Local refinement of grid-based numerical methods is an attractive option for problems with localized large gradients separating regions where the solution is relatively smooth. Such methods enhance efficiency by locally concentrating the computational effort to ensure a globally uniform level of accuracy. When the solution structure is not known a priori, as is typical for most problems, a method known as adaptive mesh refinement (AMR) can be used. Essentially, AMR employs coarse grid solutions to iteratively determine appropriate grid refinement. The technique has been successfully applied to PDE systems in numerous disciplines, and is an area of ongoing numerical methods research.
The application of AMR to particle-in-cell (PIC) methods presents a number of difficulties which arise from the breaking of symmetry due to the nonuniform mesh. Nonphysical self-forces occur near refinement boundaries which lead to incorrect solutions and loss of energy conservation. In this work we present a local correction algorithm for electrostatic PIC which redistributes the charge allocated to each grid point in order to produce a self-consistent electric field solution. Our results show that the correction can reduce self-forces to any order of accuracy, with an associated tradeoff in computational cost.