Review and Recent Advances in PIC Modeling of Relativistic Beams and Plasmas

Particle-in-Cell (PIC) simulation codes have wide applicability to first-principles modeling of multidimensional nonlinear plasma phenomena, including wake-field accelerators. This review addresses both finite difference and pseudo-spectral PIC algorithms, including numerical instability suppression and generalizations of the spectral field solver. INTRODUCTION AND SUMMARY Particle-in-Cell (PIC) simulation codes solve the Vlasov equation by following hundreds of thousands of particles and their associated electromagnetic fields as they evolve in time: Particles are advanced a time-step based on the fields, and the fields then are advanced a time-step based on the currents generated by the particles. This simple process is repeated thousands to millions of times to simulate complex, nonlinear, multidimensional plasma phenomena. Fields are evaluated either on a spatial mesh (FDTD finite difference time domain) or as a set of spatial Fourier modes (PSTD pseudo-spectral time domain). Particles, on the other hand, are distributed at arbitrary positions across the mesh, requiring interpolation between fields and particles. PIC codes with explicit temporal algorithms are inherently numerically unstable, and successfully employing them requires increasing numerical instability growth times until they are much longer that the relevant time scales of the physical phenomena simulated. Explicit vacuum field solvers themselves impose Courant limits, ∆tt < αα ∆xx, with ∆xx the characteristic dimension of a mesh cell and αα a numerical factor typically somewhat less than one. PIC methods are discussed in detail in the textbook by Birdsall and Langdon.1 Even with the Courant limit satisfied, explicit PIC codes are unstable due to field-particle interpolation coupled with disparities between the Eulerian field solver and Lagrangian particle pusher. Dispersion relations for numerical instabilities can be derived much as they are for physical instabilities and now exist for multi-dimension relativistic beam (or stationary plasma simulated in a relativistically translating frame) simulations. Solutions of numerical dispersion relations indicate that the numerical Cherenkov instability is particularly serious for relativistic beams or plasmas but also suggest various ways to ameliorate it. Several mitigation techniques that involve short wavelength digital filtering combined with minor modifications to the interpolation process have been demonstrated to work well in practice, effectively controlling numerical instabilities in most instances. This year, generalizations of the PSTD algorithm have been developed that provide variable Courant limits and other flexibilities. The utility of these generalizations currently is being investigated. Some of the techniques developed to ameliorate the numerical Cherenkov instability for the usual PSTD algorithm also work well for these generalized algorithms. BRIEF INTRODUCTION TO PIC SIMULATONS The particle equations of motion in a PIC code can be represented as