Grid-Free Plasma Simulations

Summary form only given. Many problems in plasma physics are modeled using a Lagrangian approach. One of the most common models is particle-in-cell (PIC), where the system is modeled as a collection of macro particles which interact through long range forces. To compute long range forces in fewer than (N²) operations, particles are interpolated to a mesh where the field equations are solved. Afterwards, the fields are interpolated back to the macro particles. The major problems with this approach are that it can not resolve steep plasma gradients, has cumulative interpolation errors, and has difficulty describing non-conformal domains. The current work seeks to overcome these limitations in particle plasma models by developing a grid-free approach to plasma simulations. The boundary integral/treecode (BIT) method merges boundary integral formulations with treecode algorithms. Using Green´s theorem, and setting the functions in Green´s theorem to u=G and v=Phi, we can express Poisson´s equation as an integral equation Phi(y) = intint_Omega rho(x)/epsi_o G(x/y) dOmega - conint_deltaOmega (Phi(x) grad_x G(x/y) - G(x/y) grad Phi(x)) nds where G(x/y) is the free space Green´s function for the Laplace operator. If we are modeling the system as a collection of point charges, rho is given by rho = Sigma_i-1 ^N q_idelta(x-Zi). The contribution to the field from the charge density reduces to intint_Omega rho(x)G(x/y) dOmega = Sigma_i-1 ^N q_iG(Z_i/y). Choosing an appropriate collocation method for the boundary integral formulation effectively reformulates the boundary as a set of point charges, making the field evaluation mesh-free and capable of handling complicated domains. The operation count in the field evaluation is reduced from O(N₂) to O(N log N) by using a treecode algorithm, which gains its efficiency in essence by treat- ng clusters of particles at a distance as point charges at the center of the cluster. BIT has been applied to several bounded plasmas, including 1D sheath formation in DC discharges, 1D formation of a virtual cathode, 2D planar and cylindrical ion optics and the Penning-Malmberg trap. The method has been compared with PIC for the first three examples. For static systems, such as the ion optics and DC sheath, the results of the approaches are in good agreement. In the dynamic example of a 1D virtual cathode, obvious differences between the PIC and BIT are observed. The differences are attributed to the fact that PIC does not resolve interparticle forces within a mesh cell. Currently, the method is being used to investigate a potentially new instability in Penning-Malmberg traps and is being extended to grid-free hybrid fluid-kinetic plasma simulations