Theory and appliation of microscopic approaches to macroscopic fluid-dynamic equations

本课题建立了气体动力学方程（可压缩欧拉方程、可压缩Navier-Stokes方程）的动力学格式理论，并发展了一些数值方法。动力学格式最初是作为Boltzmann方程的DSMC方法的模拟而开发的，揭示了它是经典Lax-Wendroff格式的扩展，并且表明了动力学方法在宏观状态的不连续重建管理中的意义。基于这一新发现的新开发的动力学方案可以作为激波捕获方案，并产生具有合理分辨率的精细边界层剖面。这些方案可以很容易地应用于各种分子模型；对硬球分子的NS方程和双原子气体的欧拉方程进行了推广。首席研究员与海外合作者共同开发了处理高阶稀疏效应的Burnett方程和超级Burnett方程的动力学格式。在此基础上，提出了求解（近）平衡区流体动力学方程和非平衡区动力学方程的混合方法。在这些研究过程中，研究了DSMC的时间步长截断误差。这项研究的结果反映在新方案的构建中。在项目负责人的指导下，一些研究生（硕士）参与了该研究项目，并有机会在国际会议上发言，这是本项目从教育角度来看的一个成果。

In this research project, the theory of kinetic scheme for the gas-dynamic equations (compressible Euler equations, compressible Navier-Stokes equations) is established and some numerical methods are developed. The kinetic scheme, which was originally developed as a mimic of the DSMC method for the Boltzmann equation, is revealed to be an extension of the classical Lax-Wendroff scheme and the significance of the kinetic approach is shown to be in the management of the discontinuous reconstruction of the macroscopic state. The newly developed kinetic schemes, which are based on this new finding, work as shock-capturing schemes and yield fine boundary-layer profiles with reasonable resolution. These schemes can easily be applied to various molecular models; the extensions to the NS equations for hard-sphere molecules and the Euler equation for diatomic gases are made. With the oversea research collaborator, the head investigator developed kinetic schemes for the Burnett and super-Burnett equations, which deal with higher order rarefaction effects. Furthermore, a hybrid method, which solves fluid-dynamic equations in (nearly) equilibrium regions and kinetic equation in non-equilibrium regions is developed. In the course of these studies, the time step truncation error of the DSMC was studied. The outcome of this study is reflected in the construction of new schemes. Under the supervision of the head investigator, some graduate students (master) worked in this research project and had the opportunity as speakers in an international conference, which is an outcome of the present project from the educational point of view.