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The paper presents the results of applying a scheme of very high accuracy and resolution to obtain numerical solutions of the Navier – Stokes equations of a compressible gas describing the occurrence and development of instability of a two-dimensional laminar boundary layer on a flat plate. The peculiarity of the conducted studies is the absence of commonly used artificial exciters of instability in the implementation of direct numerical modeling. The multioperator scheme used made it possible to observe the subtle effects of the birth of unstable modes and the complex nature of their development caused presumably by its small approximation errors. A brief description of the scheme design and its main properties is given. The formulation of the problem and the method of obtaining initial data are described, which makes it possible to observe the established non-stationary regime fairly quickly. A technique is given that allows detecting flow fluctuations with amplitudes many orders of magnitude smaller than its average values. A time-dependent picture of the appearance of packets of Tollmien – Schlichting waves with varying intensity in the vicinity of the leading edge of the plate and their downstream propagation is presented. The presented amplitude spectra with expanding peak values in the downstream regions indicate the excitation of new unstable modes other than those occurring in the vicinity of the leading edge. The analysis of the evolution of instability waves in time and space showed agreement with the main conclusions of the linear theory. The numerical solutions obtained seem to describe for the first time the complete scenario of the possible development of Tollmien – Schlichting instability, which often plays an essential role at the initial stage of the laminar-turbulent transition. They open up the possibilities of full-scale numerical modeling of this process, which is extremely important for practice, with a similar study of the spatial boundary layer.

This work is devoted to molecular dynamic modeling of the thermal impact processes on the metal sample consisting of nickel atoms. For the solution of this problem, a continuous mathematical model on the basis of the classical Newton mechanics equations has been used; a numerical method based on the Verlet scheme has been chosen; a parallel algorithm has been offered, and its realization within the MPI and OpenMP technologies has been executed. By means of the developed parallel program, the investigation of thermodynamic equilibrium of nickel atoms’ system under the conditions of heating a sample to desired temperature has been executed. In numerical experiments both optimum parameters of calculation procedure and physical parameters of analyzed process have been defined. The obtained numerical results are well corresponding to known theoretical and experimental data.

The paper presents a software system aimed at finding best (in some sense) parameters of an algorithm. The system handles both discrete and continuous parameters and employs massive parallelism offered by public clouds. The paper presents an overview of the system, a method to measure algorithm's performance in the cloud and numerical results of system's use on several problem sets.

The paper presents an implementation of branch and bound algorithm employing coarse grained parallelism. The system is based on CBC (COIN-OR branch and cut) open-source MIP solver and inter-process communication capabilities of Erlang. Numerical results show noticeable speedup in comparison to single-threaded CBC instance.

Molecular dynamic methods that use ReaxFF force field allow one to obtain sufficiently good results in simulating large multicomponent chemically reactive systems. Here is represented an algorithm of searching optimal parameters of molecular-dynamic force field ReaxFF for arbitrary chemical systems and its implementation. The method is based on the multidimensional technique of global minimum search suggested by R.G. Strongin. It has good scalability useful for running on distributed parallel computers.