Department of Mathematics & Statistics, Old Dominion University, Norfolk, Virginia 23529, USA
Beijing Computational Science Research Center, Beijing, China
Abstract: Traditionally, fluid flow is modeled by macroscopic hydrodynamic equations, i.e., the Euler or the Navier-Stokes equations. Naturally, computational fluid dynamics (CFD) is based on direct discretizations of the hydrodynamic equations. This traditional approach of CFD is now being challenged as new multi-scale and multi-physics problems have begun to emerge in many fields, because in nanoscale systems, the scale separation assumption does not hold; macroscopic theory is therefore inadequate, yet microscopic theory may be impractical because it requires computational capabilities far beyond our present each. Methods based on mesoscopic theories, which connect the microscopic and macroscopic descriptions of the dynamics, provide a promising approach. Besides their connection to microscopic physics, kinetic methods also have certain numerical advantages due to the linearity of the advection term in the Boltzmann equation. We will discuss two mesoscopic methods: the lattice Boltzmann equation (LBE) and the gas-kinetic scheme (GKS), their mathematical theory and their applications to simulate various complex and/or nonequilibrium flows. Examples include incompressible homogeneous isotropic turbulence, hypersonic flows, and micro-flows.References
 X. He and L.-S. Luo. A priori derivation of the lattice Boltzmann equation. Physical Review E 55:R6333 (1997).
 X. He and L.-S. Luo. Theory of lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation. Physical Review E 56(6):6811 (1997).
 K. Xu. A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method. Journal of Computational Physics 171(1):289 (2001).
 K. Xu. A unified gas-kinetic scheme for continuum and rarefied flows. Journal of Computational Physics 229(20):7747 (2010).
 Z. Guo, H. Liu, L.-S. Luo, and K. Xu. A comparative study of the LBE and GKS methods for 2D near incompressible laminar flows. Journal of Computational Physics 227(10):4955 (2008).