On a numerical scheme of exponential fitting for solving radio-frequency discharge equations in the hydrodynamic approximation

The numerical modeling of the RF discharge in the hydrodynamic approximation is widely used in studying processes of plasma-chemical etching of semiconductor materials. The integration of RF discharge equations with a small parameter at the high-order derivative is a difficult problem connected with the formation of boundary layers with big gradients of required values. In addition, the equations are described by a high nonlinearity and strong interrelation, which needs using special highly stable numerical methods. The basic methodological approach in developing absolutely monotonous finite-difference schemes is their regularization a special case of which is the method of exponential fitting used in this work. The proposed finitedifference scheme is considered by an example of continuity equations. The integration by the control volume of differential expressions for particle flows yields their finite-difference approximations of convective and diffusion components simultaneously, which provides their stable calculation at big gradients of the potential. The expressions for the flows received for the continuity equations are generalized for the electron energy equation. The introduced implicit exponential finite-difference scheme guarantees to get the solution for big Peclet numbers with keeping positive values of electron temperature and concentrations of plasma components.