UC San Diego

Geometric numerical integration has repeatedly proven useful for modeling dynamical systems, but attempts to apply this technique to kinetic plasmas face unique challenges. Traditional particle-in-cell (PIC) methods, though computationally efficient, introduce statistical noise that precludes strict structure-preservation. On the other hand, grid-based (Eulerian) methods, which constitute the other main category of kinetic plasma integrators, have degrees of freedom that scale with the size of the domain, meaning that the computational costs they incur can quickly exceed the memory and processing limitations of local workstations. Nevertheless, Eulerian methods are, in principle, compatible with a geometric approach, and how one ought to discretize, in a structure-preserving way, the infinite-dimensional Lie group of symplectomorphisms remains an interesting, and nontrivial, open question. We give a partial answer, extending a structure-preserving approach - previously applied to incompressible fluids - to handle plasmas governed by the Vlasov-Poisson equations. Then, we demonstrate the method's validity by modeling the two-stream instability on a two-dimensional velocity phase space. Finally, we discuss additional complications that arise when dealing with magnetized plasmas evolving in a six-dimensional space.