On unified preserving properties of kinetic schemes

Numerical modeling of fluid flows based on kinetic equations provides an alternative approach for the description of complex flows simulations, and a number of kinetic methods have been developed from different points of view. A particular challenge for kinetic methods is whether they can capture the correct hydrodynamic behavior of the system in the continuum limit without enforcing kinetic scale resolution. At the current stage, asymptotic preserving (AP) kinetic methods, which keep the same algorithm in different flow regimes, have been constructed. However, the detailed asymptotic properties of these kinetic schemes are indistinguishable under the AP framework. In order to distinguish different characteristics of kinetic schemes, in this paper we will introduce the concept of unified preserving (UP) which can be used to assess the real governing equations solved in the asymptotic process. Unlike the general analysis of AP property in the hydrodynamic scale, the current UP analysis is able to find the asymptotic degree of the scheme employing the modified equation approach. Generally, the UP properties of a kinetic scheme depend on the spatial/temporal accuracy and closely on the inter-connections among the three scales (kinetic scale, numerical scale, and hydrodynamic scale), and the concept of UP attempts to distinguish those scales with clear orders. Specifically, the numerical resolution and specific discretization determine the numerical flow behaviors of the scheme in different regimes, especially in the near continuum limit with a large variation of the above three scales. The UP analysis will be used in the Discrete Unified Gas-kinetic Scheme (DUGKS) to evaluate its underlying governing equations in the continuum limit in terms of the kinetic, numerical, and hydrodynamic scales.