Keywords: high-performance computing, kinetic schemes, numerical simulation, quasi-gasdynamic equations.

Computer systems with performance above 100 TFLOPS are accessible to researchers now. This opens big opportunities for mathematical modelling, allowing the simulation of processes as close as possible to the object under investigation. First of all it concerns industrial applications which are, as a matter of fact, multidisciplinary and possess a complicated geometry. It makes mathematical modelling using the HPC systems an important factor of scientific and technical progress. But a considerable part of CPU-time of most HPC systems is not used. The reasons for such a situation are the fundamental problems connected with the use of HPC systems. These problems are associated with the adaptation of algorithms and an existing program toolkit to the architecture of multiprocessor multicore systems and with the question of correctness of the algorithms used and mathematical models. We concentrate on the problem of correctness of algorithms and mathematical models - a topic concerning "pure" mathematics. The earlier systems with low performance "allowed" discretization on coarse grids only. These grids did not enable one to describe small-scale processes, but, at the same time, due to the presence of large scheme viscosity stabilized the results of numerical computations. Scheme viscosity is a novelty on fine grids appropriate for HPC systems. So there is a question - what is observed as a result of computational experiment: actual process or artificial instability? The presented examples of HPC calculations illustrate the importance of the use of proper algorithms and models in order to ensure adequacy of calculation results to the real-life process. The original approach to the simulation of gas-dynamic flows - kinetic schemes (KS) and their differential approximation - quasi-gas dynamic system of equations (QGDE) is considered. This approach is classified with algorithms which can be easily adapted to the multiprocessor architecture on one hand and reliably simulate diverse physical instabilities arising in gas media on the other hand. The essence of KS consists of using discrete kinetic models, which describe the one-particle distribution function, as a basis for constructing algorithms modelling gas dynamic flows in dense media. The QGDE, which describes explicitly macroscopic parameters of the medium and is a consequence of KS, widens the choice of algorithms. KS and QGDE ensure smoothing of the solution over distances at least on the order of the free path length and, thereby, the physical correctness of the descriptions based on them. The mathematical form of QGDE system, which differs from that of Navier-Stokes equations, enables one to construct efficient (in particular, parallel) computational algorithms.

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