Abstract
On the example of a simple and clear, but far from being trivial, model of hard sphere gas, we will try to show the main stages in constructing the mathematical formalization of a complex physical system.
We are considering a set of about \({10}^{25}\) solid balls that just fly and collide. A mathematical description of the evolution of such a system inevitably leads to the necessity of using the apparatus of the theory of random processes. To identify the mathematical and computational features of the problem under study it is important to write it in a dimensionless form. This procedure leads to the appearance of the Knudsen number, the physical meaning of which is the ratio of the mean free path of molecules to the characteristic size of the problem. The hierarchy of micro-macro models is constructed in accordance with the change in this parameter from values of the order of unity (micro) to magnitudes of the order of 0.1 (meso) and further to 0.01 (macro). Accurate movement along this path leads to more accurate mathematical models, in comparison with traditional ones, which affects their greater computational fitness - nature pays for a careful attitude towards it. In particular, obtained macroscopic equations are softer for simulations than the classical Navier-Stokes equations.
This hierarchy of mathematical statements generates a corresponding chain of computational methods. Microscopic problems are most often solved using Monte Carlo methods, although there are research groups that are committed to nonrandom methods for solving the Boltzmann equation. Recently, much attention has been paid to mesomodels based on modeling the Brownian motion or solving the deterministic Fokker - Planck - Kolmogorov equations. To solve the problems of a continuous medium, difference methods, finite element methods, and particle methods are used. The latter ones, in our opinion, are particularly promising for the entire hierarchy, uniting different statements with a single computational ideology. A discontinuous particle method is particularly effective.
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Bogomolov, S.V., Esikova, N.B., Kuvshinnikov, A.E., Smirnov, P.N. (2019). On Gas Dynamic Hierarchy. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods. Theory and Applications. FDM 2018. Lecture Notes in Computer Science(), vol 11386. Springer, Cham. https://doi.org/10.1007/978-3-030-11539-5_17
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