Mathematical Sciences: The Numerical Analysis of Nonlinear Vlasov Kinetic Equations

9622690胜利本研究的主要目标是对描述无碰撞和碰撞等离子体模型的Vlasov动力学方程进行数值分析。研究工作将集中在粒子方法上，结合有限元或有限差分方法来确定场，并结合隐式时间积分方案来推进粒子。因为在现实（三维）问题中有七个自变量，基于经典的有限差分、有限元或谱方法进行数值模拟是不现实的。实际的模拟是基于粒子方法的，其实现需要对Vlasov动力学模型的存在性、唯一性和规律性有深刻的理解。特别建议从以下两个方向进行调查：(A)。无碰撞等离子体动力学方程的数值分析。特别地，在Vlasov-Poisson系统的细胞内粒子格式的数值分析中，引入了隐式时间积分方法。同时，将Vlasov- poisson系统粒子逼近的收敛理论推广到处理Vlasov- maxwell和Vlasov- Einstein问题。(B)。等离子体碰撞Vlasov动力学方程的数值分析。特别地，半导体物理中出现的朗道（福克-普朗克）型和玻尔兹曼型的非线性碰撞算符被添加到底层的Vlasov-Poisson或Vlasov-Maxwell系统中。分析涉及随机和确定性粒子方法，包括算子分裂为非线性对流和碰撞部分。人们对动力学方程的兴趣很大程度上来自于将动力学方程的知识应用于激光聚变、等离子体物理和半导体中出现的问题所带来的实际和经济效益。弗拉索夫动力学方程的数学模型的现象，如激光与物质的相互作用发生在控制和激光聚变，并在半导体量子环境中的电子传输存在。这项由美国国家科学基金会支持的研究工作的目标是对弗拉索夫动力学模型的数值近似进行严格的调查，目的是使科学家能够有效地进行计算，以确定能量、力矩等。这里开展的工作将影响目前等离子体和半导体技术的基础，为研究半导体中的电子传输、激光物质相互作用和电子等离子体中不稳定性的开始提供扩展的定性分析和强大的数值程序。此外，非线性数学模型的数值分析传统上在应用学术和工业数学家中有着广泛的受众。这一事实将对决定攻读动力学型非线性方程研究高级学位的学生的学术或工业职业前景产生重大影响。

9622690 Victory The primary goal of this research is the numerical analysis of Vlasov kinetic equations which describe collisionless and collisional plasma models. Research efforts will focus on particle methods used in conjunction with finite element or finite-difference methods for determining the fields, and with implicit time-integrating schemes for advancing particles. Because there are seven independent variables in realistic (three-dimensional) problems, it is not practical to base numerical simulations on classical finite-differences, finite elements, or spectral methods. Practical simulations are based on particle methods, whose implementation requires a solid understanding of the existence, uniqueness, and regularity properties of Vlasov kinetic models. In particular, it is proposed to carry out investigations in the following two directions: (A). Numerical analysis of the Vlasov kinetic equations modeling collisionless plasmas. In particular, implicit time-integrating methods for advancing particles are incorporated into the numerical analysis of particle-in-cell schemes for Vlasov-Poisson systems. Also the convergence theory of particle approximations for Vlasov-Poisson systems is extended to treat Vlasov-Maxwell and Vlasov- Einstein problems. (B). Numerical analysis of Vlasov kinetic equations modeling plasmas with collisions. In particular, nonlinear collision operators of Landau (Fokker-Planck) type and of Boltzmann type arising in semiconductor physics are added to the underlying Vlasov-Poisson or Vlasov-Maxwell system. The analyses involve stochastic and deterministic particle methods involving operator splitting into nonlinear convective and collisional portions. Much of the interest in kinetic equations is derived from the practical and economic benefits that would accrue from applying knowledge of such equations to problems arising in laser fusion, plasma physics, and semiconductors. Vlasov kinetic equation s mathematically model such phenomena as diverse as laser-matter interactions occurring in controlled and laser fusion, and the transport of electrons in quantum environments present in semiconductors. The research efforts being supported by the National Science Foundation have as objective the rigorous investigation of numerical approximations to Vlasov kinetic models, with the goal of enabling scientists to perform computations efficiently in determining energies, moments, etc. The work undertaken here would affect the current base of plasma and semiconductor technology in providing extended qualitative analysis and robust numerical procedures for the study of transport of electrons in semiconductors, laser-matter interactions, and the onset of instabilities in an electron plasma. In addition, the numerical analysis of nonlinear mathematical models has had a traditionally wide audience among applied academic and industrial mathematicians. This fact will have a significant impact on the academic or industrial career prospects of students who decide to pursue advanced degrees in the study of nonlinear equations of kinetic type.