Invariance correction to Grad's equations: Where to go beyond approximations?

TL;DR

This paper reviews recent advances in Grad's method for solving the Boltzmann equation, emphasizing invariant manifold techniques to improve approximations and proposing new kinetic models for comprehensive spatial descriptions.

Contribution

It introduces the method of invariant manifold as a unified framework for correcting Grad's equations and develops a new class of kinetic models for full-space descriptions.

Findings

Derivation of regularized Grad's equations using invariant manifold

Development of kinetic models extending finite-moment descriptions

Discussion of relations to entropic lattice Boltzmann methods

Abstract

We review some recent developments of Grad's approach to solving the Boltzmann equation and creating reduced description. The method of invariant manifold is put forward as a unified principle to establish corrections to Grad's equations. A consistent derivation of regularized Grad's equations in the framework the method of invariant manifold is given. A new class of kinetic models to lift the finite-moment description to a kinetic theory in the whole space is established. Relations of Grad's approach to modern mesoscopic integrators such as the entropic lattice Boltzmann method are also discussed.