Review of unified gas-kinetic scheme and its wave-particle approach

Unlike traditional computational fluid dynamics (CFD), which is constrained by the conventional framework of discretizing partial differential equations (PDEs), the core idea of direct modeling approach is the integration of the cell-grid size as the observational scale into the construction of discrete numerical schemes, allowing for accurate simulation of physical problems at any intermediate scale between the molecular mean-free-path scale described by the Boltzmann equation and the macroscopic fluid mechanics scale described by the Navier-Stokes equations. Based on this idea, Xu and his team have proposed the unified gas-kinetic Scheme (UGKS) and the unified gas-kinetic wave-particle method (UGKWP). The key lies in constructing the time-dependent evolution fluxes, thereby overcoming the constraints imposed by molecular mean free path and collision time on grid size and time step. UGKS ensures accurate capture of non-equilibrium transport across all flow regimes by co-updating the distribution function and macroscopic conserved quantities; in contrast, UGKWP, the efficient wave-particle implementation of UGKS, preserves multiscale physical features while achieving velocity-space adaptive features through particle evolution. Initially developed for hypersonic and microscale gas flows, UGKS and UGKWP have subsequently been extended to other problems with non-equilibrium transport feature, including multi-component plasma, radiation transport, particle-laden two-phase flows, and non-equilibrium turbulence modeling, etc. This paper systematically reviews the latest progress of UGKS/UGKWP methods in the aforementioned fields, aiming to demonstrate the generalizability of the direct modeling approach in addressing multiscale problems and highlights the importance of multiscale non-equilibrium flow in fluid mechanics and related interdisciplinary fields.