Vol 11, No 1 (2019)

The concept of the strong monotonicity of the two-time-level CABARET scheme is introduced. It suggests that the difference scheme does not increase the number of the generalized local extrema in the difference solution when passing from one time level to another. The special modification of the two-time-level CABARET scheme having a strong monotonicity property is proposed. The test calculations are presented that illustrate this property of the modified CABARET scheme.

This paper presents a brief description of the Zhukovsky Central Aerohydrodynamic Institute (TsAGI) code based on the Galerkin method of a high order of accuracy with discontinuous basic functions. The functions is reconstructed for conservative variables. Gradients of the variables are determined using the Bassi–Rebay 2 method. For integrating, the Gaussian quadrature rules are used. The coordinates are transformed by serendipian elements. In computing by schemes of an order higher than the second order, the curvature of grid lines is taken into account. A comparison with finite volume methods is performed, including the WENO method with constant weights and a single quadrature point on a cell face. The classical tests are used, namely, a subsonic flow around a circular cylinder in an ideal gas, the diagonal convection of a two-dimensional isentropic vortex, and the decay of the Taylor–Green vortex.

The monotonicity of the CABARET scheme approximating the quasi-linear scalar conservation law with a convex flux is analyzed. The monotonicity conditions for this scheme are obtained in the areas where the propagation velocity of the characteristics has a constant sign, as well as in the areas of sonic lines, sonic bands, and shock waves, where the propagation velocity of the characteristics of the approximated divergent equation changes sign. The test computations are presented that illustrate these properties of the CABARET scheme.

The paper presents a technique for the numerical modeling of unsteady turbulent flows. A special feature of the technique consists of applying the immersed boundary method to ensure the fulfilment of the boundary conditions on the surface of the bodies in the flow for the resulting numerical solution. It is used for the numerical solution of the problem on a turbulent flow past a three-dimensional cylinder. The obtained numerical solution is compared with numerous experimental and calculated reference data and calculation results using boundary-fitted grids.

The principle of pressure maximum is proved for a stationary three-dimensional vortex ideal gas flow (without the assumption of barotropicity). Based on the fact that in regions where the solution is modeled with a high degree of accuracy within the boundary value problem for the Euler equations, the consequences of the Euler equations must also hold, and the obtained subsonic principle is proposed to be used for verification of the numerical solutions of the boundary value problems for Euler equations for an ideal gas and for the Navier-Stokes equations for viscous gas. The conditions of the maximum principle include the value of the Q-parameter, whose surface level image is currently widely used to visualize the flow pattern. The proposed principle of the maximum pressure reveals the meaning of the surface Q = 0. It divides the flow region into the subdomain Q > 0 in which there can be no local pressure maximum and subdomain Q < 0 in which there can be no local pressure minimum. A similar meaning of parameter Q was known for incompressible fluid (H. Rowland, 1880; G. Hamel, 1936). The expression for the Q-parameter contains only the first derivatives of the velocity components, which allows determining the sign (+/–) of Q even for numerical solutions obtained by the low-order methods. An example of the numerical solution’s verification using the subsonic principle of the pressure maximum is presented. Analysis of the results of the numerical calculation of the flow around a moving aircraft carrier in the presence of atmospheric winds showed that if the calculation results are used for the simulation of complex flight modes and analyze the state of the atmosphere from the point of view of safe air traffic, visualizing the flow pattern by Q = const surfaces is not informative. In particular, these surfaces do not reflect the true picture of the wind shear perceived by the aircraft directly entering it. To verify the numerical method, it is sufficient to provide only a surface Q = 0 which has a clear physical meaning.

Toroidal magnetic traps for plasma confinement make up an extended object of controlled nuclear fusion investigations. Mathematical simulation of equilibrium plasma configurations in the traps often deals with their analogues straightened into a cylinder. This paper presents a comparative analysis of their numerical investigations in both geometry variants. Mathematical tool of the models use two-dimensional boundary problems with the Grad-Shafranov differential equation for the magnetic flux function. As the investigation result, we present some quantitative characteristics of differences between toroidal and cylindrical configurations by two examples: a plasma torus with longitudinal electrical current and the Galathea-Belt toroidal trap with two ring-shaped current-carrying conductors immersed into the plasma.

A model of the evolution of behavioral strategies in social networks with an arbitrary topology is proposed. The model is formalized as a discrete dynamical system on a graph, which defines the scheme of possible interactions between the system’s elements. Typical evolutionary scenarios are described at the qualitative level. A simple generalization of the model, which allows modeling the evolution of the graph’s topology induced by the system’s dynamics, is discussed. Applications of the proposed model to the problem of simulating corruption (unethical behavior) are considered.