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

Mor a standard Pearson-IV distribution_{∞}=exp(-c^{2})

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_{m}=(1+c^{2})^{(-m)}

S(c) = s(ν,c•n)= exp(-ν*arctan(c•n)).

From the core distibution

core(c;ν,n) = s(ν,c•n)*M_{m}(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