New approach in computational fluid dynamics on the basis of kinetic theory

在本研究中，首先对流体动力学方程的混合方法进行了基础研究。我们建立了一个简单的可压缩Navier-Stokes方程的高分辨率激波捕获方案理论。该理论依赖于动力学方程特征的简单结构。动力学方程中对流项的线性化极大地简化了特性理论，可以很自然地进行数值通量的拆分。通过气体分子的分布函数进行流体动力变量的重建，使得在分辨率高的区域具有激波捕获性和较小的耗散性。这些结果总结为研究生和本科生的课堂讲稿。在二维射流膨胀到真空的问题中，我们证明了混合方法的有效性。就确定性杂化而言，两种解之间的连接可以毫不困难地完成。接下来，我们研究了基于动力学理论的不可压缩流的简单数值计算方法。晶格玻尔兹曼方法是一种众所周知的动力学不可压缩求解方法。然后，我们对该方法进行了系统的渐近分析，发现该方法是众所周知的人工压缩性方法的一种变体。人工可压缩性方法现在被广泛认为是获得稳定解的一种工具。然而，系统渐近分析显示了它在时变情况下作为高阶精确解算器的潜力。这个问题一直在研究。作为相关工作，我们开发了粘性Burgers方程的精确数值计算方法。利用局部著名的Cole-Hopf变换，我们可以得到一个精确的解的公式，它被表示为一个有理多项式。

In the present study, we first carried out basic research of hybrid method of fluid-kinetic equations. We established a simple theory of high-resolution shock capturing scheme for compressible Navier-Stokes equations. This theory relies on the simple structure of characteristics of kinetic equation. The linearity of the convection term in kinetic equation drastically simplifies the theory of characteristics and the splitting of numerical flux can naturally be done. The reconstruction of fluid-dynamic variables is done via the distribution function of gas molecules, which enables the shock capturing and less dissipative nature in well-resolved region. These results are summarized as lecture notes for graduate and under-graduate students. In the problem of two dimensional jet expansion into vacuum, we demonstrated the usefulness of hybrid method. As far as the deterministic hybridization, the connection between two solutions can be done without any serious difficulties. Next, we proceeded to the study on simple numerical method for incompressible, flows based on kinetic theory. The Lattice Boltzmann method is well-known as a kinetic incompressible solver. Then, we carried out a systematic asymptotic analysis of this method and found that this method is a variant of well-known artificial compressibility method. The artificial compressibility approach is now widely believed as a tool for obtaining steady solutions. However, the systematic asymptotic analysis reveals its potential as a high order accurate solver in time-dependent case. This subject is continuously studied. As a related work, we developed an accurate numerical method for viscous Burgers equation. By making use of well-known Cole-Hopf transformation locally, we could derived an accurate formula of solution, which is expressed as a rational polynomial.