A numerical method for a nonlinear system is presented. Formulation of the electromagnetic behavior of the shielding current density in high-Tc superconductors (HTS) gives a system of time-dependent integro-differential equations. The behavior can be determined by solving the initial-boundary-value problem of the system using the element-free Galerkin (EFG) method and the complete implicit difference method. After discretizing the problem, we obtain a nonlinear equation. In the present study, the shielding current density in HTS is calculated by applying the deaccelerated Newton method (DNM) and adaptively deaccelerated Newton method (ADNM) for the solution of the nonlinear system. The results of computation show that the DNM does not give a convergent solution in some cases. On the other hand, the ADNM gives a convergent solution in a few iterations.