclusions about the order of approximation depending on the type of local solutions. Stability analysis is carried out on the basis of the principle of frozen coefficients. The cases of a smooth and discontinuous behavior of solutions in the field of linearization with frozen coefficients on the faces of the final volume and spectra of the schemes are analyzed for these cases. We prove the linear stability conditions for a variety of RungeKutta methods applied to WENO schemes. As a result, our research provides guidance on choosing the best possible stability parameter, which has the smallest effect on the nonlinear properties of the schemes. The convergence of the schemes is followed from the analysis.

On the convergence of the implicit iterative linebyline recurrence method for solving difference elliptical equations
Computer Research and Modeling, 2017, v. 9, no. 6, pp. 857880Views (last year): 15. Citations: 1 (RSCI).In the article a theory of the implicit iterative linebyline recurrence method for solving the systems of finitedifference equations which arise as a result of approximation of the twodimensional elliptic differential equations on a regular grid is stated. On the one hand, the high effectiveness of the method has confirmed in practice. Some complex test problems, as well as several problems of fluid flow and heat transfer of a viscous incompressible liquid, have solved with its use. On the other hand, the theoretical provisions that explain the high convergence rate of the method and its stability are not yet presented in the literature. This fact is the reason for the present investigation. In the paper, the procedure of equivalent and approximate transformations of the initial system of linear algebraic equations (SLAE) is described in detail. The transformations are presented in a matrixvector form, as well as in the form of the computational formulas of the method. The key points of the transformations are illustrated by schemes of changing of the difference stencils that correspond to the transformed equations. The canonical form of the method is the goal of the transformation procedure. The correctness of the method follows from the canonical form in the case of the solution convergence. The estimation of norms of the matrix operators is carried out on the basis of analysis of structures and element sets of the corresponding matrices. As a result, the convergence of the method is proved for arbitrary initial vectors of the solution of the problem.
The norm of the transition matrix operator is estimated in the special case of weak restrictions on a desired solution. It is shown, that the value of this norm decreases proportionally to the second power (or third degree, it depends on the version of the method) of the grid step of the problem solution area in the case of transition matrix order increases. The necessary condition of the method stability is obtained by means of simple estimates of the vector of an approximate solution. Also, the estimate in order of magnitude of the optimum iterative compensation parameter is given. Theoretical conclusions are illustrated by using the solutions of the test problems. It is shown, that the number of the iterations required to achieve a given accuracy of the solution decreases if a grid size of the solution area increases. It is also demonstrated that if the weak restrictions on solution are violated in the choice of the initial approximation of the solution, then the rate of convergence of the method decreases essentially in full accordance with the deduced theoretical results.

Twostage single ROW methods with complex coefficients for autonomous systems of ODE
Computer Research and Modeling, 2010, v. 2, no. 1, pp. 1932Citations: 1 (RSCI).The basic subset of twostage Rosenbrock schemes with complex coefficients for numerical solution of autonomous systems of ordinary differential equations (ODE) has been considered. Numerical realization of such schemes requires one LUdecomposition, two computations of right side function and one computation of Jacoby matrix of the system per one step. The full theoretical investigation of accuracy and stability of such schemes have been done. New Astable methods of the 3rd order of accuracy with different properties have been constructed. There are high order Ldecremented schemes as well as schemes with simple estimation of the main term of truncation error which is necessary for automatic evaluation of time step. Testing of new methods has been performed.

CABARET scheme implementation for free shear layer modeling
Computer Research and Modeling, 2017, v. 9, no. 6, pp. 881903Views (last year): 17.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 twodimensional 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 $128^22048^2$ nodes. An attention is paid to the problem of underresolved layers generating a spurious vortex during the vorticity layers rollup. 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 largescale 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 smallscale 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 twodimensional 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 highorder accuracy methods

Bicompact schemes for gas dynamics problems: introducing complex domains using the free boundary method
Computer Research and Modeling, 2020, v. 12, no. 3, pp. 487504This work is dedicated to application of bicompact schemes to numerical solution of evolutionary hyperbolic equations. The main advantage of this class of schemes lies in combination of two beneficial properties: the first one is spatial approximation of high even order on a stencil that always occupies only one mesh cell; the second one is spectral resolution which is better in comparison to classic compact finitedifference schemes of the same order of spatial approximation. One feature of bicompact schemes is considered: their spatial approximation is rigidly tied to Cartesian meshes (with parallelepipedshaped cells in threedimensional case). This feature makes rather challenging any application of bicompact schemes to problems with complex computational domains as treated in the framework of unstructured meshes. This problem is proposed to be solved using wellknown methods for treating complexshaped boundaries and their corresponding boundary conditions on Cartesian meshes. The generalization of bicompact schemes on problems in geometrically complex domains is made in case of gas dynamics problems and Euler equations. The free boundary method is chosen as a particular tool to introduce the influence of arbitraryshaped solid boundaries on gas flows on Cartesian meshes. A brief description of this method is given, its governing equations are written down. Bicompact schemes of fourth order of approximation in space with locally onedimensional splitting are constructed for equations of the free boundary method. Its compensation flux is discretized with second order of accuracy. Time stepping in the obtained schemes is done with the implicit Euler method and the third order accurate $L$stable stiffly accurate threestage singly diagonally implicit Runge–Kutta method. The designed bicompact schemes are tested on three twodimensional problems: stationary supersonic flows with Mach number three past one circular cylinder and past three circular cylinders; the nonstationary interaction of planar shock wave with a circular cylinder in a channel with planar parallel walls. The obtained results are in a good agreement with other works: influence of solid bodies on gas flows is physically correct, pressure in control points on solid surfaces is calculated with the accuracy appropriate to the chosen mesh resolution and level of numerical dissipation.

