Local discontinuous Galerkin method for modeling the nonplanar structures (solitons and shocks) in an electronegative plasma

Abstract

Using the hydrodynamic equations of cold inertial positive ions with the Maxwellian distribution for light negative ion and electron densities and the Poisson equation, the family of nonplanar (cylindrical and spherical) Korteweg-de Vries (KdV) equations, i.e., the KdV, modified KdV, and extended KdV (EKdV), are obtained for small but finite amplitude ion-acoustic waves. The nonplanar EKdV equation is used to analyze the time-dependent planar and nonplanar soliton and shock structures. It is well-known that the exact solutions of the family of nonplanar KdV equations are not possible. Therefore, a local discontinuous Galerkin (LDG) method is developed for solving the nonplanar EKdV equation numerically. According to this method, the initial and boundary conditions for the solitary and shock waves are accurately identified. The L2 stability of the LDG method is proved for the general nonlinear case. The existence regions for both solitary and shock excitations have been defined precisely depending on the laboratory plasma parameters. Moreover, the effects of the negative-ion temperature ratio and the negative ion concentration on the profiles of the nonlinear structures (solitons and shocks) are examined. The effect of the geometrical divergence on the pulse profile is also reported which indicates that the localized pulses deform as time goes on. Furthermore, it is found that the amplitude of cylindrical structures (solitons and shocks) is larger than that of planar ones but smaller than that of the spherical ones. Moreover, in cylindrical geometry, the nonlinear structures travel slower than in the spherical ones. The implications of our results agree with the experimental observations.