  We conduct global galactic--scale magnetohydrodynamical (MHD) simulations of
the cosmic--ray driven dynamo. We assume that exploding stars deposit
small--scale, randomly oriented, dipolar magnetic fields into the
differentially rotating ISM, together with a portion of cosmic rays,
accelerated in supernova shocks. Our simulations are performed with the aid of
a new parallel MHD code PIERNIK. We demonstrate that dipolar magnetic fields
supplied on small SN--remnant scales, can be amplified exponentially by the
CR--driven dynamo to the present equipartition values, and transformed
simultaneously to large galactic--scales by an inverse cascade promoted by
resistive processes.
