An atomistic Kinetic Monte Carlo (KMC) model for diffusion is found to exhibit classical behaviour in the limit of small gradients of slope (for surface diffusion) and concentration (for lattice diffusion). Equivalent continuum models are derived in this limit and the two systems are shown to be compatible. A mixed KMC-continuum model is proposed which allows discrete atomistic simulations to be embedded within a smooth continuum at points where the gradients are large. A blending region is introduced between the two model descriptions, which allows mass to be transferred between them in a consistent manner. This allows the model to simultaneuously work across multiple length and time scales. The implementation of this multiscale modelling approach is illustrated with a number of simulations, including the decay of a rough surface.

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