Statistical mechanics for truncations of the Burgers-Hopf equation: a model for intrinsic stochastic behavior with scaling
Models for one-dimensional dynamics approximating the Hopf, or inviscid Burgers, equation \(u_t + \frac{1}{2}(u^2)_x = 0\) are investigated both analytically and numerically. Continuing their paper [Proc. Natl. Acad. Sci. USA 97, No. 23, 12413--12417 (2000; Zbl 0969.35117)] the authors study the Fourier Galerkin truncated spectral approximations resulting in systems of nonlinear ODE's allowing for an equilibrium statistical mechanics formulation. Computer simulation, in several parameter regimes and with random initial data, confirms the statistical predictions and scaling behaviour of correlation times. Other discrete approximations involving spatially discrete finite-difference schemes are analyzed as well. A new simple 5-points scheme is introduced and compared with those of \textit{J. Goodman} and \textit{P. D. Lax} [Commun. Pure Appl. Math. 41, No. 5, 591--613 (1988; Zbl 0647.65062)] and \textit{N. J. Zabusky} and \textit{M. D. Kruskal} [Phys. Rev. Lett. 15, 240--243 (1965)]. The new one and the latter one have intrinsic stochastic dynamics with numerical evidence for ergodicity and mixing. These methods and the Fourier-Galerkin truncation determine different nonlinear dynamics at lower wave numbers. The models demostrate that predictions can be made by simple mathematical arguments and scaling theory due to longer correlation times on the larger scales. This fits the pattern of stochastic modelling of complex systems by reduced systems involving only the degrees of freedom with longer correlation time. The style of the paper avoids proving theorems, it is a discussion of research results mingling mathematical theory and scientific computing.
Fourier-Galerkin truncated spectral approximation
scientific article; zbMATH DE number 2099228