  Numerical geodynamo simulations with parameters close to an Earth-like regime
would be of great interest for understanding the dynamics of the Earth's liquid
outer core and the associated geomagnetic field. Such simulations are far too
computationally demanding owing to the large range in spatiotemporal scales.
This paper explores the application of a multi-scale timestepping method to an
asymptotic model for the generation of magnetic field in the fluid outer core
of the Earth. The method is based on the heterogeneous multiscale modelling
(HMM) strategy, which incorporates scale separation and utilizes several
integrating models for the fast and slow fields. Quantitative comparisons
between the multi-scale simulations and direct solution of the asymptotic model
in the limit of rapid rotation and low Ekman number are performed. The
multi-scale method accurately captures the varying temporal and spatial
dynamics of the mean magnetic field at lower computational costs compared to
the direct solution of the asymptotic model.
