Applied Mathematics Seminar
"Simulations of Plasma Dynamics using a Grid Free Technique"Abstract: When a sufficient amount of energy has been put into a gas, the gas breaks down into its constituent electrons and ions, commonly known as the plasma state. 99% of all matter is in the plasma state, e.g., stars, interstellar material, etc. Therefore understanding fundamental behavior of plasmas can tell us a great deal about the Universe.
The fundamental set of equations describing a plasma is the Boltzmann-Maxwell system. This system is a kinetic model and it can be reduced to the more familiar fluid description through the Chapman-Enskog procedure. However, a wide range of observable phenomena can only be described by a kinetic model.
Limiting our attention to the simpler kinetic model of the Vlasov-Poisson (VP) system, two examples of purely kinetic phenomena are Landau damping and the bump-on-tail instability. The VP system is a nonlinear PDE-integral model for a collisionless electrostatic plasma. Due to the nonlinearity of the VP system, a natural approach to investigating the detailed dynamics of the system is through numerical simulations.
In this talk I will highlight aspects of the derivation of the kinetic description, discuss what is known about this system, and give an overview of Landau damping and the bump-on-tail instability. In addition, I will discuss existing approaches to simulating this system and the shortcomings of these methods.
Finally, I will discuss a grid-free alternative currently being developed and consider its application to the the bump-on-tail instability and the problem of crystal formation in plasmas. Novel aspects of the grid-free approach include the combination of fast summation algorithms with numerical Green's function methods and systematic refinement techniques in a fully Lagrangian framework.