A way to obtain a hyperbolic fluid closure is to calculate moments of an assumed family of distributions. An expensive numerical method for such a model would be to solve the kinetic equation for a time step and project the distribution back on to the assumed family of distributions (in a way that preserves evolved moments). You could call this Vlasov with instantaneous relaxation to an assumed family.
A chief difficulty with this approach is moment realizability. A set of moments is "physically realizable" if it is the moments of a positive distribution. A set of moments is "model-realizable" if it is the moments of a distribution from the assumed family. Positivity of the assumed family of distributions is desirable, in which case the model-realizable distributions are a subset of the physically realizable distributions. A critical question is whether model realizability is maintained.
The Pearson13 closure is obtained by calculating moments of a "Pearson-IV" family of distributions which approximates a skewed Gaussian distribution. It is defined in a special reference frame (approximately the center-of-mass frame) and in transformed velocity coordinates.
Specifically, a member of the Pearson13 family can be defined as a transformation of a member of a "core" family. To define a member of the core family, take a standard Maxwellian
M∞=exp(-c2)
or a standard Pearson-IV distribution
Mm=(1+c2)(-m)
(with m a fixed parameter)
and multiply it by a skewing function S(c) which is homogeneous
orthogonal to a direction n and depends on a skewness parameter
ν. Manuel's choice of skewing function is
S(c) = s(ν,c•n)= exp(-ν*arctan(c•n)).
From the core distibution
core(c;ν,n) = s(ν,c•n)*Mm(c)
a member of the Pearson13 family is generated by a remapping of the domain and range of the distribution: a positive-definite affine remapping of velocity space (rescaling velocity space by a positive-definite linear transformation A, then shifting velocity space by a vector lambda) and rescaling of the distribution by a normalization constant K. Moments transform analytically under this remapping.
The core distribution is defined in terms of a single parameter ν, so moments of the core distribution are a function of this single parameter. Thus, moments of the Pearson13 family are expressed analytically in terms of single-variable functions of ν. Manuel's choice to require the skewing function to be homogeneous orthogonal to the direction n allows moments to be calculated as iterated one-dimensional integrals (thanks to properties of the Maxwellian or Pearson-IV distribution — note that without loss of generality n is aligned with a coordinate axis). His specific choice of the skewing function allows him to get an analytical expression for the moment integrals in the case of a Pearson-IV distribution.
Manuel thus obtains an analytical mapping from the parameters of the distribution to the moments. This provides an implicit closure. To compute the highest moments from the evolved moments one has to invert the mapping from parameters to moments, which he does by an iterative method.
References:
[CCP10] M. Torrilhon, Hyperbolic Moment Equations in Kinetic Gas Theory Based on Multi-Variate Pearson-IV-Distributions, Comm. Comput. Phys., 7(4), (2010), p.639-673
[KRM12] M. Torrilhon, H-Theorem for Nonlinear Regularized 13-Moment Equations in Kinetic Gas Theory, Kinetic and Related Models 5/(1), (2012), p.185-201