Continuous upscaling and averaging

Abstract The procedure of upscaling/averaging of the values and equations of continuum mechanics appears in many scientific areas. Traditionally, upscaling is carried out by a “jump” between the two distant scales. We suggest an alternative procedure for gradual upscaling, with a continuous transition from a finer to a coarser scale along the scale axis, and gradual averaging on the way. It is shown, in natural assumptions, that this procedure is similar to a Markov process described by the Ornstein-Uhlenbeck equation. This is the only option, unlike traditional weight averaging, where multiple weight functions are possible. An interpretation of the procedure is provided, and a solution of the upscaling equation is obtained. We demonstrate also that, under upscaling of the conservation laws in the flow equations expressed in the divergent form, fluxes are upscaled by the same rules as densities. As an example, an equation for upscaling the diffusion coefficient is derived.