Meshless algorithm for calculating the interaction of large particles with a shock layer in supersonic heterogeneous flows
Computer Research and Modeling, 2022, v. 14, no. 5, pp. 10071027The work is devoted to numerical modeling of twophase flows, namely, the calculation of supersonic flow around a blunt body by a viscous gas flow with an admixture of large high inertia particles. The system of unsteady Navier – Stokes equations is numerically solved by the meshless method. It uses the cloud of points in space to represent the fields of gas parameters. The spatial derivatives of gas parameters and functions are approximated by the least square method to calculate convective and viscous fluxes in the Navier – Stokes system of equations. The convective fluxes are calculated by the HLLC method. The thirdorder MUSCL reconstruction scheme is used to achieve high order accuracy. The viscous fluxes are calculated by the second order approximation scheme. The streamlined body surface is represented by a model of an isothermal wall. It implements the conditions for the zero velocity and zero pressure gradient, which is also modeled using the least squares method.
Every moving body is surrounded by its own cloud of points belongs to body’s domain and moving along with it in space. The explicit threesage Runge–Kutta method is used to solve numerically the system of gas dynamics equations in the main coordinate system and local coordinate systems of each particle.
Two methods for the moving objects modeling with reverse impact on the gas flow have been implemented. The first one uses stationary point clouds with fixed neighbors within the same domain. When regions overlap, some nodes of one domain, for example, the boundary nodes of the particle domain, are excluded from the calculation and filled with the values of gas parameters from the nearest nodes of another domain using the least squares approximation of gradients. The internal nodes of the particle domain are used to reconstruct the gas parameters in the overlapped nodes of the main domain. The second method also uses the exclusion of nodes in overlapping areas, but in this case the nodes of another domain take the place of the excluded neighbors to build a single connected cloud of nodes. At the same time, some of the nodes are moving, and some are stationary. Nodes membership to different domains and their relative speed are taken into account when calculating fluxes.
The results of modeling the motion of a particle in a stationary gas and the flow around a stationary particle by an incoming flow at the same relative velocity show good agreement for both presented methods.

Bicompact schemes for the HOLO algorithm for joint solution of the transport equation and the energy equation
Computer Research and Modeling, 2023, v. 15, no. 6, pp. 14291448The numerical solving of the system of hightemperature radiative gas dynamics (HTRGD) equations is a computationally laborious task, since the interaction of radiation with matter is nonlinear and nonlocal. The radiation absorption coefficients depend on temperature, and the temperature field is determined by both gasdynamic 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 nonlinearity 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 quasidiffusion 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 highorder equation (HO) is the radiative transport equation solved in the energy multigroup approximation, the system of quasidiffusion equations in the multigroup approximation (LO_{1}) is obtained by averaging HO equations over the angular variable. The next step is averaging over energy, resulting in an effective onegroup system of quasidiffusion equations (LO_{2}), 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 LO_{1} and LO_{2} 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.

On highorder approximation of transparent boundary conditions for the wave equation
Computer Research and Modeling, 2014, v. 6, no. 1, pp. 4556Views (last year): 1. Citations: 1 (RSCI).The paper considers the problem of increasing the approximation order of transparent boundary conditions for the wave equation while using finite difference schemes up to the sixth order of accuracy in space. As an example, the problem of wave propagation in a semiinfinite rectangular waveguide is formulated. Computationally efficient and highly accurate formulas for discretizing operator of transparent boundary conditions are proposed. Numerical results confirm the accuracy and stability of the obtained difference algorithms.

A numerical method for solving twodimensional convection equation based on the monotonized Zscheme for Earth ionosphere simulation
Computer Research and Modeling, 2020, v. 12, no. 1, pp. 4358The purpose of the paper is a research of a 2nd order finite difference scheme based on the Zscheme. This research is the numerical solution of several twodimensional differential equations simulated the incompressible medium convection.
One of real tasks for similar equations solution is the numerical simulating of strongly nonstationary 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 midscale irregularities having characteristic sizes 1–50 km create conditions for development of the smallscale instabilities. The last lead to the Fspread phenomenon which significantly influences the accuracy of positioning satellite systems work and also other space and groundbased radioelectronic systems.
The difference schemes used for simultaneous simulating of such multiscale 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 threedimensional 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 Zscheme which is one of 2nd order schemes. At this monotonized procedure a nonlinear correction with socalled “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 nonincrease 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.

Numerical simulation of flow in a twodimensional flat diffuser based on two fluid turbulence models
Computer Research and Modeling, 2021, v. 13, no. 6, pp. 11491160The article presents the results of a numerical study of the flow structure in a twodimensional 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 twofluid turbulence model was used to study the flow in a flat diffuser. This model is developed on the basis of a twofluid approach to the problem of turbulence. In contrast to the Reynolds approach, the twofluid 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 twofluid model the equations used are exact equations of dynamics. One of the main advantages of the twofluid 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 twofluid model, a nonstationary system of equations was used, the solution of which asymptotically approached the stationary solution. For this purpose, a finitedifference 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 twofluid 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.
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