On Nonlinear Closures for Moment Equations Based on Orthogonal Polynomials

Abstract. In the present work, an approach to the moment closure problem on the basis of orthogonal polynomials derived from Gram matrices is proposed which is inspired by [R. O. Fox and F. Laurent, SIAM J. Appl. Math., 82 (2022), pp. 750–771]. The new theory comes with an explicit formulation of the closure relation and satisfies a generalized gauge theory in consistency with affine transformations of the phase space. It also allows us to explicitly prove global hyperbolicity of the resulting moment equations for an arbitrary number of moments. Numerical studies are carried out on local distributions to investigate different variants of the new closures in comparison to other moment closure methods, such as Grad’s closure and the maximum-entropy method. The proposed “Gramian” closure is shown to be able to provide very accurate predictions of the next higher moment for a wide range of distribution functions.