S. V. Polyakov

Exponential difference schemes with double integral transformation for solving convection-diffusion equations

Convection-diffusion equations are studied. These equations are used for describing many nonlinear processes in solids, liquids, and gases. Although many works deal with solving them, they are still challenging in terms of theoretical and numerical analysis. In this work, the grid approach based on the method of finite differences for solving equations of this kind is considered. In order to make it easier, the one-dimensional version of such an equation was chosen. However, the equation preserves its principal properties; i.e., it is non-monotonic and non-linear. To solve boundary-value problems for such equations, a special variant of the non-monotonic sweep procedure is proposed.

Molecular dynamic simulation of thermodynamic equilibrium establishment in nickel

This work is devoted to the molecular dynamic simulation of heat action on a metal sample consisting of nickel atoms. For the decision of this problem a continuous mathematical model based on equations of the Newton classical mechanics and a numerical method with the Verlet scheme; are used a parallel algorithm is proposed that is implemented within the MPI and OpenMP technologies. The developed program was used for the investigation the process of establishing thermodynamic equilibrium for nickel atoms at heating the sample up to the specified temperature. In the numerical experiment, both the optimal parameters of the calculation procedure and the physical parameters of the simulated process have been determined. The obtained numerical results are in good agreement with the known theoretical and experimental data.

Matrix method for simulating the tunneling transfer

We develop an efficient approach for computer simulation of stationary scattering and tunneling transfer across an arbitrary one-dimensional potential barrier. New algorithms and programs were worked out and tested and the convergence of the method in question was examined.

Quasihydrodynamic model and small-scale turbulence

A model of the averaged quasihydrodynamic system of equations governing small-scale turbulence is proposed. In the case of plane geometry the model is tested against the problem of the flow past a rectangular cylinder. For this problem conservative finite-difference schemes on a rectangular grid are constructed and the algorithms of their numerical implementation are developed. Numerical experiments show that the model describes well the turbulent flow, both qualitatively and quantitatively. The results obtained are in good agreement with the results of other authors and the experimental data.

Modeling the simultaneous distribution of legal and counterfeit copies of innovative products

The basic mathematical model for the dynamics of the number of legal and counterfeit customers of computer games and other innovative products is presented. The model is based on information propagation and information warfare models. The analysis is conducted using the theory of ordinary differential equations. It is shown that the results are in qualitative agreement with the empirical data by StarForce.

Transformation-matrix method for tunnel-effect simulation

The problem of numerically simulating steady-state scattering and the tunneling transfer of electrons was considered for a 1D potential barrier with an arbitrary shape. An effective numerical approach to solving this problem was developed on the basis of the transformation-matrix method. To test this approach, a computer program was written and applied to calculations of the tunneling processes. The convergence of the method proposed was further investigated in numerical experiments.

Calculation of radiation in gas around a descent space vehicle

The problem of numerical modeling of the effect of radiation processes on gas dynamic parameters around a descent space vehicle in the Earth’s atmosphere is studied. A relatively new approach is suggested of direct modeling of radiation processes oriented to the use of high performance multiprocessor computer systems. The physical model of radiation transfer is described by a diffusion approach with a discretization of radiation and absorption spectra and a set of quasigasdynamic equations. The computations were carried out on both triangular and hybrid locally refined grids. The gas-flow modeling results taking into account radiation processes are represented for bodies of different shapes.

Calculation of gas flow parameters around a reentry vehicle

An original numerical algorithm for calculating the gas parameters around a reentry vehicle was described and tested. It is based on a system of Quasi Gas Dynamics (QGD) equations and its approximations on hybrid meshes, which suppose adaptive, locally thickening meshes including both rectangular and triangular cells. The results of the calculations on a sequence of condensed meshes for Mach number values of 2, 5, and 12 are presented. These results correspond to well-known data and demonstrate the high efficiency of the algorithm.

Critical depinning current of elastic vortex strings in superconductors with extended linear defects

The problem of the depinning of an Abrikosov elastic vortex string from an extended linear defect in the plate of a 3D anisotropic superconductor of thickness d > 2λ (where λ is the London penetration depth) under the action of an inhomogeneously distributed transport current that flows in the surface shielding layer has been solved using classical mechanics approach. Conditions for the appearance of the instability of the pinned state of a vortex have been investigated and calculations of the corresponding critical current density on the surface at which the depinning of the vortex string occurs have been carried out. The dependence of the average of the critical current density on the thickness of the plate has been determined.

Parallel Software Package for Simulation of Continuum Mechanics Problems on Modern Multiprocessor Systems

A parallel software package designed for numerical simulation of continuum mechanics problems is presented. In order to illustrate the capabilities of the package, two problems on simulation of supersonic gas flows, around a descent space vehicle and in the vicinity of a micro nozzle, were chosen. A system of equations of quasi-gas dynamics was used as a mathematical model for the dynamics of gas. Inasmuch as radiation transfer in gas is taken into account in the first problem, the multigroup approach and diffusion approximation are employed to describe radiation processes. The numerical algorithm is based on explicit in time finite volume schemes using nonregular locally condensed grids of different types. Parallel implementation of numerical schemes was executed in the context of the MPI+OpenMP technology and optimized for computations performed by means of modern clusters with hybrid architecture.