The 35-Moment System with the Maximum-Entropy Closure for Rarefied Gas Flows

Abstract

This paper presents a robust implementation of the maximum-entropy closure in the context of rarefied gas dynamics. Moment systems supplied with the maximum-entropy closure have attractive mathematical properties: They are hyperbolic in the interior of the domain of definition of the dual minimization problem and endowed with an entropy law. In contrast to Grad’s classical closure theory, the maximum-entropy closure allows for applications to strongly non-equilibrium gas flows. The 35-moment system studied in this paper includes as basis functions all monomials up to order four, so that evolution equations for important non-equilibrium quantities, such as the stress tensor and heat flux vector, are contained in the system. To remove the singularity in the maximum-entropy closure, we consider a bounded underlying velocity domain and approximate moments of the reconstructed maximum-entropy distribution with a fixed, block-wise Gauss–Legendre quadrature rule. The convex dual minimization problem is solved with a Newton type algorithm. We show that the Hessian matrix used in the Newton iteration can become ill-conditioned even for equilibrium states if monomial basis functions are used. To improve the robustness of the Newton iteration, we consider partially and fully adaptive basis algorithms and demonstrate that the 35-moment system allows for accurate and robust simulations of non-equilibrium rarefied gas flows in the transition regime by applying the model to one-dimensional gas processes, including a continuous shock-structure problem.