On a 1D-electrostatic test problem for the PIC method

A test problem for the 'particle-in-cell' method is proposed which allows one to check individual errors on each stage of numerical implementation of the method. General testing usually controls only total (final) error. Using the test problem, we analyze errors of a difference method of MacCormack type and the CIC method being the most popular version of the 'particle-in-cell' method. It is shown that the CIC method having also the second formal order of accuracy is seriously inferior to the difference method under the same given error and same grid parameters. In particular, the results of the particle method lose their sense after approximately one calculation period, although the time interval of confidence in simulation data for the difference method is practically unlimited. The testing method proposed here is suitable for an arbitrary one-dimensional version of the 'particle-in-cell' method, and it allows one not only to compare numerical implementations of individual stages with each other, but also to verify available theoretical results.