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

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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.