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Time Discontinuous Galerkin methods require the factorization of a matrix larger than that exploited in standard implicit schemes. Therefore, they lend themselves to implementations based on predictor-multicorrector solution algorithms. In this paper, various convergent and computationally efficient iterative methods implemented in the unknown displacements for determining the solution of non linear systems are proposed. The iterative solutions presented here differ from those implemented in the unknown velocities in that they are computationally superior. The results of numerical simulations relevant to Duffing oscillators and to a stiff spring pendulum discretized with finite elements which are designed to evaluate the efficacy of these iterative methods with non-linear systems, show a low-computational expense when compared to earlier iterative schemes. [ABSTRACT FROM AUTHOR]