On errors in the PIC-method when modeling Langmuir oscillations

A test problem is constructed that simulates nonlinear Langmuir oscillations excited by a short powerful laser pulse. The problem has an analytical solution in Lagrangian coordinates, which can be transformed into Eulerian coordinates, however, in a specific implicit form. For various variants of the particle-in-cell method (PIC-method), analytical estimates of the error that occurs when the assembly of the charge from the macroparticles into the centers of the cells. In addition, numerical experiments have been carried out to illustrate the quality of these estimates on a model problem. Additionally, a new difference method for the model problem is proposed and its accuracy is compared with the charge assembly stage from the PIC-method.