Mathematical Sciences: The Numerical Analysis of Nonlinear Vlasov Kinetic Equations

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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.