The Particle-in-Cell methods (PIC) are widely used in the numerical simulation. The media under study in these methods are represented with a sufficiently large number of model particles with definite characteristics such as mass, charge, velocity. The evolution of a system of particles at each time step is evaluated in two steps. First, the Euler step computes the impact of particles on a medium with fixed particles in grid values. Second, the Lagrange step computes the positions and velocities of particles from the equation of motion with the right-hand side computed at the Euler stage.

Since it is possible to trace trajectories of model particles, the PIC method allows studying the evolution of a medium. But it is necessary to keep in mind that model particles do not exactly correspond the real prototypes. Thus, it would be a mistake to directly compare them. The number of e model and the real particles significantly differ.

The PIC method has a relatively low precision due to several sources of errors. One of them is the interpolation of forces from the Euler grid nodes into the position of particles as well as approximation of grid functions. Another source of an error is the so-called self-force. It is the impact of the particle field on the particle itself through a spatial grid. To decrease errors of the above two given sorts, various particle form-factors are used .

One more source of errors is statistical fluctuations and noise that arise due to a difference of model particles and real particles. It was proved that the precision of computation depends not only on the time and the spatial steps but, also, on the number of model particles. Thus, the only universal way to reduce the noise is to increase the number of particles. But it is not always possible, especially for the 2D and the 3D computations. Moreover, this greatly increases computer costs.

Nevertheless, the PIC method is capable of simulating many physical effects that are unreachable for other computational schemes.

Due to this reason, new methods are being developed that enable us to control the number of particles in a cell to add new particles if their number is lower than some definite number, or to remove particles if their number is too high. The main characteristics such as density, momentum, the center of mass and energy in a cell, must not be changed in the course of adding or removing model particles. At this point, model particles start having different masses. This may improve the precision of computation when the process involves masses or numbers of atoms less than the size of one model particle. This situation may take place in the following problems: gas dynamics at the boundary with vacuum, chemical processes with a low concentration of reagents, multistream flows.

There was found that not only the method of inserting or deleting particles has a meaning, but also the choice of particle number limits (the threshold functions). Three types of threshold functions are given and compared: constant functions, spatial averaged functions and local averaged functions.

In Section 2, the PIC method with an adaptive mass is presented. In Section 3, threshold functions for inserting and deleting particles in a cell are described. In Section 4, the model problem statement and methods of its solution are described. In Section 5, computing experiments for the PIC method with adaptive mass with different threshold functions and an adaptive mass are given and their results are compared with results of the PIC method with a constant mass.