Abstract
A complete characterisation of the moment space corresponding to the Levermore basis is given here, through constraints on the moments. The necessary conditions are obtained thanks to classical tools, similar to Hankel determinants. In the mono-variate case, it is well-known that these conditions are sufficient. To generalize this result to multi-variate case, a non-classical constructive proof is given here reducing the problem to several mono-variate ones. However, it is also shown here on an example that the obtained multi-variate closure does not necessarily inherit of the good properties of the mono-variate closure.
Highlights
Moment closure methods, consisting in transforming a kinetic equation into a system of equations on moments of the number density function (NDF), is used in several applications such as rarefied gas dynamics [8] or dispersed phases of a multiphase flow [1]
The moment space is the space of the moment sequences corresponding to any finite Borel measure
The objective of this paper is to express the realizability conditions in the case of the moment space corresponding to the Levermore basis
Summary
Moment closure methods, consisting in transforming a kinetic equation into a system of equations on moments of the number density function (NDF), is used in several applications such as rarefied gas dynamics [8] or dispersed phases of a multiphase flow [1]. The moment space is the space of the moment sequences corresponding to any finite Borel measure It is characterized by some constraints on the moments, called realizability conditions. The objective of this paper is to express the realizability conditions in the case of the moment space corresponding to the Levermore basis The choice of this basis allows to consider moments till order four and to respect Galilean invariance (in particular, it does not single out any direction). For any positive measure dμ, let us defined the following moments of order 0 to 4, assuming that they exist and are finite: M0(μ) := They are scalars, vectors and matrix and correspond to the Levermore basis m(v) = 1, v, v ⊗ v, v2v, v4 [4], even if they are written slightly differently, using only products of matrices and column vectors, v being a column vector, like M1 and M3, whereas M2 is a d -by-d matrix and M0 and M4 are some scalars. We will focus on the central moment space to characterize the moment space
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