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

本研究建立了气体动力学方程（可压缩Euler方程、可压缩Navier-Stokes方程）的动力学格式理论，发展了相应的数值计算方法。动力学计划，这是最初开发的模拟DSMC方法的玻尔兹曼方程，被发现是一个经典的Lax-Wendroff计划的扩展和动力学方法的意义被证明是在管理的宏观状态的不连续重建。新开发的动力学方案，这是基于这一新的发现，工作作为激波捕获计划，并产生良好的边界层轮廓与合理的分辨率。这些方案可以很容易地应用于各种分子模型，扩展到NS方程的硬球分子和欧拉方程的双原子气体。与海外研究合作者一起，首席研究员开发了Burnett和super-Burnett方程的动力学格式，该格式处理高阶稀疏效应。在此基础上，提出了一种在（近）平衡区求解流体动力学方程，在非平衡区求解动力学方程的混合方法。在这些研究的过程中，时间步长截断误差的DSMC进行了研究。这项研究的结果反映在新计划的建设中。在首席研究员的监督下，一些研究生（硕士）参与了本研究项目，并有机会在一次国际会议上发言，从教育的角度来看，这是本项目的一个成果。

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.