稳态玻尔兹曼方程的高性能数值矩方法与应用

The non-Maxwellian steady-state solution of the Boltzmann equation has a special significance, not only that it helps to better understand the Boltzmann equation as well as the kinetic theory, but also itself has wide range of applications in a number of high-tech fields such as rarefied gas dynamics, microflows, and semiconductor devices. Since the analytical solution is generally unavailable, numerical solutions are highly needed. However, due to the complexity of the Boltzmann equation, numerical solution is very challenging, especially for the steady-state solution, which is usually achieved after a long time simulation. Numerical moment method is one of the important numerical methods for the Boltzmann equation. It integrates the computable modeling and the numerical algorithm of kinetic problems into a uniform numerical framework, which makes it become attractive to use the high-order moment models in practical applications. Motivated by these, this project is concentrated on the development and applications of high-performance numerical moment method for the steady-state solution, directly from the steady-state Boltzmann equation. Specifically, the research of this project will involve the following aspects. Multilevel iterative methods, using acceleration techniques such as the coarse grid correction and the lower-order model correction, will be developed to improve the convergence rate. Adaptive methods including adaptive mesh and adaptive moment model will also be investigated to improve the efficiency by reducing the redundant degrees of freedom. Combining these two methods together, a comprehensively efficient and robust numerical moment method will then be developed. This project will certainly improve the ability of the numerical moment method to solve practical problems, and promote large-scale applications of numerical moment method in the related high-tech fields.

玻尔兹曼方程的非麦克斯韦稳态解在稀薄气体动力学、微流、半导体器件电子输运等众多高科技领域均有重要的应用。由于玻尔兹曼方程自身的复杂性，其非麦克斯韦稳态解难以通过解析方式获得，而高效的数值求解也面临着巨大的挑战。数值矩方法是求解玻尔兹曼方程的重要数值方法之一，它将动理学问题的可计算建模与数值算法有效融合起来，为高阶矩模型的实际应用打下了良好的基础。本项目将直接从稳态玻尔兹曼方程出发，研究粗网格校正和低阶模型校正等加速收敛技术，发展具有高收敛性的多水平迭代数值矩方法；研究网格自适应和模型自适应等技术，发展既能有效减少自由度总数又不失精度要求的自适应数值矩方法；结合上述两种方法，为玻尔兹曼方程的稳态解发展综合性的高效而稳健的数值矩方法。通过本项目的研究，将显著提高数值矩方法求解实际问题的能力，推动其在前述诸多领域中的大规模应用。

动理学理论是介于流体力学和分子动力学之间，以统计物理方法研究微观粒子运动状态的理论，在稀薄气体动力学、微流、半导体器件、等离子体等众多高科技领域有着重要应用。玻尔兹曼方程是其中最基本的动理学方程，它的求解方法不仅具有重要的理论价值，也具有重要的实际应用价值。然而玻尔兹曼方程自身的复杂性，使其高效的数值求解面临着巨大的挑战。本项目致力于研究和发展求解玻尔兹曼方程及其相关动理学模型的数值矩方法，特别是发展稳态问题的高效而稳健的数值矩方法。具体地，项目发展和完善了低阶模型校正加速收敛技术，提出了基于该技术和数值矩方法一般框架的、稳态高阶模型的非线性多水平迭代新方法；发展了基于空间粗网格校正加速收敛技术的非线性多重网格方法，提出了稳态模型的非线性多重网格新方法，并推广应用于具有二元碰撞的原始玻尔兹曼方程的数值求解；发展了一个基于区域分解的移动网格方法和一个简单有效的二维移动网格方法，并在移动热源的热传导现象的数值模拟中得到应用；针对具有二元碰撞的原始玻尔兹曼方程，研究了数值矩方法在该模型中的推广应用与高效数值方法的发展，提出了保持Maxwellian的碰撞项计算方法。项目取得的上述研究成果，以及持续研究开发的、具有统一框架和接口的算法程序，显著提高了数值矩方法求解实际问题的能力，为推动其在各个重要领域中的实际应用提供了有力的理论指导和技术支持。