Computational coarse graining of a randomly forced 1-D Burgers equation

The behavior of physical systems is frequently observed, and modeled, at different levels of complexity. Laminar Newtonian fluid flow is a case in point: it is modeled at the atomistic level through molecular dynamics simulators; at a "mesoscopic" level through Lattice Boltzmann models; and (in physical and engineering practice) through continuum macroscopic equations for density, momentum and energy: the Navier-Stokes. What makes the latter, coarse grained description possible is an appropriate closure, in this case Newton's law of viscosity, which accurately models the effect of higher order, unmodeled quantities (such as the stresses) on the variables ("observables") in terms of which the model is written (the velocity gradients). Such closures are often known from physical and engineering observation and practice long before their mathematical justification becomes available (in this case, through Chapman-Enskog expansions and kinetic theory). It is, of course, tempting to consider a situation in which the fine scale description is the Navier-Stokes equations themselves, and the coarse grained description is an evolution equation for some observables of turbulent velocity fields, or possibly of ensembles of such fields. Discovering the appropriate observables and deriving such evolution equations has been a long-standing ambition in both physics and mathematics. Our goal here is much more modest: we present a very simple illustration of a problem for which a fine scale description is available, and for which (based on extended observations of direct simulation) we have reason to believe that a coarse grained evolution equation exists -yet it is not available in closed form. We illustrate how (with a good guess of the appropriate coarse grained observables) we circumvent the derivation of a closed form coarse grained model, but still perform scientific computing tasks at the coarse grained level.

The numerical tasks we demonstrate are coarse projective integration, which accelerate the (coarse grained) computations of the system evolution, and coarse dynamic renormalization, which (when the coarse grained evolution is self-similar, as the case appears to be here) targets the computation of the self-similar solution shape and the corresponding exponents. This is an illustration of the so-called Equation-Free framework for coarse grained scientific computation in the absence of explicit quantitative closures and the resulting coarse grained evolution equations [21,22,29].

We consider the one dimensional in space, randomly forced Burgers equation, subject to periodic boundary conditions over the domain x ∈ [0, 2π]. We start with random very low energy initial conditions. We are interested in the relatively long term and large scale properties of this system in the inviscid limit, when subject to a random forcing at the small scales. In order to achieve this inviscid limit numerically, we consider a general dissipation term of the form ν hyp (-1) n+1 ∂ 2n u/∂ x 2n following [3]. The coefficients ν hyp and n are chosen so that the dissipation acts only at the highest wavenumbers; here, their values are ν hyp = 10 -54 and n = 7. This choice of dissipation is based on the assumption that the universal ranges of the system are not sensitive to a particular choice of the parameters ν hyp and n [3]. The white-in-time random force f (x,t) is defined in Fourier space by

where k and ω are the spatial and temporal frequencies. The forcing term has a peak at wavenumber k = k s = 1000 and dies off at large wavenumbers; the dissipation term is essentially zero up to around wavenumber k = 3000 and only becomes important at much larger wavenumbers; see Fig. 1.

Before we start our computations, we briefly recall certain known features of the equations and their dynamics. It is quite remarkable that equations (1,2) have non-trivial asymptotic behavior in both the ultraviolet (UV, k > k s ) and the infrared (IR, k < k s ) limits. Attempts to use (1,2) as a simplified 1-D model for the investigation of small scale features of 3-D turbulence, aimed at recovering E(k) ∝ k -5/3 in the small scale limit, were based on deep similarities between the two: the total energy in this system is u 2 /2 ≡ K ≈ (PL) 2/3 and, as in 3-D turbulence, the inertial

where k d is the dissipation wave number, is characterized by a constant energy flux J E = P, which, in the interval r ≪ L = 1/k s , is reflected in the analytic behavior of the third-order structure function

Even in decaying high Re-turbulence this equation is correct: ∂ E/∂t is very small. At first glance, due to the shock formation leading to the E(k) ∝ k -2 energy spectrum (rather than E(k) ∝ k -x with the exponent x ≈ 5/3), these attempts failed. However, the model revealed a non-trivial bi-scaling behavior of the structure functions S n (r) = (u(x + r)u(x)) n ∝ r n for n ≤ 1 and S n (r) ∝ r for n ≥

1 [12]. The model also generated asymmetric probability densities with algebr

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