Two-dimensional simulations of flow instability at the interface of a three-layer, density-matched, viscosity-stratified Poiseuille flow are performed using a front-tracking/finite difference method. This is an extension of the study for the stability of two-layer viscosity-stratified flow of Cao et al., Int. 1. Multiphase Flow, 30, 1485-1508 (2004). We present results for large-amplitude non-linear evolution of the interface for varying viscosity ratio m, Weber number We, and phase difference between the perturbations of the two interfaces. Strong non-linear behaviour is observed for relatively large initial perturbation amplitude. The higher viscosity fluid is drawn out as a finger that penetrates into the lower viscosity layer. The finger originates at the crest of the perturbation at the interface. The simulated interface shape compares well with previously reported experiments. Increasing interfacial tension retards the growth rate of the interface as expected, whereas increasing the viscosity ratio enhances it. The sinuous instability appears to evolve faster than the varicose one. For certain flow parameters the high-viscosity finger displays a bulbous tip, which is also seen in our previously conducted experiments and two-layer results, although it is less pronounced. The low-viscosity intruding finger does not display this curious bulbous tip. Drop formation is precluded by the two-dimensional nature of the calculations.