A kinetic generator for classical field theories with conservation laws

¹ Department of Mathematics, Tufts University - Medford, MA 02155, USA
² Center for Life Nanoscience at La Sapienza, Italian Institute of Technology - viale Regina Margherita, 285, Roma, Italy
³ John A. Paulson School of Engineering and Applied Sciences, Harvard University - Cambridge, MA 02138, USA

Received: 17 September 2020
Accepted: 11 November 2020

Abstract

In this study, we demonstrate that, by suitably modifying the equilibrium distribution, the Bhatnagar-Gross-Krook form of the Boltzmann equation can serve as a generator of a broad class of nonlinear partial differential equations via a suitable generalization of the Chapman-Enskog method. We achieve this by properly choosing the relaxation time and moments of the equilibrium state in such a way as to suppress the dynamics of the zeroth moment, density, at increasingly faster time scales. As concrete examples, explicit analytical formulations of the Burgers', Korteweg-de Vries and Kuramoto-Sivashinsky equations are presented, along with a discussion of the associated conservation laws of the corresponding moments. The concept is numerically demonstrated by solving the BGK Boltzmann equation with the finite-element method. Because the Boltzmann-BGK equation can be equipped with a H-theorem, the framework developed in this work indicates a path towards the development of new algorithms for the numerical solution of the aforementioned equations with enhanced stability.