On Nonlinear Closures for Moment Equations Based on Orthogonal Polynomials

TL;DR

This paper introduces a new moment closure method based on orthogonal polynomials derived from Gram matrices, demonstrating high accuracy and favorable mathematical properties in gas kinetic theory applications.

Contribution

The paper proposes a novel Gramian-based orthogonal polynomial approach for moment closure, with proven mathematical advantages and improved accuracy over existing methods.

Findings

The Gramian closure exhibits superior accuracy for various distribution functions.

Numerical comparisons show the approach outperforms Grad's and maximum-entropy closures.

Mathematical properties of the Gramian closure are thoroughly analyzed.

Abstract

In the present work, an approach to the moment closure problem on the basis of orthogonal polynomials derived from Gram matrices is proposed. Its properties are studied in the context of the moment closure problem arising in gas kinetic theory, for which the proposed approach is proven to have multiple attractive mathematical properties. Numerical studies are carried out for model gas particle distributions and the approach is compared to other moment closure methods, such as Grad's closure and the maximum-entropy method. The proposed ``Gramian'' closure is shown to provide very accurate results for a wide range of distribution functions.