Vol 11, No 6 (2019)

A difference scheme for the transfer problem, constructed as a linear combination of the Cabaret scheme and the central difference scheme, is proposed. The stability and dispersion properties of the scheme are studied. It is shown that the constructed scheme has the best dispersion properties for high-frequency harmonics at small Courant numbers compared with the known Cabaret scheme for the transport equation. A comparison is made of the errors of this scheme and the two-parameter third-order difference accuracy scheme based on the numerical experiments on the previously used test problem sets. It is shown that in the normal grid space L₁ the developed scheme has smaller errors and also uses a more compact template (in calculating the inode the node values i – 1, i, and i + 1 are used), and the transition to the next time layer is carried out for a smaller number of arithmetic operations.

The parasite–host system is considered with two different variants of a pathogen agent with complete cross protection, where one of the variants may transform into another with a certain probability. This system corresponds to the epidemic process initiated by the antibiotic-resistant variant of the pathogen agent. The behavior of solutions is studied. It is proved that the main case has a unique nontrivial steady-state solution that is a global attractor. The speed of the exponential approach of small deviations to the steady-state solution is obtained.

Algorithms of the optimal arrangement of heat sources with volumetric heat release within regions of a complex geometric shape are considered. The distribution found has the minimum total power and provides the temperature in the given temperature corridor. Finite-dimensional approximations of the original problem are constructed in the form of a linear programming problem. A method is given for constructing a finite-difference scheme for solving the heat equation, as well as a brief description of the developed software modules for constructing grids and solving equations. Several computer experiments have been carried out using the developed programs.

The papers dedicated to the procedures of solving direct problems in seismic exploration of fractured formations are considered in this review article. The fractured formations are known to be potential hydrocarbon reservoirs, which are being studied actively at the present time. Due to high cost of field prospecting works, the numerical simulation is an important part in such research, leading to a significant decrease in financial and time expenditures. The papers focusing on conventional popular practical modeling methods based on the use of effective models are considered. A significant part of this work also deals with the papers that use the procedures developed by the authors to solve the formulated range of problems. These procedures are based on a grid-characteristic numerical method with interpolation on unstructured triangular (in the 2D-case) and tetrahedral (in the 3D-case) meshes. The grid-characteristic method most accurately describes the dynamical processes in exploration seismology problems, because it considers the nature of wave phenomena. The used approach allows making the correct computational algorithms at the boundaries and contact boundaries of the integration domain. An important part of this article describes the different models of fracturing. The results of mathematical modeling by the developed procedure, which are presented in the authors’ papers, are also considered. The important practical conclusions made in the considered papers are presented.

We develop a mathematical model of a dynamical testbed for testing accelerometer assemblies created at the Keldysh Institute of Applied Mathematics. It has a movable platform with one degree of freedom. It can rotate about the axis fixed in the prescribed direction. The tested accelerometer assemblies are placed in the platform that rotates arbitrarily. Its real motion is reconstructed a posteriori by measurements of the vision system. The reconstruction is carried out in digital form and allows us to calculate the real acceleration felt by the assemblies. The calculated acceleration is compared with the measured one. The comparison results are used to test and calibrate sensors, electronic units, etc.

In this paper, we consider a new numerical method for solving the transport equations for a multicomponent heterogeneous system on fixed Eulerian grids. The system consists of an arbitrary number of components. Any two components are separated by a boundary (interface). Each component is determined by a characteristic function, i.e., a volume fraction that is transported in a specified velocity field and determines the spatial instantaneous component distribution. A feature of this system is that its solution requires two conditions to be met. Firstly, the volume fraction of each component should be in the range [0, 1], and, secondly, any partial sum of volume fractions should not exceed unity. To ensure these conditions, we introduce special characteristic functions instead of volume fractions and propose solving transport equations with respect to them. It is proved that the fulfillment of these conditions is ensured when using this approach. In this case, the method is compatible with various TVD schemes (MINMOD, Van Leer, Van Albada, and Superbee) and interface-sharpening methods (Limited downwind, THINC, Anti-diffusion, and Artificial compression). The method is verified by calculating a number of test problems using all of these schemes. The numerical results show the accuracy and reliability of the proposed method.

Glaciation and thawing models of the outer surface of an offshore gas pipeline in the northern seas are presented. The glaciation model suggests a modification of the Stefan condition that allows us to account for the features of sea-ice growth in salt water. A numerical algorithm for solving an unsteady problem of glaciation (thawing) of a multilayer cylindrical area by an explicit front-tracking method and some calculation results for different versions of these problems of practical interest are given. Qualitative estimates of the feasibility of the transition to a quasi-stationary version of the glaciation (thawing) model of multilayer areas are obtained. The quantitative condition for the admissibility of using the quasi-stationary approximation in the calculations of glaciation (thawing) of a certain multilayer area is given. These estimates are important for developing effective numerical algorithms to calculate unsteady regimes of gas transportation through offshore gas pipelines. For the problems of the thawing of the surface of an offshore gas pipeline, an equation is proposed that allows us to find the minimal ice thickness under the conditions studied.

The paper presents a brief review of computational approaches for the simulation of turbulent flows and shows that the correct calculation of large-scale vortex structures necessarily requires the use of eddy-resolving methods, while the applied numerical schemes must be stable and accurately describe the spatial evolution of vortices. The dissipative properties and stability of most numerical schemes implemented in the OpenFOAM package are analyzed by solving problems on the decay of the homogeneous isotropic turbulence and scalar transfer. It is established that the considered schemes are not suitable for correctly calculating the propagation and dissipation of vortices in space; therefore, they are refined to eliminate oscillations and maintain an acceptable level of dissipation. An algorithm for combining the URANS and LES methods using zoning of the computational domain is described and implemented. To test the implemented calculation method of unsteady turbulent flows, we simulate the flow around a maneuvering aircraft with an installed airbrake deflection. The flow patterns of the aircraft and its aerodynamic characteristics are obtained and compared with the experimental data.

Based on the discrete sources method, a mathematical model is developed that enables to analyze the fluorescence process in the presence of a plasmonic structure taking into consideration the nonlocal screening effect. It is shown that the plasmonic structure’s quantum yield can be represented analytically omitting integration procedures. The influence of the effect on the quantum yield and the fluorescence enhancement factor is investigated depending on the plasmonic structure geometry. It is demonstrated that accounting for the nonlocal screening effect leads to a shift of the maximum position towards the long-wave region and a decrease in the amplitude of the fluorescence enhancement factor.

This paper aims to focus on the implementation of a new approach, quintic trigonometric B-spline basis functions in differential quadrature method to find numerical solution of generalized fourth order partial differential equations with nonlinearity involved. The obtained results using this approach are presented in comparison with available exact and numerical solutions obtained by other researchers. The obtained solutions are in agreement with the available exact solutions and even better than the solutions proposed by the other schemes in the literature. The applicability of the scheme are demonstrated by different eight test problems for the discussed equations that are simulated, for calculating errors like L₂, \({{L}_{\infty }}\), RMS and GRE. To visualize the same, solutions are also presented graphically along with the exact solutions. Scheme is shown to be unconditionally stable with the help of eigenvalues, which demonstrates the consistency of the proposed numerical scheme.