Hierarchical Moment Closure Approximation of the Boltzmann - Michael Abdel Malik
Abstract
This work applies the moment method onto a generic form of kinetic equations, given by Boltzmann’s equation (BE), to simplify kinetic models of particle systems. This leads to a hierarchy of moment systems and the correspond- ing moment closure problem. Grad [1] conceived of moment closure approximations based on the expansion of the one-particle distribution in Hermite polynomials. However, Grad’s moment systems are impaired by two essential deficiencies, viz., the potential occurrence of inadmissible locally negative phase-space distributions and potential loss of hyperbolicity [2, 3]. Levermore [4] has developed a moment-closure procedure based on constrained entropy mini- mization that leads to an exponential closure. Levermore’s moment systems retain the fundamental structural proper- ties of BE. Moreover, the moment systems form a hierarchy of symmetric hyperbolic systems and the corresponding distributions are non-negative. It was later shown by Junk [5], however, that Levermore’s moment-closure procedure suffers from a realizability problem, in that there exist moments for which the minimum-entropy distribution is non- existent. Moreover, the fluxes in Levermore’s moment systems can become arbitrarily large in the vicinity of (local) equilibrium when they exist on the boundary of the domain of realizable moments. Practically, the non-tractability of Levermore’s moment-closure system poses a formidable challenge to numerical implementation.
In this work we consider alternative moment-closure relations for BE, based on non-negative approximations of the exponential function. We propose a generalization of the setting of the moment-closure problem from Kullback- Leibler divergence [6] (i.e relative entropy) to the class of ϕ-divergences [7]. The considered ϕ-divergences con- stitute an approximation to the Kullback-Leibler divergence in the vicinity of (local) equilibrium. It will be shown that the approximate-exponential closure relation can be derived via constrained minimization of a corresponding ϕ- divergence. The proposed description encapsulates as special cases Grad’s closure relation and Levermore’s entropy- based closure. Moreover, the corresponding moment systems are symmetric hyperbolic, tractable and retain the fun- damental properties of BE. Finally, the opportunities pertaining to goal-oriented adaptive modeling provided by the hierarchical structure exhibited by the resulting closed systems of moment equations will be discussed.
References
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