Keywords: non-equilibrium, extended thermodynamics, rarefied gas dynamics, method of moments, Boltzmann equation, lattice Boltzmann method, Chapman-Enskog expansion.

Non-equilibrium flows occur in many different applications and across a wide range of Mach numbers, from very low-speed flows in micro-devices through to hypersonic flows associated with re-entry vehicles. In contrast to the highly successful predictions of equilibrium fluid flow, the conventional Navier-Stokes-Fourier equations often fail to capture the physics associated with non-equilibrium flow and their predictive capability is significantly reduced. The challenge of simulating such flows has stimulated research into many aspects of kinetic theory, constantly seeking ways to efficiently solve the Boltzmann equation or develop extended thermodynamic methods that can exploit the numerical advances made in computational fluid dynamics. In this review paper, we discuss some of the emerging techniques related to modelling non-equilibrium flow such as directly solving the Boltzmann equation, particle-based approaches using the direct simulation Monte Carlo method, and recent progress in extended thermodynamics using Grad's method of moments.

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