多组分格子波尔兹曼方法的数值分析

项目名称： 多组分格子波尔兹曼方法的数值分析

项目编号： No.11471185

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 雍稳安

作者单位： 清华大学

项目金额： 70万元

中文摘要： 格子波尔兹曼方法是一种基于动理学理论的计算流体方法，已被成功用于涉及到不可压流体的许多实际工程问题的数值模拟。尽管该方法已被用来获得了许多重要的结果，它的使用在很大程度上还依赖于经验。 本项目旨在从数学上严格分析多组分格子波尔兹曼方法的稳定性、收敛性以及收敛阶。它将基于我们已有的关于单组分方法的工作，这些工作已得到国际同行的广泛认可（本项目负责人于2013年在牛津大学举办的第十届介观方法国际会议（ICMMES2013）上做了大会的开场邀请报告）。与单组分模型相比较，多组分模型具有迥然不同的稳定性结构以及相当复杂的耦合碰撞项。这样的严格分析不仅能为该方法建立坚实的理论基础，也会为方法的具体落实给出重要的指导和启示。比如，可以给出分布函数初始值的解析公式、确立时间步长与空间步长的关系、指出如何把外力正确地纳入计算过程等。

中文关键词： 格子Boltzmann方法；多组分；稳定性；渐进分析；收敛性

英文摘要： The lattice Boltzmann method is a promising alternative to conventional numerical methods for simulating fluid flows and has proved especially effective for simulating incompressible flows in complicated geometries, and for exploiting massively parallel computer architectures. It is very simple for use and has its root in the kinetic theory. In spite of great successes, its implementation is often heavily based on experience. The goal of this project is to study the stability, convergence and convergence-rate of the lattice Botzmann method for multi-component fluids mathematically, which will be an extension of the applicant's previous results for common fluids. Such analyses will not only establish a solid mathematical basis for the method, it will also give some important hints for implementing the method. Because mathematical proofs can only be carried out when the method is precisely defined, we expect,through the proofs, to derive analytical formulae for the initial data of the distribution functions, to obtain a precise relation between the time step and spatial step, to gain some details of introducing the external forces into the method, and so on. These hints will reduce the dependence of the method on the experience considerably and thereby may make the method more rational.

英文关键词： lattice Boltzmann method;multi-component;stability;asymptotic analysis;convergence