Abstract
We describe a hierarchy of formal expansions that represent the Fourier transform of a solution of the Boltzmann equation. The constructed approximations are based on the family of weighted Taylor expansions. The first two representations correspond to the Maxwellian and to the Gaussian expansions. The third representation has a weight that generalizes the Gaussian and it depends on the first 13 moments of the Boltzmann density f. It can be shown that this weight is Galilean invariant and it is close to the Gaussian, providing that the heat fluxes are not too large. The 13 moment weight yields a revised form of Grad’s 13 moment expansion for the Boltzmann equation. In search for the entropy dissipation inequality, we also examine the relation between Levermore’s 14 moment density and Grad’s 13 moment expansion. First, we show that the coefficients of the Godunov potential are described by a system of partial differential equations, with coefficients that depend on the Fourier transform of the Levermore’s density f Λ itself. Then, we argue that the same Taylor expansion exploited in the Grad’s scheme, can be used to approximate Levermore’s 14 moment density. We also show that the weighted Taylor expansions are related to a formal solution of the Hamburger problem.
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Karwowski, A. Grad’s 13 Moment Equations in a Modified Form. J Stat Phys 128, 667–698 (2007). https://doi.org/10.1007/s10955-006-9216-6
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DOI: https://doi.org/10.1007/s10955-006-9216-6