Transitions of "correct-incorrect" numerical calculations for solutions of some problems: "phase transitions" in computer turbulence

Анотація

Many effects that generally characterise the phenomenon of turbulence, in particular, the emergence of structures (including the cascade process of birth of coherent structures of decreasing scales) and self-stochasticity can be observed in "simple" examples of dynamical systems which are generated by one- and two-dimensional boundary value problems (BVP) consisting of linear partial differential equations and nonlinear boundary conditions. But mathematical models using these BVP (without certain additional conditions or assumptions) do not describe adequately the behaviour of real physical objects when time is sufficiently large, at least, because the diameters of structures appearing in these BVPs can decrease up to zero, which is not in agreement with discrete nature of time and space. Thus these models can describe adequately a real process only within certain time-space limits. Most likely the models have to be adjusted beginning with certain time-space scales. At the same time, results obtained by numerical investigation of these models are probably more close (in some sense) to the reality than exact solutions not in spite but owing to "incorrect" calculations.

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