Numerical modeling of plasmas often demands a kinetic approach to handle extreme nonlinearities, wave-particle interactions and other non-Maxwellian phenomena. Mathematically this requires the ab initio solution of the relativistic Vlasov-Boltzmann equation for the plasma constituents together with the appropriate Maxwell equations for the electromagnetic fields. Currently the model of choice is the particle-in-cell (PIC) code (or equivalently gyrokinetics for magnetized plasmas) a highly versatile, robust, finite-difference discretization of the Vlasov equation for the particle distribution function f(x,p). State of the art three dimensional PIC simulations involve up to 10^12 particles on 10^5 cores.
Even when performed on modern supercomputers, PIC simulation still has its limitations: the necessity of transferring information to and from the spatial grid makes it inherently noisy, collisional regimes are only accessible with the help of ad-hoc extensions, and some form of adaptive mesh refinement is required to handle geometrically complex problems. Recently a new modeling paradigm has been established %Gâ%@ mesh-free plasma simulation %Gâ%@ in which fast summation techniques replace the solution of the field equations on the mesh. The key innovation behind this development is the hierarchical tree algorithm, a rapid O(N log N) technique for evaluating mutual (Coulomb) forces due to an ensemble of charged particles. At JSC we have developed a parallel tree-code (PEPC) capable of running on the entire 458k processors of the BlueGene/Q supercomputer JUQUEEN, a milestone which makes it feasible to perform first principles simulation of collective and collisional plasma phenomena in a variety of physical settings using well over 10^9 particles.
After a brief introduction to the tree method and PEPC framework we examine its performance and scalability on the BG/Q, highlighting potential avenues for effective operation on a future exascale machine. Recent work with the code in various application areas are then shown, including plasma-wall interactions in ITER-like environments, determination of the collision frequency in strongly coupled plasmas, and a fully kinetic treatment of the Kelvin-Helmholtz instability in a magnetized plasma sheath.