Non-standard finite difference schemes for multi-dimensional second-order systems in non-smooth mechanics

This work is an extension of the paper (Proc. R. Soc. London 2005; 461A:1927-1950) to impact oscillators with more than one degree of freedom. Given the complex and even chaotic behaviour of these non-smooth mechanical systems, it is essential to incorporate their qualitative physical properties, such as the impact law and the frequencies of the systems, into the envisaged numerical methods if the latter is to be reliable. Based on this strategy, we design several non-standard finite difference schemes. Apart from their excellent error bounds and unconditional stability, the schemes are analysed for their efficiency to preserve some important physical properties of the systems including, among others, the conservation of energy between consecutive impact times, the periodicity of the motion and the boundedness of the solutions. Numerical simulations that support the theory are provided.