Mathematical Sciences: The Analysis of Particle Methods for Solving Vlasov Kinetic Equation

该项目的主要目的是研究和 分析Vlasov-Poisson-Fokker-Planck随机粒子方法 系统. 它建议看看是否一个随机的粒子在细胞 方法可以被实现，该方法考虑到以下事实： 等离子体粒子（离子和电子）经历布朗运动， 与介质或背景粒子的碰撞。 的 研究工作将主要致力于融合 这些方法的分析和数值实现。 同时，调查人员应继续调查 粒子方法求解Vlasov-Poisson方程组 他们的 研究将致力于提高收敛速度， 仿真方法，引入时间离散化， 细胞中的粒子方法， 算法 对于受控聚变和激光聚变问题， 激光与物质相互作用的物理学仍然不是很好 明白 因此，科学家们越来越多地 到基于动力学方程的微观模型， 理解这些现象。 用于模拟 稀薄等离子体的时间小于双星 碰撞时间，使用Vlasov-Poisson方程组 对于相关的动力学模型，当磁场变化时， 慢慢 粒子模拟方法包括替换 等离子体由一个大的一组带电的超粒子，遵守 经典和现代物理学的一般定律，然后计算 这些超粒子之间的相互作用 的透彻 了解这些方法的准确性， 使科学家能够使用不同的近似参数 在数值模拟中有效地（例如，最小数目 以获得给定的精度）。 弗拉索夫-泊松-福克-普朗克动力学方程组 结合了等离子体与背景的碰撞效应 材料（例如，在热浴或“储液器”中的等离子体系统） 通过把粒子的运动看作是布朗运动， 与背景的碰撞。 该模型类似于 不规则或布朗运动的经典描述 表现为浸在液体中的胶体大小的颗粒。 在恒星动力学中，恒星之间相遇的影响 在重力的作用下， 布朗运动。 随机粒子法 提出了Vlasov-Poisson-Fokker-Planck模型， 更好地了解碰撞的影响， 数值格式中的布朗运动。

The primary objective of this project is to study and analyze random particle methods for Vlasov-Poisson-Fokker-Planck systems. It is proposed to see if a random particle-in-cell method can be implemented that takes into account the fact that plasma particles (ions and electrons) undergo Brownian motion due to collisions with the medium or background particles. The research efforts will be mainly devoted to the convergence analysis and numerical implementation of these methods. Simultaneously, the investigators shall continue investigations on particle methods for solving Vlasov-Poisson systems. Their research will be addressed to improving convergence rates for simulation methods, introducing time discretization of particle-in-cell methods, and analyzing more efficient algorithms. For problems of controlled fusion and laser fusion, the physics of laser-matter interactions is still not well understood. Consequently, scientists are turning more and more to microscopic models based on kinetic equations to better understand these phenomena. For modeling the evolution of rarefied plasmas for those times less than the binary collision times, the Vlasov-Poisson system of equations is used for the relevant kinetic model, when the magnetic fields vary slowly. Particle simulation methods consist of replacing the plasma by a large set of charged superparticles that obey the usual laws of classical and modern physics, and then computing the interactions of these superparticles. A thorough understanding of the accuracy properties of these methods will enable the scientist to use different approximate parameters efficiently in a numerical simulation (e.g., the minimum number of particles to obtain a given accuracy). The Vlasov-Poisson-Fokker-Planck system of kinetic equations incorporates collisional effects of a plasma with the background material (e.g., a plasma system in a thermal bath or "reservoir") by regarding the motion of a particle as Brownian motion caused by collisions with the background. This model is analogous to the classical description of the irregular or Brownian motion exhibited by a particle of colloidal size immersed in a fluid. In stellar dynamics also, the effect of encounters between stars under gravitational forces influences their motion in the manner of Brownian motion. The random particle method for Vlasov-Poisson-Fokker-Planck models is proposed in order to better understand the effects of collisions by incorporating the Brownian motion in the numerical schemes.