Keywords: high-performance computing, resolution limits, kinetic scheme.

Ultra-high performance multiprocessor computer systems (1PFlops and above) provide new opportunities for mathematical modeling, including those applications that require a detailed space-time description of the process. It is important, for example, for detailed modelling of various physical instabilities, which in many cases are revealed at relatively small scales. However, in the realization of computation system functionalities to the aforementioned problems a number of serious difficulties are encountered [1,2].

We omit the important problems of both the computational algorithms adaptation to the multiprocessor system architecture and software development at the moment, and discuss the high performance computing data verification problem. The latter problem is of the most acute and is directly related to the correctness of numerical algorithms and mathematical models used as their basis.

In this paper, by the example of some continuum mechanics problems we show that the construction of such smoothing terms can be done easily on physical grounds. The emergence of these terms is due to the existence of minimal spatial scales, beyond which any artificial reduction of scales with the goal to obtain more detailed solutions no longer makes sense.

It should be noted that the appearance of this paper is closely related both to the increased capabilities of mathematical modeling, and to an unprecedented growth of computing performance capabilities. Modern supercomputers in many cases no longer put any limitations on the computing level of the solution detail in space and time. These trends will only strengthen in the near future. Therefore, the lower limitations on the solution spatial-time details seems reasonable for problem formulation.

It is important to point out that the additional terms that arise due to limitations on the minimal scales do not really change solutions, based on classical formulations. These terms appear only as substantiated regularizers smoothing physically inconsistent effects that arise in numerical solutions. In this regard particular values of variables that enter in regularization terms as coefficients no longer play any significant role. The important point is that they fall only in the right range of order of magnitude. This fact greatly facilitates the practical use of the approach.

The proposed theory formed the basis for high-performance computing of incompressible fluid and underground hydrodynamics problems. To approximate the three-dimensional problems over 10⁹ spatial nodes were involved. Hence up to a computer performance of 700 TFlops was needed to solve the one alternate version.

1

B.N. Chetverushkin, "High-Performance Computing: Fundamental Problems in Industrial Applications", in B.H.V. Topping, P. Iványi, (Editors), "Parallel, Distributed and Grid Computing for Engineering", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 17, pp 369-388, 2009. doi:10.4203/csets.21.17

2

B.N. Chetverushkin, "Applied mathematics and the problem of utilization of high-performance computer systems", MIPT Proceedings, 3(4), pp. 96-108, 2011 (in Russian).

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