  We study charge diffusion in relativistic resistive second-order dissipative
magnetohydrodynamics. In this theory, charge diffusion is not simply given by
the standard Navier-Stokes form of Ohm's law, but by an evolution equation
which ensures causality and stability. This, in turn, leads to transient
effects in the charge diffusion current, the nature of which depends on the
particular values of the electrical conductivity and the charge-diffusion
relaxation time. The ensuing equations of motion are of so-called stiff
character, which requires special care when solving them numerically. To this
end, we specifically develop an implicit-explicit Runge-Kutta method for
solving relativistic resistive second-order dissipative magnetohydrodynamics
and subject it to various tests. We then study the system's evolution in a
simplified 1+1-dimensional scenario for a heavy-ion collision, where matter and
electromagnetic fields are assumed to be transversely homogeneous, and
investigate the cases of an initially non-expanding fluid and a fluid initially
expanding according to a Bjorken scaling flow. In the latter case, the scale
invariance is broken by the ensuing self-consistent dynamics of matter and
electromagnetic fields. However, the breaking becomes quantitatively important
only if the electromagnetic fields are sufficiently strong. The breaking of
scale invariance is larger for smaller values of the conductivity. Aspects of
entropy production from charge diffusion currents and stability are also
discussed.
