Non-standard finite-difference methods for vibro-impact problems

Impact oscillators are non-smooth systems with such complex behaviours that their numerical treatment by traditional methods is not always successful. We design nonstandard finite-difference schemes in which the intrinsic qualitative parameters of the system-the restitution coefficient, the oscillation frequency and the structure of the nonlinear terms-are suitably incorporated. The schemes obtained are unconditionally stable and replicate a number of important physical properties of the involved oscillator system such as the conservation of energy between two consecutive impact times. Numerical examples, including the Duffing oscillator that develops a chaotic behaviour for some positions of the obstacle, are presented. It is observed that the cpu times of computation are of the same order for both the standard and the non-standard schemes.