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

在本研究中，我们首先开展了流体-动力学方程混合方法的基础研究。建立了求解可压缩N-S方程的高分辨率激波捕捉格式的简单理论。这一理论依赖于动力学方程结构简单的特点。动力学方程中对流项的线性度极大地简化了特征理论，数值通量的分裂是很自然的。通过气体分子的分布函数对流体动力学变量进行重构，使激波在分辨率较好的区域内具有捕获性和较小的耗散性。这些结果被总结为研究生和本科生的课堂讲稿。在二维射流膨胀为真空的问题中，我们论证了混合方法的有效性。就确定性杂交而言，两个解之间的连接可以不存在任何严重困难。接下来，我们对基于动量理论的不可压缩流动的简单数值方法进行了研究。格子Boltzmann方法是众所周知的动力学不可压缩求解器。然后，我们对该方法进行了系统的渐近分析，发现该方法是著名的人工可压缩方法的一种变形。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，系统的渐近分析揭示了它在含时情况下作为高阶精确求解器的潜力。这一课题正在不断地研究中。作为相关工作，我们发展了一种求解粘性Burgers方程的精确数值方法。利用局部熟知的Cole-Hopf变换，我们可以得到用有理多项式表示的精确解公式。

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.