This paper is concerned with the development of a high-order numerical scheme for the modelling of two-phase Newtonian flows. The companion paper, herein referred to as Part 2, extends the scheme to two-phase viscoelastic flows. The particular problem of the collapse of a two-dimensional bubble in the vicinity of a rigid boundary is considered. The governing equations are discretized using the spectral element method, and the two phases are modelled using a marker particle method. The marker particle scheme is validated using the Zalesak slotted disk rotation test problem. A comprehensive set of results is presented for the problem of bubble collapse near a rigid wall, and qualitative agreement is obtained with other numerical studies and experimental observations. Viscous effects are shown to inhibit bubble collapse and prevent jet formation and are therefore likely to have a mitigating effect on cavitation damage.Copyright © 2011 John Wiley & Sons, Ltd.