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We design and implement two non‐standard finite difference methods (NSFDMs) to solve singularly perturbed advection–reaction equations (SPARE). Our methods constitute a big plus to the class of those ‘rare’ fitted operator methods, which can be extended to singularly perturbed partial differential equations. Unlike the standard finite difference methods (SFDMs), the NSFDMs designed in this paper allow the time and the space step sizes to vary independently of one another and of the parameter ε in the SPARE under consideration. The NSFDMs replicate the linear stability properties of the fixed points of the continuous problem. Furthermore, these methods preserve the positivity and boundedness properties of the exact solution. Numerical simulations that confirm the theoretical results are presented. Copyright © 2007 John Wiley & Sons, Ltd.

This work is an extension of the paper (Proc. R. Soc. London2005; 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. Copyright © 2006 John Wiley & Sons, Ltd.